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Characterization and suppression of noise in superconducting quantum systems
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Characterization and suppression of noise in superconducting quantum systems
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Content
Characterization and Suppression of noise in Superconducting Quantum systems
by
Vinay Tripathi
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(PHYSICS)
August 2024
Copyright 2024 Vinay Tripathi
In loving memory of my late mother
ii
Acknowledgments
I would like to extend my deepest gratitude to my advisor, Professor Daniel Lidar, for his
mentorship and support throughout these years. Daniel has taught me to focus on the
bigger picture and has shown me what it means to be truly dedicated and hardworking. His
guidance on writing papers with utmost clarity has been instrumental in my growth as a
researcher.
Special thanks to Professor Sasha Korotkov for introducing me to the world of superconducting systems. During my master’s thesis project at UC Riverside, I learnt a great
deal about superconducting systems under Sasha’s guidance, which was immensely helpful
throughout my PhD journey. I am also grateful to my collaborator and mentor Prof. Eli
Levenson-Falk, for always been available to answer any questions related to experiments or
superconducting device physics.
Discussions with Mostafa Khezri were instrumental during the early stages of my PhD.
Huo Chen introduced me to his open-source package, HOQST, designed for simulating open
quantum system dynamics, which has been an essential toolkit throughout my PhD. I am
also thankful to my USC friends Amy, Arian, Arjun, Daria, Haimeng, Humberto, Jenia,
Joey, Mario, Nic, Pat, Saurav, Tong, and Victor for making my stay at USC both happy
and memorable. I would also like to thank my friends from India, Radhe and Vikash, who
have always motivated and inspired me.
My gratitude extends to all my external collaborators, especially Zlatko Minev, Derek
Wang, Elisa B¨aumer, Alireza Seif, Ali Javadi, Haoran Liao, Eyob Sete, Noah Goss, and Long
Nguyen, who have helped me become a better researcher.
Finally, I would like to thank my family members, who have always believed in me and
supported me in whatever I wanted to do. A special thanks to Monika for always being
beside me in stressful situations and for keeping me sane over the years.
iii
List of Publications
The work presented in this thesis contains materials from the following publications and
preprints:
1. Suppression of crosstalk in superconducting qubits using dynamical decoupling
Vinay Tripathi, Huo Chen, Mostafa Khezri, Ka-Wa Yip, EM Levenson-Falk and
Daniel A. Lidar, Phys. Rev. Applied 18, 024068 (2022)
2. Modeling low- and high-frequency noise in transmon qubits with resourceefficient measurement
Vinay Tripathi, Huo Chen, E. M. Levenson-Falk, Daniel A. Lidar, PRX Quantum
5, 010320 (2024)
3. Qudit dynamical decoupling on a superconducting quantum processors
Vinay Tripathi ∗
, Noah Goss ∗
, Arian Vezvaee, Irfan Siddiqi, Daniel A. Lidar, arXiv:
2407.04893 (2024)
4. Deterministic benchmarking of quantum gates
Vinay Tripathi ∗
, Daria Kowsari ∗
, Kumar Saurav, Haimeng Zhang, EM LevensonFalk, Daniel A. Lidar, arXiv: 2407.09942 (2024)
5. Role of the virtual Z gates on open qunatum dynamics
Arian Vezvaee ∗
, Vinay Tripathi ∗
, Daria Kowsari, EM Levenson-Falk, Daniel A.
Lidar, arXiv: 2407.14782 (2024)
*These authors contributed equally to this work.
iv
Other publications and preprints completed during the PhD but not part of the thesis:
6. Efficient Long-Range Entanglement using Dynamic Circuits
Elisa B¨aumer, Vinay Tripathi, Derek S. Wang, Patrick Rall, Edward H. Chen,
Swarnadeep Majumder, Alireza Seif, Zlatko K. Minev, arXiv:2308.13065 (2023)
7. Error budget of parametric resonance entangling gate with a tunable coupler
Eyob A. Sete, Vinay Tripathi, Joseph A. Valery, Daniel A. Lidar, Josh Y. Mutus,
arXiv:2402.04238(2024)
8. Dynamically Generated Decoherence-Free Subspaces and Subsystems on
Superconducting Qubits
Gregory Quiroz, Bibek Pokharel, Joseph Boen, Lina Tewala, Vinay Tripathi, Devon
Williams, Lian-Ao Wu, Paraj Titum, Kevin Schultz, Daniel Lidar, arXiv:2402.07278
(2024)
9. Suppressing Correlated Noise in Quantum Computers via Context-Aware
Compiling
Alireza Seif, Haoran Liao, Vinay Tripathi, Kevin Krsulich, Moein Malekakhlagh,
Mirko Amico, Petar Jurcevic, Ali Javadi-Abhari, arXiv:2403.06852 (2024)
10. Quantum Fourier Transform using Dynamic Circuits
Elisa B¨aumer, Vinay Tripathi, Alireza Seif, Daniel Lidar, Derek S. Wang, arXiv:
2403.09514 (2024)
11. Simulating nonlinear optical processes on superconducting quantum device
Yuan Shi, Bram Evert, Amy F Brown, Vinay Tripathi, Eyob Sete, Vasily Geyko, Yujin Cho, Jonathan DuBois, Daniel Lidar, Ilon Joseph, Matt Reagor, arXiv: 2406.13003(2024)
v
Table of Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
Chapter 1:
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction to Quantum computing . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Superconducting quantum systems . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Dynamical decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Chapter 2:
Crosstalk Suppression with dynamical decoupling . . . . . . . . . . . . . . . 20
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Experimental results for state protection . . . . . . . . . . . . . . . . . . . . 22
2.3 Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 DD for gate operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Chapter 3:
Qudit Dynamical Decoupling on a Superconducting Quantum Processor 45
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Qudit dynamical decoupling theory . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Single qudit dXd experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Cross-Kerr suppressing DD (CKDD) . . . . . . . . . . . . . . . . . . . . . . 51
3.5 Experimental validation of CKDD . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6 Qutrit entanglement preservation via CKDD . . . . . . . . . . . . . . . . . . 54
3.7 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
iv
3.8 Experimental Device Characterization . . . . . . . . . . . . . . . . . . . . . 56
Chapter 4:
Modeling low- and high-frequency noise on a transmon qubit . . . . . . . 59
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Numerical model of transmons . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Open quantum system simulation . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Methodology and Fitting Results . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Model Prediction Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.6 Calibration-independent learning . . . . . . . . . . . . . . . . . . . . . . . . 82
4.7 Extension to two qubits noise model . . . . . . . . . . . . . . . . . . . . . . 86
4.8 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Chapter 5:
Deterministic benchmarking of quantum gates . . . . . . . . . . . . . . . . . 93
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Qubit model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3 Deterministic Benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.5 Fidelity with coherent errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.6 Open system model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.7 Sensitivity to coherent errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.8 Interplay of T1 asymmetry and gates . . . . . . . . . . . . . . . . . . . . . . 104
5.9 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Chapter 6:
Virtual Z gates and symmetric gate compilation . . . . . . . . . . . . . . . . 107
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2 VZ gate in an open quantum system . . . . . . . . . . . . . . . . . . . . . . 110
6.3 Experimental verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Chapter 7:
Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Appendix A: General theory of qudit dynamical decoupling . . . . . . . . 125
A.1 Application to single-axis, pure dephasing noise . . . . . . . . . . . . . . . . 130
A.2 Qutrit (d = 3) dynamical decoupling and 3X3 . . . . . . . . . . . . . . . . . 132
A.3 Proof of first order suppression of cross-Kerr coupling by the CKDD sequence 134
Appendix B: Fidelities of Y Y and XX in open quantum system . . . . . . 137
B.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
B.2 Y Y sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
v
B.3 XX sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
vi
List of Tables
2.1 Specifications of the Ourense and Yorktown devices along with the access
dates of our experiments. The sx (√
σ
x) gate forms the basis of all the single
qubit gates and any single qubit gate of the form U3(θ, ϕ, λ) is composed of
two sx and three rz(λ) = exp(−i
λ
2
σ
z
) gates (which are error-free and take zero
time, as they correspond to frame updates). Figure 2.2 uses qubit 1 (Q1) of
Ourense and qubit 3 (Q3) of Yorktown as the main qubit. . . . . . . . . . . 43
2.2 Specifications of the Quito device along with the access dates of our experiments. Data for Fig. 2.6 was acquired over a period of six days from
12/30/2021 to 01/02/2022 and from 01/07/2022 to 01/08/2022 and the data
for Fig. 2.7 was acquired on 11/30/2021. . . . . . . . . . . . . . . . . . . . . 44
3.1 Transition frequencies ωij = (Ej − Ei)/ℏ up to d = 4 of the qudits employed
in our DD experiments. Additionally, we provide the two-level subspace mean
T1 and T2 echo times of the device calculated from 100 repetitions of each
coherence experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 The qutrit cross-Kerr crosstalk rates present in the spectator DD experiment
in Fig. 3.3(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 The ququart cross-Kerr crosstalk rates present in the spectator DD experiment
in Fig. 3.3(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1 System-bath parameter values extracted using the fitting procedure of Section 4.4.2, and corresponding to the minima indicated by the green circles in
Fig. 4.3 for Quito (top row) and Lima (bottom row). . . . . . . . . . . . . . 82
4.2 Specifications of the Quito (top row) and Lima (bottom row) devices accessed
on September 1, 2021, and January 1, 2023, respectively. The sx (√
σ
x) gate
forms the basis of all the single qubit gates, and any single qubit gate of the
form U3(θ, ϕ, λ) is composed of two sx and three rz(λ) = exp(−i
λ
2
σ
z
) gates
(which are error-free and take zero time, as they correspond to frame updates). 85
vii
List of Figures
1.1 Left: Schematic of a transmon qubit, consisting of a Josephson junction in
parallel with a capacitor. Ic is the Josephson junction critical current, φ is the
superconducting phase across the junction, n is the number of Cooper pairs
transferred across junction, and C is the capacitance of the capacitor. Right:
Cosine potential of transmon with different energy levels. Here, |0⟩ and |1⟩
form the qubit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Energy-level diagrams of (a) capacitive coupling between transmons, and (b)
transmon-transmon effective coupling via a bus resonator. Here, we show only
the bottom two levels per transmon. . . . . . . . . . . . . . . . . . . . . . . 8
2.1 Schematics of the layout of the (a) Yorktown and (b) Ourense, Lima, and
Quito devices. Thin circles indicate the main qubits used in our experiments.
The other qubits are spectators. . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Fidelity of the |+⟩ state of the main qubit, for different spectator qubits’ initial
states {|0⟩, |1⟩, |+⟩}, obtained experimentally [(a) and (b)] and by solving
the Redfield master equation [(c) and (d)]. (a) Results for Ourense. The
free evolution curves all exhibit oscillations with nearly equal periods but
distinct amplitudes. These effects disappear under the application of the XY4
sequence just to the spectator qubits, leaving only a common fidelity decay.
(b) Results for Yorktown. The free evolution curves range from monotonic
to oscillatory. The differences disappear under DD applied to the spectator
qubits, leaving only a common damped oscillation. Error bars denote 95%
confidence intervals. (c) Redfield equation simulation results for a multi-qubit
system with ωd = ωq1
. (d) Simulation results for ωd = ωq1−2J. All qualitative
features observed in (a) and (b) are reproduced in (c) and (d), respectively.
The crosstalk strength considering a two-qubit model is (c) J/2π = 51.55 KHz
and (d) J/2π = 52.63 KHz, obtained by fitting the periods of (a) and (b). . 24
2.3 Results of Fig. 2.2 averaged over all the three spectator qubit states for
Ourense (a) and Yorktown (b). Note that the simulations did not account
for state preparation and measurement errors. . . . . . . . . . . . . . . . . . 25
viii
2.4 Energy level diagram of two coupled transmons with qubit frequencies ωq1
and ωq2 and anharmonicities ηq2 and ηq1
, coupled linearly with strength g.
The solid lines represent the bare energy levels and dashed lines represent
the eigenlevels. |k, l⟩ represents levels k and l in the main and spectator
transmons, respectively. Only 6 levels of the infinite-dimensional Hilbert space
formed by both transmons are shown. . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Circuit diagram for single qubit gate experiments. The top circuit shows a
sequence of random gates of length 4 chosen from the set G = {Rx(±π/8),
Rx(±π/4), Ry(±π/8), Ry(±π/4)} of 8 single qubit gates applied to the main
qubit, and a sequence of Identity operations applied to all the spectator qubits.
U represents the gate applied to prepare any predefined initial state on the
main qubit. In Fig. 2.6, U = Ry(π/2) which prepares a |+⟩ state on the main
qubit. Si represents the gate applied to prepare i = |0⟩, |1⟩ and |+⟩ on all
the spectator qubits, where S|0⟩ = I (Identity). In the bottom circuit, the
XY4 sequence is applied to the spectator qubits in the gaps between the gates
applied to the main qubit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.6 Experimental fidelity results for random sequences of single qubit gates consisting of elements of the set G = {Rx(±π/8), Rx(±π/4), Ry(±π/8), Ry(±π/4)},
averaged over the three spectator states and 100 different experimental runs,
with the main qubit initialized in the |+⟩ state. The fidelity is shown as a
function of time, with and without DD applied to the spectator qubits. The
exponential fit to both the free and the DDPGs case shows a clear improvement in the decay rate, by a factor of 0.0172/0.0081 = 2.12. Error bars
represent 2σ confidence intervals obtained by bootstrapping. Data was acquired over a period of six days from 12/30/2021 to 01/02/2022 and from
01/07/2022 to 01/08/2022. See Table 2.2 for device parameters. . . . . . . . 39
ix
2.7 (a) Optimized pulse and DD sequence placement in the CR-based CNOT gate.
D0-D4 denote drive channels for qubits Q0-Q4 of IBMQ Quito processor. U3
represents CR pulses acting on Q1 at the Q3 frequency. The control and target
qubits are Q1 and Q3, respectively; the rest are the spectator qubits. We apply
the XY4 (or palindromic XY4 or UDD4) sequence to Q0 and Q2, and the pureX sequence to Q4, with pulses placed in gaps between the CR pulses. Note
that one X gate in the DD sequence applied to Q0 and Q2 has been replaced
by the pre-existing X gate on Q1. VZ denotes the virtual Z gate. (b) QST
results after applying 15 (top) and 19 (bottom) CNOT gates. Left: without
DD. Right: with XY4. Clearly, the XY4 results are significantly closer to the
expected Bell state, i.e., equal corner peaks of 0.25. (c) Fidelity of Bell state
preparation after a repeated odd number of up to 29 CNOT gates, averaged
over 5 separate runs with the spectator qubits initialized in |0⟩ (we checked
and found the effect of different initial spectator states to be insignificant).
Error bars represent 95% confidence intervals. CNOT fidelity< 1 at 0 gates
is due to preparation and measurement errors. CNOT with DD takes longer
than without DD since to avoid overlap we inserted delays to accommodate the
two pure-X sequences on D4. The fidelity with DD is statistically significantly
higher than fidelity without DD for all DD sequences we tried after ∼ 3µs, or
∼ 9 consecutive CNOT gates. . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1 Schematic illustration of two transmon qudits with quantized energy levels
affected by relaxation and dephasing errors, along with the qudit-qudit crossKerr couplings αij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Experimental results showing the fidelity of the qudit uniform superposition
state |+⟩d as a function of total time under free evolution (No DD) and dXD,
for (a) qutrits and (b) ququarts. The minimum pulse interval is τmin = 120
ns. For qutrits, we implement 1, 2, and 3 repetitions of the 3X3 sequence,
with corresponding pulse intervals 3τ , 3τ/2, and τ , respectively. The total
evolution time is always 9τ . Universal DD is a 9-pulse long sequence formed
from cycling over the full HWG, applied once with a pulse interval of τ and
τmin = 180 ns. For ququarts, we similarly implement 1, 2, 3, and 4 repetitions
of the 4X4 sequence, with corresponding pulse intervals of 4τ , 2τ ,
4
3
τ , and τ
respectively, where the total time is always 16τ . . . . . . . . . . . . . . . . . 50
x
3.3 Experimental results showing suppression of cross-Kerr interactions using the
CKDD sequence. (a) Fidelity of the qutrit superposition state |+⟩3, with
CKDD (solid) and without (dashed). CKDD is a 9τ -long sequence, with
τmin = 120 ns. (b) Fidelity of the ququart superposition state |+⟩4, with
CKDD (solid) and without (dashed). CKDD is a 16τ -long sequence, with
τmin = 180 ns. As in Fig. 3.2, longer evolution times correspond to a single
repetition of CKDD with increased τ . CKDD removes the cross-Kerr oscillations and improves the fidelity in both cases, converging to the fully mixed
state fidelity baseline (dashed horizontal line) more slowly than the free evolution (No DD) curves. See text for further details. . . . . . . . . . . . . . . 52
3.4 Left: Fidelity of the qutrit Bell state over time with and without CKDD.
See text for details. Right: real (upper) and imaginary (lower) components
of quantum state tomography results for the final time point in the left plot
(red circles). Ideally, ℜ(ρik,jl) = 1
3
δikδjl and ℑ(ρik,jl) = 0 for i, j, k, l ∈ Z3,
where ρ is the density matrix. Left: free evolution. The nine red-colored
bars at the ideal positions are of varying magnitude and some contain large
imaginary components, indicating deviations from the ideal qutrit Bell state.
Right: with CKDD. In contrast, the nine red-colored bars are nearly uniform
in height are have negligible imaginary components, indicating proximity to
the ideal qutrit Bell state. Note that here CKDD is a 9τ -long sequence with
τmin = 180 ns, the X3 gate duration; this differs from the previous figures
where the X3 gate duration is 120 ns due to a different calibration. . . . . . 53
4.1 Gaussian pulse envelope (solid, orange) [see Eq. (4.8)] and its Fourier transform (dashed, blue) with amplitude ε chosen to keep both in the range [0, 1].
(a) and (b) show the pulse with gate time tg = 70 and 10 ns, respectively,
and σ = tg/6. The bottom horizontal axis represents time in ns, and the
top horizontal axis represents frequency in GHz. Shorter gate times result
in a larger frequency spread of the spectrum, with associated larger leakage,
as illustrated in (c), which shows the frequency spectrum corresponding to
a tg = 10 ns gate (left), compared to the energy levels (right) |0⟩, |1⟩ and
|2⟩ of the transmon. The energy levels are shown in the rotating frame such
that E|0⟩ = E|1⟩ and E|2⟩ = −ηq = −200 MHz. As indicated by the dashed
horizontal line, the spectrum overlaps with level |2⟩, resulting in leakage into
this level from the {|0⟩, |1⟩} qubit subspace. The sampling frequency used to
compute the Fourier transform is 10GS/s, which is state-of-the-art in experiments; the pulses that control the IBM processors used in this work have a
sampling frequency of ∼ 5GS/s. . . . . . . . . . . . . . . . . . . . . . . . . . 65
xi
4.2 The circuit schematics for the free-evolution and DD-evolution types of experiments. For the free-evolution case, we apply N cycles of the XY4 dynamical
decoupling (DD) sequence on all the spectator qubits and 2N cycles of the I4
sequence (here I4 means four identity gates) on the main qubit, which suppresses crosstalk errors. Note that an X or Y gate is twice as long as an
identity gate on the IBM cloud quantum devices, hence the extra factor of 2.
For the DD-evolution case, we apply the DD sequence only to the main qubit
and apply identity gates to all the spectator qubits. This suppresses both
crosstalk and environment-induced noise. We measure only the main qubit. . 69
4.3 Top: Results for the Quito processor. Bottom: Results for the Lima processor. Left: The cost function defined as the l2 norm distance between the
experimental and simulation results [Eq. (4.23)], averaged over N = 70 time
instants, as a function of the bath parameters ω
c
x
and gx for free-evolution of
the |1⟩ initial state. Middle: The average of the cost function over the six
Pauli states for DD-evolution as a function of ω
c
z
and gz. Right: The cost
function for free-evolution of the |+⟩ initial state, as a function of γmax and b.
The green circles indicate the positions of the global minima in all the panels. 73
4.4 Results for the Quito (top row) and Lima (bottom row) processors. Left: Box
plots showing the relative error of our model for the free-evolution experiments
as a function of time for 16 different initial states containing six Pauli states
and ten Haar-random states. Right: the same as on the left, but for the
DD-evolution experiments. We measured a total of 70 time instants, up to
a total evolution time of 19.6 µs, but only display every other instant to
avoid overcrowding. Green triangles indicate the mean over the 16 initial
states, black horizontal lines are the median, gray boxes represent the [25, 75]
percentiles, the whiskers (black lines extending outside the boxes) represent
the [0, 25] and [75, 100] percentiles, and circles are outliers. . . . . . . . . . . 77
4.5 Relative error results for the Quito processor. We display a comparison of
the relative errors between the full model (which uses a three-step learning
procedure and consists of four energy levels per transmon and realistic pulses)
with the simplified models SM1 and SM2 (which are based on a two-step
learning procedure and use just two energy levels and instantaneous pulses)
for free-evolution and DD-evolution experiments. SM1 (SM2) is trained on
the DD (free) evolution experiments. Each box contains a total of 16 initial
states and all 70 time instants varying from 0 to 19.6 µs. . . . . . . . . . . . 78
xii
4.6 Lindblad equation simulation results for the Quito processor. Left: Box plots
showing the relative error of the Lindblad simulations for the free-evolution
experiments as a function of time for 16 different initial states containing six
Pauli states and ten Haar-random states. Right: the same as on the left, but
for the DD-evolution experiments. We measured a total of 70 time instants,
up to a total evolution time of 19.6 µs, but only display every other instant
to avoid overcrowding. Green triangles indicate the mean over the 16 initial
states, black horizontal lines are the median, gray boxes represent the [25, 75]
percentiles, the whiskers (black lines extending outside the boxes) represent
the [0, 25] and [75, 100] percentiles, and circles are outliers. . . . . . . . . . . 82
4.7 Integrated relative error results for the Lima processor. We compare the relative errors between the free-evolution and DD-evolution experiments. Each
box contains 16 initial states and all 70 time instants varying from 0 to 19.6 µs.
The color bar indicates the time evolved for the outliers. All the outliers correspond to times longer than 11 µs. . . . . . . . . . . . . . . . . . . . . . . 84
4.8 Fidelity results for the Quito processor, for the 16 different initial states of
the main qubit. The caption of each of the panels gives (θ, ϕ, λ) in degrees,
parametrizing the initial state |ψ⟩ = U3(θ, ϕ, λ)|0⟩ (panels are arranged in
increasing order of θ, the polar angle with the z-axis). These are the 6
Pauli states (panels 1,9-12,16) and the 10 Haar-random states. Blue curves
(squares): no DD is applied, resulting in coherent oscillations due to crosstalk.
Orange curves (diamonds): DD (XY4) is applied just to the spectator qubits;
the resulting suppression of crosstalk between the main qubit and the spectator qubits removes the oscillations. These are what we call the free-evolution
experiments in the main text. Green curves (circles): DD (XY4) is applied just
to the main qubit, suppressing both crosstalk and errors due to environmentinduced noise. Results are averaged over three different runs of experiments.
All of the data was acquired on Sep. 1, 2021. Error bars are smaller than the
markers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.9 Fidelity results for the Lima processor, for the 16 different initial states of
the main qubit. The caption of each of the panels gives (θ, ϕ, λ) in degrees,
parametrizing the initial state |ψ⟩ = U3(θ, ϕ, λ)|0⟩ (panels are arranged in
increasing order of θ, the polar angle with the z-axis). These are the 6
Pauli states (panels 1,9-12,16) and the 10 Haar-random states. Blue curves
(squares): no DD is applied. In contrast to Fig. 4.8, there are no cross-talk oscillations. This is because all the spectator qubits are kept in the ground state
|0⟩, and calibration for Lima is done in the same spectators’ state. These are
what we call the free-evolution experiments for Lima in the main text. Green
curves (circles): DD (XY4) is applied just to the main qubit, suppressing errors due to environment-induced noise. All of the data was acquired on Jan.
1, 2023. Error bars are smaller than the markers. . . . . . . . . . . . . . . . 92
xiii
5.1 Schematic representations of the gate sequences for (a) randomized benchmarking (RB) and (b) deterministic benchmarking (DB), respectively. (c)
Experimental fidelity measurement of |+⟩ state for various gate sequences, revealing coherent oscillations with XX and Y Y sequences. These oscillations
indicate the presence of phase errors δϕ and rotation errors δθ. In contrast,
RB exhibits an average Clifford gate infidelity rClifford = 0.262±0.004%. Solid
curves are fits obtained using Eq. (5.3), whereas dashed curves denote the
Lindblad master equation based simulation results. . . . . . . . . . . . . . . 97
5.2 Experimental results showing the sensitivity of DD sequences and RB to coherent errors (a) δθ = 0◦
, δϕ = 0◦
, (b) δθ = 0◦
, δϕ = 0.893 ± 0.002◦
, (c)
δθ = 0.932 ± 0.007◦
, δϕ = 0◦
, (d) δθ = 0.995 ± 0.009◦
, δϕ = 0.90 ± 0.002◦
.
UR6 is robust even to large coherent errors and hence is included as a reference. Note that δϕ is controlled by varying the DRAG parameter. . . . . . . 102
5.3 Asymmetry in the decay pattern of DD sequences due to the interplay of
gates and asymmetric T1. (a) Fidelity decay as a function of repetition number
(shown in units of time) for the XX sequence applied to two sets of orthogonal
initial states. Applying XX to |±i⟩ states results in an asymmetric decay,
highlighting the impact of the relaxation errors throughout the Bloch sphere.
This pattern is also observed for orthogonal initial states defined by angles
(θ, ϕ) = (45◦
, 45◦
) and (45◦
, −135◦
). Dashed curves represent the Lindblad
master equation simulation results. (b) Similar to (a), but with Y Y sequences
applied to |±⟩ states and the same additional orthogonal states as in (a). . . 105
6.1 The open system effect of the symmetric and asymmetric compilation of the Y
gate with respect to VZ gates. (a) The |±i⟩ state follows different Bloch sphere
trajectories under Y
asym, which consists of an instantaneous VZ gate followed
by a physical X gate. This causes |−i⟩ (|+i⟩) to go through a stable (unstable)
ground (excited) state which leads to the asymmetry in the fidelity of the two
states. The symmetric decomposition Y
sym overcomes this asymmetry, similar
to a physical Y gate. (b) Experimental demonstration of the symmetric and
asymmetric effects of the Y -gate decomposition on the MUNINN processor. The
fidelity of the states |±i⟩ is shown under both Y
asym and Y
sym, as a function
of time (bottom axis) or number of Y Y sequence cycles (top axis). The
symmetric decomposition results in similar fidelities (black and red) for the
initial states |±i⟩. The asymmetric decomposition results in very different
fidelities (yellow and green) for the same two initial states. Error bars denote
two standard deviation of the mean. . . . . . . . . . . . . . . . . . . . . . . . 108
xiv
6.2 Effect of the asymmetric Y
asym and symmetric Y
sym gates on state fidelity
for various qubits of the ibm sherbrooke processor. We applied different
repetitions of the Y Y sequence and observe that subject to Y
sym the fidelities
of the |±i⟩ states (black and red) are much closer than subject to Y
asym
(yellow and green). Different subfigures correspond to different qubits on
ibm sherbrooke: (a) qubit 0, (b) qubit 13, (c) qubit 81, and (d) qubit 89. . 112
6.3 The |+⟩ state fidelity as a function of total sequence time (or number of sequence cycles; top axis), subject to the XY4sym, XY4asym, UR4, Y Y sym, and
XX sequences applied to a single qubit (qubit 37) on the ibm sherbrooke device. Data points for the two-pulse-long Y Y sym and XX sequences are shown
for every second cycle (i.e., their total number of cycles is 640). The XY4asym
and UR4 sequences exhibit nearly identical fidelity decay behavior, clearly
distinct from that of the XY4sym sequence, confirming that the asymmetric
Y gates transform XY4 into the UR4 sequence. All sequences shown exhibit
oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4 As in Fig. 6.3 (except for the absence of XY4asym) with the pulse interval
doubled from τ = 56.8 ns to 2τ = 113.6 ns. The oscillation periods of Y Y sym
and XX increase significantly, and the difference between the now decaying
XY4sym and UR4 fidelities is nearly eliminated. . . . . . . . . . . . . . . . . 118
6.5 Experimental study of the fidelities of the XY4sym (blue) and UR4 (red) sequences with different pulse intervals (1τ , 2τ , and 3τ ) for the whole set of
127 qubits on ibm sherbrooke. (a), (b) and (c) show the results for 123 of
the qubits for the noted pulse intervals. (c), (d) and (e) show the corresponding mean and standard deviations. The oscillations in fidelities vanish as we
increase τ due to mitigation of the pulse interference effects. . . . . . . . . . 120
A.1 Numerical simulation of the the universality of the DD sequence generated
by cycling over the HWG for 2 ≤ d ≤ 10. Here we plot the infidelity of the
resulting unitary evolution as a function of the pulse interval time τ . Since the
sequence is expected to cancel the errors to the first order O(τ
2
), the infidelity
should scale as O(τ
4
), as confirmed by our simulations. . . . . . . . . . . . . 131
xv
Abstract
Superconducting quantum systems, particularly those based on transmon qubits, have become one of the leading platforms for quantum computing research. Essential to the advancement of these systems are the challenges of crosstalk errors, accurate noise modeling,
reliable benchmarking of quantum gates, and the development of efficient multi-level quantum processors, or qudits. This dissertation explores these critical areas to enhance the
performance and scalability of superconducting quantum processors.
We begin by addressing crosstalk errors, one of the major challenges in superconducting
qubits. By exploring the underlying physics and developing strategies using dynamical decoupling, we demonstrate effective methods to suppress qubit-qubit ZZ crosstalk errors. Both
experimental results and theoretical analyses, showcasing significant reductions in crosstalk,
and thus, enhanced state protection and gate performance in coupled qubit systems. Extending the study to transmon qudits, we introduce a general framework for utilizing dynamical
decoupling to suppress low-frequency noise and qudit-qudit crosstalk errors. This work generalizes the ZZ suppression technique for qubits and is supported by experimental results.
We then focus on characterization of both low- and high-frequency noise components in
transmon qubits. Using hybrid Redfield model, we learn the noise parameters and elucidate
the effects of environmental noise on qubit performance. Furthermore, we develop deterministic benchmarking techniques for single qubit gates providing strategies to quantify gate
infidelities in terms of coherent and incoherent contributions. Finally, we investigate the
role of virtual Z gates in gate compilation strategies within open quantum systems, offering
explanations for several previously observed phenomena in dynamical decoupling studies.
xvi
Chapter 1
Introduction
1.1 Introduction to Quantum computing
The idea of a quantum computer was first proposed by Richard Feynman in 1982 [1]. He highlighted the limitations of classical computers in modeling quantum phenomena, and hence
new computing platforms using the quantum mechanical systems are needed to perform
tasks that rely on the fundamental quantum mechanical principles. It laid the groundwork
for several future discoveries and the development of the whole field of quantum information
and computation.
In 1984, Bennett and Brassard introduced the field of quantum cryptography, which
turned out to be a pivotal moment in quantum information science [2]. Shortly thereafter,
Deutsch conceptualized the universal quantum computer [3]. For a period, quantum computing remained a field of theoretical interest, which changed with the introduction of Shor’s
factoring algorithm. This algorithm demonstrated the potential for efficient prime factorization using a quantum computer [4, 5]. This breakthrough captured widespread interest
and showcased the practical implications of quantum computing. Another seminal contribution came from Grover who developed a quantum algorithm for database search [6], further
illustrating the unique capabilities of quantum computers.
The development of quantum error correction codes by Shor, Calderbank, and Steane
1
propelled the field forward, since error-correcting codes are critical for realizing scalable
and fault-tolerant quantum computers [7, 8, 9]. Kitaev’s proposal of topological quantum
computing using anyons opened new avenues for robust quantum computations [10].
A comprehensive framework for quantum computing implementation was put forward
by DiVincenzo, which outlined essential criteria that any quantum computer must meet,
now popularly know as the DiVincenzo criteria [11]. These criteria emphasize scalability,
qubit readout, reliable initialization, prolonged coherence times, and the ability to perform
a universal set of quantum gates.
Among the various implementations of qubits, superconducting circuits have emerged
as one of the leading candidates. They not only meet DiVincenzo’s requirements but also
benefit from established semiconductor fabrication techniques, making them relatively easy
to construct and scale. In the next section, we discuss the superconducting quantum systems.
1.2 Superconducting quantum systems
Superconducting quantum circuits typically consist of Josephson junctions [12], which are
formed by placing a thin insulating barrier between two superconductors. This setup allows
for the tunneling of Cooper pairs across the barrier, resulting in supercurrent. The macroscopic quantities such as the current flowing through the junction or the voltage developed
across it, are governed by a macroscopic order parameter. This order parameter is a complex
quantity that describes the collective state of superconducting electrons. It encapsulates the
collective properties of the Cooper pairs that form the superconducting state that governs
the dynamics of the Josephson junction and determines how macroscopic quantities like
current and voltage behave. These qualities enable the precise control and long coherence
times necessary for the reliable manipulation of the qubits, thus making it ideal for quantum
computing and advanced quantum technologies.
The first experimental demonstration of macroscopic quantum effects in superconducting
2
systems was reported by Clarke et al. [13] in 1988. In 1999, time-resolved coherent oscillations were observed in a superconducting qubit [14], which was followed by the demonstration
of coupled qubits in 2003 Pashkin et al. [15] and Yamamoto et al. [16]. The idea of circuit
quantum electrodynamics (cQED) was introduced in 2004 [17], followed by the first experimental implementations with gate times smaller than coherence times [18, 19]. Circuit QED
is the study of the interaction between light and matter in superconducting circuits, where
microwave photons interact with superconducting qubits. It extends the principles of cavity
QED to integrated circuits, enabling strong coupling and coherent manipulation of quantum
states.
The first quantum algorithm on superconducting systems was performed by DiCarlo et
al. [20]. Significant strides have been made in error correction using superconducting systems, with recent contributions from both academic [21, 22, 23] and industrial labs [24, 25].
The landmark experiment of quantum supremacy was first performed on superconducting
processors [26], followed by recent developments in quantum utility experiments [27].
1.2.1 LC oscillator
We begin with an oscillating LC circuit, and explore how quantum effects manifest in this
system. The classical Hamiltonian of the LC oscillator is given by:
HLC =
Q2
2C
+
Φ
2
2L
, (1.1)
where Q is the charge on the capacitor and Φ is the flux threading the inductor. In the
quantum regime, the charge and flux variables are promoted to non-commuting observables
that satisfy the commutation relation
h
Φˆ, Qˆ
i
= iℏ. (1.2)
3
Introducing the creation and annihilation operators ˆa and ˆa
† of the harmonic oscillator, we
define
Φ = Φ ˆ
zpf(ˆa + ˆa
†
), Qˆ = iQzpf(ˆa
† − aˆ) (1.3)
with Φzpf =
p
ℏ/2ωrC and Qzpf =
p
ℏωrC/2, where ωr = 1/
√
LC. The Hamiltonian of the
LC oscillator then reduces to
HLC = ℏωr
nˆ +
1
2
, (1.4)
where ˆn = ˆa
†aˆ. Here, ˆa
†
creates a quantized excitation of the flux and charge degrees of
freedom of the oscillator, or equivalently, of the magnetic and electric fields. In other words,
aˆ
†
creates a photon of frequency ωr stored in the circuit.
To observe this quantization effect, the oscillator must have minimal decoherence, meaning its energy levels should be well-separated and narrower than the separation between
them. This is quantified in terms of the quality factor Q =
ωr
κ
, where κ is the oscillator
linewidth or the photon loss rate. The quantity 1/κ represents the lifetime of the photon
in a given energy level. A higher Q indicates a higher quality oscillator, making it easier to
observe quantization. Additionally, thermal excitations must be minimized, i.e., ℏωr ≫ kBT.
Therefore, it is crucial to maintain the temperature T much lower, typically in the range of
10 − 20 mK. Currently, 2D coplanar waveguide resonators are used, to perform circuit QED
experiments, which involves coupling the LC type oscillators to non-linear elements (qubits)
to manipulate and measure them. Interested readers can refer to detailed descriptions in
Refs. [17, 19]. A comprehensive review of circuit QED can be found here [28].
1.2.2 Transmon circuit
In a linear LC oscillator circuit, the energy levels are evenly spaced, which is not suitable for
performing quantum computation. This is because control pulses at a specific frequency can4
Figure 1.1: Left: Schematic of a transmon qubit, consisting of a Josephson junction in parallel
with a capacitor. Ic is the Josephson junction critical current, φ is the superconducting phase
across the junction, n is the number of Cooper pairs transferred across junction, and C is
the capacitance of the capacitor. Right: Cosine potential of transmon with different energy
levels. Here, |0⟩ and |1⟩ form the qubit.
not distinguish between different transitions. To resolve this issue, we introduce non-linearity
to the circuit using Josephson tunnel junction [12]. As discussed earlier, a Josephson junction (JJ) consists of two superconducting electrodes separated by a thin insulating barrier.
Josephson showed that in this configuration, a supercurrent can flow between the electrodes,
described by the relation I = Ic sinφ, where Ic is the critical current (a property of the
junction) and φ is the phase difference between the superconducting islands on either side
of the junction. The phase difference φ is also related to the voltage across the junction by
dφ/dt = 2πV/Φ0, where Φ0 = h/2e is the flux quantum. The phase φ can also be written
in terms of the flux variable as φ(t) = 2πΦ(t)/Φ0 (mod 2π), where “mod 2π” signifies that
the superconducting phase φ is a compact variable on the unit circuit, while the flux Φ can
take arbitrary real values. Using these relations, we can define the Josephson inductance as:
LJ (Φ) =
∂I
∂Φ
=
Φ0
2πIc
1
cos(2πΦ/Φ0)
, (1.5)
5
which shows the non-linear inductance effect in JJ and thus a non-linear potential energy
given by:
E =
Z
dtV (t)I(t) = Z
dt
dΦ
dt
Ic sin
2πΦ
Φ0
= −EJ cos
2πΦ
Φ0
, (1.6)
where EJ = Φ0Ic/2π is the Josephson energy.
By replacing the inductor in the LC oscillator with a JJ, as shown in Fig. 1.1, the
quantized Hamiltonian takes the form:
Htransmon = 4EC(ˆn − ng)
2 − EJ cos ˆφ, (1.7)
where EJ is the Josephson energy as discussed above, EC = e
2/2C is the charging energy
due to the capacitor, ˆφ is the quantum operator for superconducting phase across the JJ,
and ˆn is the canonically conjugate operator corresponding to number of Cooper pairs passing
through the JJ. The circuit representing the above Hamiltonian is shown in Fig. 1.1, and is
called a transmon [29], when the Josephson energy is much larger than the capacitive energy
(EJ /EC ≈ 102
).
In contrast to the parabolic potential for LC oscillator, the cosine potential of transmon
results in a small anharmonicity between the energy levels, making it suitable to use as
qubit. Approximately, nine levels exist in the potential well of the transmon and the charge
dispersion due to fluctuations in ng increases with higher levels. However, the lowest two
levels are almost unaffected by this charge noise (due to the large ratio of EJ and EC chosen
in designing transmons), making them suitable for forming qubits. Recent advancements in
design and control have helped explore higher levels of transmons [30, 31, 32]. In Chapter 3,
we explore qutrits (d = 3) and ququarts (d = 4) of transmon circuit and design dynamical
decoupling schemes to suppress the effects of the low-frequency charge noise on the qudit
level.
In Eq. (1.7), since EJ ≫ EC, the phase ˆφ would be confined to a small value with a much
6
smaller variance i.e., ∆φ ≪ 1. Therefore, under the EJ /EC ≫ 1 limit, we can expand the
cosine term in Eq. (1.7) around φ = 0 as:
Htransmon ≈ 4Ec(ˆn − ng)
2 − EJ
1 −
φˆ2
2
−
1
4!φˆ4
!
. (1.8)
Combining the first and the third terms on the right-hand side of the above equation and
again introducing the Harmonic oscillator raising and lowering operators such that:
φ =
8EC
EJ
1
4 aˆ + ˆa
†
√
2
, (1.9)
nˆ − ng =
8EC
EJ
− 1
4 aˆ − aˆ
†
√
2i
, (1.10)
we get a harmonic oscillator with a correction that gives the anharmonicity. Making rotating
wave approximation (RWA) where we only keep the terms with net zero excitations, we get:
Htransmon = ωqaˆ
†
aˆ +
η
2
aˆ
†
aˆ(ˆa
†
aˆ − 1), (1.11)
where the qubit frequency ωq =
√
8ECEJ − EC and the anharmonicity η = −EC. Note this
form is also called the Duffing (Kerr) oscillator model. The RWA is valid because in a frame
rotating with frequency ωq, any terms with an unequal number of ˆa and ˆa
† will oscillate.
If the frequency of these oscillations is larger than the prefactor of the oscillating term, i.e.
ωq ≫ |η|/2, the term rapidly averages out and can be neglected. The above equation can be
diagonalized to give:
Htransmon =
X
k∈Z≥0
h
kωq +
η
2
k(k − 1)i
|k⟩⟨k| (1.12)
Transmon spectrum can be made more accurate by including the next order correction of
cos ˆφ i.e., O( ˆφ
6
) in Eq. (1.8).
7
0,0
1,0
0,1
1,1
�! �"
�"
0,0,0
1,0,0
0,0,1
1,0,1
�"
�! �"
�"
�!
0,1,0
�#
(a) (b)
J Je↵
Figure 1.2: Energy-level diagrams of (a) capacitive coupling between transmons, and (b)
transmon-transmon effective coupling via a bus resonator. Here, we show only the bottom
two levels per transmon.
1.2.3 Coupling two transmons
The next step in building a universal quantum computer is to couple two transmons to
perform entangling operations. Transmons can be coupled directly through capacitive coupling or via a resonator. In the simpler case of capacitive coupling, the interaction can
be thought of as dipole-dipole (transversal) interaction, and the total Hamiltonian of two
coupled transmons is given as:
Hcoupled =
X
i=1,2
Htransmon,i + Hcoupling, (1.13)
=
X
k,l
h
kω1 + lω2 +
η1
2
k(k − 1) + η2
2
l(l − 1)i
|k, l⟩⟨k, l| + g nˆ1nˆ2. (1.14)
Using the harmonic oscillator basis ˆn ∝ i(−aˆ + ˆa
†
), the coupling term reduces under RWA
to:
Hcoupling = −J(ˆa
†
1 − aˆ1)(ˆa
†
2 − aˆ2) ≈ J(ˆa
†
1aˆ2 + ˆa
†
2aˆ1), (1.15)
where g and the normalization constant are absorbed in J. Since J is much smaller than
∆ = (ω1−ω2), the coupling J hybridizes the bare states |01⟩ and |10⟩, as shown in Fig. 1.2(a).
8
The dressed states (denoted with a bar on top) in the first-order perturbation theory become:
|10⟩ = |10⟩ +
J
∆
|01⟩, |01⟩ = |01⟩ − J
∆
|10⟩ (1.16)
Furthermore, |11⟩ is repelled by both |20⟩ and |02⟩, and hence the energy of the qubit
computational states are:
E00 = E00, (1.17)
E01 = E01 −
J
2
∆
, (1.18)
E10 = E10 +
J
2
∆
, (1.19)
E11 = E11 +
2J
2
∆ − η2
−
2J
2
∆ + η1
. (1.20)
The computational subspace consisting of |00⟩, |01⟩, |10⟩, and |11⟩ can be equivalently described by an effect Hamiltonian of the form
Hcoupled =
1
2
(ωZIZI + ωIZIZ + ωZZZZ) (1.21)
where
ωZI = −ω1 − ωZZ −
J
2
∆
, (1.22)
ωIZ = −ω2 − ωZZ +
J
2
∆
, (1.23)
ωZZ =
J
2
(η1 + η2)
(∆ + η1)(∆ − η2)
. (1.24)
This represents the origin of the ZZ crosstalk in coupled transmon qubits, which is further
explored in Chapter 2. For transmon qudits, this leads to multiple ZZ type crosstalk errors
in different qubit-qubit subspaces, also known as cross-Kerr interactions, which are studied
in Chapter 3.
9
In practice, transmons are typically coupled via a bus resonator. This scenario is depicted
in Fig. 1.2 (b) where each transmon is coupled to the same resonator, thus providing an
effective coupling between the transmons. The total Hamiltonian is described by:
Hcoupled =
X
i=1,2
Htransmon,i + ωraˆ
†
aˆ +
X
i=1,2
gi
aˆ
†
iaˆr + ˆaiaˆ
†
r
(1.25)
Considering the transmon-resonator-transmon energy level diagram shown in Fig. 1.2(b),
the energy levels |k, l, m⟩ represent the levels k, l, and m of the first transmon, resonator and
second transmon, respectively. In the dispersive regime gi/|ωi − ωr| ≪ 1 and ωr > ω1, ω2,
coupling g1 between the first transmon and resonator hybridizes the two states |100⟩ and
|010⟩, resulting into the dressed state |010⟩
g1
as:
|010⟩
g1
= |010⟩ +
g1
ωr − ω1
|100⟩. (1.26)
If we now consider the coupling between the dressed state |010⟩
g1
and the excited state of
the second transmon |001⟩ via the coupling g2, we get:
|001⟩ = |001⟩ − g2
ωr − ω2
|010⟩
g1
(1.27)
= |001⟩ − g2
ωr − ω2
|010⟩
g1
ωr − ω1
|100⟩
. (1.28)
Comparing the above equation with Eq. (1.16), we find:
Jeff
∆eff
=
g1g2
(ωr − ω1)(ωr − ω2)
. (1.29)
Since the interaction between |100⟩ and |001⟩ is through a virtual transition to state |010⟩,
the detuning ∆eff should be either ωr − ω1 or ωr − ω2. If we consider ωr ≫ ω1, ω2, we can
consider the detuning to be the average of the two cases, ∆eff = ωr −(ω1 +ω2)/2. Therefore,
10
we have:
Jeff =
g1g2(ω1 + ω2 − 2ωr)
2(ωr − ω1)(ωr − ω2)
(1.30)
Eq. (1.25) thus reduces to:
Hcoupled =
X
i=1,2
Htransmon,i + ωraˆ
†
aˆ + Jeff(ˆa
†
1aˆ2 + ˆa1aˆ
†
2
). (1.31)
In summary, the coupling of the two transmons via a resonator can be reduced to an effective
direct coupling between the two transmons whose strength depends on the frequency of the
resonator and the transmons, and coupling between each transmon and the resonator.
1.2.4 Quantum control via microwave drives
The single qubit gates are mostly implemented via the microwave drive, whereas two-qubit
gates can be implemented in various ways. These include tuning the coupling between the
two transmons either by modifying their frequencies or the bare coupling strength [33, 34].
For fixed coupling, microwave pulses can also create entangling gates [35, 36]. Another
approach involves parametric modulation [37, 38, 39] of the system parameters such as qubit
frequencies or coupling strengths. Discussing all these implementations is beyond the scope
of this thesis, and readers are encouraged to refer to excellent reviews on the topic [28, 40].
Here, we focus only on the single qubit gates.
Transmons are driven externally (with a microwave source) via a capacitive coupling with
a resonator (see e.g., Fig. 12 in Ref. [40]). The Hamiltonian of the transmon under such a
drive (applied via charge degree of freedom) is given by:
Hˆ = Hˆ
transmon + Hˆ
drive (1.32)
11
where
Hˆ
drive = ε(t) cos(ωdt + ϕ0)ˆn, (1.33)
We perform a change of basis to the energy eigenbasis of the transmon (which diagonalizes
the transmon Hamiltonian Hˆ
transmon). In that basis, we have:
Hˆ eigen
transmon =
X
k≥0
ωk|k⟩⟨k|, (1.34)
Hˆ eigen
drive = ε(t) cos(ωdt + ϕ0)
X
k,l≥0
⟨k|nˆ|l⟩|k⟩⟨l|. (1.35)
Defining the charge matrix elements gk,l = ⟨k|nˆ|l⟩ and keeping the excitation preserving
couplings under the RWA, we get:
Hˆ eigen
drive = ε(t) cos(ωdt + ϕ0)
X
k≥0
gk,k+1(|k + 1⟩⟨k| + |k⟩⟨k + 1|). (1.36)
To simplify the analysis, we move into the rotating frame of the drive by using the transformation Uˆ
d = e
iωdN t ˆ
, where Nˆ =
P
k≥0
k|k⟩⟨k| is the transmon number operator (different
from ˆn, the Cooper pairs number operator). Note that Nˆ, when restricted to the computational subspace, equals ( ˆI − Zˆ)/2, and the transformation reduces to Uˆ
d = e
−iωdtZ/ˆ 2
. The
effective Hamiltonian under this transformation is:
H˜ = Uˆ
d
Hˆ eigen
transmon + Hˆ eigen
drive
Uˆ†
d + i
˙
Uˆ
dUˆ†
d
(1.37)
=
X
k≥0
ωke
iωdN t ˆ
|k⟩⟨k|e
−iωdN t ˆ
+ Uˆ
dHˆ eigen
driveUˆ†
d −
X
k≥0
kωd|k⟩⟨k| (1.38)
=
X
k≥0
(ωk − kωd)|k⟩⟨k| + Uˆ
dHˆ eigen
driveUˆ†
d
. (1.39)
Driving off-resonantly with the transition frequency of the computational subspace (ωd ̸= ω1,
where ωq = ω1−ω0 = ω1) will cause additional coherent phase rotations in the rotating frame.
12
Taking ϕ0 = 0, we get:
Uˆ
dHˆ eigen
driveUˆ†
d = ε(t) cos(ωdt)
X
k≥0
gk,k+1
e
iωdN t ˆ
|k + 1⟩⟨k|e
−iωdN t ˆ
+ e
iωdN t ˆ
|k⟩⟨k + 1|)e
−iωdN t ˆ
(1.40)
= ε(t)
e
iωdt + e
−iωdt
2
X
k≥0
gk,k+1
e
iωdt
|k + 1⟩⟨k| + e
−iωdt
|k⟩⟨k + 1|
(1.41)
≈
ε(t)
2
X
k≥0
gk,k+1 (|k + 1⟩⟨k| + |k⟩⟨k + 1|) (1.42)
where we have made RWA in the last equation. Eq. (1.42) represents a rotation about the
X-axis on the Bloch sphere. If we choose ϕ0 = π/2 instead, we would obtain:
Uˆ
dHˆ eigen
driveUˆ†
d ≈
iε(t)
2
X
k≥0
gk,k+1 (|k + 1⟩⟨k| − |k⟩⟨k + 1|) (1.43)
which is a rotation about the Y -axis in the computational subspace. The drive also induces
excitations to higher levels, causing leakage out of the computational subspace. Additionally,
it can induce a large ac Stark shift by driving the higher level transitions off-resonantly,
leading to the repulsion of the computational levels, leading to an off-resonant drive during
the single-qubit gates. Both the leakage and the ac Stark shift can be addressed using
derivative removal by adiabatic gate (DRAG) [41, 42].
1.3 Noise
Several factors contribute to errors in a transmon-based quantum computing circuit. These
can be divided into two categories: coherent and incoherent. Coherent errors arise from the
intrinsic flaws in the system and the control circuitry and are systematic in nature. ZZ
crosstalk (see Chapter 2) and systematic errors in the gates (see Chapter 5) are the usual
sources of coherent errors.
The decoherence sources include thermal effects, fabrication impurities, and material de1
fects, such as two-level systems (TLSs) residing in the bulk dielectric, etc. While minimizing
the coupling of the system to all the environmental degrees of freedom is crucial to reduce
decoherence, it is equally essential to control and measure the system effectively; otherwise,
a high qubit lifetime is of limited practical use.
The readout circuitry introduces additional coupling, which consequently introduces new
sources of noise into the system. Readout-related errors primarily arise from thermal photons
in the resonator that is dispersively coupled to the transmon, resulting in qubit dephasing.
1.3.1 Modeling open quantum system
Superconducting circuits, including transmons, are coupled to environmental degrees of freedom. This coupling leads to interactions between the quantum system and its surrounding
environment, resulting in decoherence and hence the loss of quantum information. The
Hamiltonian describing the interaction between a single quantum system and its environment can be expressed as:
HSB =
X
i
gi Si ⊗ Bi
, (1.44)
where Si and Bi represent the dimensionless system and bath coupling operators, respectively, and the coupling strengths gi have dimensions of energy. Eq. (1.44) is a general form
that apply for both qubits as well as qudits. Additionally, if Bi are not operators but scalars,
the above equation captures the interaction with a classical bath. In Chapter 4, we explore
a detailed bath consisting of both quantum elements and classical two-level fluctuators.
We now describe the most popular method to model the decoherence effects in quantum systems, i.e., the Gorini–Kossakowski–Sudarshan–Lindblad equation (GKSL) or simply
Lindblad equation (LE). This is one of the general forms of Markovian master equation describing the open quantum systems. Given a system with quantum state ρ and Hamiltonian
14
Hsys, undergoing a coupling to the bath with its own Hamiltonian HB, where the coupling
HSB as given in Eq. (1.44), total Hamiltonian is:
Htot = Hsys + HSB + HB. (1.45)
The dynamics of the reduced system, after tracing out the bath completely, can be described
as:
dρ
dt = −
i
ℏ
[H, ρ] +X
k
γk
LkρL†
k −
1
2
{L
†
kLk, ρ}
, (1.46)
where Lk are the Lindblad operators representing different decoherence processes and γk
represent their strengths. The first term on the right-hand side represents the unitary evolution of the system (in the absence of the bath), while the second term accounts for the
non-unitary effects due to interactions with the bath. Lindblad master equation ensures that
the density matrix ρ remains positive and trace-preserving, which are essential for a valid
physical state. This makes the Lindblad formalism widely applicable to quantum optics,
quantum information, and other fields where accurate modeling of open quantum systems is
crucial.
Despite these advantages, LE has its limitations due to its dependence on Born-Markov
approximation and the secular or rotating wave approximation (RWA), which discards some
the underlying physical processes, providing no insights into how specific modes affect the
system. This means that LE cannot capture non-Markovian effects and can become invalid when the system Hamiltonian Hsys has fast time-dependence. An alternative could
be Nakajima-Zwanzig equation which does not involve such approximations or the postMarkovian master equation (PMME) [43], both of which are computationally expensive. A
simpler alternative is the Redfield master equation [44], which requires the Born-Markov
approximation but not the secular approximation, making it useful to capturing some of the
complicated noise phenomenon under weak-coupling assumption. It depends on defining a
15
noise spectrum and also includes temperature effects.
Using Redfield master equation requires first defining the standard bath correlation function:
Cij (t) = Tr{UB(t)BiU
†
B
(t)BjρB}, (1.47)
where UB(t) = e
−iHBt
is the unitary evolution operator generated by the bath Hamiltonian
HB, and the reference state ρB is the Gibbs state of HB:
ρB =
e
−βHB
Tr (e
−βHB )
, (1.48)
where β = 1/(KBT) is the inverse Temperature. Assuming that the bath operators are
uncorrelated, Cij = δijCii(t) ≡ Ci(t), the Redfield equation takes the form:
∂ρ
∂t = −i[Hsys, ρ] + LR(ρ) (1.49)
where LR is the Redfield Liouville superoperator
LR = −
X
i
[Si
,Λi(t)ρ(t)] + h.c., (1.50)
and
Λi(t) = Z t
0
Ci(t − τ )Usys(t, τ )SiU
†
sys(t, τ )dτ. (1.51)
Here, Usys(t) is the unitary evolution operator generated by the system Hamiltonian Hsys.
The bath correlation function Ci(τ ) are the Fourier transforms of the bath noise spectra:
γi(ω) = Z ∞
−∞
Ci(τ )e
iωτdτ. (1.52)
16
A standard choice for the bath’s noise spectra is Ohmic, given by:
γi(ω) = 2πηg2
i
ωe−|ω|/ωc
i
1 − e
−βω . (1.53)
In Chapter 4, we use a hybrid version of the above Redfield master equation, where we add
classical fluctuators along with the quantum baths with Ohmic noise spectra.
1.4 Dynamical decoupling
In this section, we provide a brief introduction to the concept of dynamical decoupling
(DD). Throughout this thesis, we will use ideas of DD both for suppressing errors in the
superconducting systems as well as for characterizing and benchmarking quantum gates.
DD is a technique used to suppress the effects of decoherence in quantum systems by
applying a series of control pulses. The primary goal is to average out unwanted interactions
between the quantum system and its environment. Consider a simple example of a single
qubit coupled with the pure dephasing system-bath coupling Hamiltonian:
Hdeph = σ
z ⊗ B
z
(1.54)
and the system Hamiltonian:
HS = λ(t)σ
x
, (1.55)
where λ(t) represents time-dependent control pulses applied at different intervals, on the
qubit. Note that the pure bath Hamiltonian HB is not considered here for simplicity. We
turn on the pulses for a small duration δ and with fixed amplitude λ and then turn it off,
and again repeat the same process with period τ , such that λδ = π/2, then the control
Hamiltonian HS results in a unitary operation denoted by X = e
−iδλσx
= −iσx
. In the
17
case of ideal pulses, i.e., δ → 0 and λ → ∞, the system-bath unitary evolution after two
repetitions of the pulses with total time 2τ is given by:
f
′
2τ = XfτX
†
fτ = σ
x
e
−iτHdeph σ
x
e
−iτHdeph = e
−iτσxHdephσ
x
e
−iτHdeph
, (1.56)
where fτ is the unitary of the joint system and bath interaction at time τ we have used
UeAU
† = e
UAU†
for an operator A and unitary U. The above equation can further be
simplified as:
f
′
2τ = e
−iτσxσ
zσ
x
e
−iτσz
= e
+iτσz
e
−iτσz
= I, (1.57)
which shows that the bath has no effect on the system at the instant t = 2τ . If we continue
applying these pulses then the system would stroboscopically decouple from the bath every
2τ .
Now if we include a more general bath given by:
HSB =
X
i=x,y,z
σ
i ⊗ B
i
, (1.58)
the above sequence consisting of two X gates would not be able to decouple all the terms in
HSB and thus would require an additional layer of coupling, resulting into a four pulse long
sequence given by:
f
′′
4τ = Y f′
2τY f′
2τ
(1.59)
= Y XfτXfτY XfτXfτ (1.60)
= ZfτXfτZfτXfτ (1.61)
= I + O(τ
2
), (1.62)
where fτ = e
−iτHSB and in the last step, we have used the Baker-Campell-Hausdorff (BCH)
18
formula and the commutation relations between Pauli matrices. Another equivalent sequence
Y fτXfτY fτXfτ also works and is popularly known as XY 4 sequence.
The rest of the thesis is organized as follows:
In Chapter 2, we propose and test the use of dynamical decoupling to suppress crosstalk
errors in superconducting quantum processors, demonstrating significant improvements in
quantum memory and gate operations through experiments on IBM quantum cloud processors.
In Chapter 3, we develop and experimentally verify dynamical decoupling protocols for
qudit systems, especially targeting qutrits and ququarts on a transmon processor. These
protocols significantly improve the fidelity of time-evolved qutrit Bell states, showcasing the
potential of dynamical decoupling for scalable qudit-based quantum processors.
Chapter 4 models the noise effects on transmon qubits using Redfield master equation
with a hybrid bath consisting of both low- and high-frequency components. In this work, we
develop an iterative fitting procedure, which helps extracting the relevant noise parameters.
Chapter 5 introduces deterministic benchmarking, a protocol to identify both coherent
and incoherent errors overlooked by traditional randomized benchmarking. Through experiments performed on transmon qubit, we test the protocol and find that it reveals gate
performance asymmetries due to strong relaxation error in transmons.
In Chapter 6, we demonstrate the significant impact of virtual Z gate compilation methods in open quantum systems. By highlighting the need for symmetric compilation of single
qubit gates with respect to the virtual Z gate, we show that improper gate decomposition
can lead to unintended operations, underperformance of dynamical decoupling sequences etc.
We finally summarize and conclude the thesis in Chapter 7.
19
Chapter 2
Crosstalk Suppression with dynamical
decoupling
Note: This chapter is adapted from [45].
Main results.— Currently available superconducting quantum processors with interconnected
transmon qubits are noisy and prone to various errors. The errors stems from open quantum
system effects and spurious inter-qubit couplings (crosstalk). Static ZZ-coupling between
qubits in transmon architectures is always present and contributes to both coherent and
incoherent crosstalk errors. Its suppression is therefore a key step towards enhancing the
fidelity of quantum computation using transmons. Here we propose the use of dynamical
decoupling to suppress the crosstalk, and demonstrate the success of this scheme through
experiments performed on several IBM quantum cloud processors. In particular, we demonstrate improvements in quantum memory as well as the performance of single-qubit and
two-qubit gate operations. We perform open quantum system simulations of the multi-qubit
processors and find good agreement with the experimental results. We analyze the performance of the protocol based on a simple analytical model and elucidate the importance of
the qubit drive frequency in interpreting the results. In particular, we demonstrate that the
XY4 dynamical decoupling sequence loses its universality if the drive frequency is not much
larger than the system-bath coupling strength. Our work demonstrates that dynamical decoupling is an effective, practical and scalable way to suppress crosstalk and open system
20
effects, thus paving the way towards higher-fidelity logic gates in transmon-based quantum
computers.
2.1 Introduction
As discussed in Chapter 1, among the leading implementations of quantum computing are
superconducting transmon qubits [46, 47], which were designed to have a suppressed sensitivity to charge noise and thus possessing a higher coherence and lifetime than most other
superconducting qubit types [48]. Transmons have been used to demonstrate quantum information processing in a series of recent experiments [49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59]. In order to construct a large scale quantum computer (QC) capable of performing useful
tasks it must be possible to both store and process quantum information with sufficiently low
error rates so as to perform fault-tolerant quantum computation [60, 61, 62]. These requirements are affected by different types of errors which afflict quantum processors. Primary
error sources are open quantum system effects resulting in decoherence, and spurious coupling between qubits, control lines, and readout apparatus, resulting in crosstalk. For fixed
frequency transmon processors, the ZZ-coupling between any two neighboring qubits is always present and contributes to both coherent and incoherent errors, making the suppression
of ZZ-crosstalk one of the most important challenges for such processors.
Various schemes have been proposed and demonstrated toward this end, e.g., combining a
capacitively shunted flux qubit and a transmon qubit with opposite-sign anharmonicity [63,
64], combining multiple coupling paths and additional drive tones [65, 66], or probabilistic error cancellation [67]. For qubits coupled via tunable couplers, it is possible to adjust
the coupler frequency such that the ZZ interactions from each coupler destructively interfere [68], or to detune neighboring qubits [69]. However, all these approaches require extra
calibration or circuit elements, and may be difficult to maintain in large systems. A simple
and universal scheme for ZZ-crosstalk suppression based purely on transmons is still lacking.
21
Here we propose and experimentally demonstrate such a scheme using three different 5-qubit
IBM Quantum Experience (IBMQE) processors [70]. Our scheme is based on dynamical decoupling (DD) [71, 72, 73, 74] – the simplest of all quantum error correction or suppression
protocols [75].
We demonstrate that DD is highly effective at suppressing ZZ-crosstalk, while at the
same time also suppressing unwanted system-bath interactions responsible for decoherence.
Previous work on the use of DD to protect transmon qubit states did not separate these two
different contributions to fidelity decay [76, 77, 78], and we show here that fidelity oscillations
that were previously interpreted as a possible sign of a non-Markovian bath are in fact
attributable to crosstalk. We demonstrate these results experimentally and numerically, and
provide an analytical basis for the choice of the pertinent DD pulse sequences. We highlight
the important role played by gate calibration procedures in interpreting the outcomes of our
crosstalk suppression experiments. We then use our DD scheme to improve the performance
of both single-qubit and two-qubit gate operations by suppressing the crosstalk originating
from the spectator qubits. Since DD uses the well-calibrated single qubit gates already
available in the native set of gates for any given platform, it does not add any extra calibration
complexity and is a scalable approach that can be applied to arbitrarily large quantum
computers. Moreover, it can be readily combined with quantum error correction to reduce
the resource requirements of the latter [79, 80].
2.2 Experimental results for state protection
We consider Ramsey-like experiments, where we prepare an initial state, let it evolve, and
then undo the state preparation. Our experiments were conducted on the ibmq ourense
(Ourense), ibmq 5 yorktown (Yorktown), and ibmq lima (Lima) five-qubit IBMQE processors (see Fig. 2.1). In each case we selected one “main” qubit and consider the other four
to be “spectator” qubits. We performed two types of experiments: free and DD-protected
22
Figure 2.1: Schematics of the layout of the (a) Yorktown and (b) Ourense, Lima, and Quito
devices. Thin circles indicate the main qubits used in our experiments. The other qubits are
spectators.
evolution.
In both cases the main qubit was initialized in the |0⟩ state, then an Ry(π/2) gate was
applied to prepare it in the |+⟩ = (|0⟩ + |1⟩)/
√
2 state. All spectator qubits were initialized
in the same state, which we varied. In the free evolution case we then applied a series of identity gates (separated by barriers) on all the qubits, followed by Ry(−π/2) on the main qubit
to undo the |+⟩ state preparation. In the DD-protected case we applied the universal XY4
sequence [81, 82, 83] to all the spectator qubits, but only identity gates to the main qubit.
The ideal XY4 sequence comprises repetitions of XfτY fτXfτY fτ , where fτ denotes free evolution for a duration of τ (in our experiments τ = 71.1 ns), and X and Y are instantaneous
π pulses about the x and y axes, respectively (in reality, the pulses have a finite duration; we
spaced them without any delay, so τ is their peak-to-peak spacing). This was done primarily
in order to decouple the ZZ-crosstalk term; it also suppresses undesired system-bath interactions, as explained below. After each gate sequence, we measured the main qubit in the
computational (0/1) basis. Each circuit was repeated 8192 times and we used the fraction
of 0 outcomes on the main qubit, F
(e)
+ , as the empirical fidelity measure (augmented by
23
Yorktown
Ourense
Figure 2.2: Fidelity of the |+⟩ state of the main qubit, for different spectator qubits’ initial states {|0⟩, |1⟩, |+⟩}, obtained experimentally [(a) and (b)] and by solving the Redfield
master equation [(c) and (d)]. (a) Results for Ourense. The free evolution curves all exhibit
oscillations with nearly equal periods but distinct amplitudes. These effects disappear under
the application of the XY4 sequence just to the spectator qubits, leaving only a common
fidelity decay. (b) Results for Yorktown. The free evolution curves range from monotonic
to oscillatory. The differences disappear under DD applied to the spectator qubits, leaving
only a common damped oscillation. Error bars denote 95% confidence intervals. (c) Redfield
equation simulation results for a multi-qubit system with ωd = ωq1
. (d) Simulation results for
ωd = ωq1 −2J. All qualitative features observed in (a) and (b) are reproduced in (c) and (d),
respectively. The crosstalk strength considering a two-qubit model is (c) J/2π = 51.55 KHz
and (d) J/2π = 52.63 KHz, obtained by fitting the periods of (a) and (b).
bootstrapping), i.e., as a proxy for F+ ≡ Tr{Ry(−π/2)E[Ry(π/2)|0⟩⟨0|Ry(−π/2)]Ry(π/2)},
where E is the quantum map corresponding to either free or DD-protected evolution of the
main qubit. Ideally Ry(π/2)|0⟩ = |+⟩ and E corresponds to the identity channel; in reality
Ry(π/2) prepares a slightly different state due to gate errors, and E corresponds to a noisy
channel.
Fig. 2.2 shows the results of the free evolution and DD-protected evolution experiments
on Ourense (panel a) and Yorktown (panel b). We plot the empirical fidelity F
(e)
+ (t) of the |+⟩
state of the main qubit for different spectator qubits’ initial states {|j⟩}j=0,1,+. The envelope
24
Figure 2.3: Results of Fig. 2.2 averaged over all the three spectator qubit states for Ourense
(a) and Yorktown (b). Note that the simulations did not account for state preparation and
measurement errors.
of the DD-protected evolution decays more slowly than that of the free evolution, for both
processors as shown in Fig. 2.3. A glance at the Ourense and Yorktown results reveals striking
differences for two identical sets of experiments. We explain below how these arise due to
the different choice of qubit drive frequency for the two processors. Conversely, a striking
qualitative similarity between the two is that the DD-protected evolution essentially erases
the difference between the three different spectator initial states, whereas the differences are
pronounced in the free evolution case.
Having confirmed that the DD protocol works as intended to preserve single qubit coherence, we next provide a theoretical explanation for our findings.
2.3 Theoretical analysis
2.3.1 Model
Consider for simplicity a system of just two nominally uncoupled qubits which, however, have
an undesired and always on ZZ-coupling of strength J ̸= 0, just like coupled transmons.
The generalization to n > 2 qubits is straightforward and is considered in Sec. 2.3.4. The
effective Hamiltonian can be written as
25
HS = −
ωq1
2
Z1 −
ωq2
2
Z2 + JZZ (2.1)
where ωq1 and ωq2 are the qubit frequencies of the main and the spectator qubit respectively, where Z2 ≡ IZ ≡ I ⊗ σ
z
, etc. The ZZ-coupling term dresses the qubit frequencies
such that the frequency of the main qubit changes and depends on the state of the spectator
qubit. Correspondingly, we define the eigenfrequencies of the main qubit by the spectral gap
of HS after fixing the state of the spectator qubit to either |0⟩, |1⟩, or |+⟩ (i.e., replacing
the spectator qubit operators in HS by their expectation values {−1, 1, 0}). This yields, respectively, ω
0
eig = ωq1 − 2J, ω
1
eig = ωq1 + 2J, and ω
+
eig = ωq1
, which is also the bare frequency
of the main qubit. We later show that this conclusion is identical to one derived from a
first-principles model of transmons as multi-level systems in Sec. 2.3.1.1.
We now explain below that these different eigenfrequencies explain the difference between
the Ourense and Yorktown processors seen in Fig. 2.2. In the open system settings, the total
Hamiltonian can be written as H = HS + HSB, where we assume a general system-bath
interaction HSB =
P
α,β∈{0,x,y,z}
gαβσ
α ⊗ σ
β ⊗ Bαβ, where σ
0 = I, gαβ = g
∗
βα is the strength
of the coupling to the bath, and Bαβ are Hermitian bath operators.
Let us now move to a rotating frame defined by the number operator Nˆ = I −
1
2
(Z1 +Z2)
for the main and spectator qubits. Ignoring the overall energy shift, we write the unitary
transformation operator U(t) = e
iωdN t ˆ
, where ωd ̸= 0 is the drive frequency, as U(t) =
e
−iωd(Z1+Z2)t/2
. The rotating frame Hamiltonian H˜ (t) = UHU† + iUU˙ † becomes
H˜ (t) = X
2
i=1
ΩiZi + JZZ + H˜
SB(t) , Ωi ≡
ωd − ωqi
2
(2.2)
where H˜
SB(t) = P
αβ gαβ
U(t)
σ
α ⊗ σ
β
U
†
(t)
⊗Bαβ. The free evolution unitary generated
by the rotating frame Hamiltonian is U˜
f (tf , ti) = T+ exp
−i
R tf
ti
dt H˜ (t)
, with T+ denoting
forward time-ordering.
2
Figure 2.4: Energy level diagram of two coupled transmons with qubit frequencies ωq1 and ωq2
and anharmonicities ηq2 and ηq1
, coupled linearly with strength g. The solid lines represent
the bare energy levels and dashed lines represent the eigenlevels. |k, l⟩ represents levels k and
l in the main and spectator transmons, respectively. Only 6 levels of the infinite-dimensional
Hilbert space formed by both transmons are shown.
2.3.1.1 Circuit model description of ZZ coupling and its implications on rotating frame analysis
In practice, transmons are not perfect two-level systems but anharmonic oscillators consisting
of multiple levels. Here we show how ZZ coupling arises in the multi-level model of coupled
transmons, and verify its implications on free evolution.
The energy level diagram in the lab frame of two capacitively coupled transmons with
an always-on coupling strength g is shown in Fig. 2.4 [84]. Because of the coupling g, levels
|1, 0⟩ and |0, 1⟩ are repelled and form the dashed lines representing the eigenstates |1, 0⟩ and
|0, 1⟩ with energies E|1,0⟩ = ωq1 + g
2/∆ and E|0,1⟩ = ωq2 − g
2/∆ where ∆ = ωq1 − ωq2 and
we have assumed g/∆ ≪ 1. Therefore, the main qubit eigenfrequency when the spectator
27
qubit is in |0⟩ is:
ω
0
eig = E|1,0⟩ − E|0,0⟩ = ωq1 +
g
2
∆
. (2.3)
Similarly, |1, 1⟩ is pushed downward by |2, 0⟩ and upward by |0, 2⟩. Therefore, we have:
E|1,1⟩ = ωq1 + ωq2 −
2g
2
∆ − η
+
2g
2
∆ + η
, (2.4)
where we have assumed that ηq1 = ηq2 = η. Thus, the main qubit eigenfrequency when the
spectator qubit is in |1⟩ is:
ω
1
eig = E|1,1⟩ − E|0,1⟩ = ωq1 −
2g
2
∆ − η
+
2g
2
∆ + η
+
g
2
∆
. (2.5)
Now the ZZ coupling strength can be defined as [84]:
2J = ωzz =
ω
1
eig − ω
0
eig
2
=
g
2
∆ + η
−
g
2
∆ − η
. (2.6)
Using Eqs. (2.3) and (2.5), we can also define the eigenfrequency of the main qubit when
the spectator qubit is in |+⟩ and is given as
ω˜
+
eig =
ω
0
eig + ω
1
eig
2
= ωq1 +
g
2
∆
−
g
2
∆ − η
+
g
2
∆ + η
. (2.7)
Note that unlike the two-level system case discussed in Sec. 2.3.1, this eigenfrequency (˜ω
+
eig)
is not same as the bare qubit frequency ωq1
, and this is one sense in which the two-level
system model is oversimplified. We now choose the drive frequency as ωd = ˜ω
+
eig and move
into a rotating frame about the number operator Nˆ =
P
k,l(k+l)|k, l⟩⟨k, l|. With this choice,
the eigenfrequencies of the main qubit for the spectator qubit in |0⟩ and |1⟩ are, respectively:
ω˜
0
q1 = ω
0
eig − ωd = −ωzz (2.8)
ω˜
1
q1 = ω
1
eig − ωd = ωzz . (2.9)
28
Therefore, in the s = + frame, we have oscillations with frequency ωzz = 2J irrespective
of the state of the spectator qubit, which is exactly what we showed in the Sec. 2.3.1 for
the simplified two-level system model. We can similarly verify the results for the s = 0 and
s = 1 frames.
2.3.2 Free evolution and extraction of the crosstalk frequency
The choice of ωd gives rise to different rotating frame Hamiltonians. Consider the rotating
frames corresponding to the different eigenfrequencies mentioned above: ωd ∈ {ω
0
eig, ω1
eig, ω+
eig}.
Let ∆ = ωq1−ωq2 denote the detuning between the two qubit frequencies, and s ∈ {0, 1, +} (s
for spectator) the rotating frame according to the choice of ωd. Up to a constant, the systemonly Hamiltonians obtained from H˜ (t) [Eq. (2.2)] in the two frames that are experimentally
realized in the IBMQE devices we used are
H˜ +
S = −(∆ + 2J)|01⟩⟨01| − 2J|10⟩⟨10| − ∆|11⟩⟨11| (2.10a)
H˜ 0
S = −∆|01⟩⟨01| + (4J − ∆)|11⟩⟨11| (2.10b)
Let us furthermore assume a simple Markovian dephasing model, with a Lindbladian of
the form
L
s = −i[H˜ s
S
, ·] +X
α
γα(Lα · L
†
α −
1
2
{L
†
αLα, ·}) (2.11)
where the Lindblad operators are {L1 = ZI, L2 = IZ, L3 = ZZ}. Let γ = γ1 + γ3. Using
Eq. (2.10), under free evolution the probability p
s
+s
′(t) of the main qubit’s final state being
|+⟩ if its initial state is |+⟩ (i.e., F+), is, for the three different initial state |s
′
⟩ s
′ ∈ {+, 0, 1}
29
of the spectator qubit:
p
+
+s
′(t) = 1
2
1 + e
−2γt cos(2J t)
∀s
′ ∈ {+, 0, 1} (2.12a)
p
0
+s
′(t) = 1
2
1 + e
−2γtfs
′(t)
(2.12b)
f+(t) = cos2
(2J t) , f0(t) = 1 , f1(t) = cos(4J t) (2.12c)
In the s = + frame we thus expect to observe damped fidelity oscillations with a period of
τ
+ = 2π/2J for all spectator states, consistent with our data in Fig. 2.2(a). Likewise, in the
s = 0 frame we expect no oscillations in the |0⟩ case, but damped fidelity oscillations with a
period of τ
0 = 2π/4J when the spectator qubit is prepared in the |1⟩ or |+⟩ states, consistent
with Fig. 2.2(b). The larger amplitude oscillations observed in Fig. 2.2(b) for the |1⟩ case are
also in agreement with Eq. (2.12c). We thus conclude that the Ourense and Yorktown main
qubit drive frequencies are ωq1 and ωq1 − 2J, respectively, i.e., the devices were calibrated
with the spectator qubits in |+⟩ and |0⟩, respectively. Moreover, the oscillations predicted
by Eq. (2.12) are entirely crosstalk-induced (they disappear when J = 0). In principle
this coherent crosstalk could be compensated for by keeping track of each spectator state;
however, this amounts to computing the outcome of a quantum algorithm and so for large
systems is computationally infeasible. In other words, calibration by itself can never solve the
problem of ZZ crosstalk and thus DD is an essential alternative. Note that while Eq. (2.12a)
incorrectly predicts equal amplitude oscillations for all three initial spectator states, this can
be remedied by including a lowering operator |0⟩⟨1| as an additional Lindblad operator,
because the transmon qubit suffers from strong relaxation process; however, this model fails
to capture the observed ordering of the fidelity amplitudes.
2.3.3 Suppression of ZZ-crosstalk and system-bath interactions
using dynamical decoupling
Having established that crosstalk (and not environmentally induced non-Markovian dynamics) suffices to explain the fidelity oscillations observed in our free evolution experiments,
we now analyze its suppression using DD. For simplicity, consider a DD sequence consisting
purely of ideal (i.e., zero-width) X-pulses of the form iI ⊗e
−i
π
2 X = X2 (henceforth we ignore
global phases) applied just to the spectator qubit. In the rotating frame, the time evolution
at the end of one cycle of such a “pure-X” DD sequence with pulse interval τ is given by
U˜X(2τ ) = X2U˜
f (2τ, τ )X2U˜
f (τ, 0) (2.13)
Using UeAU
† = e
UAU†
(A arbitrary, U
†U = I), we may write
X2U˜
f (2τ, τ )X2 = T+ exp
−i
Z 2τ
τ
dt X2 H˜ (t)X2
(2.14)
Using the Magnus expansion (see, e.g., Refs. [82, 79]) one can show that, to first order in τ ,
this sequence cancels every term in H˜ (t) that anticommutes with X2. Using Eq. (2.2), we
are thus left with
U˜X(2τ ) = U˜′
(2τ ) + O(τ
2
) (2.15)
where
U˜′
(2τ ) = exp
− iτ (ωd − ωq1
)Z1
− i
Z τ
0
dt[H˜
SB(t) + X2H˜
SB(t + τ )X2]
(2.16)
31
In the XY4 case, the integral also contains Y2H˜
SB(t + 2τ )Y2 + Z2H˜
SB(t + 3τ )Z2. In both
the pure-X and XY4 cases the integral always vanishes for the terms Z2 and ZZ, as these
terms anticommute with X2 both in the lab frame and the rotating frame. I.e., it follows
from the form of U˜′
(2τ ) that the ZZ-crosstalk and the Z2 and ZZ bath-coupling terms
are all suppressed to O(τ ) by both the ideal pure-X and XY4 sequences. This explains the
suppression of crosstalk observed in our experiments. The fact that in the s = + frame
ωd = ωq1 = ω
+
eig means that the Z1 term in U˜′
(2τ ) vanishes. This explains the absence of
oscillations in the Ourense DD results [Fig. 2.2(a)]. Likewise, the Z1 term in U˜′
(2τ ) remains
in the s = 0 frame, when ωd = ω
0
eig. This explains the remaining oscillations in the Yorktown
DD results [Fig. 2.2(b)]. Moreover, the suppression of the Z2 and ZZ bath-coupling terms
explains, via its effect on second and higher order terms in the Magnus expansion, why the
fidelity of the main qubit under DD is generally higher than for free evolution, as can be
seen in Fig. 2.2. Any incoherent interaction which couples via a Z2 term (or an X2 or Y2
term in the case of XY4) will be suppressed, thus preserving the coherence of the main qubit
and therefore the fidelity.
Now note that a term in H that anticommutes with X2 (e.g., Y Y ) may transform to a
term in H˜ (t) that does not; this causes many terms to not cancel to first order in τ under
pure-X DD. As a consequence the XY4 sequence loses its exact universality in the rotating
frame. This may adversely affect the performance of DD sequences which are designed for
high-order cancellation [85, 86]. Since the pure-X sequence is shorter by a factor of 2, it is
preferred in the present setting.
We emphasize that our DD protocol suppresses any term involving a spectator qubit
operator X2, Y2, or Z2 (for an XY4 sequence; pure-X suppresses only Z2). Thus we see
suppression of both the coherent ZZ crosstalk and any bath interactions (i.e. decoherence)
mediated by a spectator qubit operator. We thus see additional improvement in main qubit
coherence and operation fidelity beyond the cancellation of crosstalk.
32
2.3.4 Numerical results
The theoretical analysis above was oversimplified since it missed features such as the unequal decay rates associated with different initial spectator states [Eq. (2.12) predicts the
same decay rate γ for all three such states], and the fidelity amplitude ordering. Thus we
now complement this analysis with a numerical study. We simulated the transmon system
(modeled as qubits) via the Redfield master equation using the HOQST package [87].
We consider a system of n coupled qubits with only the linear system-bath interaction
given by
HSB =
Xn−1
i=0
X
α∈{x,y,z}
gασ
α
i ⊗ Biα
where σ
α
i
and Biα represents the system and bath coupling operators and giα are coupling
strengths. In our simulations, we included the main qubit and all the spectator qubits which
are directly coupled to the main qubit. Therefore, our simulations of the Ourense device
included four qubits (see Fig. 2.1(b), where Q1 is the main qubit, coupled to three spectator
qubits), and similarly, we included only three qubits in the Yorktown simulations, where Q3
is the main qubit. The system Hamiltonian in this case is given by:
HS = −
Xn−1
i=0
ωqi
2
Zi +
Xn−1
j>i=0
JijZiZj
. (2.17)
We again move to a rotating frame defined by the number operator Nˆ given by
Nˆ =
X
i0,..,in−1∈{0,1}
(i0 + · · · + in−1)|i0 . . . in−1⟩⟨i0 . . . in−1| , (2.18)
and solve the Redfield (or TCL2) master equation [88, 87] for the rotated system Hamiltonian. Introducing a superindex {m} instead of {iα}, we define the standard bath correlation
function:
Cmn(t − τ ) = gmgnTr{UB(t − τ )BmU
†
B
(t − τ )BnρB} , (2.19)
33
where UB(t) = e
−iHBt
is the unitary generated by the pure-bath Hamiltonian HB, and the
reference state ρB is the Gibbs state of HB:
ρB = e
−βHB /Tr
e
−βHB
, (2.20)
where β = 1/T is the inverse temperature. Assuming the bath operators Bm and Bn are
uncorrelated, i.e., Cmn(t) = Cnm(t) = δmnCn(t), the Redfield equation is
∂ρS
∂t = −i[HS, ρS] + L(ρS) , (2.21)
where L is the Redfield Liouvillian
L(ρS) = −
X
m
[Am,Λm(t)ρS(t)] + h.c. , (2.22)
and
Λm(t) = Z t
0
Cm(t − τ )US(t, τ )AmU
†
S
(t, τ )dτ , (2.23)
where US(t) = e
−iHSt
is the unitary operator generated by the system Hamiltonian HS. Note
that the Am used in our simulations are the coupling operators defined in the rotating frame
of the number operator Nˆ [Eq. (2.18)].
We choose the bath to be Ohmic, which means that the noise spectrum,
γm(ω) = Z ∞
−∞
Cm(τ )e
iωτdτ , (2.24)
has the following form
γm(ω) = 2πηg2
m
ωe
−|ω|/ωm
1 − e
−βω , (2.25)
where ωm = 2πfm is the cutoff frequency for bath operator Bm, and η is a positive constant with dimensions of time squared that arises in the specification of the Ohmic spectral
function.
3
We work in units such that ℏ = 1 and assume the bath temperature T = 20 mK. For all
the simulations shown in Figs. 2.2(c) and (d), we used
η = 10−4 GHz−2
, fm = 2 GHz ∀m (2.26a)
gi,σz = 0.1175 GHz (2.26b)
gi,σx = gi,σy =
1
2
gi,σz (Ourense)
3
4
gi,σz (Yorktown)
∀i . (2.26c)
For the Ourense device, the Jij are provided in the IBMQE device backend information ([89])
and take the following values:
J01 = 25.48 KHz, J12 = 18.24 KHz, J13 = 8.77 KHz. (2.27)
The Jij are not provided for the Yorktown device; therefore we assume that all the non-zero
Jij are equal and we extract Jij = 24.27 KHz by tuning it to match the oscillations in
Fig. 2.2(b).
Finally, we remark that the values of the bath parameters reported here are chosen to
provide a qualitative agreement with the experimental data. In subsequent work, a more
rigorous optimization scheme is employed to adjust bath parameters between theoretical and
simulations. This is discussed in detail in Chapter 4.
We use the open system model described here to obtain Fig. 2.2(c) and Fig. 2.2(d).
Figure 2.2(c) shows the simulation results for the s = + frame. The effect of the bath is
to induce an overall exponential decay envelope due to dephasing , as already suggested
by Eq. (2.12a). When DD is applied to the spectator qubit, the ZZ-induced oscillations
are entirely suppressed, independently of the initial state of the spectator qubit, as in the
experimental data in Fig. 2.2(a). Additionally, the DD sequence can be seen to suppress the
coupling of the spectator qubit to the bath, in the sense that the maximum amplitude for
35
the |1⟩ and |+⟩ spectator qubit states (at 10µs) is lower in the free evolution case than in
the DD case. The small but noticeable difference between the DD-protected curves is due
to the fact that the DD sequence only generates first order suppression.
Figure 2.2(d) shows the simulation results for the s = 0 frame. As already expected from
Eq. (2.12), the oscillation period for the free evolution cases with the spectator qubit prepared
in the |1⟩ or |+⟩ states is 2π/4J ≈ 5µs, while the |0⟩ case exhibits no oscillations. These
results are entirely consistent with the experimental data in Fig. 2.2(b). In the presence
of DD pulses applied to the spectator qubit the three cases again collapse onto a single
curve. However, this time oscillations persist with a frequency of 2J. As we discuss in
the Sec. 2.3.3, these are due to the presence of the uncanceled Z1 term in U˜′
(2τ ), with
ωd = ωq1 − 2J. Crucially, despite the dependence on J, this is a single-qubit effect, and the
goal of suppressing an unwanted two-qubit term that would interfere with proper two-qubit
gate operation has been accomplished. Similar comments as in the s = + frame apply to
the effect of DD on suppressing the effect of coupling to the bath; the envelope amplitude
of the |1⟩ and |+⟩ cases is higher in the presence of DD. The |0⟩ case is not helped by DD,
since in this case there is no relaxation of the spectator qubit.
Figure 2.2 reveals that overall, our simulations are in close qualitative agreement with the
experimental results. The |+⟩ curve has the smallest amplitude in our experiments, which
our simulations account for by setting gz > gx, gy (see Eq. 2.26c). I.e., we conclude from our
simulations that dephasing dominates over the other noise channels. This is consistent with
the data documented in Table 2.1 and 2.2, which shows that almost all spectator qubits have
T2 < T1. Our simulations qualitatively reproduce the oscillation pattern of the free evolution
of the |+⟩ initial state in Fig. 2.2(b). This required accounting for all of the spectator qubits
coupled to the main qubit (unlike in the phenomenological Lindblad model with only one
spectator). Thus, a multi-qubit description of the system is needed to fully understand and
characterize the crosstalk in these devices.
36
2.4 DD for gate operations
Having established the efficacy of DD in state preservation, we finally apply DD to counter
crosstalk and decoherence-induced errors during gate operations; we call the resulting gates
“DD-protected gates” (DDPGs) [79]. We first present the results for the single-qubit gates in
Sec. 2.4.1, which demonstrate a significant improvement, namely, a reduction of the fidelity
decay rate by more than a factor of 2. Finally, we focus on CNOT gate (based on cross
resonance (CR) [35, 84, 90]) in Sec. 2.4.2.
2.4.1 DD for single-qubit gates
In our single-qubit gate experiments, we chose Q1 of Quito as the main qubit and generated
a series of circuits consisting of random sequences of gates of varying length, where each
of the gates is taken from the set G = {Rx(±π/8), Rx(±π/4), Ry(±π/8), Ry(±π/4)} of 8
single qubit gates, where Rx/y(θ) represents a rotation about the x/y-axis by an angle θ. We
performed quantum state tomography (QST) to construct the density matrix at the end of
each of the circuits. This involved repeating the same experiment thrice, measuring each
time in a different Pauli basis. The results for each basis were again given as counts. We
compared the density matrix thus obtained with the expected state to calculate the fidelity
as a function of time or number of gates. We used the bootstrap method by resampling
with replacement over the number of counts, as discussed above, to create the resampled
density matrix and thus an average fidelity for each of the circuits. In the DD-protected
gates (DDPGs) case, a DD sequence (XY4) was simultaneously applied to all the spectator
qubits, in parallel with the gates we applied to the main qubit. The DD pulses occupy only
the gap between each of the gates on the main qubit; see the circuit diagram in Fig. 2.5. We
used this approach because the single qubit gates applied to the main qubit can also behave
as a DD sequence, which could interfere with the DD sequence we apply to the spectator
qubits.
37
|0/1⟩
|0/1⟩
Main : |0⟩ U G1 I G2 I G3 I G4 I
No DD
Spectators : |0⟩ Si I I I I I I I I
Main : |0⟩ U G1 I G2 I G3 I G4 I
DD
Spectators : |0⟩ Si I X I Y I X I Y
Figure 2.5: Circuit diagram for single qubit gate experiments. The top circuit
shows a sequence of random gates of length 4 chosen from the set G = {Rx(±π/8),
Rx(±π/4), Ry(±π/8), Ry(±π/4)} of 8 single qubit gates applied to the main qubit, and
a sequence of Identity operations applied to all the spectator qubits. U represents the gate
applied to prepare any predefined initial state on the main qubit. In Fig. 2.6, U = Ry(π/2)
which prepares a |+⟩ state on the main qubit. Si represents the gate applied to prepare
i = |0⟩, |1⟩ and |+⟩ on all the spectator qubits, where S|0⟩ = I (Identity). In the bottom
circuit, the XY4 sequence is applied to the spectator qubits in the gaps between the gates
applied to the main qubit.
For the results shown in Fig. 2.6, we first prepare the main qubit in the |+⟩ state, and
apply the above sequences of random gates with identity operations (or delay) between any
two consecutive gates, such that the distance between the center of any two consecutive
gates is twice the total duration of the gates (see Fig. 2.5). We choose the same random
circuit and repeat it for three different initial states of the spectator qubits: {|0⟩, |1⟩, |+⟩}.
We repeat the whole experiment 100 times, with each run having its own random sequence
of gates, with and without DD on the spectators. Figure 2.6 shows the result of averaging
over three spectator qubits states and 100 experimental runs (each with a different random
gate sequence). We observe a clear and statistically significant improvement in fidelity for
the DDPGs case. In particular, the exponential decay rate decreases by more than a factor
of 2, from (0.0172 ± 0.0007) MHz without DD, to (0.0081 ± 0.0002) MHz with DD.
38
0 10 20 30 40 50
time (µs)
0.0
0.2
0.4
0.6
0.8
1.0
Fidelity
Fit : A exp(−Bt) + C
Free: A = 0.466 ± 0.003, B = 0.0172 ± 0.0007, C = 0.519 ± 0.003
DD: A = 0.438 ± 0.005, B = 0.0081 ± 0.0002, C = 0.544 ± 0.005
Figure 2.6: Experimental fidelity results for random sequences of single qubit gates consisting
of elements of the set G = {Rx(±π/8), Rx(±π/4), Ry(±π/8), Ry(±π/4)}, averaged over the
three spectator states and 100 different experimental runs, with the main qubit initialized in
the |+⟩ state. The fidelity is shown as a function of time, with and without DD applied to
the spectator qubits. The exponential fit to both the free and the DDPGs case shows a clear
improvement in the decay rate, by a factor of 0.0172/0.0081 = 2.12. Error bars represent
2σ confidence intervals obtained by bootstrapping. Data was acquired over a period of six
days from 12/30/2021 to 01/02/2022 and from 01/07/2022 to 01/08/2022. See Table 2.2 for
device parameters.
2.4.2 DD for two-qubit gates
Here we use ibmq quito (Quito), a five qubit IBMQE processor. We choose two qubits as
control and target, which we prepare in the |+, 0⟩ state, and the remaining three are the spectator qubits. Significant effort has been put into suppressing ZZ interactions between main
qubits to improve CR gate fidelity [65, 91, 66, 92, 93]. Here we focus on suppressing crosstalk
from spectator qubits in CNOT gates. This has previously been explored by dividing the
CNOT gate into 4 CR pulses (along with single qubit gates) and applying an X pulse on
all spectators after the second CR pulse [94, 95]. Here, we optimize spectator-induced error
suppression in CNOT gates by exploring a wide range of pulse placements and DD sequences.
39
Figure 2.7: (a) Optimized pulse and DD sequence placement in the CR-based CNOT gate.
D0-D4 denote drive channels for qubits Q0-Q4 of IBMQ Quito processor. U3 represents
CR pulses acting on Q1 at the Q3 frequency. The control and target qubits are Q1 and
Q3, respectively; the rest are the spectator qubits. We apply the XY4 (or palindromic XY4
or UDD4) sequence to Q0 and Q2, and the pure-X sequence to Q4, with pulses placed in
gaps between the CR pulses. Note that one X gate in the DD sequence applied to Q0 and
Q2 has been replaced by the pre-existing X gate on Q1. VZ denotes the virtual Z gate.
(b) QST results after applying 15 (top) and 19 (bottom) CNOT gates. Left: without DD.
Right: with XY4. Clearly, the XY4 results are significantly closer to the expected Bell state,
i.e., equal corner peaks of 0.25. (c) Fidelity of Bell state preparation after a repeated odd
number of up to 29 CNOT gates, averaged over 5 separate runs with the spectator qubits
initialized in |0⟩ (we checked and found the effect of different initial spectator states to be
insignificant). Error bars represent 95% confidence intervals. CNOT fidelity< 1 at 0 gates
is due to preparation and measurement errors. CNOT with DD takes longer than without
DD since to avoid overlap we inserted delays to accommodate the two pure-X sequences on
D4. The fidelity with DD is statistically significantly higher than fidelity without DD for all
DD sequences we tried after ∼ 3µs, or ∼ 9 consecutive CNOT gates.
We find that the DDPG solution depicted in Fig. 2.7(a) significantly improves performance,
as evidenced in Fig. 2.7(b),(c). More specifically, we apply an odd number of CNOT gates,
then perform quantum state tomography (QST) and compute the fidelity with respect to
the expected Bell state (|00⟩ + |11⟩)/
√
2. We then compare to gates integrating different
types of DD sequences including XY4 (XYXY), palindromic XY4 (XYXYYXYX) and the
4th order Uhrig DD sequence (UDD4) [86] on the spectator qubits. Figures 2.7(b) and (c),
respectively, show QST results and the fidelity with and without DD. Clearly, incorporation
of XY4 yields a notable improvement in CNOT performance, as a function of the number
of consecutive CNOT gates. The results for palindromic XY4 and UDD4 are statistically
40
indistinguishable from XY4. Note that, in contrast to other techniques using DD to improve
CNOT gates, our method does not increase gate duration beyond the ordinary echo CNOT
gate. Thus we are not subject to additional decoherence during a prolonged gate, and we
see only the benefits of crosstalk cancellation. These QST-based results suffice to establish
the improvement of DD-protected CNOT gates. While QST is resource-intensive, it is an
important check to confirm the validity of our method. Now that we have strong evidence
that our method can improve gate fidelity, we expect future work in this direction, which will
more efficiently quantify the improvement using advanced characterization methods such as
gate set tomography [96].
2.5 Discussion
We have formulated and experimentally implemented a simple, effective DD scheme to suppress ZZ-crosstalk in a multi-qubit transmon processor, that unlike other approaches [63, 64,
68] does not require any hardware redesign. The same scheme also suppresses interactions
with the ambient bath, resulting in a significant improvement in quantum memory and gate
performance. Our procedure provides a scalable method that uses only the pre-existing set
of gate operations. One might worry that if there is non-negligible classical crosstalk between
qubit drive lines, then the DD pulses on one set of qubits will appear as stray drive pulses on
other nearby qubits. If the pulses were not calibrated this would be a valid concern; however,
DD pulses are identical to ordinary gates, which would already require calibration in any
quantum processor. Therefore our method can be used in a large-scale quantum processor
without adding any extra calibration overhead. While some residual classical crosstalk may
remain, our approach trades off these (small) errors on far-away qubits in exchange for much
lower error on nearby qubits. Given recent advances in classical crosstalk error calibration
and cancellation [97], we expect this tradeoff to be favorable even for very large systems.
We note that the ZZ crosstalk we suppress is a coherent effect and could, in principle,
41
be eliminated by keeping track of spectator qubits’ states and adjusting subsequent gates
accordingly. However, this would require predicting the states of all qubits in a system at
all points in a calculation, a computation which is equivalent to classically simulating the
quantum algorithm. Such a simulation would be impractical in any large-scale quantum
processor. Our DD method requires no such simulation and has the added advantage of
suppressing decoherence from 2-qubit bath interactions.
Our DD-based crosstalk cancellation method is completely general and can be applied to
any gate-based quantum processor where crosstalk is an issue. That is, the method is not
limited to transmon qubits, or even superconducting qubits, but can be applied to almost
any qubit system. By reducing crosstalk, it becomes possible to achieve higher quantum
logic gate fidelities and approach the requirements for fault tolerant quantum computation.
We thus expect DD to play a significant role in various quantum algorithms in the NISQ
era.
42
Processor Ourense Yorktown
Date accessed 01/18/2021 01/19/2021
Q0
Qubit freq.(GHz) 4.8203 5.2828
T1 (µs) 117.7 38.0
T2 (µs) 79.4 23.1
sx gate error [10−2
] 0.0310 0.1371
sx gate length (ns) 35.556 35.556
readout error [10−2
] 1.60 2.92
Q1
Qubit freq.(GHz) 4.8902 5.2476
T1 (µs) 96.3 52.4
T2 (µs) 29.6 23.2
sx gate error [10−2
] 0.0368 0.1563
sx gate length (ns) 35.556 35.556
readout error [10−2
] 3.35 3.00
Q2
Qubit freq.(GHz) 4.7166 5.0335
T1 (µs) 117.1 63.1
T2 (µs) 114.5 87.6
sx gate error [10−2
] 0.0668 0.0464
sx gate length (ns) 35.556 35.556
readout error [10−2
] 1.76 7.68
Q3
Qubit freq.(GHz) 4.7891 5.2923
T1 (µs) 138.4 59.3
T2 (µs) 106.7 43.8
sx gate error [10−2
] 0.0374 0.0388
sx gate length (ns) 35.556 35.556
readout error [10−2
] 3.48 5.54
Q4
Qubit freq.(GHz) 5.0238 5.0785
T1 (µs) 110.2 47.0
T2 (µs) 33.8 32.0
sx gate error [10−2
] 0.0495 0.0703
sx gate length (ns) 35.556 35.556
readout error [10−2
] 4.69 2.94
Table 2.1: Specifications of the Ourense and Yorktown devices along with the access dates of
our experiments. The sx (√
σ
x) gate forms the basis of all the single qubit gates and any single
qubit gate of the form U3(θ, ϕ, λ) is composed of two sx and three rz(λ) = exp(−i
λ
2
σ
z
) gates
(which are error-free and take zero time, as they correspond to frame updates). Figure 2.2
uses qubit 1 (Q1) of Ourense and qubit 3 (Q3) of Yorktown as the main qubit.
43
Date accessed 12/30/2021 12/31/2021 01/01/2022 01/02/2022 01/07/2022 01/08/2022 11/30/2021
Q0
Qubit freq. (GHz) 5.3006 5.3006 5.3006 5.3006 5.3006 5.3006 5.3006
T1 (µs) 55.8 84.6 96.7 79.8 114.7 95.2 85.7
T2 (µs) 87.3 137.4 77.1 178.6 118.2 112.7 149.6
sx gate error [10−2
] 0.0336 0.0309 0.0363 0.0325 0.0370 0.0257 0.0275
sx gate length (ns) 35.556 35.556 35.556 35.556 35.556 35.556 35.556
x gate error [10−2
] 0.0336 0.0309 0.0363 0.0325 0.0370 0.0257 0.0275
x gate length (ns) 35.556 35.556 35.556 35.556 35.556 35.556 35.556
readout error [10−2
] 5.71 4.72 4.51 3.75 3.40 3.11 4.31
Q1
Qubit freq. (GHz) 5.0806 5.0806 5.0806 5.0806 5.0806 5.0806 5.0806
T1 (µs) 129.2 114.4 89.4 58.6 131.7 106.3 191.1
T2 (µs) 122.4 98.2 59.8 59.9 140.9 154.0 82.1
sx gate error [10−2
] 0.04605 0.0597 0.0566 0.0283 0.0653 0.0247 0.0860
sx gate length (ns) 35.556 35.556 35.556 35.556 35.556 35.556 35.556
x gate error [10−2
] 0.04605 0.0597 0.0566 0.0283 0.0653 0.0247 0.0860
x gate length (ns) 35.556 35.556 35.556 35.556 35.556 35.556 35.556
readout error [10−2
] 2.80 3.83 3.80 1.54 3.00 2.30 7.91
Q2
Qubit freq. (GHz) 5.3222 5.3222 5.3222 5.3222 5.3221 5.3222 5.3221
T1 (µs) 90.0 87.8 76.9 120.9 94.2 114.2 77.0
T2 (µs) 143.9 170.5 129.2 158.8 26.6 121.9 103.3
sx gate error [10−2
] 0.0370 0.0333 0.0452 0.0495 0.0371 0.1668 0.0352
sx gate length (ns) 35.556 35.556 35.556 35.556 35.556 35.556 35.556
x gate error [10−2
] 0.0370 0.0333 0.0452 0.0495 0.0371 0.1668 0.0352
x gate length (ns) 35.556 35.556 35.556 35.556 35.556 35.556 35.556
readout error [10−2
] 2.23 2.20 2.19 1.78 5.77 4.46 4.58
Q3
Qubit freq. (GHz) 5.1636 5.1636 5.1636 5.1636 5.1636 5.1636 5.1636
T1 (µs) 125.1 81.7 112.7 112.2 110.5 113.3 111.3
T2 (µs) 21.4 21.4 21.4 21.4 21.7 21.7 10.3
sx gate error [10−2
] 0.0257 0.0278 0.0299 0.0260 0.0238 0.0241 0.0288
sx gate length (ns) 35.556 35.556 35.556 35.556 35.556 35.556 35.556
x gate error [10−2
] 0.0257 0.0278 0.0299 0.0260 0.0238 0.0241 0.0288
x gate length (ns) 35.556 35.556 35.556 35.556 35.556 35.556 35.556
readout error [10−2
] 2.60 2.04 2.52 2.24 2.60 2.44 2.73
Q4
Qubit freq. (GHz) 5.0524 5.0524 5.0524 5.0524 5.0524 5.0523 5.0524
T1 (µs) 145.2 106.1 117.4 104.9 87.5 97.4 104.8
T2 (µs) 171.5 187.9 113.4 160.2 110.6 180.7 203.5
sx gate error [10−2
] 0.0412 0.0291 0.0381 0.0418 0.0396 0.0299 0.0405
sx gate length (ns) 35.556 35.556 35.556 35.556 35.556 35.556 35.556
x gate error [10−2
] 0.0412 0.0291 0.0381 0.0418 0.0396 0.0299 0.0405
x gate length (ns) 35.556 35.556 35.556 35.556 35.556 35.556 35.556
readout error [10−2
] 2.23 2.15 2.28 2.01 2.19 1.87 2.19
CNOT (Q1→ Q3)
gate length (ns) 334.222
gate error [10−2
] 0.90
Table 2.2: Specifications of the Quito device along with the access dates of our experiments.
Data for Fig. 2.6 was acquired over a period of six days from 12/30/2021 to 01/02/2022 and
from 01/07/2022 to 01/08/2022 and the data for Fig. 2.7 was acquired on 11/30/2021.
44
Chapter 3
Qudit Dynamical Decoupling on a Superconducting Quantum Processor
Note: This chapter is adapted from [98].
Main results.— Multi-level qudit systems are increasingly being explored as alternatives to
traditional qubit systems due to their denser information storage and processing potential.
However, qudits are more susceptible to decoherence than qubits due to increased loss channels, noise sensitivity, and crosstalk. To address these challenges, we develop protocols for
dynamical decoupling (DD) of qudit systems based on the Heisenberg-Weyl group. We implement and experimentally verify these DD protocols on a superconducting transmon processor
that supports qudit operation based on qutrits (d = 3) and ququarts (d = 4). Specifically,
we demonstrate single-qudit DD sequences to decouple qutrits and ququarts from systembath-induced decoherence. We also introduce two-qudit DD sequences designed to suppress
the detrimental cross-Kerr couplings between coupled qudits. This allows us to demonstrate a significant improvement in the fidelity of time-evolved qutrit Bell states. Our results
highlight the utility of leveraging DD to enable scalable qudit-based quantum computing.
45
3.1 Introduction
Multilevel quantum systems, also known as qudits [99], offer potentially superior computational capabilities and denser information encoding relative to traditional, qubit-based
schemes [100, 101, 102, 103, 104, 105, 106, 107, 30, 108, 31, 109, 110, 111, 112, 113, 114,
32, 115, 116]. In addition, qudits enable resource-efficient fault-tolerant quantum computation [117, 118, 119] and the exploration of complex novel quantum applications [120, 121,
122, 123, 124] with reduced resource requirements. However, in superconducting devices,
qudits are more susceptible to low-frequency noise and correlated errors, which pose significant challenges [125]. Addressing these requires the development of scalable strategies
for the mitigation and suppression of decoherence [75]. Dynamical decoupling (DD) [126,
74, 127, 73, 128, 129] is a powerful technique designed to enhance the fidelity of quantum
states by employing carefully timed control pulses. It has been used to effectively decouple
superconducting qubits from environmental noise [76, 130, 131] and unwanted crosstalk as
shown in Chapter 2 and in the follow up works [132, 133, 134]. While DD has been studied
across a broad spectrum of qubit-based systems, its experimental application to qudits has
been limited primarily to trapped ions and nitrogen-vacancy ensembles [135, 136, 137, 138],
and very recently to enhance the fidelity of a qutrit-assisted three-qubit Toffoli gate on an
IBM transmon device [139].
In this chapter, we present a general DD framework tailored for qudits and experimentally
demonstrate its effectiveness using coupled superconducting transmon circuits [29] operated
as qutrits (d = 3) and ququarts (d = 4). Our framework employs the Heisenberg-Weyl (HW)
group, which has found many uses in the study of d-dimensional quantum systems [140, 141].
We present DD sequences for universal noise suppression and also introduce a single-axis DD
sequence designed to suppress the prevalent 1/f dephasing noise that plagues superconducting qudits. We then introduce a multi-qudit DD sequence designed to suppress unwanted
cross-Kerr interactions between coupled qudits (see Fig. 3.1), which stand in the way of
46
Figure 3.1: Schematic illustration of two transmon qudits with quantized energy levels affected by relaxation and dephasing errors, along with the qudit-qudit cross-Kerr couplings
αij .
scaling superconducting qudit systems [30, 125]. Using our DD sequences, we additionally
report a significant enhancement in preserving the fidelity of a qutrit Bell state over time.
This work serves as a proof-of-concept demonstration of the efficacy and scalability of active refocusing techniques in qudit systems and provides a stepping stone toward operating
large-scale high-dimensional architectures..
3.2 Qudit dynamical decoupling theory
Building upon the general symmetrization ideas of Refs. [74, 127], the theory of qudit DD
was developed in Refs. [142, 143, 144, 145]. We briefly introduce essential terminology and
present a detailed review in the Appendix A, where we also generalize the theory.
The decoupling group Gd is a set of unitary transformations (pulses) gj acting purely on
the system: Gd = {g0, · · · , gK}, where g0 is the d-dimensional identity operator I. Under
the instantaneous and ideal pulse assumptions, cycling over all elements of the group yields
the following DD pulse sequence [74, 127]:
47
U(T) = Y
K
j=0
g
†
j
fτ gj
. (3.1)
Here, τ is pulse interval, T = |Gd|τ = (K + 1)τ is the total time taken by the sequence, and
fτ = e
−iτH is the free-evolution unitary, where H is the total Hamiltonian of the system
and the bath. A universal DD sequence for a qubit (d = 2) is obtained by choosing the
decoupling group as the Pauli group G2 = {I, X, Y, Z}, for which U(T) simplifies into the
well-known XY4 sequence U(4τ ) = Y fτXfτY fτXfτ [81].
For d > 2, we instead use the decoupling group to be the Heisenberg-Weyl group (HWG)
of order d
2
, which generalizes the Pauli group. The HWG is generated by the following shift
and phase operators:
Xd ≡
X
d−1
k=0
|(k + 1) mod d⟩⟨k|, Zd ≡
X
d−1
k=0
γ
k
d
|k⟩⟨k|, (3.2)
where γd = e
2πi/d is the dth root of unity. The remaining HWG elements are given by
Λαβ = (−
√γd)
αβXα
d Z
β
d where α, β ∈ Zd = {0, 1, 2, ..., d − 1}.
The dominant decoherence mechanism in transmon qutrits and ququarts is dephasing
due to 1/f noise, which has been connected to charge fluctuations and higher level charge
sensitivity [29, 146]. Thus, for single qudits, we focus on single-axis DD sequences consisting only of the shift operator and its powers, i.e., the decoupling group formed by the
HW subgroup {Xk
d
}
d−1
k=0. Note that (Xk
d
)
† = X
d−k
d
. Thus, cycling over these operators, we
obtain U(T) = (X1
d
fτX
d−1
d
)...(X
d−2
d
fτX2
d
)(X
d−1
d
fτXd)(Ifτ I). Simplifying, this becomes the
sequence dXd ≡ XdfτXdfτXd...Xdfτ .
3.3 Single qudit dXd experiment
We conduct all our experiments on a superconducting transmon qudit processor with d = 3
and 4; other parameters are detailed in the Section 3.8. Since DD sequences are particularly
48
effective against low-frequency noise [147], and superconducting circuits are especially susceptible to such noise when higher excited states are targeted [30], we focus primarily on the
dXd sequence family. The underlying cycle operator Xd is compiled using 2(d − 1) native
√
σ
x
s
subspace rotations where s ∈ {(0, 1),(1, 2), ..,(d − 1, d)}, and σ
x
(i,j) = |i⟩⟨j| + |j⟩⟨i| is the
Pauli-x operator between levels i and j. Fig. 3.2 presents our experimental single qudit dXd
results. Free evolution (no DD) corresponds to the preparation of a uniform qudit superposition state |+⟩d ≡ (|0⟩ + · · · + |d − 1⟩)/
√
d, waiting for a specified delay time, unpreparing
the state, and finally measuring the qudit. Assuming ideal preparation, unpreparation, and
measurement, the fidelity of the superposition state |+⟩d is the probability of finding the
qudit back in the |0⟩ state. We then repeat the experiment with the dXd sequence applied
during the delay time and study its impact on the state fidelity.
Fig. 3.2(a) presents the results for the qutrit experiments. Crucially, all the DD curves
exhibit an improvement over the free evolution (No DD) experiment, confirming the effectiveness of our DD sequences in suppressing decoherence. In more detail, for each total time
T = 9τ we conducted a free evolution experiment and four DD experiments: 1, 2, or 3
repetitions of 3X3, and universal qutrit DD (the full order-9 HWG), with respective pulse
intervals of τ1 = 3τ , τ2 =
3
2
τ , τ3 = τ , and τuniv. = τ . DD theory predicts that for instantaneous, ideal pulses, state preservation fidelity increases monotonically as the pulse interval
decreases for a fixed total evolution time [148, 149]. Moreover, universal DD is expected to
outperform single-axis DD. Our results exhibit the opposite of both expectations: the single
repetition 1 × 3X3 experiment, with the longest pulse interval τ1 = 3τ , yields the highest
fidelity, while the universal sequence, with the shortest interval τ (as for 3 × 3X3) yields
the lowest DD fidelity. The reason for these results is likely to be the presence of coherent
pulse errors, which accumulate more detrimentally the longer the pulse sequence, and whose
effect overwhelms the benefit of shorter pulse intervals [150, 151]. While X3 gates can be
decomposed in terms of four native √
σ
x
s
gates, the remaining HW pulses require six native
√
σ
x
s
gates, so that τmin = 180 ns for universal DD compared to τmin = 120 ns for the 3X3
49
0 10 20 30 40 50
time (µs)
0.0
0.2
0.4
0.6
0.8
1.0
Fidelity of |+i3
(a) d = 3 No DD
1× 3X3
2× 3X3
3× 3X3
Universal DD
0 10 20 30 40 50
time (µs)
0.0
0.2
0.4
0.6
0.8
1.0
Fidelity of |+i4
(b) d = 4 No DD
1× 4X4
2× 4X4
3× 4X4
4× 4X4
Figure 3.2: Experimental results showing the fidelity of the qudit uniform superposition
state |+⟩d as a function of total time under free evolution (No DD) and dXD, for (a) qutrits
and (b) ququarts. The minimum pulse interval is τmin = 120 ns. For qutrits, we implement
1, 2, and 3 repetitions of the 3X3 sequence, with corresponding pulse intervals 3τ , 3τ/2,
and τ , respectively. The total evolution time is always 9τ . Universal DD is a 9-pulse long
sequence formed from cycling over the full HWG, applied once with a pulse interval of τ
and τmin = 180 ns. For ququarts, we similarly implement 1, 2, 3, and 4 repetitions of the
4X4 sequence, with corresponding pulse intervals of 4τ , 2τ ,
4
3
τ , and τ respectively, where
the total time is always 16τ .
sequences. This additional opportunity for the accumulation of coherent errors explains why
the universal sequence underperforms the 3 × 3X3 sequence. The superior performance of
the 3X3 sequences also confirms that the dominant source of noise is dephasing.
Similar improvements are observed with DD for ququarts, as shown in Fig. 3.2(b). Since
ququarts are more susceptible to charge noise due to the involvement of the third excited
state [29], the free evolution fidelity is significantly lower than in the qutrit case, and the
improvement with DD is even more pronounced. Note that the difference between the DD
sequences is much smaller than in the qutrit case, except for the 4 × 4X4 case at times
< 25 µs. These ququart results highlight the effectiveness of DD in higher dimensions and
its critical role in suppressing noise in more complex systems.
50
3.4 Cross-Kerr suppressing DD (CKDD)
Having shown significant improvements with single-qudit DD against decoherence, we now
deploy DD to mitigate qudit crosstalk. Recent work has demonstrated the efficacy of DD
in suppressing coherent crosstalk errors in qubit systems [45, 132, 152]. In our fixed linear
coupling qudit processor, single qudit operations suffer from always-on crosstalk, which is a
generalization of the ZZ interaction between transmon qubits [36]. This type of crosstalk is
commonly referred to as cross-Kerr interactions, which describe the spectator-state dependent shifts of the relevant qudit transition frequencies. For two coupled qubits, in the lab
frame where H2q =
P1
i,j=0 Eij |ij⟩⟨ij| (in the eigenbasis), Tr(ZZH2q) = E11+E00−E10−E01.
To model cross-Kerr interactions, consider the rotating frame that nullifies all bare transmon energy terms, leaving only the diagonal interaction terms. Then the Hamiltonian for
the two-coupled transmon qudits simplifies to [30, 31]:
HCK =
X
d−1
i,j=1
αij |ij⟩⟨ij|. (3.3)
Here, αij = ωij + ω00 − ωi0 − ω0j (taking ℏ = 1), where i, j ∈ Zd, are the qudit frequency
shifts.
Although these cross-Kerr interactions, along with off-resonantly applied microwave drives,
have been shown to facilitate entangling operations [31, 30], even minor cross-Kerr interactions during idle periods can introduce significant coherent errors. Building on the qudit
DD formalism developed above, we now propose a DD sequence for coupled qudits. This
DD sequence effectively suppresses all the cross-Kerr interactions, thereby enhancing system
stability and operational fidelity.
The evolution operator due to dXd applied only to the first qudit is given by U
(1)
dτ ≡
dXd ⊗ I. By concatenating this sequence with the same dXd sequence applied to the second
51
0 10 20 30 40 50
time (µs)
0.0
0.2
0.4
0.6
0.8
1.0
Fidelity of |+i3
|00i
|01i
|02i
|10i
|11i
|12i
|20i
|21i
|22i
No DD DD
0 5 10 15 20 25
time (µs)
0.0
0.2
0.4
0.6
0.8
1.0
Fidelity of |+i4
|0i
|1i
|2i
|3i
No DD DD
Figure 3.3: Experimental results showing suppression of cross-Kerr interactions using the
CKDD sequence. (a) Fidelity of the qutrit superposition state |+⟩3, with CKDD (solid) and
without (dashed). CKDD is a 9τ -long sequence, with τmin = 120 ns. (b) Fidelity of the
ququart superposition state |+⟩4, with CKDD (solid) and without (dashed). CKDD is a
16τ -long sequence, with τmin = 180 ns. As in Fig. 3.2, longer evolution times correspond to
a single repetition of CKDD with increased τ . CKDD removes the cross-Kerr oscillations
and improves the fidelity in both cases, converging to the fully mixed state fidelity baseline
(dashed horizontal line) more slowly than the free evolution (No DD) curves. See text for
further details.
qudit, we obtain the total evolution Ud
2τ ≡ U
(2)
dτ ◦ U
(1)
dτ , i.e.,
Ud
2τ = (I ⊗ Xd)U
(1)
dτ (I ⊗ Xd)· · · U
(1)
dτ (I ⊗ Xd)U
(1)
dτ . (3.4)
Concatenation of DD sequences was originally introduced in order to obtain high-order
suppression [85]; here, it serves the purpose of staggering the sequences on the two qudits,
thus generalizing the idea of robust qubit-crosstalk suppression via staggering [132]. We show
in Appendix A that Ud
2τ = e
iθI ⊗ I + O(T
2
), where θ is a global phase, and T = d
2
τ . Thus,
we expect that, to first order in the pulse interval, the d
2
τ -long cross-Kerr DD (CKDD)
sequence in Eq. (3.4) suppresses all crosstalk between two coupled qudits.
52
0 2 4 6 8 10
time (µs)
0.0
0.2
0.4
0.6
0.8
1.0
Qutrit Bell State Fidelity
No DD
with CKDD
Figure 3.4: Left: Fidelity of the qutrit Bell state over time with and without CKDD. See
text for details. Right: real (upper) and imaginary (lower) components of quantum state
tomography results for the final time point in the left plot (red circles). Ideally, ℜ(ρik,jl) =
1
3
δikδjl and ℑ(ρik,jl) = 0 for i, j, k, l ∈ Z3, where ρ is the density matrix. Left: free evolution.
The nine red-colored bars at the ideal positions are of varying magnitude and some contain
large imaginary components, indicating deviations from the ideal qutrit Bell state. Right:
with CKDD. In contrast, the nine red-colored bars are nearly uniform in height are have
negligible imaginary components, indicating proximity to the ideal qutrit Bell state. Note
that here CKDD is a 9τ -long sequence with τmin = 180 ns, the X3 gate duration; this differs
from the previous figures where the X3 gate duration is 120 ns due to a different calibration.
3.5 Experimental validation of CKDD
To validate the CKDD sequence, we conduct experiments on coupled transmon qutrits and
ququarts. Fig. 3.3(a) shows the results for a linear chain of three coupled qutrits, where we
prepare the set of nine initial states |i⟩ ⊗ |+⟩3 ⊗ |j⟩, i, j ∈ Z3. We trace out the states of the
left and right “spectator” qutrits, and display the fidelity of the middle (main) qutrit’s state
with respect to |+⟩3 as a function of delay time. As depicted by the dashed light-color curves,
different initial spectator states exhibit distinct curves that oscillate at different frequencies.
The high frequency oscillations and the differences between the curves are attributable to
the cross-Kerr interactions between the main qutrit and the two spectators. Next, we apply
the CKDD sequence as U
(2)
dτ ◦ (U
(1)
dτ ⊗ U
(3)
dτ ), that is, an inner sequence where dXd is applied
simultaneously to the spectators, and an outer sequence where dXd is applied to the main
qutrit. The spectators’ pulses are synchronized, so the sequence still takes a total time of
53
d
2
τ . The resulting solid, bold-color curves exhibit a higher fidelity and none of variation
of the free evolution curves, highlighting the efficacy of the CKDD sequence in suppressing
cross-Kerr interactions and stabilizing the system dynamics for different initial spectator
states.
In Fig. 3.3(b), we present the fidelity results for two coupled ququarts, for the set of four
initial states |i⟩ ⊗ |+⟩4, i ∈ Z4 for the spectator. We again trace out the spectator state.
Similarly to the qutrit case, under free evolution we observe different curves corresponding
to different states of the spectator, but when CKDD is applied, all four curves exhibit a
similar exponential decay, indicating suppression of the cross-Kerr interactions. The CKDD
sequence consists of the ququart shift operators X4 comprising three σ
x
i,i+1 gates in the
two-level subspaces spanned by {|i⟩, |i + 1⟩}, i ∈ Z4. In the presence of large cross-Kerr
interactions, driving the two-level subspaces is prone to large detuning errors. This results
in somewhat lower CKDD fidelities for the ququart experiments compared to qutrits. Despite
this, the suppression of the cross-Kerr interactions is clearly evident from the results, thus
providing a proof-of-principle demonstration of the scalability of the CKDD protocol to
higher dimensions.
3.6 Qutrit entanglement preservation via CKDD
To demonstrate the effectiveness of CKDD beyond product state preservation, we prepare
the qutrit Bell state (|00⟩+|11⟩+|22⟩/
√
3 and measure its fidelity over time with and without
CKDD. The preparation of this state involves a qutrit controlled-phase (CZ) gate [31]. We
employ quantum state tomography to compute the fidelity of the Bell state. As depicted
in Fig. 3.4, the experimental results for free evolution (No DD) and with CKDD applied to
both qutrits contrast sharply. In the absence of DD, the state fidelity suffers significantly
due to strong coherent errors arising from large cross-Kerr interactions. The fidelity drops
to near zero in ∼ 1 µs, meaning that the state evolves to an orthogonal qutrit Bell state,
54
then oscillates around the fully mixed state fidelity baseline of 1/9 (dashed horizontal line),
highlighting the importance of suppressing cross-Kerr interactions. In contrast, with CKDD
we observe a marked improvement; the oscillations are nearly eliminated and the fidelity
remains > 50% even after 10 µs. The state tomography histograms in Fig. 3.4 further
highlight CKDD’s ability to maintain the integrity of qutrit Bell states.
3.7 Conclusions and outlook
Building on the theory of qudit DD [142, 143, 144, 145], we have demonstrated the suppression of decoherence in transmon-based qutrits and ququarts. Our experimental results
exhibit a substantial improvement in the preservation of the fidelity of superposition states of
such qudit systems. Beyond decoherence, a significant challenge in scaling superconducting
qudit systems arises from the persistent cross-Kerr interactions between coupled qudits. To
address this, we introduced cross-Kerr DD as a protocol aimed at suppressing these spurious
interactions. Our experimental results demonstrate that the CKDD sequence successfully
suppresses cross-Kerr interactions in both qutrits and ququarts, which suggests that CKDD
can be employed in yet higher-dimensional systems as well. Furthermore, we have shown
that CKDD significantly improves the fidelity of maximally entangled qutrit states.
Our findings broaden the scope of dynamical decoupling used in the service of the suppression of decoherence and crosstalk beyond the traditional setting of qubits to qudit systems.
While our focus here was on transmons, our findings can be directly applied to other quantum computing platforms with access to qudits, such as fluxonium systems operated at half
flux [153, 154]. This addition to the quantum noise suppression toolkit will hopefully benefit
the development of scalable qudit-based quantum processors.
55
3.8 Experimental Device Characterization
The superconducting qudit device employed in this work consists of 8 fixed-frequency transmon qudits coupled together by coplanar waveguide resonators in a ring topology. For all
the experiments in this work, we employ a susbset of the device consisting of a 3-qudit line.
Q2 was used for both single-qudit DD experiments reported in Fig. 3.2. We report the basic
single qudit parameters of the subset of the device used in this work in Table. 3.1. For
further, more extensive characterization of the device, including gate fidelities and readout
fidelities, see Refs. [155, 125, 31].
Parameters Q1 Q2 Q3
ω01/2π (GHz) 5.333 5.396 5.572
ω12/2π (GHz) 5.061 5.124 5.303
ω23/2π (GHz) 4.757 4.821 5.005
Avg. T
01
1
(µs) 50(4) 49(4) 60(5)
Avg. T
12
1
(µs) 35(2) 35(4) 31(8)
Avg. T
23
1
(µs) 24(4) 26(3) 23(4)
Avg. T
01
2e
(µs) 78(5) 85(9) 90(6)
Avg. T
12
2e
(µs) 57(4) 57(4) 56(9)
Avg. T
23
2e
(µs) 26(2) 27(2) 24(3)
Table 3.1: Transition frequencies ωij = (Ej − Ei)/ℏ up to d = 4 of the qudits employed in
our DD experiments. Additionally, we provide the two-level subspace mean T1 and T2 echo
times of the device calculated from 100 repetitions of each coherence experiment.
Qudit Cross-Kerr Coupling Rates.— As discussed in the Section 3.4, the fixed-linear
coupling between superconducting qudits generates a longitudinal hybridization mediated
largely by the higher levels present in each transmon well. These longitudinal interactions
impart entangling phases on all qudit states |i, j⟩ (i, j ∈ Zd) and lead to significant coherent errors. In the doubly rotating frame of the two-qudit system, the effective cross-Kerr
Hamiltonian defines the accumulation of all non-local entangling phases and is given by
HCK =
Pd−1
i,j=1 αij |ij⟩⟨ij| [30, 31] [Eq. (3.3)]. In the cross-Kerr DD experiments reported in
Fig. 3.3, we demonstrate that local pulses are sufficient to provide effective suppression of
56
the d = 3 and d = 4 cross-Kerr interaction as well as the system-bath decoupling.
d = 3 cross-Kerr DD Experiment.— For our d = 3 cross-Kerr DD experiment [see
Fig. 3.3(a)], we prepared the |+⟩3 on Q2 using a qutrit Hadamard gate and simultaneously
prepared the two spectator qutrits Q1 and Q3 in states |i, j⟩ (i, j ∈ Z3). We then allowed
the system to time-evolve with and without our cross-Kerr DD sequence, and assessed the
time-evolved state fidelity by remapping |+⟩3 to |0⟩ via a final qutrit Hadamard gate. The
relevant four qutrit cross-Kerr rates for this experiment measured via conditional Ramsey
experiments are presented in Table 3.2.
Parameters Q1-Q2 Q2-Q3
α11 (MHz) 0.112 0.212
α12 (MHz) 0.623 0.465
α21 (MHz) -0.515 -0.162
α22 (MHz) 0.341 0.615
Table 3.2: The qutrit cross-Kerr crosstalk rates present in the spectator DD experiment in
Fig. 3.3(a).
d = 4 cross-Kerr DD Experiment.— For our d = 4 cross-Kerr DD experiment [see
Fig. 3.3(b)], we prepared the |+⟩4 state on Q2 using a ququart Hadamard gate and simultaneously prepared the spectator qudit Q1 in |i⟩ (i ∈ Z4). We then allowed the system to
time-evolve with and without our cross-Kerr DD sequence, and assessed the time-evolved
state fidelity by remapping |+⟩4 to |0⟩ via a final ququart Hadamard gate. The relevant nine
ququart cross-Kerr rates for this experiment measured via conditional Ramsey experiments
are presented in Table 3.3.
57
Parameters Q1-Q2
α11 (MHz) 0.112
α12 (MHz) 0.623
α13 (MHz) 0.021
α21 (MHz) -0.515
α22 (MHz) 0.341
α23 (MHz) 0.730
α31 (MHz) 0.226
α32 (MHz) -0.442
α33 (MHz) 0.345
Table 3.3: The ququart cross-Kerr crosstalk rates present in the spectator DD experiment
in Fig. 3.3(b)
58
Chapter 4
Modeling low- and high-frequency noise
on a transmon qubit
Note: This chapter is adapted from [156].
Main results.— Transmon qubits experience open system effects that manifest as noise at
a broad range of frequencies. We present a model of these effects using the Redfield master equation with a hybrid bath consisting of low- and high-frequency components. We
use two-level fluctuators to simulate 1/f-like noise behavior, which is a dominant source of
decoherence for superconducting qubits. By measuring quantum state fidelity under free evolution with and without dynamical decoupling (DD), we can fit the low- and high-frequency
noise parameters in our model. We train and test our model using experiments on quantum devices available through IBM quantum experience. Our model accurately predicts
the fidelity decay of random initial states, including the effect of DD pulse sequences. We
compare our model with two simpler models and confirm the importance of including both
high-frequency and 1/f noise in order to accurately predict transmon behavior.
4.1 Introduction
Starting with the first implementations of superconducting qubits [157, 158], the field has
developed several flavors of qubits, broadly classified as charge, flux, and phase qubits [47].
59
However, the real workhorse behind many of the recent critical developments [49, 50, 51,
52, 53, 54, 55, 56, 57, 58] in gate-based quantum computing is the transmon qubit [159].
Transmons are designed by adding a large shunting capacitor to charge qubits, the result
being that they are almost insensitive to charge noise. Transmon-based cloud quantum
computers (QCs) are now widely available to the broad research community for proof of
principle quantum computing experiments [160, 161, 162, 163, 164, 76, 165, 166, 167, 168,
169].
Quantum computers in their current form have high error rates. This includes coherent errors (originating from imperfect gates), state preparation and measurement errors
(SPAM), and incoherent errors (environment-induced noise) [170]. The latter, which results
in dephasing and relaxation errors, is a pernicious problem in quantum information processing. Characterizing and modeling these open quantum system effects is crucial for advancing
the field and improving the prospects of fault-tolerant quantum computation [60, 61, 62].
Various procedures for modeling decoherence and control noise affecting idealized qubits have
been discussed [77, 171, 172]. Still, modeling noise effects from first principles, i.e., starting
at the circuit level of transmons and including 1/f noise, is relatively unexplored [173, 174].
In this work, we develop a framework to model environment-induced noise effects on a
transmon qubit using the master equation formalism. We use a hybrid quantum bath with
an Ohmic-like noise spectrum to model dephasing and relaxation processes in multi-level
transmons. We also include classical fluctuators and use a hybrid Redfield model [44] to
account for both low- (1/f) and high-frequency noise. We develop a simple noise learning
procedure relying on dynamical decoupling (DD) [71, 73, 72, 74, 82] to obtain the noise
parameters (see Ref. [175] for early experimental work in this area). Our procedure relies
only on measurements of quantum state fidelity with and without a single type of DD sequence, and so is quite resource-efficient compared to protocols requiring full quantum state
tomography or DD-based spectral analysis. We test our noise model via fidelity preservation
experiments on IBMQE processors [70] for random initial states and find that the model can
60
correctly capture these experiments. The model is, moreover, capable of reproducing the effects of time-dependent dynamical decoupling pulses on the main qubit. Finally, we compare
the predictions based on our model with two simpler models using ideal two-level qubits,
excluding the fluctuators and assuming ideal, zero-width DD pulses. In contrast to our
complete model, these simpler models fail to capture noise simultaneously in both the lowand high-frequency regimes. As a result, whether with or without DD, they underperform
in capturing fidelity preservation experiments. Our model is tailored to transmon qubits;
however, our approach is extremely general and can be adapted to any qubit experiencing
the ubiquitous combinations of high-frequency and 1/f noise.
This paper is organized as follows. In Section 4.2, we develop our numerical method
focusing on simulating multi-level transmon qubit and single-qubit gates, which form the
DD sequences. Next, we discuss our open quantum system in Section 4.3 and describe our
noise learning method in Section 4.4. We then test our learned model on Quito using DD
experiments with random initial single-qubit states in Section 4.5. We extend our method
to Lima, which relies on a different calibration procedure, in Section 4.6. We conclude in
Section 4.8. The appendix provides additional details and calculations in support of the
main text.
4.2 Numerical model of transmons
In this section, we focus on the circuit-level description of the transmon qubit that we use
in our model. We start with the transmon Hamiltonian and find an effective Hamiltonian to
simulate single-qubit time-dependent microwave gates. We include the Derivative Removal
of Adiabatic Gates (DRAG) [41] technique in our numerical model. By including the DRAG
technique (used in the IBMQE devices) and considering the residual errors it is unable to
suppress, we more accurately model the transmon behavior.
61
4.2.1 Transmon Hamiltonian
The Hamiltonian of a fixed-frequency transmon qubit is [159]:
Htrans = 4EC (ˆn − ng)
2 − EJ cos ˆφ . (4.1)
We work in units where ℏ = 1. EC = e
2/(2C) is the charging energy (C is the capacitance,
and e is the electron charge), EJ = IC/(2e) is the potential energy of the Josephson junction
(IC is the critical current of the junction) and ng represents the charge offset number which
can result in charge noise. In the operating regime of a transmon qubit, i.e., EJ /EC ≫ 1,
the lowest few energy levels of the transmon are almost immune to charge noise, in which
case ng can be safely ignored. The two operators ˆn and ˆφ are a canonically conjugate pair
analogous to momentum and position. They satisfy the commutation relation [ˆn, φˆ] = i; ˆn is
the number operator for the Cooper pairs transferred between the superconducting islands
of the Josephson junction, and ˆφ is the gauge invariant phase difference across the Josephson
junction, i.e., between the islands. Note that this commutation relation is exact only if we
restrict ˆφ to a single 2π range; this is a good approximation for the lower energy states of
transmons which are almost completely confined to a single well.
4.2.2 Time-dependent drives
To numerically simulate the time-dependent drive-pulses or gates, we start with Eq. (4.1)
and write it in the charge basis (the eigenbasis of ˆn) such that the number of Cooper pairs
takes values from −nmax to nmax. Eq. (4.1) thus reduces to
Htrans = 4EC
nXmax
−nmax
n
2
|n⟩⟨n|
−
EJ
2
nXmax
−nmax
(|n⟩⟨n + 1| + |n + 1⟩⟨n|) (4.2)
62
where we have taken ng = 0 since we are in the transmon regime (our EJ /EC ≈ 47).
Note that the charge dispersion ϵng ∝ e
−
√
8EJ /EC [159]. We truncate to nmax (later we set
nmax = 50) and diagonalize the resulting Hamiltonian:
H
eigen
trans = SHtransS
† =
X
k≥0
ωk|k⟩⟨k| , (4.3)
where ωk for k = 0, 1, ... represents the energy of the k
th level in the transmon eigenbasis,
and S is the unitary similarity transformation. The eigenfrequencies are ωij ≡ ωi − ωj
. The
bare qubit frequency is ωq ≡ ω10 and the anharmonicity is ηq ≡ ω10 − ω21. Since ωq and ηq
are the two quantities accessible via experiments, we use these values to obtain the fitting
parameters EC and EJ in Eq. (4.1), which is the starting point of our transmon model.1
Next, we add the coupling to the microwave drive, which couples to the transmon charge
operator. The total system Hamiltonian can then be written as
Hsys = H
eigen
trans + ε(t) cos (ωdt + ϕd) ˆn , (4.4)
where ε(t) is the pulse envelope, ωd is the drive frequency, and ϕd is the phase of the drive.
We can simplify Eq. (4.4) by first writing the charge operator in the transmon eigenbasis of
Eq. (4.3), i.e., ˆn =
P
k,k′⟨k|nˆ|k
′
⟩|k⟩⟨k
′
| and considering the charge coupling matrix elements.
Only nearest-level couplings ⟨k|nˆ|k ± 1⟩ are found to be non-negligible, allowing us to ignore
all higher order terms:
nˆ ≈
X
k≥0
⟨k|nˆ|k + 1⟩|k⟩⟨k + 1| + h.c. (4.5)
Transforming into a frame rotating with the drive and employing the rotating wave approx1
In more detail, we try different values of EJ and EC in Eq. (4.1) by diagonalizing the corresponding
Htrans and comparing the result with the experimental values of ωq and ηq. Once we find the values of EJ
and EC yielding the closest match, we proceed to Eq. (4.3) to find the full spectrum.
63
imation (RWA), we obtain, for ϕd = 0, the effective Hamiltonian
H˜
sys =
X
k≥0
(ωk − kωd)|k⟩⟨k| (4.6)
+
ε(t)
2
X
k≥0
gk,k+1(|k⟩⟨k + 1| + |k + 1⟩⟨k|) ,
where gk,k+1 ≡ ⟨k|nˆ|k + 1⟩. By tuning ϕd, we can implement a rotation about any axis in
the (x, y) plane of the qubit subspace (after an additional projection). In particular, taking
ϕd = 0 or π/2 corresponds to a rotation about the x or y axis, respectively.
The pulse envelope ε(t) plays a vital role in the final implementation of the gate. Since
we are interested mainly in applying π pulses, we choose
Z tg
0
ε(t)dt = π , (4.7)
where tg is the pulse or gate duration. For our numerical simulations, we choose Gaussianshaped pulses with envelopes given by
ε(t) = ε [G(t, tg, σ) − G(0, tg, σ)] (Θ(t) − Θ (t − tg)) , (4.8)
where
G(t, tg, σ) = exp
−
(t − tg/2)2
2σ
2
!
. (4.9)
Here ε is the maximum drive amplitude during the pulse, Θ(t) is the step function, and σ is
the standard deviation of the Gaussian pulse.
An essential aspect of gate design is that population should not leak to higher levels of
the transmon, i.e., the drive pulses should be bandwidth-limited (adiabatic). An accurate
measure of these off-resonant excitations is the Fourier transform of the pulse envelope at
the detuning frequencies [176, 177]. For example, consider a Gaussian pulse with standard
deviation σ. Its Fourier transform has a standard deviation proportional to 1/σ, which
64
Figure 4.1: Gaussian pulse envelope (solid, orange) [see Eq. (4.8)] and its Fourier transform
(dashed, blue) with amplitude ε chosen to keep both in the range [0, 1]. (a) and (b) show the
pulse with gate time tg = 70 and 10 ns, respectively, and σ = tg/6. The bottom horizontal
axis represents time in ns, and the top horizontal axis represents frequency in GHz. Shorter
gate times result in a larger frequency spread of the spectrum, with associated larger leakage,
as illustrated in (c), which shows the frequency spectrum corresponding to a tg = 10 ns gate
(left), compared to the energy levels (right) |0⟩, |1⟩ and |2⟩ of the transmon. The energy
levels are shown in the rotating frame such that E|0⟩ = E|1⟩ and E|2⟩ = −ηq = −200 MHz.
As indicated by the dashed horizontal line, the spectrum overlaps with level |2⟩, resulting in
leakage into this level from the {|0⟩, |1⟩} qubit subspace. The sampling frequency used to
compute the Fourier transform is 10GS/s, which is state-of-the-art in experiments; the pulses
that control the IBM processors used in this work have a sampling frequency of ∼ 5GS/s.
means that the drive pulse applied at the qubit frequency ωq has a frequency spread close to
1/σ about ωq. If 1/σ is of the order of the anharmonicity of the transmon, the pulse spectral
width will overlap with some of the higher-level transitions. Fig. 4.1 shows the Gaussian
pulse envelope and its Fourier transform, and illustrates how choosing a shorter gate time
results in a larger frequency spread and vice versa. The use of DRAG pulses mitigates this
leakage, as discussed further below.
65
In the two-state (qubit) subspace, Eq. (4.6) reduces to
HX(t) = ε(t)
2
(|0⟩⟨1| + |1⟩⟨0|) = ε(t)
2
σ
x
, (4.10)
where g0,1 = g has been absorbed into ε(t) and σ
x
is the Pauli X matrix. When HX(t) is
evolved for a time tg such that Eq. (4.7) is satisfied, the resulting unitary is an ideal Xπ gate.
To include the effect of higher levels, we first use the full gate Hamiltonian from Eq. (4.6)
and then project the result to the qubit subspace.
The gate fidelity averaged over all input states in the qubit Hilbert space can be written
as the average over the six polar states (i.e., the six eigenstates of σ
x
, σy
, and σ
z
) [178, 41]:
Fg =
1
6
X
j=±x,±y,±z
Tr h
Uidealρ
1q
j U
†
idealΠ[ρ(tg)]i
, (4.11)
where ρ
1q
j
is the single qubit density matrix, and Uideal represents the ideal unitary corresponding to the gate we wish to study. Π[ρ(tg)] is the projection of the full density matrix
onto the single qubit subspace. Fg compares the density matrix ρ after application of the
gate (i.e., at t = tg) with the expected density matrix in the qubit subspace.
To reduce phase errors caused by the presence of additional levels, a commonly used
trick to implement single qubit gates such as Xπ is to break the gate into two halves where
each half performs a π/2 rotation, accompanied by some virtual Z rotations [179, 180].
Numerically, we observe that with four levels included in the transmon Hamiltonian and a
total gate duration tg = 70 ns, with σ = tg/6, the average single-qubit gate error 1 − Fg
can be suppressed by around 20% if we use two such pulses instead of a single long pulse.
The exact quantitative improvement depended on other model parameters. We observed this
error reduction in a closed system setting with no environmental coupling, and so any fidelity
improvement may be counteracted by open-system effects. For a detailed numerical study
on the error of time-dependent gates with transmon qubits in the open system settings, see
Ref. [181]. We also include DRAG [41, 182] in our single qubit gates. DRAG is a state66
of-the-art technique used to improve the performance of single-qubit gates by suppressing
leakage and phase errors. The former refers to non-zero population of non-computational
levels at the end of a pulse, while the latter is a type of coherent error that results from the
non-zero population in non-computational levels during the pulse: the admixture of such
levels leads to a phase shift of the computational levels, resulting in a net phase error at the
end of the pulse.
4.3 Open quantum system simulation
This section describes the noise model and discusses the hybrid Redfield equation used for
the open quantum system simulations. For all simulations we truncate to the lowest four
levels of the transmon qubit.
4.3.1 Interaction Hamiltonian
The single-qubit system bath interaction Hamiltonian in the lab frame can be written as
HSB =
X
i=x,y,z
giAi ⊗ Bi
, (4.12)
where Ai and Bi represent the dimensionless system and bath coupling operators, respectively, and the coupling strengths gi have dimensions of energy. There are several contributions to decoherence and noise for a multi-level transmon circuit. With fixed-frequency
architectures, charge noise and fluctuations in the critical current contribute most to decoherence. In the flux-tunable variants of transmon qubits, flux noise becomes an additional
noise channel [159]. These considerations determine which coupling operators are needed to
describe the noise model for a given architecture. Similarly, we can define similar coupling
operators for fluxonium qubits [183]. In the IBMQE processors used, the transmons are
fixed-frequency. We therefore choose appropriate noise operators below.
67
We consider the following system-bath interaction Hamiltonian:
HSB = gxAx ⊗ Bx + Az ⊗
gzBz +
X
k
bkχk (t) IB
!
, (4.13)
where the coupling operators Ax and Az correspond to the charge coupling operator and to
the Josephson energy operator and are defined as
Ax = c1nˆ (4.14a)
Az = c2cos ˆφ , (4.14b)
where c1 and c2 are fixed constants that depend on the charge energy EC and the Josephson energy EJ of the transmon qubit, respectively. We expect, based on the discussion in
Section 4.2.2 – and observe in our simulations – that Ax and Az act like σ
x and σ
z when
projected into the qubit subspace. We find numerically that Eq. (4.13) is an adequate model
accounting for the nearly equal decay of the |+⟩ and |i⟩ states, which is why we do not include a separate σ
y
coupling term. Note, however, that a (dependent) σ
y
component appears
when we transform Eq. (4.13) from the lab frame into a frame rotating with the drive.
Previous studies have found that noise in the superconducting circuit can be separated
into high and low-frequency components [184]. To account for this observation, we combine
two noise models. We choose the bath operators Bx and Bz in Eq. (4.13) to be bosonic bath
operators, which generally represent the high-frequency component of the noise. However,
this is not always the case, as we argue in Section 4.4.
To account for the low-frequency noise component, which is a dominant noise source for
superconducting qubits [185], we include a sum over classical fluctuators in Eq. (4.13), via
the term proportional to the bath identity operator IB. This semiclassical contribution, when
parameterized properly, can simulate the behavior of 1/f noise. We model the fluctuators
as having equal coupling strengths, i.e., we set bk = b (with dimensions of energy) for
68
|0/1⟩
|0/1⟩
Main : |0⟩ U (−I − I − I − I−)
2N U
−1
Free
Spectators : |0⟩ U (−X − Y − X − Y −)
N U
−1
Main : |0⟩ U (−X − Y − X − Y −)
N U
−1
DD
Spectators : |0⟩ U (−I − I − I − I−)
2N U
−1
Figure 4.2: The circuit schematics for the free-evolution and DD-evolution types of experiments. For the free-evolution case, we apply N cycles of the XY4 dynamical decoupling
(DD) sequence on all the spectator qubits and 2N cycles of the I4 sequence (here I4 means
four identity gates) on the main qubit, which suppresses crosstalk errors. Note that an X
or Y gate is twice as long as an identity gate on the IBM cloud quantum devices, hence the
extra factor of 2. For the DD-evolution case, we apply the DD sequence only to the main
qubit and apply identity gates to all the spectator qubits. This suppresses both crosstalk
and environment-induced noise. We measure only the main qubit.
k = 1, · · · , 10. Each fluctuator can be characterized by a stochastic process χk(t) that
switches between ±1 with a frequency γk, which is log-uniformly distributed between γmin
and γmax [186]. We find 10 fluctuators is enough to reproduce the 1/f spectrum ubiquitous
in these devices.
4.3.2 Hybrid Redfield model
To simulate the reduced system dynamics of the interaction Hamiltonian in Eq. (4.13), we
use a hybrid form of the Redfield (or TCL2) master equation [44, 87]. We first define the
standard bath correlation function
Cij (t) = Tr{UB(t)BiU
†
B
(t)BjρB} , (4.15)
69
where UB(t) = e
−iHBt
is the unitary evolution operator generated by the bath Hamiltonian
HB, and the reference state ρB is the Gibbs state of HB:
ρB = e
−βHB /Tr
e
−βHB
, (4.16)
where β = 1/T is the inverse temperature. Assuming the bath operators Bx and Bz are
uncorrelated, i.e., Cxz(t) = Czx(t) = 0, we construct the following hybrid Redfield equation
∂ρS
∂t = −i[Hsys + Az
X
10
k=1
bkχk(t), ρS] + LR(ρS) , (4.17)
where LR is the Redfield Liouville superoperator
LR(ρS) = −
X
i=x,z
[Ai
,Λi(t)ρS(t)] + h.c. , (4.18)
and
Λi(t) = Z t
0
Ci(t − τ )Usys(t, τ )AiU
†
sys(t, τ )dτ , (4.19)
where Cj (τ ) ≡ Cjj (τ ) [from Eq. (4.15)] and Usys(t) is the unitary evolution operator generated
by the system Hamiltonian Hsys. The reduced system dynamics are obtained by averaging
the solution of Eq. (4.17) over all the realizations of χk(t) for k = 1, ..., 10.
The bath component correlation functions Cj (τ ) are the Fourier transforms of the bath
component noise spectra
γj (ω) = Z ∞
−∞
Cj (τ )e
iωτdτ . (4.20)
We choose the bath to be Ohmic, which means that the component noise spectra have the
form
γj (ω) = 2πηg2
j
ωe
−|ω|/ωc
j
1 − e
−βω , (4.21)
where ω
c
j
is the cut-off frequency for bath operator Bj
, and η is a positive constant with
dimensions of time squared arising in the specification of the Ohmic spectral function.
7
Lastly, the hybrid Redfield equation (4.17) can be transformed into a frame rotating
with the drive frequency ωd by replacing every operator with the interaction-picture one
[specifically, the Ai operator in Eq. (4.19) needs to be replaced by Ai(τ )]. We simulate the
Redfield master equation in this rotating frame in the methodology and results we discuss
next.
4.4 Methodology and Fitting Results
This section discusses our methodology for modeling a transmon qubit’s open quantum
system behavior in a multi-qubit processor. We refer to the qubit of interest as the main
qubit and all the others as spectator qubits. The goal is to extract the bath parameters
in our open quantum system model and then use this model to predict the outcomes of
experiments on the main qubit, including dynamical decoupling sequences. We treat qubit
1 (Q1) of the Quito processor as our main qubit [see Fig. 2.1(b)]. We are interested only in
the main qubit’s behavior here; hence, we measure only the main qubit.
4.4.1 Free and DD Evolution Experiments
Our procedure involves two types of experiments, as shown in Fig. 4.2. The first type, which
we call a free-evolution experiment, consists of initializing all the qubits in a given state by
applying a particular unitary operation U3(θ, ϕ, λ) [187] (denoted as U in Fig. 4.2) to each
of the qubits and is given as
U3(θ, ϕ, λ) =
cos
θ
2
−e
iλ sin
θ
2
e
iϕ sin
θ
2
e
i(ϕ+λ)
cos
θ
2
(4.22)
We then apply a sequence of identity gates on the main qubit, which we vary in number.
Simultaneously we also apply the XY4 DD sequence to all the other (spectator) qubits (i.e.,
XfτY fτXfτY fτ , where fτ denotes free-evolution in the absence of pulses for a duration of τ
[82]) for the same total duration as that of the identity gates on the main qubit. As shown
in Chapter 2, DD sequences applied to spectator qubits suppress unwanted ZZ-interactions,
i.e., ZZ-crosstalk between the main qubit and the spectator qubits. Without crosstalk
suppression, we observe oscillations in the probability decay as a function of time [76, 77];
see experimental results in Fig. 4.8. With the crosstalk suppression scheme, i.e., DD applied
to the spectator qubits, these oscillations disappear, and the main qubit is now primarily
affected only by environment-induced noise. Finally, we apply the inverse of U3(θ, ϕ, λ) and
measure in the Z-basis. The result is how we compute the initial states’ decay probability.
Everything remains the same in the second type of experiment, which we call DDevolution, except that we now apply the XY4 sequence to the main qubit and identity gates
on the spectator qubits. As discussed in Chapter 2, when we apply the XY4 sequence only to
the main qubit, we suppress the ZZ-crosstalk between the latter and all the spectator qubits
and also decouple unwanted interactions between the main qubit and the environment. We
note that, in contrast to experiments using DD to perform noise spectroscopy, here we use
only a single type of DD sequence and do not vary any of its parameters.
4.4.2 Fitting Procedure
We perform the free-evolution and DD-evolution experiments for the six Pauli states as initial
states, i.e., we choose U3(θ, ϕ, λ) to prepare |0⟩, |1⟩, |+⟩, |−⟩, |i⟩ and |−i⟩, and use the hybrid
Redfield equation described in Section 4.3.2 to simulate the dynamics of these experiments.
We sweep over different values of the bath parameters and obtain the simulated probability
decay as a function of time. To identify the simulation parameters that optimally match the
experimental results, we define a cost function C for a given initial state |ψ⟩ as the l2 norm
distance between the experimental probabilities P
Exp
|ψ⟩,s
(ti) and the simulation probabilities
72
0.5 1.0 1.5 2.0 2.5 3.0
ωc
x/2π (GHz)
0.002
0.004
0.006
0.008
0.010
g
x/2
π (GHz)
(a)
0.0000
0.0056
0.0112
0.0168
0.0224
0.0280
0.0336
0.0392
0.0448
0.0504
C|1i, free
0.02 0.04 0.06 0.08 0.10
ωc
z/2π (GHz)
0.002
0.004
0.006
0.008
0.010
g
z/2
π (GHz)
(b)
0.0006
0.0014
0.0022
0.0030
0.0038
0.0046
0.0054
0.0062
0.0070
Cavg, DD
0.05 0.10 0.15 0.20 0.25 0.30
γmax/2π (GHz)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
b/2
π (GHz)
×10
−
3
(c)
0.0016
0.0048
0.0080
0.0112
0.0144
0.0176
0.0208
0.0240
0.0272
C|+i, DD
0.5 1.0 1.5 2.0 2.5 3.0
ωc
x/2π (GHz)
0.002
0.004
0.006
0.008
0.010
g
x/2
π (GHz)
(d)
0.0008
0.0064
0.0120
0.0176
0.0232
0.0288
0.0344
0.0400
0.0456
0.0512
0.02 0.04 0.06 0.08 0.10
ωc
z/2π (GHz)
0.002
0.004
0.006
0.008
0.010
g
z/2
π (GHz)
(e)
0.00048
0.00096
0.00144
0.00192
0.00240
0.00288
0.00336
0.00384
0.00432
0.00480
0.05 0.10 0.15 0.20 0.25 0.30
γmax/2π (GHz)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
b/2
π (GHz)
×10
−
3
(f)
0.0030
0.0065
0.0100
0.0135
0.0170
0.0205
0.0240
0.0275
0.0310
0.0345
Figure 4.3: Top: Results for the Quito processor. Bottom: Results for the Lima processor.
Left: The cost function defined as the l2 norm distance between the experimental and simulation results [Eq. (4.23)], averaged over N = 70 time instants, as a function of the bath
parameters ω
c
x
and gx for free-evolution of the |1⟩ initial state. Middle: The average of the
cost function over the six Pauli states for DD-evolution as a function of ω
c
z
and gz. Right:
The cost function for free-evolution of the |+⟩ initial state, as a function of γmax and b. The
green circles indicate the positions of the global minima in all the panels.
P
Sim
|ψ⟩,s
(ti) for every instant ti
:
C|ψ⟩,s =
1
N
vuut
N
X−1
i=0
P
Sim
|ψ⟩,s
(ti) − P
Exp
|ψ⟩,s
(ti)
2
, (4.23)
where s ∈ {free, DD} and N is the total number of instants. Note that we compensate for
state preparation and measurement (SPAM) errors by shifting the experimental results such
that in all cases, P
Exp
|ψ⟩,s
(0) = 1.
We limit the number of free parameters requiring fitting to six: the coupling strengths
gx, gz, and bk ≡ b [Eq. (4.13)], and the cut-off frequencies γmax (for 1/f noise), ω
c
x
, and
ω
c
z
[Eq. (4.21)]. We set the bath temperature T = 20 mK (∼ the fridge temperature),
γmin = 10−4 GHz, and η = 10−4 GHz−2
; these values are the same as in our previous
study in Chapter 2, which showed strong agreement between open system simulations and
73
experiments using other IBMQE devices, and remain fixed throughout our fitting procedure.
This procedure consists of three steps, which we detail next.
4.4.2.1 Step I: free-evolution for |1⟩
We first focus on the free-evolution experiment for the initial state |1⟩, the first excited state
in the transmon eigenbasis. Since the free-evolution for this state is only affected by charge
noise, i.e., noise along the x-axis, the only contribution to the decay of |1⟩ should come
from the gxAx ⊗ Bx term in Eq. (4.13). Thus, we consider only this term in our numerical
simulations for this initial state. For a given set of values of the coupling strength gx and
bath cut-off frequency ω
c
x
, we compute the cost function C|1⟩,free using Eq. (4.23), and obtain
the contour plot shown in Fig. 4.3(a).
In our simulations, we vary ω
c
x
/(2π) from 0.5 to 3 GHz and gx from 0 to 10−2 GHz, each
with 20 equidistant points so that the contour plot has a total of 400 data points. We take the
position of the global minimum of the cost function on this grid as the optimal set of bath parameter values. To reduce the resulting discretization uncertainty, we interpolate the contour
plot and use the Nelder-Mead optimization method to locate the minima. We numerically
find the global minimum at ω
c
x
/(2π) = 1.948 GHz and gx/(2π) = 0.573×10−2 GHz, denoted
by the green circle in Fig. 4.3(a). With this, we have two out of the six bath parameters,
and we use these learned parameters in the subsequent steps.
4.4.2.2 Step II: DD-evolution for all six Pauli states
The second step involves the DD-evolution experiment for all six Pauli states. This requires
including the term gzAz ⊗Bz in Eq. (4.13), along with the first term whose bath parameters
we already obtained. We do not include the semiclassical term in Eq. (4.13) consisting
of fluctuators since it is expected to be strongly suppressed when DD is applied to the
main qubit. We simulate time-dependent gates with DRAG corrections to model the DD
pulses, as discussed in Section 4.2. We are again left with just two bath parameters to
74
optimize: ω
c
z
and gz. Fig. 4.3(b) shows the average of the cost function [Eq. (4.23)] over
the six Pauli states. The global minimum is found at ω
c
z
/(2π) = 0.569 × 10−2 GHz and
gz/(2π) = 0.441 × 10−2 GHz.
4.4.2.3 Step III: free-evolution for |+⟩
The final step requires optimizing the two remaining free parameters associated with the
fluctuators: γmax and b. Here we focus on the free-evolution experiment for initial state |+⟩.
We now employ the full system-bath Hamiltonian in Eq. (4.13) with the optimal parameters
found in Steps I and II. Fig. 4.3(c) shows the contour plot for the cost function [Eq. (4.23)],
where, as in Step I, we again use 20 different values of γmax and b each. The global minimum
is found at γmax/(2π) = 0.051 GHz and b/(2π) = 0.598 × 10−3 GHz.
4.4.3 Methodology wrap-up
Let us briefly summarize our methodology and add a few technical details. As explained
above, we extract the bath parameters by performing free-evolution experiments for two
initial states (|1⟩ and |+⟩) and DD-evolution experiments for up to six initial states (the
Pauli states). Our optimization procedure is iterative and is thus not guaranteed to yield
the globally optimal values of all the bath parameters, but this is by design: we choose initial
states that allow us to isolate the bath parameters one pair at a time, which renders the
optimization problem tractable.
This methodology is quite general and can be used to characterize all the transmon qubits
on a given transmon processor or, much more broadly, to characterize single qubits on any
quantum information processing platform capable of supporting individual qubit gates and
measurements, provided a sufficiently accurate and descriptive model of the qubits and
the system-bath interaction is available. Our procedure inherently suppresses the effects
of crosstalk due to the neighboring qubits via DD applied either to the spectator qubits
(free-evolution experiments) or the main qubit (DD-evolution experiment), which reduces
75
the number of free parameters of the noise model by eliminating the need to model crosstalk.
To obtain the contour plots shown in Fig. 4.3, we solve the Redfield master equation
(Section 4.3.2) for each point (i.e., each set of model parameters), requiring a total of 400
simulation runs for each optimization. In the final step, including classical fluctuators to
obtain Fig. 4.3(c), we use the trajectory version of the Redfield model introduced in Section 4.3.2 to simulate a total of 600 trajectories at each point. This is large enough to yield
negligible error bars (< 2×10−2
). The experimental results are obtained using the standard
bootstrap method. In defining the cost function [Eq. (4.23)], we use the mean value of the
experimental fidelity obtained after bootstrapping and ignore the associated tiny error bars
(≤ 6 × 10−3
). These error bars are much smaller than the error induced by the discrete
nature of our 40 × 40 parameter grid, and so we can safely ignore them. We confirmed
that varying the probabilities to the extremes of the error bars does not affect the values
of the bath parameters we have extracted to the least significant digit we report. Table 4.1
summarizes the extracted values and the parameters we have fixed.
The accuracy of our results depends on the number of points in the contour plots in
Fig. 4.3 (we used a 20 × 20 grid for each panel). Even though we interpolate the otherwise
discrete contour plots and find the minima over the resulting smooth surface, the limited
number of points affects the precision of the learned bath parameters. Increasing this precision requires more sophisticated optimization techniques to speed up the process of obtaining
the bath parameters. This becomes especially acute when extending the model to learning
a multi-qubit system-bath Hamiltonian with correlated noise, as in this case, the number of
bath parameters increases significantly. Here, we aim to demonstrate the model and methodology and illustrate both via the example of a single transmon qubit, and so we perform a
simple brute-force search of the parameter space. Note that our methods for extracting the
bath parameters also work with density matrices (from state tomography) instead of just
probabilities. In that case, the l2 norm distance in the cost function of Eq. (4.23) can be
replaced by the trace-norm distance between the density matrices obtained from the simula76
10 20 30 40 50 60 70
time instants
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
relative error
(a)
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
time (µs)
10 20 30 40 50 60 70
time instants
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
relative error
(b)
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
time (µs)
10 20 30 40 50 60 70
time instants
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
relative error
(c)
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
time (µs)
10 20 30 40 50 60 70
time instants
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
relative error
(d)
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
time (µs)
Figure 4.4: Results for the Quito (top row) and Lima (bottom row) processors. Left: Box
plots showing the relative error of our model for the free-evolution experiments as a function
of time for 16 different initial states containing six Pauli states and ten Haar-random states.
Right: the same as on the left, but for the DD-evolution experiments. We measured a total of
70 time instants, up to a total evolution time of 19.6 µs, but only display every other instant
to avoid overcrowding. Green triangles indicate the mean over the 16 initial states, black
horizontal lines are the median, gray boxes represent the [25, 75] percentiles, the whiskers
(black lines extending outside the boxes) represent the [0, 25] and [75, 100] percentiles, and
circles are outliers.
tion and the experiment. However, quantum state tomography imposes a much higher cost
in terms of the number of required experiments and is thus less practical to scale up with
a larger number of qubits. Our protocol requires only fidelity measurements and so is more
resource-efficient.
77
free
full model
free
SM 1
free
SM 2
DD
full model
DD
SM 1
DD
SM 2
0.00
0.05
0.10
0.15
0.20
relative error
Free evolution
DD evolution
Figure 4.5: Relative error results for the Quito processor. We display a comparison of the
relative errors between the full model (which uses a three-step learning procedure and consists
of four energy levels per transmon and realistic pulses) with the simplified models SM1 and
SM2 (which are based on a two-step learning procedure and use just two energy levels
and instantaneous pulses) for free-evolution and DD-evolution experiments. SM1 (SM2) is
trained on the DD (free) evolution experiments. Each box contains a total of 16 initial states
and all 70 time instants varying from 0 to 19.6 µs.
4.5 Model Prediction Results
4.5.1 Full Model
We now test our model for different initial states of the main qubit of the Quito processor.
Since we always apply the DD sequence to the spectator qubits during the free-evolution
experiments, the initial states of the latter do not matter due to the suppressed ZZ coupling.
This section considers a total of 16 initial states, consisting of the six Pauli states and ten
Haar-random states. We model the experimental results using the bath parameters we
extracted in the previous section (Table 4.1). This serves as a stringent test of the model:
we now use the previously fitted model to predict the outcome of experiments not included
78
in Steps I-III of Section 4.4.2, i.e., the results with different initial states. The data for all
the experiments (both fitting and testing) was obtained in one batch. We used only the data
for the six Pauli initial states needed for Steps I-III to perform the fitting. We used the data
for all 16 initial states in the testing phase.
We consider the same two kinds of experiments: free-evolution and DD-evolution. Fig. 4.4
(top row) shows our model’s prediction accuracy for the ten random and six Pauli states.
The top left panel [Fig. 4.4(a)] corresponds to the free-evolution case, where we present
the relative error in the prediction of our model as a function of time compared with the
experimental results. The relative error is defined as |meanexp − meansim|/meanexp, where
meanexp is the bootstrapped average over 8192 experimental repetition and meansim is the
average over 600 trajectory simulations of the hybrid Redfield model for any given time
instant. The box plot contains the spread in the relative error over all 16 states, showing
that the relative error of the model is always below 8% over the total time considered here.
The median and the mean over the 16 states are confined well below 3% for every instant.
The performance of our model for the DD-evolution experiments is shown in the top
right panel [Fig. 4.4(b)]. Here the relative error is always below 2%. The median and the
mean are below 1%. The closer agreement of the model with the DD-evolution experiments
is expected, given that in contrast to the free-evolution experiments, DD suppresses the
low-frequency noise affecting the main qubit, and the limitations of our fluctuator model of
this noise is a likely source of modeling error.
The first and fourth columns of Fig. 4.5 show the relative error of our model over the 16
states and all 70 instants of the free and DD-evolution experiments, respectively. The results
of the latter are better, as expected from Fig. 4.4. In both cases, however, we observe that
the model has a relative prediction error of just a few percent.
79
4.5.2 Simplified Models
As discussed in Section 4.2, our numerical simulations use the circuit model Hamiltonian of a
transmon qubit truncated to the four lowest transmon eigenstates. The gates are applied with
time-dependent pulses of non-zero duration. To test the robustness of our detailed model
and learning procedure, we compare it with two simpler models, SM1 and SM2, derived
from our detailed model. The simpler models use a more straightforward qubit description
where we truncate the transmon Hamiltonian to only two levels. The time-dependent gates
are replaced with instantaneous (zero-duration), ideal gates. Moreover, we focus only on the
Ohmic bath terms in Eq. (4.13), thus simplifying the noise model by removing the classical
fluctuators. To test these simpler models’ predictive power, we follow the same procedure
as in Section 4.4, but using only Steps I and II. The difference between SM1 and SM2 lies
in Step II, where SM1 uses the DD-evolution experiments for the six Pauli states, whereas
SM2 uses the free-evolution experiments for the same states. In both cases, we extract the
model parameters and then use the resulting learned models to predict the outcomes of both
the free-evolution and the DD-evolution experiment.
Fig. 4.5 shows the comparison between our detailed model and the simpler models SM1
and SM2. We observe that SM1 has the largest relative error for the free-evolution experiments, whereas SM2 has the largest relative error for the DD-evolution case. Our full model
has the smallest relative error among the three models considered here for both the free
and DD evolution experiments. However, the performance of SM1 and SM2 is essentially
indistinguishable from the full model results in the DD and free-evolution cases, respectively.
This is not unexpected, given that SM1 (SM2) is trained on the DD (free) evolution experiments and predicts these well. In other words, SM1 (SM2) captures the high (low)-frequency
noise well, as expected since for SM1, the use of DD suppresses most of the low-frequency
noise, while for SM2, the use of free-evolution means that the low-frequency noise remains a
dominant source of decoherence. The added value of the detailed model and the use of Step
80
III is that this provides enough information to capture both the low- and high-frequency
components of the noise, which yields a more complete model with better predictive power.
We do note that taking the qubit approximation and treating DD pulses as instantaneous
does not seem to appreciably worsen the performance of the simple models in their regime of
accuracy, as SM1 (SM2) are roughly as accurate as the full model in the DD (free) evolution
case. This suggests that an intermediate model, taking the qubit and instantaneous-pulse
approximations but retaining the fluctuators, may be accurate and computationally efficient.
4.5.3 Lindblad equation model
As a sanity check, we also conducted a simple test for the Quito processor using the backendreported T1 and T2 values (see Table Table 4.2). Using the Lindblad equation in the standard
form
ρ˙ = −i[H, ρ] +X
i
γi
LiρL†
i −
1
2
n
L
†
iLi
, ρo
, (4.24)
we consider two Lindblad operators corresponding to relaxation (Lrelax = σ
− = |0⟩⟨1|)
and dephasing (Ldephase = σ
z/2). We take the corresponding rates to be 1/T1 and 1/Tϕ,
respectively. Here T1 is the relaxation time and Tϕ is the dephasing time obtained via
a Hahn-echo based Ramsey experiment. We treat the transmons as two-level systems and
assume the DD pulses to be instantaneous. We then simulate both the free and DD evolution
dynamics starting with all 16 states considered in the main text in the single-qubit case. The
results, given in terms of the relative error as a function of time, are presented in Fig. 4.6.
We observe that the relative error of both the free and DD evolution cases is substantially
larger than for our detailed simulation model results [see Fig. 4.4(a),(b)]. Qualitatively, the
relative error using the Lindblad equation is as high as 20% and 14% for the free and DD
evolutions, respectively, compared to our detailed model results which always have corresponding relative errors below 8% and 2% for the Quito processor. This observation underscores the need for a methodology that goes beyond a simple Markovian model to capture
81
10 20 30 40 50 60 70
time instants
0.00
0.03
0.05
0.08
0.10
0.12
0.15
0.18
0.20
relative error
(a)
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
time (µs)
10 20 30 40 50 60 70
time instants
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
relative error
(b)
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
time (µs)
Figure 4.6: Lindblad equation simulation results for the Quito processor. Left: Box plots
showing the relative error of the Lindblad simulations for the free-evolution experiments as
a function of time for 16 different initial states containing six Pauli states and ten Haarrandom states. Right: the same as on the left, but for the DD-evolution experiments. We
measured a total of 70 time instants, up to a total evolution time of 19.6 µs, but only
display every other instant to avoid overcrowding. Green triangles indicate the mean over
the 16 initial states, black horizontal lines are the median, gray boxes represent the [25, 75]
percentiles, the whiskers (black lines extending outside the boxes) represent the [0, 25] and
[75, 100] percentiles, and circles are outliers.
noise in transmon qubits.
4.6 Calibration-independent learning
For multi-qubit superconducting processors, calibrating single-qubit drive frequencies is crucial for gate operations. In the presence of ZZ coupling, the state of the spectator qubits
modifies the eigenfrequency of the main qubit. This results in different choices of calibration
Params/(2π) Quito Lima
gx [MHz] 5.734 4.782
gz [MHz] 4.413 9.393
ω
c
x
[GHz] 1.948 2.340
ω
c
z
[MHz] 5.690 5.979
b [MHz] 0.598 0.323
γmax [GHz] 0.051 0.083
Table 4.1: System-bath parameter values extracted using the fitting procedure of Section 4.4.2, and corresponding to the minima indicated by the green circles in Fig. 4.3 for
Quito (top row) and Lima (bottom row).
82
frequencies depending on the spectator qubits’ state [Chapter 2]. So far, we have focused on
one particular device (Quito), which is calibrated while keeping the spectator qubits in the
|+⟩ state. The all-|+⟩ or all-|0⟩ are usually the two preferred choices for the spectators’ state
while calibrating a given qubit in a multi-qubit processor. When we perform state protection
experiments on the main qubit initialized in the |+⟩ state while keeping all the spectator
qubits in |0⟩, there exists a frequency mismatch which results in ZZ crosstalk oscillations
(see Fig. 4.8); to remove these oscillations, we applied DD to the spectator qubits before
starting our noise learning procedure. When device calibration is performed while keeping
the spectators in the |0⟩ state, a similar state protection experiment does not result in any
oscillations, as evidenced by our Lima results (see Fig. 4.9).
Therefore, we extend our noise learning method to Lima in this section. Following our
procedure from Section 4.4, we again perform free-evolution (no DD is applied to any qubit)
and DD-evolution (XY4 is applied just to the main qubit) experiments. The only difference
from the Quito case is that, for the reasons explained above, the free-evolution experiment
does not require the application of DD to the spectator qubits to suppress crosstalk oscillations. Fig. 4.3 (bottom row) shows the contour plots for each of the three steps involved
in our learning methodology as described in Section 4.4.2. We find the global minima at
the parameter values given in Table 4.1. Comparing the Quito and Lima parameters in
Table 4.1, we observe that the coupling strength gz of the Ohmic bath along the z-axis is
roughly double for Lima, whereas the strength of the fluctuators is roughly double for Quito.
This indicates that Quito is more prone to low-frequency (1/f) noise.
Fig. 4.4 (bottom row) shows the Lima prediction results for the 16 different initial states
described above, using the learned noise parameters. The bottom left panel [Fig. 4.4(c)]
shows the relative error in the prediction of our model of the free-evolution experiments as
a function of time compared to the experimental results for all 16 states. The bottom right
panel [Fig. 4.4(d)] shows the same for the DD-evolution experiments. The relative error is
always below 17% and 4% for free-evolution and DD-evolution, respectively. Similar to the
83
free DD
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
relative error
12
13
14
15
16
17
18
19
time (µs)
Figure 4.7: Integrated relative error results for the Lima processor. We compare the relative
errors between the free-evolution and DD-evolution experiments. Each box contains 16 initial
states and all 70 time instants varying from 0 to 19.6 µs. The color bar indicates the time
evolved for the outliers. All the outliers correspond to times longer than 11 µs.
Quito case, the relative error is significantly lower for the DD-evolution experiments. For
longer evolution times (≳ 14 µs), the agreement worsens for the free-evolution experiments.
As for the Quito results, the closer agreement of the model with the DD-evolution experiments is likely because DD suppresses the low-frequency noise affecting the main qubit,
which dominates the free-evolution case’s simulation error.
Fig. 4.7 shows the time-integrated version of the Lima results of Fig. 4.4 (c,d), where we
have combined all 16 states and 70 time instants into one box each for the free-evolution
and DD-evolution experiments. Except for a few outliers, almost all the data points for the
free-evolution case have relative errors below 9%. The outliers all correspond to evolution
times longer than 11 µs, as indicated by the color bar. Similarly, for DD-evolution, all data
84
Quito Q0 Q1 Q2 Q3 Q4
Qubit freq. (GHz) 5.3006 5.0806 5.3220 5.1637 5.0524
ηq (MHz) 331.5 319.2 332.3 335.1 319.3
T1 (µs) 86.7 98.6 61.5 111.5 85.7
T2 (µs) 132.5 149.0 78.9 22.7 136.7
sx gate error [10−2
] 0.0302 0.0243 0.1042 0.0629 0.0884
sx gate length (ns) 35.556 35.556 35.556 35.556 35.556
readout error [10−2
] 3.91 2.10 6.42 2.28 2.00
Lima Q0 Q1 Q2 Q3 Q4
Qubit freq. (GHz) 5.0297 5.1277 5.2474 5.3026 5.0920
ηq (MHz) 335.7 318.3 333.6 331.2 334.5
T1 (µs) 125.2 105.7 88.1 59.9 23.2
T2 (µs) 194.3 136.2 123.3 16.8 21.1
sx gate error [10−2
] 0.0230 0.0189 0.0308 0.0251 0.0578
sx gate length (ns) 35.556 35.556 35.556 35.556 35.556
readout error [10−2
] 1.73 1.40 1.69 2.42 4.820
Table 4.2: Specifications of the Quito (top row) and Lima (bottom row) devices accessed on
September 1, 2021, and January 1, 2023, respectively. The sx (√
σ
x) gate forms the basis of
all the single qubit gates, and any single qubit gate of the form U3(θ, ϕ, λ) is composed of
two sx and three rz(λ) = exp(−i
λ
2
σ
z
) gates (which are error-free and take zero time, as they
correspond to frame updates).
points have relative errors below 3%, except for a few outliers. There are two main reasons
for the larger errors at longer evolution times. First, the Redfield equation is based on the
weak coupling approximation, and its accuracy degrades as we increase the simulation time
(for rigorous error bounds, see Ref. [188]). Second, as time increases, the effect of distant
fluctuators is increasingly felt, thus reducing the accuracy of our fluctuator model, which
relies on a fixed number of fluctuators. Adding more fluctuators and exploring a distribution
of fluctuator strengths, as opposed to our assumption of a fixed fluctuator strength b, is
expected to improve the predictive power of our model.
Finally, as another test of our learning procedure, we computed T1 from the learned
models, using the values we report in Table 4.2. We did this by simulating the fidelity decay
of the |1⟩ state for 19.6 µs and fitting exp(−t/T1) to estimate T1. We find T1 = 92.5 µs for
Quito and 86.5 µs for Lima, compared to the reported T1 = 98.6 µs and 105 µs, respectively.
Our result gives the correct order of magnitude and is particularly reasonable for Quito.
The discrepancy may be in part due to the relatively short simulation time of 19.6 µs (larger
times become prohibitively expensive). In addition, the discrepancy for Quito is smaller
85
because we use three rounds of experiments within the same calibration to model the noise,
which removes short-time fluctuations via bootstrap averaging, whereas for Lima, we used
only one repetition. As T1 often drifts significantly over hour-long time scales, we would not
expect the Lima prediction to line up exactly with the reported T1.
4.7 Extension to two qubits noise model
Following an in-depth analysis and validation of our model for characterizing interactions of
single qubits with the bath, we now extend our methodology to propose how to investigate
two-qubit couplings with the bath. For simpler models relying on Lindblad equations based
on T1 and T2 information, noise modeling for two-qubit systems has been studied [189]. For a
Hamiltonian based noise learning approach like presented here, it is non-trivial. For system
comprising two qubits, the interaction term that delineates the system-bath dynamics can
be formulated as Htot
SB = H1
SB + H2
SB + H
2q
SB where Hi
SB represents the single qubit systembath interaction for the i
th qubit as described by Eq. (4.13). The two-qubit system bath
interaction term can be expressed as
H
2q
SB =gzx (Az ⊗ Ax + Ax ⊗ Az) ⊗ Bxz (4.25a)
+ gxxAx ⊗ Ax ⊗ Bxx + gzzAz ⊗ Az ⊗ Bzz (4.25b)
+ Az ⊗ Az ⊗
X
k
bk,12χk(t)IB (4.25c)
Recall that Ax and Az corresponds to the σx and σz coupling operator respectively when
reduced to two-level systems. Note that we have assumed that the qubits are symmetric
with respect to combination of one Ax and one Az type couplings to the bath. Further we
have considered a mix of Ohmic-like (Eq. 4.25a and 4.25b) and fluctuator bath (4.25c).
As with the single qubit noise learning method, we can conduct both free-evolution and
DD-evolution experiments, initializing with two-qubit states. In scenarios where one of the
86
primary qubits is linked to spectator qubits, the free-evolution experiments would necessitate
the application of DD sequences to these spectator qubits. This is done to suppress the ZZ
coupling between the main qubits and the spectator qubits (see Chapter 2). For the DDevolution experiments on the primary qubits, we utilize the ZZ-crosstalk robust variant of
the XY4 sequences, represented as (XI)fτ (IX)fτ (Y I)fτ (IY )fτ (XI)fτ (IX)fτ (Y I)fτ (IY )fτ
[190], where fτ denotes free-evolution in the absence of pulses for a duration of τ . This
sequence aims to suppress the ZZ crosstalk between both the main qubits and between the
main and spectator qubits. Additionally, it mitigates the low-frequency noise resulting from
the fluctuator term in the two-qubit system-bath interaction Hamiltonian, as described by
Eq. (4.25c). It is worth noting that the ZZ crosstalk occurring between the two transmons
can be measured directly. Consequently, this should be incorporated into the closed system
Hamiltonian.
Within our Hamiltonian, as described by Eq. (4.25), there exist four distinct bath couplings that necessitate learning. Specifically, these encompass:
1. Three couplings associated with the Ohmic baths, characterized by strengths gzx, gxx,
and gzz. These have corresponding cut-off frequencies ω
c
zx, ω
c
xx, and ω
c
zz.
2. A coupling related to the fluctuators, defined by strengths bk,12. For simplification, we
can postulate that bk,2q = b2q holds true across all fluctuators, with a high-frequency
cut-off γ
max
zz .
This culminates in a requirement to discern eight parameters in total: four are coupling
strengths, and the remaining four are their associated cut-off frequencies. To achieve this,
we undertake a structured approach involving a series of four iterative steps. Each step is
tailored to determine the parameters associated with each of the bath terms in Eq. (4.25).
The methodology involves leveraging specific initial states, combined with either free or DD
evolution, ensuring that each term is distinctly accentuated.
1. Step I: We commence with the free-evolution experiment for the initial state |00⟩+|11⟩
√
2
.
87
This state remains unaffected by Ax ⊗ Ax or Az ⊗ Az couplings to the bath. Predominantly, the system’s dynamics are influenced by the noise stemming from the combination of Ax and Az coupling to the bath. For this, we consider Az ⊗Ax +Ax ⊗Az as our
coupling operator, with strength gzx under an Ohmic-like bath with a cut-off frequency
ω
c
zx. By contrasting the Redfield simulations (with the above-mentioned bath incorporated) to the actual experiment, and using the cost function from Eq. (4.23), we can
deduce two pivotal parameters: the coupling strength gzx and the cut-off frequency
ω
c
zx.
2. Step II: For this step, we leverage the |11⟩ initial state in the free-evolution experiment. We simulate the Redfield dynamics, taking into account the bath coupling term
determined in Step I and an additional term proportional to Ax ⊗Ax. The initial state
|11⟩ remains unaffected by the other two bath terms, which couple through Az ⊗ Az.
This aids in extracting the coupling strength gxx and the bath’s cut-off frequency ω
c
xx.
3. Step III: We turn our attention to the nine computational states |ψi⟩|ψi⟩, where |ψi⟩
can be |0⟩, |1⟩, or |+⟩, and subject them to DD evolution. The gzz term, along with the
terms previously identified, becomes more pronounced. We employ the crosstalk-robust
DD, which curtails the low-frequency bath in the z-direction. The amalgamation of
the selected initial state and DD evolution distinctly emphasizes the gzz term within
the system’s dynamics, aiding in the extraction of gzz and ω
c
zz.
4. Step IV: As the concluding step, we utilize the |+⟩|+⟩ initial state and expose it to
free evolution, emphasizing the low-frequency term. Given that this state would be
influenced by all terms in Eq. (4.25) and considering that we have already discerned
the bath parameters of all the Ohmic-like baths in the prior steps, our focus now shifts
to understanding the fluctuators’ contribution. This involves learning about b2q and
γ
max
zz which can be done easily by comparing the simulations including all the bath
terms and following the cost function approach.
88
Employing this structured iterative approach, we achieve a clear demarcation and understanding of each system-bath coupling term, ensuring a comprehensive comprehension of the
system’s inherent dynamics. In step I, the free-evolution experiment with a select initial state
allowed us to isolate the effects of the combined Ax and Az coupling to the bath, proving
instrumental in discerning the coupling strength gzx and the cut-off frequency ω
c
zx. Step II
built upon the initial findings, using the |11⟩ initial state under free evolution to further differentiate and extract the parameters associated with the Ax ⊗ Ax coupling, namely gxx and
ω
c
xx. Step III expanded our understanding by emphasizing the gzz term through a judicious
combination of initial state selection and DD evolution, facilitating the extraction of gzz and
ω
c
zz. Finally, step IV employed the |+⟩|+⟩ initial state to emphasize the low-frequency term,
enabling us to holistically extract the parameters of the fluctuator contribution, specifically
b2q and γ
max
zz . Through this meticulous progression, each step illuminated distinct facets of
the two-qubit couplings to the bath, culminating in a robust and detailed characterization
of the system-bath interactions.
4.8 Summary and Conclusions
In this chapter, we present a detailed noise model for transmon qubits consisting of both
low- (1/f-like) and high-frequency noise components based on a hybrid Redfield master
equation. We designed an iterative three-step procedure to extract the unknown system-bath
and bath parameters from a few simple “free-evolution” and “DD-evolution” experiments,
as illustrated in Fig. 4.2, using the six Pauli matrix eigenstates. In both cases, we used
dynamical decoupling (DD) to suppress diagonal (ZZ) qubit crosstalk so that the remaining
dominant noise effect on the main qubit (the qubit of interest) is decoherence. Our model
treats the transmon qubit as a four-level system based on the circuit model description of
transmons (Section 4.2) and treats the DD pulses as realistic time-dependent gates subject
to quantum control (DRAG).
89
Once the unknown system-bath and bath parameters are extracted, we compare the
model predictions with new experiments and a larger set of initial states and demonstrate
that the model predicts the experimental results of free-evolution and DD-evolution with a
relative error below 8% and 2% for Quito, and below 9% and 3% for Lima, respectively. This
is based on a test with the six Pauli matrix eigenstates and ten random states for a total
duration of up to 19.6 µs. The relative errors are higher for larger times (see Fig. 4.4), as
expected because the Redfield model is based on the weak-coupling approximation, and its
accuracy degrades as the simulation time is increased [188].
To test the robustness of our model, we performed a comparison with two simpler, twolevel models with instantaneous pulses; while these models capture either the low- or the
high-frequency noise, the full model captures both types of noise. Furthermore, our method is
applicable independently of the particular device-calibration procedure followed, as witnessed
by the agreement we find for both Quito and Lima – devices with different drive-frequency
calibrations.
The low relative error we find in the case of DD-evolution experiments (< 2% and <
3% for Quito and Lima, respectively) suggests that our full noise model helps model gate
dynamics under the influence of decoherence. The model can also be used to study several
qubits in parallel as long as there is no direct crosstalk between these qubits.
Finally, we propose a simple extension of our methodology to learn the two-qubit couplings to the bath by performing simple to implement free- and DD- evolution experiments.
We leave the experimental implementation of this scheme as a future direction. Extending
our noise model beyond weak coupling and using it to analyze and improve entangling gates
are promising future directions.
We hope this work will benefit experimental groups working with superconducting qubits
by helping them understand and learn experimental noise using a first principles approach,
which only requires a set of straightforward experiments.
90
0.0
0.5
1.0
F
(e)
|0i
[1] (0.0, 0.0, 0.0) [2] (5.89, 132.88, 38.35) [3] (19.52, 131.69, -50.24) [4] (51.95, -83.84, 244.53)
0.0
0.5
1.0
F
(e)
[5] (53.01, -165.21, -186.79) [6] (67.6, -316.36, 3.4) [7] (76.44, 119.73, 58.95) [8] (86.03, 31.32, 23.58)
0.0
0.5
1.0
F
(e)
|+i
[9] (90.0, -90.0, 90.0)
|−i
[10] (90.0, 0.0, 0.0)
|ii
[11] (90.0, 90.0, -90.0)
0 10 20
| − ii
[12] (90.0, 180.0, -180.0)
0 10 20
time (µs)
0.0
0.5
1.0
F
(e)
[13] (94.47, 133.1, -116.54)
0 10 20
time (µs)
[14] (116.65, -0.53, 138.38)
0 10 20
time (µs)
[15] (142.29, -47.45, -86.98)
0 10 20
time (µs)
|1i
[16] (180.0, 90.0, -90.0)
0.0
0.5
1.0
F
(e)
No DD DD on Spec: free evo DD on Main: DD evo
Figure 4.8: Fidelity results for the Quito processor, for the 16 different initial states of the
main qubit. The caption of each of the panels gives (θ, ϕ, λ) in degrees, parametrizing the
initial state |ψ⟩ = U3(θ, ϕ, λ)|0⟩ (panels are arranged in increasing order of θ, the polar
angle with the z-axis). These are the 6 Pauli states (panels 1,9-12,16) and the 10 Haarrandom states. Blue curves (squares): no DD is applied, resulting in coherent oscillations
due to crosstalk. Orange curves (diamonds): DD (XY4) is applied just to the spectator
qubits; the resulting suppression of crosstalk between the main qubit and the spectator
qubits removes the oscillations. These are what we call the free-evolution experiments in the
main text. Green curves (circles): DD (XY4) is applied just to the main qubit, suppressing
both crosstalk and errors due to environment-induced noise. Results are averaged over three
different runs of experiments. All of the data was acquired on Sep. 1, 2021. Error bars are
smaller than the markers.
91
0.0
0.5
1.0
F
(e)
|0i
[1] (0.0, 0.0, 0.0) [2] (5.89, 132.88, 38.35) [3] (19.52, 131.69, -50.24) [4] (51.95, -83.84, 244.53)
0.0
0.5
1.0
F
(e)
[5] (53.01, -165.21, -186.79) [6] (67.6, -316.36, 3.4) [7] (76.44, 119.73, 58.95) [8] (86.03, 31.32, 23.58)
0.0
0.5
1.0
F
(e)
|+i
[9] (90.0, -90.0, 90.0)
|−i
[10] (90.0, 0.0, 0.0)
|ii
[11] (90.0, 90.0, -90.0)
0 10 20
| − ii
[12] (90.0, 180.0, -180.0)
0 10 20
time (µs)
0.0
0.5
1.0
F
(e)
[13] (94.47, 133.1, -116.54)
0 10 20
time (µs)
[14] (116.65, -0.53, 138.38)
0 10 20
time (µs)
[15] (142.29, -47.45, -86.98)
0 10 20
time (µs)
|1i
[16] (180.0, 90.0, -90.0)
0.0
0.5
1.0
F
(e)
No DD: free evo DD on Main: DD evo
Figure 4.9: Fidelity results for the Lima processor, for the 16 different initial states of the
main qubit. The caption of each of the panels gives (θ, ϕ, λ) in degrees, parametrizing the
initial state |ψ⟩ = U3(θ, ϕ, λ)|0⟩ (panels are arranged in increasing order of θ, the polar angle
with the z-axis). These are the 6 Pauli states (panels 1,9-12,16) and the 10 Haar-random
states. Blue curves (squares): no DD is applied. In contrast to Fig. 4.8, there are no crosstalk oscillations. This is because all the spectator qubits are kept in the ground state |0⟩,
and calibration for Lima is done in the same spectators’ state. These are what we call the
free-evolution experiments for Lima in the main text. Green curves (circles): DD (XY4) is
applied just to the main qubit, suppressing errors due to environment-induced noise. All of
the data was acquired on Jan. 1, 2023. Error bars are smaller than the markers.
92
Chapter 5
Deterministic benchmarking of quantum gates
Note: This chapter is adapted from [156].
Main results.— We introduce deterministic benchmarking (DB), a protocol designed to identify the interplay of coherent and incoherent errors overlooked by randomized benchmarking
(RB) and related benchmarking methods. Our DB protocol provides a set of four parameters
that characterize both the incoherent and coherent errors in the single-qubit gate set. We
experimentally demonstrate DB using a superconducting transmon qubit, and support these
results with a simple analytical model and numerical simulations. Additionally, DB reveals
asymmetries in gate performance induced by strong relaxation errors (T1). Remarkably, a
simple Lindblad master equation can comprehensively model all the experimental intricacies. These findings not only uncover the critical errors missed by conventional RB, but also
inspire strategies to mitigate these errors.
5.1 Introduction
Fault-tolerant quantum error correction is essential for the ultimate scalability of quantum
computing but requires high-fidelity operations exceeding an accuracy threshold [191, 192,
193, 60, 62]. Coherent errors arising from systematic deviations in quantum gate operations
93
pose a significant challenge in this context [194, 195]. The reason is that while stochastic
errors add up as probabilities, coherent errors add up as amplitudes, generally leading to
a quadratically smaller accuracy threshold [196]. Traditional error characterization methods such as Randomized Benchmarking (RB) [197, 198, 199], which deliberately randomize
noise channels, do not capture coherent error accumulation unless appropriately modified
[200]. This can result in an underestimation of the gate performance needed to achieve accuracy thresholds, critical for realizing fault-tolerant quantum computing [201, 202]. Various
techniques exist that can mitigate the accumulation of coherent errors by converting them
into stochastic noise but only at the expense of a significant additional cost in the number
of required circuits [203, 204, 205, 206]. Significant advantages can be accrued if, instead,
coherent gate errors can be accurately and efficiently characterized, and subsequently eliminated or at least suppressed at the control level, rather than placing the burden for doing
so on fault-tolerant quantum error correction using large codes [207].
To address these characterization and control challenges, here we introduce a method we
call deterministic benchmarking (DB) and validate its utility through experiments conducted
on a superconducting transmon qubit [29].
Unlike RB, which averages over many and lengthy random Clifford gate sequences to
approximate all errors as a depolarizing channel and lacks the sensitivity needed to detect
small changes in decoherence times and qubit frequencies [208, 209], DB is highly sensitive
to such changes; as illustrated in Fig. 5.1(a,b), it utilizes a carefully chosen, small set of
two-gate-long sequences that specifically target the real noise affecting actual single-qubit
gates.
The DB protocol is significantly more efficient than RB or other well-established methods
such as process tomography [210] or gate set tomography [96]. Indeed, we show below
that DB requires only four sets of state-fidelity measurements to accurately estimate all
incoherent and coherent errors relevant to single-qubit gate operations. Additionally, our
DB protocol can also be used to study the temperature of the baths, providing further
94
insights into the environmental influences on quantum systems, which cannot be obtained
by other benchmarking techniques. Moreover, we show that DB also enables the exploration
of how coherent and incoherent errors interact to cause biased noise, whose impact on fault
tolerance has been thoroughly studied [211, 212].
The gate sequences used in DB resemble the simplest dynamical decoupling (DD) pulse
sequences introduced decades ago in nuclear magnetic resonance [213, 214, 215], but whereas
DD is nowadays typically used to suppress low-frequency noise during idling—whether between gates [216, 217, 129, 218, 168, 219] or as a result of crosstalk [45, 132, 220, 221,
152]—the goal of DB is instead to characterize the noise arising during gate operation.
5.2 Qubit model
We model the transmon qubit as a two-level system and consider it in the drive frame under
the rotating wave approximation (e.g., Ref. [222]). Let Rα(θ) ≡ exp[−i(θ/2)σα], where {σα}
is the set of Pauli matrices. The Hamiltonian for the single qubit gates Rx(θ) and Ry(θ),
including both phase and rotation errors, is given by:
Hα = (ε + εerr)
σα
2
+ ∆err
σz
2
, α ∈ {x, y}. (5.1)
Ideally, εerr = ∆err = 0. In reality, both are present and give rise to rotation and phase errors
δθ ≡ εerrtg , δϕ ≡
∆err
ε¯
(5.2)
respectively, with tg denoting the gate duration and ¯ε the effective pulse amplitude. We
expect to see pulse-shape dependence in the phase error δϕ through ¯ε which can be computed
numerically. For instance, for a square pulse Rα(π) gate , ¯εtg = εtg = π. We assume
square-shaped pulses for our analytical theory and cosine-shaped pulses for our numerical
calculations.
95
Eq. (6.1) is an effective model, where ∆err captures a phase error resulting from both
detuning—a discrepancy between the drive and qubit frequencies—and the ac-Stark shift
induced by driving the higher levels of the transmon off-resonantly during the gate [223].
The two-level model is justifiable given that leakage to higher levels of the transmon during our gates is very small (see Supplementary Material (SM) [224]). In our experiments,
the Hx Hamiltonian is fully calibrated and the Hy Hamiltonian is then obtained by simply
applying a phase shift via an arbitrary waveform generator (AWG), which is equivalent to
Hy = Rz(−
π
2
)HxRz(
π
2
), leaving εerr and ∆err unchanged.
5.3 Deterministic Benchmarking
Any completely positive trace-preserving (CPTP) map acting on a d-dimensional Hilbert
space can be characterized using no more than d
4 − 1 parameters, i.e., 15 for a single qubit.
RB reduces this to a single effective parameter, which, as we argue here, is too drastic.
Instead, we advocate for the primacy of four parameters: the relaxation time T1, coherence
time T2 (which contains information about pure dephasing Tϕ), rotation error δθ, and phase
error δϕ. Assuming a Markovian noise model, we show below that these parameters can both
be straightforwardly extracted and at the same time, they provide just the right amount of
information about the errors affecting single-qubit gates to enable such gates to be accurately
calibrated.
We denote X ≡ Rx(π), X ≡ Rx(−π), similarly for Y and Y . To define DB, we need twogate-long sequences of the form P1P2. In particular, we consider P1P2 ∈ {XX, XX, XX, Y Y,
Y Y , Y Y }. Note that in the absence of any errors, i.e., when εerr = ∆err = 0 and without
coupling to a bath, P1P2 = I (identity operator) for this set. In reality, P1P2 ̸= I and we
consider the corresponding fidelity FP1P2
(n) = |⟨ψ|(P1P2)
n
|ψ⟩|2 where |ψ⟩ is the initial state
and n is the number of repetitions.
96
= 0.262 ± 0.004%
(b)
Evolution time
Readout ( )
(a)
Readout
...
Evolution time (µs)
0 20 40 60 80
0.0
0.2
0.4
0.6
0.8
1.0
Fidelity
XX;
YY;
YY;
YY;
XX;
Free;
(c)
|+
|+
|+
|+
|+
|1
Figure 5.1: Schematic representations of the gate sequences for (a) randomized benchmarking
(RB) and (b) deterministic benchmarking (DB), respectively. (c) Experimental fidelity measurement of |+⟩ state for various gate sequences, revealing coherent oscillations with XX and
Y Y sequences. These oscillations indicate the presence of phase errors δϕ and rotation errors
δθ. In contrast, RB exhibits an average Clifford gate infidelity rClifford = 0.262 ± 0.004%.
Solid curves are fits obtained using Eq. (5.3), whereas dashed curves denote the Lindblad
master equation based simulation results.
The core DB protocol consists of the following set of four “learning” experiments designed
to extract {T1, T2, δθ, δϕ}, followed by tests involving new experimental data. In each case,
the notation {P1P2; |ψ⟩} means preparation of the state |ψ⟩ from the |0⟩ state, varying
number of repetitions of the P1P2 sequence, unpreparation of |ψ⟩ and a measurement in the
z-basis. Each such series of operations is a circuit. Finally, the empirical fidelity is found for
each experiment as the ratio of |0⟩ outcomes to the total number of experimental shots (800
in our case). Remarkably, the behavior of these experiments, both for learning and testing,
97
can be accurately described using a simple analytical expression:
F(tn, TD, ω, a) = 1
2
(1 + a) + (1 − a)e
−tn/TD
cos2
(ωtn) −
1
2
(5.3)
where tn = 2ntg (n ∈ N; hence F is defined stroboscopically) represents the total evolved
time, with tg being the gate time and n the number of repetition of the P1P2 gate sequence.
The parameter TD denotes a characteristic timescale associated with incoherent noise processes, while ω captures the frequency of coherent errors. The parameter a serves as an
offset, indicative of the bath temperature. Eq. (5.3) effectively captures the interplay between coherent and incoherent noise sources. Derivations of the above fidelity expression for
some of the relevant sequences are provided in the SM [224]. The experiments in the DB
protocol are detailed below:
1. {free; |1⟩}: This is a standard T1 measurement, where we do not apply any pulses (free
evolution). Eq. (5.3) reduces to (1 − a)e
−t/T1 + a in this case which we fit to determine
T1. This yields the relaxation component of the Lindbladian.
2. {XX; |+⟩}: Fitting this experiment with Eq. (5.3) yields TD = T2, the coherence time
from which we deduce the pure dephasing time Tϕ = 2T1T2/(2T1 − T2). This is analogous to an echo-Ramsey measurement [40]. This yields the dephasing component of the
Lindbladian.
3. {Y Y ; |+⟩}: This yields fidelity oscillations which we fit to find the rotation error δθ where
δθ ≈ 2tgω in Eq. (5.3).
4. {XX; |+⟩}: Similarly, this yields fidelity oscillations which we fit to find the phase error
δϕ where δϕ = tgω in Eq. 5.3. The decay times TD here and in previous step are a
combination of the incoherent noise and coherent errors and hence would be different from
T2 obtained in step 2. We now have all four parameters, which we use to numerically
calculate predictions for the test experiments.
98
5. Test 1: {Y Y ; |+⟩} and {Y Y ; |+⟩}: This highlights the interplay of asymmetric relaxation
noise and gate operations and can also be used to characterize the qubit temperature (see
SM [224]).
6. Test 2: Various other experiments, e.g., dynamical decoupling. In this work we report
tests using the UR6 sequence [225].
We now demonstrate that this protocol enables a detailed assessment of both coherent
and incoherent errors affecting quantum gate operations.
5.4 Experimental results
We conduct all our experiments using a superconducting transmon qubit dispersively coupled
to its readout resonator (see SM [224]). We implement Ry(θ) gates by adjusting the phase
of the microwave pulses used for Rx(θ) gates, thereby eliminating the need for additional
calibrations. All single-qubit gates are calibrated using the Derivative Removal of Adiabatic
Gates (DRAG) technique [41] to minimize leakage and phase errors induced by higher levels of
the transmon. To emulate drifts in experimental parameters, we first perform measurements
on an outdated set of gate calibration parameters. The results are shown in Fig. 5.1(c).
We observe fidelity oscillations for the Y Y sequence, indicating rotation errors δθ. Similarly, the XX sequence exhibits oscillations due to phase errors δϕ. The XX sequence is
insensitive to the coherent errors and instead exhibits an exponential decay characterizing
the T2 time during time-dependent gate operations. Note that this may differ from conventionally reported T2 times (e.g., Ref. [39]) but is more relevant to decoherence during gate
operations. Additionally, the TD from other DD experiments cannot be used as the effective
T2 because all these sequences cause significant deviations from the initial state |+⟩, either
due to the intended trajectory over the Bloch sphere or unintended trajectories resulting
from coherent errors. Also shown in Fig. 5.1(c) are results from the Y Y and Y Y sequences.
Surprisingly, they exhibit a pronounced difference; the reason is discussed in detail later.
99
We use Eq.(5.3) to fit each of the experimental curves in Fig. 5.1(c), as denoted by the
solid curves, and find a good match. This fitting process enables us to extract the parameters
(T1 = 23.36±0.40 µs, T2 = 44.13±2.49 µs, δθ = 0.398±0.004◦
, δϕ = 0.426±0.004◦
) from the
four types of experiments according to our DB protocol. We contrast these detailed metrics
with the single metric rClifford derived from the RB protocol. The RB protocol necessitates
averaging over multiple Clifford gate sequences (30 in our case, each consisting of 700 Clifford
gates). This process yields an average gate infidelity of 0.262 ± 0.004% for the single-qubit
Clifford gate set. Note that, on average, our RB Clifford gate set consists of approximately
80% of π/2 and 20% of π gates.
As we proceed to show next, all observed features can be understood using either a
simple, analytically solvable closed-system model or a numerical model based on a Lindblad
master equation with time-dependent gates. Here we provide the analytical solutions of the
fidelity in the closed system case, highlighting the oscillations in certain sequences. In SM
[224], we derive the fidelity expressions in the open quantum system setting, justifying the
choice of Eq. (5.3).
5.5 Fidelity with coherent errors
Assuming the system is closed, Eq. (6.1) can be used to compute the unitary operation
analytically and subsequently the fidelity of the gate sequences comprising our DB protocol.
The fidelity FP1P2
(n) then reflects only the coherent errors contained within P1 and P2. For
the Y Y and XX sequences, we find:
FY Y (n) = cos2
(nθerr) (5.4a)
FXX(n) = 1 −
πδϕ sin(nθerr)
θerr 2
, (5.4b)
100
where θerr ≡
p
(εtg + δθ)
2 + (πδϕ)
2
. Expanding to leading order in δϕ and δθ, with εtg = π,
FY Y (n) simplifies to cos2
(nδθ), indicating its dependence is dominated by rotation errors
δθ. As for FXX(n), in the small-error limit the second term in Eq. (5.4b) is suppressed and
FXX(n) ≈ 1. Consequently, there is no dependence on either the rotation or the phase error.
Finally, the fidelity of the XX sequence with |+⟩ as the initial state is (see [224]):
FXX(n) = cos2
(nφerr) (5.5a)
φerr ≡ tan−1
2π
δϕ
θerr
sin
θerr
2
p
1 − Λ/2
1 − Λ
!
(5.5b)
Λ ≡
π
δϕ
θerr2
(1 − cos θerr) (5.5c)
Expanding to leading order again, with εtg = π, we have FXX(n) ≈ cos2
2nδϕ
1 −
1
π
δθ,
showing that this fidelity is dominated by phase errors.
These results justify our assignment of the oscillations observed in the Y Y and XX
experiments shown in Fig. 5.1(c) to rotation and phase errors, respectively.
5.6 Open system model
The empirical fidelity expression from Eq. (5.3), is well motivated by the coherent error
fidelity expressions derived in the previous section. The choice of incoherent parameters in
the DB protocol, specifically T1 (associated with free evolution) and T2 (associated with XX
sequence), warrants further investigation through detailed numerical simulation to better
understand their contributions to the parameters in Eq. (5.3). We use a simple Markovian Lindblad master equation to capture the effects of an open quantum system, using
the Lindbladian of the form L
s = −i
h
H, ˜ ·
i
+
P
α=1,ϕ γα
Lα · L
†
α −
1
2
L
†
αLα, ·
. The Lindblad operators L1 = σ
− = |0⟩⟨1| and Lϕ = σz/
√
2 with rates γ1 = 1/T1 and γϕ = 1/Tϕ,
corresponding to the relaxation and dephasing rates, respectively.
Using the fitted parameters (T1, T2, δθ, δϕ) from DB protocol, we simulate the Lindblad
1
Evolution time (µs)
0 20 40 60 80
Evolution time (µs)
0 20 40 60 80 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0 (a) (b)
± = 0.247 ± 0.014%
(c) (d)
= 0.264 ± 0.018% = 0.265 ± 0.013%
Fidelity |+ Fidelity |+
UR6
YY
XX
Figure 5.2: Experimental results showing the sensitivity of DD sequences and RB to coherent
errors (a) δθ = 0◦
, δϕ = 0◦
, (b) δθ = 0◦
, δϕ = 0.893±0.002◦
, (c) δθ = 0.932±0.007◦
, δϕ = 0◦
,
(d) δθ = 0.995 ± 0.009◦
, δϕ = 0.90 ± 0.002◦
. UR6 is robust even to large coherent errors and
hence is included as a reference. Note that δϕ is controlled by varying the DRAG parameter.
equation and find good agreement with experimental results, as shown by the dashed lines in
Fig. 5.1(c). Moreover, the parameters obtained from the DB protocol allow us to predict the
outcomes of other experiments, specifically Y Y and Y Y , as demonstrated in Fig. 5.1(c). This
supports the choice of T1 from free evolution and T2 from XX sequence as the appropriate
incoherent parameters.
5.7 Sensitivity to coherent errors
Having confirmed that DB effectively identifies various errors, we next assess the sensitivity
of both DB and RB to coherent errors. Initially, we calibrate the gates to minimize both δϕ
and δθ, and immediately follow this calibration by a series of DD experiments, as depicted in
Fig. 5.2(a). These experiments focus on XX and Y Y sequences, which are sensitive to the
phase and rotation errors. We include as a control a Universally Robust (URn) DD sequence
[225] engineered to suppress coherent errors. We also conduct RB on these gates, recording
an average gate error of rClifford = 0.247 ± 0.018%. We then deliberately introduce coherent
102
errors with (δθ, δϕ) values of (0◦
, 0.893◦
), (0.932◦
, 0
◦
) and (0.995◦
, 0.90◦
) for Fig. 5.2(b),
(c), and (d) where these angles are obtained from fitting with the analytical formula from
Eq. (5.3). The larger increase in rotation error δθ from Fig. 5.2(c) (in the absence of δϕ) to
(d) (in the presence of δϕ) can be attributed to the second-order dependence on δϕ, which
is equal to (δϕ)
2/(2π + 2δθ). The phase error δϕ, is introduced by mis-tuning the DRAG
weighting parameter α, which effectively varies the detuning error term ∆err in our model.
As anticipated, immediately following the gate calibration, neither the XX nor the Y Y
sequences exhibit oscillations, as shown in Fig. 5.2(a). However, upon introducing phase
errors, significant oscillations appear in the XX sequence as depicted in Fig. 5.2(b), while
the Y Y sequence simply decays exponentially without exhibiting any oscillations. In contrast, Fig. 5.2(c) reveals clear oscillations in the Y Y sequence but none in the XX sequence
when an over-rotation error is introduced. In the case where both the errors are introduced, we observe clear oscillations for both sequences [Fig. 5.2(d)]. The UR sequence never
exhibits oscillations, confirming that coherent errors are their source. We again compare
the experimental results from Fig. 5.2 with the previously described numerical open system
model, using the DB parameters, to accurately pinpoint these errors, and find a good agreement between experimental results and simulations, as shown by dashed curves. This again
strengthens our choice of parameters included in the DB protocol. The RB-derived error
rate rClifford is unchanged when a phase error (δϕ ≈ 0.90◦
) is introduced, and increases only
slightly (∼0.02%) upon introducing an over-rotation error (δθ ≈ 0.93◦
). We see that small
coherent errors can cause large, rapid oscillations in sequence fidelity even when RB fidelity
is almost unchanged. Our results show RB’s insensitivity to phase errors, underscoring its
limitations in detecting specific types of errors. Recall that, as mentioned earlier, our RB
experiments consist of, on average, 80% of π/2 and 20% of π pulses, with π/2 pulses expected
to exhibit half the rotation errors and one-fourth of the phase errors.
103
5.8 Interplay of T1 asymmetry and gates
An intriguing observation from the DB analysis in Fig. 5.1(c) is the asymmetry in the decay
of the fidelity curves for Y Y and Y Y sequences. Transmon qubits operate at frequencies
in the 3 − 6 GHz range and typically exhibit temperatures below 50 mK (∼ 1 GHz). In
this regime the thermal energy is negligible and the ground state |0⟩ is approximately the
thermal state of the transmon qubit—excitation from |0⟩ to |1⟩ is negligible and relaxation
dominates. Tracing the trajectories on the Bloch sphere reveals that the Y Y sequence
confines the state to the southern (excited state) hemisphere, while Y Y limits it to the
northern (ground state) hemisphere. Therefore, the former is more susceptible to relaxation
(T1) errors, which explains the faster decay and lower saturation fidelity of Y Y compared to
Y Y . This asymmetry is also well-captured by the time-dependent Lindblad-based simulation
in Fig. 5.1(c) as denoted by the dashed curves.
To illustrate this effect, we vary the initial state and DD sequence. We performed these
experiments in the same outdated gate calibration cycle used in Fig. 5.1 and perform the XX
sequence on two sets of orthogonal quantum states, |+i⟩ and |−i⟩ states (eigenstates of σy),
as well as the |ψ45◦,135◦ ⟩ and |ψ135◦,−135◦ ⟩ states, where |ψθ,ϕ⟩ = cos(θ/2)|0⟩ + e
iϕ sin(θ/2)|1⟩.
We observe a clear asymmetry between these orthogonal states under the XX sequence
as shown in Fig. 5.3(a), which is almost exactly reproduced in simulations as denoted by
the dashed curves. We further confirm this asymmetry using the Y Y sequence on the |+⟩
and |−⟩ states (eigenstates of σx), as well as the same set of orthogonal states previously
mentioned, finding the same asymmetric decay patterns both experimentally (markers) and
numerically from the simulations (dashed curves) in Fig. 5.3(b).
This asymmetry arises from the interplay of time-dependent gates with incoherent relaxation noise. It is not detected by traditional random-sequence-based benchmarking schemes,
but successfully identified through a DB protocol. Understanding these asymmetric noise
behaviors could significantly impact the development of compilation strategies, where choices
104
(a)
0.0
0.2
0.4
0.6
0.8
1.0
Fidelity
(b)
Evolution time (µs)
0 20 40 60 80
Evolution time (µs)
0 20 40 60 80
θ=45˚, φ=45˚
θ=135˚, φ=-135˚
θ=45˚, φ=45˚
θ=135˚, φ=-135˚
θ=45˚, φ=45˚
θ=135˚, φ=-135˚
θ=45˚, φ=45˚
θ=135˚, φ=-135˚
θ=45˚, φ=45˚
θ=135˚, φ=-135˚
θ=45˚, φ=45˚
θ=135˚, φ=-135˚
θ=45˚, φ=45˚
θ=135˚, φ=-135˚
θ=45˚, φ=45˚
θ=135˚, φ=-135˚ θ=45˚, φ=45˚
θ=135˚, φ=-135˚
|+
|-
θ=45˚, φ=45˚
θ=135˚, φ=-135˚ θ=45˚, φ=45˚
θ=135˚, φ=-135˚
θ=45˚, φ=45˚
θ=135˚, φ=-135˚
θ=45˚, φ=45˚
θ=135˚, φ=-135˚
θ=45˚, φ=45˚
θ=135˚, φ=-135˚
θ=45˚, φ=45˚
θ=135˚, φ=-135˚
θ=45˚, φ=45˚
θ=135˚, φ=-135˚
θ=45˚, φ=45˚
θ=135˚, φ=-135˚
θ=45˚, φ=45˚
θ=135˚, φ=-135˚ θ=45˚, φ=45˚
θ=135˚, φ=-135˚
|+i
|-i
θ=45˚, φ=45˚
θ=135˚, φ=-135˚
Figure 5.3: Asymmetry in the decay pattern of DD sequences due to the interplay of gates
and asymmetric T1. (a) Fidelity decay as a function of repetition number (shown in units of
time) for the XX sequence applied to two sets of orthogonal initial states. Applying XX to
|±i⟩ states results in an asymmetric decay, highlighting the impact of the relaxation errors
throughout the Bloch sphere. This pattern is also observed for orthogonal initial states
defined by angles (θ, ϕ) = (45◦
, 45◦
) and (45◦
, −135◦
). Dashed curves represent the Lindblad
master equation simulation results. (b) Similar to (a), but with Y Y sequences applied to
|±⟩ states and the same additional orthogonal states as in (a).
can be made to keep the single qubit states confined to northern hemisphere of Bloch sphere,
analogous to ideas of quantum refrigerator [226]. This can have significant advantages for
the resource estimates for fault-tolerant quantum algorithms [227, 228, 229]. Furthermore,
this insight could advance our understanding of the noise biases in various qubit modalities,
thereby facilitating progress in biased-noise error correction [212]. Finally, numerical modeling of this asymmetry enables calibration of qubit temperature, although it is not strictly
necessary as the standard 4-step DB is also sensitive to temperature, as discussed in SM [224].
5.9 Conclusion and outlook
We have developed and experimentally implemented a deterministic benchmarking (DB)
protocol on a superconducting transmon qubit. Unlike traditional randomized benchmarking
schemes that rely on random Clifford gates, our method employs simple two-pulse dynamical
decoupling sequences. These sequences not only elucidate the presence of incoherent errors,
but also expose various types of coherent errors that RB is incapable of capturing. This
105
approach eliminates the need to average over various random circuits, thereby enhancing
resource efficiency and providing a more direct measure of error types, which facilitates a
straightforward benchmarking protocol. Moreover, our DB protocol highlights the bias in
noise, potentially aiding in the development of more effective compilation strategies and
bias-focused error correction codes. The implementation of this method is straightforward
and adaptable to quantum systems beyond superconducting qubits. We anticipate that our
findings will significantly contribute to the benchmarking and enhancement of the quantum
gates across various platforms.
Looking ahead, future efforts will aim to extend deterministic benchmarking to twoqubit gates, necessitating a deeper exploration of gate dynamics and the intricate nature of
potential coherent error types.
106
Chapter 6
Virtual Z gates and symmetric gate compilation
Note: This chapter is adapted from [230].
Main results.—The virtual Z gate has been established as an important tool for performing
quantum gates on various platforms, including but not limited to superconducting systems.
Many such platforms offer a limited set of calibrated gates and compile other gates, such
as the Y gate, using combinations of X and virtual Z gates. Here, we show that the
method of compilation has important consequences in an open quantum system setting.
Specifically, we experimentally demonstrate that it is crucial to choose a compilation that is
symmetric with respect to virtual Z rotations. This is particularly pronounced in dynamical
decoupling (DD) sequences, where improper gate decomposition can result in unintended
effects such as the implementation of the wrong sequence. Our findings indicate that in
many cases the performance of DD is adversely affected by the incorrect use of virtual Z
gates, compounding other coherent pulse errors. In addition, we identify another source of
coherent errors: interference between consecutive pulses that follow each other too closely.
This work provides insights into improving general quantum gate performance and optimizing
DD sequences in particular.
107
(a)
X
X
Instantaneous
Rz(−π)
0 100 200 300 400 500
sequence number N
0 20 40 60 80
time (µs)
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
fidelity
YYasym |°ii
YYsym |°ii
YYasym |+ii
YYsym |+ii
Instantaneous
Rz(−π)
0 100 200 300 400 500
sequence number N
0 20 40 60 80
time (µs)
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
fidelity
YYasym |°ii
YYsym |°ii
YYasym |+ii
YYsym |+ii
0 100 200 300 400 500
number of cycles N
0 20 40 60 80
time (µs)
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
fidelity
YYasym |°ii
YYsym |°ii
YYasym |+ii
YYsym |+ii
(b)
Figure 6.1: The open system effect of the symmetric and asymmetric compilation of the Y
gate with respect to VZ gates. (a) The |±i⟩ state follows different Bloch sphere trajectories
under Y
asym, which consists of an instantaneous VZ gate followed by a physical X gate. This
causes |−i⟩ (|+i⟩) to go through a stable (unstable) ground (excited) state which leads to the
asymmetry in the fidelity of the two states. The symmetric decomposition Y
sym overcomes
this asymmetry, similar to a physical Y gate. (b) Experimental demonstration of the symmetric and asymmetric effects of the Y -gate decomposition on the MUNINN processor. The
fidelity of the states |±i⟩ is shown under both Y
asym and Y
sym, as a function of time (bottom
axis) or number of Y Y sequence cycles (top axis). The symmetric decomposition results in
similar fidelities (black and red) for the initial states |±i⟩. The asymmetric decomposition
results in very different fidelities (yellow and green) for the same two initial states. Error
bars denote two standard deviation of the mean.
6.1 Introduction
Any quantum computing processor is inherently an open quantum system that interacts
with its environment, leading to decoherence and errors, which adversely affect quantum
computations [170]. Various error correction, suppression, and mitigation techniques are
employed to suppress these effects [231, 232, 62, 233]. There has been a great interest in
demonstrations of overcoming decoherence, which have recently become possible with the
availability of commercial cloud-based quantum processors [234, 235, 236, 237, 238]. These
quantum processors usually have a native set of calibrated gates from which all other gates
can be constructed. An important part of the native gate set is the Virtual-Z (VZ) gate,
which is an instantaneous, error-free operation that plays a central role in gate compilation.
Ref. [239] demonstrated that VZ gates can be implemented by simply adding a phase offset
in software, unlike physical Z-gates that involve physical rotations around the z-axis of the
108
Bloch sphere. Moreover, they showed, by manipulating the phases of pulses driving the
qubits, VZ gates can be effectively combined with two √
X gates to construct any SU(2)
gate, thus achieving universality when combined with a two-qubit entangling gate [240,
241]. This approach simplifies the gate decomposition and circuit compilation procedure,
and its applicability extends beyond qubits to qudits as well as beyond superconducting
systems [111, 242, 243, 244, 245, 246, 247]. Compiling an arbitrary SU(2) operation using
VZ gates provides flexibility, but ensuring accuracy in the presence of open quantum system
effects is essential for reliable computations. Although different compilations involving VZ
gates can be equivalent in closed systems, discrepancies may arise if open-system effects are
not correctly accounted for during compilation.
In this work, we investigate the role of VZ gates in gate compilation within an open
quantum system dynamics framework. We find that even slight variations in the compilation of quantum gates using VZ gates reveal significant detectable effects. Specifically, an
asymmetrical compilation of the Y gate relative to VZ gates introduces fidelity discrepancies
between the Y eigenstates |±i⟩, which can be completely mitigated with proper compilation techniques. This observation has important consequences. Asymmetric compilation
influences the implementation of dynamical decoupling (DD) sequences [71, 248, 74, 73, 72,
249, 250, 85, 86, 150, 251, 252, 253, 254], potentially leading to misimplementation and
misidentification of commonly used sequences. For example, DD implementations using
cloud quantum processors reveal unexpected pulse-interval effects, as well as unexplained
significant oscillations in single-qubit experiments [254]. In addition, our findings uncover
previously unrecognized oscillations, even in DD sequences designed to be robust to coherent
errors [150, 251, 252]. The investigations into the VZ gate we report here reveal that interference between consecutive pulses explains these oscillations in robust sequences. Given the
recent critical role DD has played in improving the fidelity of quantum states [253, 255, 45,
254, 256, 257, 258], circuits [52, 218, 259], and even entire algorithms [260, 261, 262, 263], we
expect these findings to contribute to further improvement of quantum error suppression via
109
pulse-based methods such as DD. However, the impact extends beyond DD to any quantum
algorithm or error-correction method that requires high-fidelity single-qubit gates.
6.2 VZ gate in an open quantum system
We conduct all our experiments using two superconducting transmon quantum processors:
the IBM cloud quantum processor ibm sherbrooke and our in-house quantum processor
MUNINN [264].
We model the transmon qubit as a driven two-level system and consider it in the drive
frame under the rotating wave approximation (e.g., Chapter 4). Let {σα} denote the set
of Pauli matrices. The time-dependent system Hamiltonian that generates single-qubit X
rotation gates is given by:
H(t) = εtot(t)
σx
2
+ Herr (6.1a)
Herr = εerr
σx
2
+ ∆err
σz
2
. (6.1b)
Here, εtot(t) is the intended time-dependent control field and εerr and ∆err are errors. Ideally,
εerr = ∆err = 0. In reality, both are present and give rise to rotation and phase errors
δθ ≡ εerrtg , δϕ ≡
∆err
ε¯
, (6.2)
respectively, with tg denoting the gate duration and ¯ε the effective pulse amplitude [see
Chapter 5]. An open system single-qubit gate includes both rotation and phase errors,
as well as a system-bath interaction term that is always present while the gate is being
generated.
110
6.2.1 Gate compilation
VZ gates eliminate the need for performing physical rotations about the Bloch z-axis, allowing us to focus solely on rotations in the (x, y) plane. We denote by Rϕ(θ) a rotation by
an angle θ about an arbitrary axis in the (x, y) plane, making an angle ϕ with the x-axis:
Rϕ(θ) ≡ exp[−i(θ/2)(cos(ϕ)σx + sin(ϕ)σy)]. (6.3)
We also denote
Rx(θ) ≡ Rϕ=0(θ) , Ry(θ) ≡ Rϕ=π/2(θ). (6.4)
The physical implementation of Rϕ(θ) involves applying an on-resonance microwave pulse of
the form εtot(t) = ε(t) cos(ωt + ϕ) to the qubit, where the integrated pulse amplitude (for
a given pulse duration) determines θ, and the pulse phase determines ϕ. The phase ϕ is
arbitrary, as is the choice of the (x, y) coordinate system, both set by the initial pulse. This
illustrates how the VZ gate is implemented simply by updating the definition of which pulse
phase corresponds to ϕ = 0 (usually set to be the x-axis, as above). However, this adjustment
has tangible physical effects on subsequent gates: after a virtual Rz(φ) ≡ exp[−i(φ/2)σz]
gate [note that Rz(π) ≡ Z = −iσz
], the phase of each of the rotations that follow is shifted
by φ. For example, when φ = π, then the next operation Rx(θ) becomes Rπ(θ) = R−x(θ),
i.e., a rotation about the x-axis becomes a rotation about the −x axis, in the sense that
R−x(θ) = R†
z
(π)Rx(θ)Rz(π).
For most commercial cloud-based quantum processors not all rotations are natively
available. For example, for the IBMQ devices, the calibrated single-qubit native gate set
typically consists of the operations G = {Rz(φ),
√
X, X}, where √
X ≡ Rx(π/2) and
X ≡ Rx(π) = −iσx, which are generated using H(t) given in Eq. (6.1). However, these
are not the only Clifford operations necessary for universal quantum computation. All other
Clifford gates must be decomposed into these operations. Specifically, a Y ≡ Ry(π) = −iσy
111
0 180 360 540 720
number of cycles N
0.75
0.80
0.85
0.90
0.95
1.00
fidelity
YYasym |°ii
YYsym |°ii
YYasym |+ii
YYsym |+ii
0 180 360 540 720
number of cycles N
0.75
0.80
0.85
0.90
0.95
1.00
0 20 40 60 80
time (µs)
0.75
0.80
0.85
0.90
0.95
1.00
fidelity
0 20 40 60 80
time (µs)
0.75
0.80
0.85
0.90
0.95
1.00
(a) (b)
(c) (d)
Figure 6.2: Effect of the asymmetric Y
asym and symmetric Y
sym gates on state fidelity for
various qubits of the ibm sherbrooke processor. We applied different repetitions of the Y Y
sequence and observe that subject to Y
sym the fidelities of the |±i⟩ states (black and red)
are much closer than subject to Y
asym (yellow and green). Different subfigures correspond
to different qubits on ibm sherbrooke: (a) qubit 0, (b) qubit 13, (c) qubit 81, and (d) qubit
89.
gate requires an X gate combined with VZ gates, which can be done in different ways. One
method of compiling a Y gate is asymmetrical:
Y
asym = XRz(−π). (6.5)
Alternatively, a symmetric compilation of the Y gate with respect to the VZ gates is:
Y
sym = Rz(π/2)XRz(−π/2). (6.6)
Although these methods are theoretically equivalent in the sense that Y
asym = Y
sym = Y is
a mathematical identity, this is no longer the case when one accounts for deviations from
unitary dynamics due to open quantum system effects, as we discuss in detail below.
112
6.2.2 Trajectories matter: asymmetry between |+i⟩ and |−i⟩
To demonstrate how the two compilation strategies result in different outcomes, we consider
a simple experiment, in which we apply sequences with a varying number of Y Y pulses to
the two orthogonal initial states |±i⟩. Ideally, the fidelity of Y Y applied to |+i⟩ or |−i⟩
should be identical. However, with the asymmetric decomposition the two states follow
different Bloch sphere trajectories and leave the (x, y) plane. That is, in the case of Y
asym,
the virtual Rz(−π) gate instantaneously interchanges |+i⟩ and |−i⟩ (up to a global phase of
i) before the physical X gate is applied. This has the effect of |−i⟩ following a trajectory
through the stable ground state |0⟩ during the X gate, while |+i⟩ passes through the unstable
excited state |1⟩ [see Fig. 6.1(a)]. The second Y gate leads to a reversal of this trajectory,
again passing through the ground/excited, as the virtual Rz(−π) reverses the direction of
rotation. Consequently, |−i⟩ experiences a lower relaxation rate and maintains a higher
fidelity compared to |+i⟩ over the course of repeated applications of the Y Y sequence.
Conversely, using the symmetric decomposition Y
sym, the first VZ gate, Rz(−π/2) transforms |−i⟩ to |−⟩ and |+i⟩ to |+⟩ (up to a global phase of e
iπ/4
). These states then undergo
an X gate, which leaves them unchanged (up to a global phase). The next VZ gate, Rz(π/2),
transforms the state back to its original position on the y-axis. Therefore, with this compilation, |±i⟩ both remain in the (x, y) plane at all times during the Y
sym gate, and do not
experience different relaxation rates. As a result, the fidelities of |±i⟩ should be similar under
Y Y , as for a physical Y gate. By linearity, this extends to any state in the (x, y) plane, i.e.,
to any superposition of |±i⟩ or |±⟩. Another way to see this result is that symmetric Y Y
compiles to two repetitions of Eq. (6.6), which is then equal to Rz(π/2)XXRz(−π/2). The
interior XX sequence traces out a full 2π rotation and thus always leads to trajectories that
are symmetric about the (x, y) plane, as expected by the Y Y sequence.
We verified the effects predicted above through various experiments, using both the
ibm sherbrooke and MUNINN processor. As described above, we first prepare the initial
113
states |±i⟩, apply a series of Y Y sequences, unprepare the initial state, and measure the
system in the σz eigenbasis. We define the empirical fidelity as the frequency of favorable
outcomes, i.e., the number of |0⟩ outcomes divided by the total number of experimental shots
(800). Fig. 6.1(b) shows the results on the MUNINN processor, where we demonstrate that
the asymmetric compilation of the Y gate leads to the predicted asymmetry in the fidelity
of the |±i⟩ states. In contrast, the two states decay almost identically when the symmetric
compilation of Y gate is used. We observe the same effect after repeating the experiments
on different qubits of the ibm sherbrooke processor, as highlighted in Fig. 6.2.
6.2.3 Impact on DD sequences
Next, we consider the impact of asymmetric compilation on DD sequences. Specifically, we
consider asymmetric and symmetric versions of the XY4 sequence [265]:
XY4 ≡ Y fτXfτY fτXfτ , (6.7)
where fτ = e
−iτH denotes the free evolution unitary generated by the total system-bath
Hamiltonian H. We also consider how to correctly implement the X gate.
6.2.3.1 Asymmetric Y yields UR4
Using the asymmetric definition of the Y gate [Eq. (6.5)], the XY4 sequence becomes:
XY4asym = (6.8)
Rx(π)Rz(−π)fτRx(π)fτRx(π)Rz(−π)fτRx(π)fτ .
As discussed earlier, the VZ gates enact a frame transformation for all subsequent gates. In
the present context, this manifests as the identity
Rx(π)Rz(±π) = −Rz(∓π)Rx(−π), (6.9)
114
which allows us to commute the VZ gate to the left. Since it is a virtual gate implemented
via phase offsets in software, the VZ gate commutes with the free evolution operator fτ .
Thus, dropping overall phase factors, we can rewrite XY4asym as follows:
XY4asym = Rx(π)Rz(−π)fτRx(π)Rz(π)fτRx(−π)fτRx(π)fτ
= Rx(π)Rz(−2π)fτRx(−π)fτRx(−π)fτRx(π)fτ
= Rx(π)fτRx(−π)fτRx(−π)fτRx(π)fτ . (6.10)
This sequence is, in fact, the fourth order “universally robust” sequence [252],
UR4 = XfτXfτXfτXfτ , (6.11)
where X ≡ Rx(−π), rather than the intended XY4. I.e.,
XY4asym = UR4. (6.12)
6.2.3.2 Symmetric Y yields XY4
We first note the identity
Rx(π)Rz(−π/2) = Rz(−π/2)Ry(π), (6.13)
115
i.e., Rz(−π/2) changes the rotation axis of the subsequent gates by π/2, transforming
Rx(π) → Ry(π). Therefore, using Y
sym [Eq. (6.6)], we have:
XY4sym = Rz(π/2)Rx(π)Rz(−π/2)fτRx(π)fτ×
Rz(π/2)Rx(π)Rz(−π/2)fτRx(π)fτ
= Rz(π/2)Rz(−π/2)Ry(π)fτRx(π)fτ×
Rz(π/2)Rz(−π/2)Ry(π)fτRx(π)fτ
= Ry(π)fτRx(π)fτRy(π)fτRx(π)fτ , (6.14)
which is indeed the XY4 sequence [Eq. (6.7)].
6.2.3.3 Correct X ≡ Rx(−π)
We note that the frame transformation defined by Eq. (6.9) can be reinterpreted as a way to
create a correct X gate, which plays an important role in robust DD sequences [252, 251].
Namely, we perform the symmetric version of the gate as:
X = Rz(−π)XRz(π) or Rz(π)XRz(−π). (6.15)
As we show in the next section, performing the correct X gate is critical for understanding
the oscillations in the fidelity of the robust DD sequences that have been observed on IBM
devices [254]. In particular, it is essential that the physical rotation implemented is Rx(π)
and not only Rx(π) plus some later frame updates.
6.3 Experimental verification
Next, we report on experiments with various DD sequences to test our predictions about the
role of VZ gates in open quantum systems.
116
0 40 80 120 160 200 240 280 320
number of cycles N
0 10 20 30 40 50 60 70
time (µs)
0.0
0.2
0.4
0.6
0.8
1.0
fidelity |+i
XXsym
XY4sym
YYsym
XY4asym
UR4
Figure 6.3: The |+⟩ state fidelity as a function of total sequence time (or number of sequence
cycles; top axis), subject to the XY4sym, XY4asym, UR4, Y Y sym, and XX sequences applied
to a single qubit (qubit 37) on the ibm sherbrooke device. Data points for the two-pulselong Y Y sym and XX sequences are shown for every second cycle (i.e., their total number
of cycles is 640). The XY4asym and UR4 sequences exhibit nearly identical fidelity decay
behavior, clearly distinct from that of the XY4sym sequence, confirming that the asymmetric
Y gates transform XY4 into the UR4 sequence. All sequences shown exhibit oscillations.
6.3.1 Symmetric vs asymmetric sequence implementations
To test our prediction that when using Y
asym, XY4 is effectively the same as UR4, Fig. 6.3
presents the results of measuring the fidelity of the |+⟩ state as a function of time for a variety
of different pulse sequences, each of which is applied repeatedly. Specifically, we apply the
following sequences to a single qubit on ibm sherbrooke: UR4 using the symmetric definition
for X given in Eq. (6.15), and two versions of XY4 using the symmetric and asymmetric Y
gates. As shown in Fig. 6.3, the UR4 and XY4asym sequences are almost indistinguishable,
as expected. In contrast, XY4sym is distinct.
It is important to note that Qiskit [266] natively compiles the Y gate in the asymmetric
form of Eq. (6.5). Therefore, caution is necessary when interpreting previously reported DD
117
0 20 40 60 80 100 120 140 160
number of cycles N
0 10 20 30 40 50 60 70
time (µs)
0.0
0.2
0.4
0.6
0.8
1.0
fidelity |+i
XXsym
XY4sym
YYsym
UR4
Figure 6.4: As in Fig. 6.3 (except for the absence of XY4asym) with the pulse interval doubled
from τ = 56.8 ns to 2τ = 113.6 ns. The oscillation periods of Y Y sym and XX increase
significantly, and the difference between the now decaying XY4sym and UR4 fidelities is
nearly eliminated.
results involving transmon qubits that did not use the Y
sym gate, including numerous studies
involving the XY4 sequence.
6.3.2 Pulse interference
Fig. 6.3 also displays the Y Y and XX sequences, constructed using the symmetric definitions
given in Eqs. (6.6) and (6.15), respectively. An unexpected feature observed in Fig. 6.3 is
that all five sequences shown (including the robust ones), exhibit oscillations, which typically
arise from coherent errors. We hypothesize that this phenomenon is due to an interference
effect between consecutive pulses, e.g., due to an impedance mismatch in the microwave
control lines [267].
To test this hypothesis, we repeated the same experiments as shown in Fig. 6.3 (except
that we did not repeat XY4asym since we already established its equivalence with UR4),
but with an intentional delay added between consecutive pulses, thus doubling the pulse
118
interval τ = 56.8 ns, defined as the time delay between the peaks of two consecutive pulses.
The result, shown in Fig. 6.4, is that the XY4sym and UR4 sequences no longer oscillate,
but exhibit simple decay. Moreover, the stark difference between the latter two sequences
seen in Fig. 6.3 has now almost disappeared. This is consistent with the observation that
Z (dephasing) errors are the dominant error source in transmon qubits, so that sequences
suppressing X or Y errors have little added benefit over sequences suppressing only Z errors.
The fact that the fidelities seen in Fig. 6.3 are higher for intervals of 2τ rather than
τ also helps to explain why previous studies involving transmon qubits [253, 254] have
observed that, in contrast to the DD theory for ideal, zero width pulses (e.g., Ref. [149]), the
optimal pulse interval is not always the shortest possible (the same phenomenon has also been
observed in other platforms, e.g., nuclear magnetic resonance [151] and trapped ions [268]).
The pulse interference effect, with pulses applied consecutively with the minimum shortest
possible pulse interval τ , can introduce additional coherent errors that result in inferior
DD performance even with sequences (such as UR4) that are robust against small coherent
errors. Our results confirm the conclusion of Ref. [254] that it is essential to optimize the
pulse interval for a given quantum processor, with the added insight that this optimization
can reduce or (depending on the pulse sequence) even eliminate coherent errors due to pulse
interference.
The reason we include the XX and Y Y sequences in Fig. 6.3 is that XX is susceptible
to phase errors, while Y Y is susceptible to rotation errors [Eq. (6.2)], as discussed in detail
in Ref. [156]. More specifically, Figs. 6.3 and 6.4 shows that the two sequences exhibit oscillations for both the τ and 2τ cases, with a period significantly shorter than that of the
other sequences shown. This is consistent with the existence of single-pulse phase and rotation errors in addition to pulse interference errors. Doubling the pulse interval significantly
increases the oscillation period, as seen in Fig. 6.4, but does not eliminate the oscillations.
Moreover, we have checked (not shown) that further increasing the pulse interval to 3τ has
little effect on the XX and Y Y fidelities, showing that coherent phase and rotation errors
119
0.2
0.4
0.6
0.8
1.0
fidelity |+i
pulse interval: 1ø pulse interval: 2ø pulse interval: 3ø
0 10 20 30 40 50 60
time (µs)
0.2
0.4
0.6
0.8
1.0
fidelity |+i
0 10 20 30 40 50 60
time (µs)
0 10 20 30 40
time (µs)
(a) (b) (c)
(d) (e) (f)
Figure 6.5: Experimental study of the fidelities of the XY4sym (blue) and UR4 (red) sequences with different pulse intervals (1τ , 2τ , and 3τ ) for the whole set of 127 qubits on
ibm sherbrooke. (a), (b) and (c) show the results for 123 of the qubits for the noted pulse
intervals. (c), (d) and (e) show the corresponding mean and standard deviations. The oscillations in fidelities vanish as we increase τ due to mitigation of the pulse interference effects.
cannot be eliminated by controlling the pulse interference effect alone.
Both Figs. 6.3 and 6.4 display results from a single qubit. To test whether the small
difference between the robust XY4sym and UR4 sequences [252] seen in Fig. 6.4 even with
a doubled pulse interval is a statistically significant feature, we performed the XY4sym and
UR4 experiments on all 127 qubits of the ibm sherbrooke device, for pulse intervals of
τ = 56.8ns, 2τ , and 3τ . The results are shown in Fig. 6.5, after removing four of the qubits
whose measurements were inadvertently performed during a calibration cycle (qubits 20, 21,
56, 63). We find that the oscillations exhibited by both XY4sym and UR4 in the 1τ case
entirely disappear in 122 out of the 123 qubits (the only exception being qubit 67) for pulse
intervals of 2τ and 3τ . After fitting the fidelities to a + be−t/TD , a small difference in the
decay constants remains for 2τ intervals: TD(UR4) ≥ TD(XY4sym) in 59% of the cases. This
difference disappears almost entirely for the 3τ intervals: TD(UR4) ≥ TD(XY4sym) in 51% of
the cases.
We may thus conclude that the pulse interference effect is significant when pulses are ap120
plied back-to-back but strongly diminishes when the pulse interval is doubled, and essentially
disappears entirely when it is tripled.
6.4 Conclusion
This work highlights the critical role of the VZ gate and its interplay with the open system
dynamics of quantum processors. We have demonstrated that a symmetric compilation of
quantum gates with respect to the VZ gate, especially the Y and X gates, significantly
improves the fidelity of these gates. In particular, it removes an undesired asymmetry between states in the (x, y) plane of the Bloch sphere that is present when an asymmetric gate
compilation is used instead. We have experimentally validated the advantage offered by a
symmetric gate compilation using our in-house processor MUNINN as well as using the IBM
cloud processor ibm sherbrooke, showing in particular the impact on commonly used DD
sequences.
Our findings highlight the need to carefully consider VZ gate compilation in future studies, as well as the impact on previous studies that used asymmetric gate compilations. Specifically, we have shown that asymmetric compilations can lead to unexpected outcomes, such
as fidelity asymmetries and incorrect implementations of DD sequences, which can result
in misleading interpretations of earlier experimental results. A case in point is that an
asymmetric compilation of the Y gate has the effect that the standard XY4 DD sequence is
actually an implementation of the UR4 sequence, which does not suppress any undesired Xtype interactions. Conversely, symmetric compilations preserve the intended gate operations
and result in a faithful implementation of the desired DD sequences.
Furthermore, we explored the impact of pulse interference, which can introduce coherent
errors even in DD sequences that are designed to be robust to such errors. We have demonstrated that by intentionally increasing the pulse interval these effects can be mitigated,
highlighting the importance of optimizing the pulse interval for a given quantum processor.
121
These results explain earlier observations where robust sequences resulted in suboptimal
performance; this effect can now be attributed to pulse interference effects.
Future studies may focus on refining gate compilation strategies and addressing pulse
interference effects to further enhance the fidelity of quantum gates.
122
Chapter 7
Conclusion and Outlook
In this dissertation, we have extensively investigated superconducting quantum processors,
with a particular emphasis on transmons. This work has addressed several crucial challenges associated with these processors, including crosstalk errors, noise characterization,
benchmarking of quantum gates, and error suppression for qudits.
The first major contribution of this work is the utilization of dynamical decoupling (DD)
to suppress ZZ-type crosstalk errors in superconducting qubits (see Chapter 2). We presented theoretical analyses and experimental results that demonstrate the effectiveness of
these methods in protecting quantum states and reducing crosstalk in coupled qubit systems. This work has paved the way for the development of crosstalk-robust DD sequences
with several follow up studies, including [21, 152, 269] building on these findings.
Expanding beyond qubits, Chapter 3 explored dynamical decoupling for transmon qudits. We provided a general framework for utilizing dynamical decoupling to suppress lowfrequency noise and qudit-qudit cross-Kerr interactions, which serve as the bottleneck for
scaling qudit processors. Experimental validation of the techniques developed in this work
highlighted their practical viability and potential for scaling large-scale high dimensional
systems.
In Chapter 4, we develop a procedure to model both low- and high-frequency noise
components in transmon qubits in a three-step iterative process. This helps us model the
environmental impacts on the qubit. This comprehensive noise modeling offers a deeper
123
understanding of the sources of decoherence and can further help inform strategies to mitigate
their effects.
The dissertation further developed deterministic benchmarking techniques for single qubit
gates in chapter 5, offering strategies to quantify gate infidelities in terms of the constituent
sources of coherent and incoherent errors. These techniques are essential for evaluating and
improving the performance of quantum gates, ensuring their reliability in practical quantum
computing applications.
Finally, the role of virtual Z gates in gate compilation strategies, within open quantum
systems was investigated in Chapter 6. This work provided explanations for previously
observed phenomena in dynamical decoupling studies and offered insights into optimizing
virtual gate operations for enhanced quantum computing performance.
We hope that the results and the tools developed in this dissertation will be instrumental
for future research in quantum technologies, bringing us closer to the realization of practical
and scalable quantum computing systems.
124
Appendix A
General theory of qudit
dynamical decoupling
Consider the total time-independent Hamiltonian of a system coupled to a bath H =
HS ⊗ IB + IS ⊗ HB + HSB where HS, HB, and HSB are the Hamiltonian terms associated with the system, the bath, and the system-bath interaction, respectively. Here, HS
represents undesired system terms such as crosstalk and stray local fields. Pulses are applied
to the system via an additional, time-dependent control Hamiltonian. Correspondingly, the
decoupling group Gd is defined by a set of unitary transformations gj acting purely on the
system: Gd = {g0, · · · , gK}, where g0 is the d-dimensional identity operator I. Under the
instantaneous (zero width) and ideal (error-free) pulse assumptions, cycling over all elements
of the group yields the following DD pulse sequence [74, 127]:
U(T) = Y
K
j=0
Dj (τ ) = Y
K
j=0
g
†
j
fτ gj = e
−iT H′
+ O(T
2
). (A.1)
Here, τ is the pulse interval (the time between consecutive pulses), T = |Gd|τ is the total
time taken by the sequence, and fτ = e
−iτH is the free-evolution unitary. The effective
Hamiltonian at the end of the sequence is
H
′ = H
′
SB + H
′
S + HB, (A.2)
125
where
H
′
SB = PGd
(HSB) , H′
S = PGd
(HS). (A.3)
Here
PGd
(Ω) = 1
|Gd|
|G
Xd|−1
j=0
g
†
jΩgj (A.4)
is the projection of the operator Ω into the commutant of Gd, i.e., the set of operators that
commute with every element of Gd. Crucially, this projection can be made proportional
to I or even vanish via a proper choice of Gd. When H′
SB = H′
S = 0, we call Gd and the
corresponding DD sequence universal. For example, a universal DD sequence for a qubit
(d = 2) is obtained by choosing the decoupling group as the Pauli group G2 = P, which
leads to the well-known XY4 sequence U = Y fτXfτY fτXfτ [81].
For d ≥ 2, we instead use the corresponding Heisenberg-Weyl group (HWG) of order d
2
,
which generalizes the Pauli group. We define shift and phase operators as in Eq. (3.2) of the
Chapter 3, repeated here for convenience:
Xd ≡
X
d−1
k=0
|(k + 1) mod d⟩⟨k|, Zd ≡
X
d−1
k=0
γ
k
d
|k⟩⟨k|, (A.5)
where γd = e
2πi/d is the d’th root of unity.
Note that Xd
d = Z
d
d = I, and that Xd and Zd are generally non-Hermitian but are both
unitary for all d and hence satisfy
(X
†
d
)
αX
β
d = X
β−α
d
, (Z
†
d
)
αZ
β
d = Z
β−α
d
. (A.6)
In particular,
X
†
d = X
−1
d = X
d−1
d
, Z†
d = Z
−1
d = Z
d−1
d
. (A.7)
126
Xd and Zd are the generators of the HWG, whose elements are
Λαβ = (−
√
γd)
αβX
α
d Z
β
d
, (A.8)
where α, β ∈ Zd. For d = 2, the HWG trivially reduces to the Pauli group.
For the two generators Xd and Zd, we have
XdZd =
X
d−1
k=0
e
2πik/d|k + 1 mod d⟩⟨k| (A.9a)
ZdXd =
X
d−1
k=0
e
2πi(k+1)/d|k + 1 mod d⟩⟨k|, (A.9b)
i.e.,
ZdXd = γdXdZd. (A.10)
Similarly, we find:
Z
†
dXd = γ
−1
d XdZ
†
d
. (A.11)
Using Eqs. (A.10) and (A.11), we can show that
Z
β
d X
α
d = γ
αβ
d X
α
d Z
β
d
(A.12a)
(Z
†
d
)
βX
α
d = γ
−αβ
d X
α
d
(Z
†
d
)
β
. (A.12b)
Proof. For α = β = 1, Eq. (A.12) reduces to Eq. (A.10). Consider β ≥ 2:
Z
β
d Xd = γdZ
β−1
d XdZd = · · · = γ
β
d XdZ
β
d
. (A.13)
127
When α ≥ 2:
Z
β
d X
α
d = (Z
β
d Xd)X
α−1
d = γ
β
d Xd(Z
β
d Xd)X
α−2
d
(A.14a)
= γ
2β
d X
2
dZdX
α−2
d = · · · = γ
αβ
d X
α
d Z
β
d
. (A.14b)
Eq. (A.12b) follows analogously.
This means that the commutation relations for two arbitrary HW operators are
ΛαβΛµν = γ
βµ−αν
d ΛµνΛαβ (A.15a)
Λ
†
αβΛµν = γ
αν−βµ
d ΛµνΛ
†
αβ. (A.15b)
Unless α = β = 0, the HW operators are non-Hermitian for d ≥ 3
Λ
†
αβ = e
−iπαβ d+1
d (X
α
d Z
β
d
)
†
(A.16a)
= (−
√
γd)
−αβ(Z
†
d
)
β
(X
†
d
)
α
̸= Λαβ, (A.16b)
but unitary for all d:
Λ
†
αβΛαβ = (Z
†
d
)
β
(X
†
d
)
αX
α
d Z
β
d = I, (A.17)
where we used the unitarity of Xd and Zd. Combining unitarity with Eq. (A.15a), we obtain
the identity
Λ
†
αβΛµνΛαβ = γ
αν−βµ
d Λµν, (A.18)
which will prove to be crucial below for demonstrating that the HWG is a universal DD
group.
The operators {Λαβ}α,β∈Zd
form an irreducible, unitary, and projective representation of
the HWG over the d-dimensional system Hilbert space when d is a prime power (d = p
k
for
prime number p and positive integer k). This implies, by Schur’s Lemma, that both H′
SB
and H′
S
are proportional to I or vanish. Therefore, the unitary U(T) defined in Eq. (A.1)
128
reduces (up to a global phase) to the identity operation on the system, i.e., the condition
for first-order decoupling is satisfied. The latter (or an equivalent one using group character
tables) was the argument used in Refs. [142, 143, 144, 145]; going beyond the case of prime
powers, we now show that, in fact, first-order decoupling holds for arbitrary d.
The HW operators also form an operator basis for the d-dimensional system Hilbert
space. Thus, we can expand H = HS ⊗ IB + HSB + IS ⊗ HB as
H =
d
X2−1
µ,ν=0
Λµν ⊗ Bµν, (A.19)
where Bµν is either zero, proportional to IB (to account for HS ⊗ IB), or is a non-identity
bath operator. The term with µ = ν = 0 corresponds to the pure-bath term IS ⊗ HB.
Now recall that cycling over the decoupling group Gd yields Eq. (A.1). Choosing the
decoupling group as the HWG {Λαβ} means that the effective Hamiltonian becomes
H
′ = PGd
(H) (A.20a)
=
1
d
2
d
X2−1
α,β=0
Λ
†
αβ
d
X2−1
µ,ν=0
ΛµνΛαβ ⊗ Bµν (A.20b)
=
1
d
2
d
X2−1
µ,ν=0
fµνΛµν ⊗ Bµν, (A.20c)
where, using Eq. (A.18),
fµν =
d
X2−1
α,β=0
γ
αν−βµ
d
. (A.21)
Let us now show that
fµν = d
4
δµ0δν0. (A.22)
Intuitively, this follows from the zero-sum property of the roots of unity: Pd−1
k=0 γ
k
d = 0.
129
Proof. Note that fµν = h
∗
µhν, and
hν =
d
X2−1
α=0
γ
να
d =
X
d−1
k=0
X
d−1
j=0
e
2πi(kd+j)ν/d (A.23a)
=
X
d−1
k=0
X
d−1
j=0
(e
2πiν/d)
j = dS, (A.23b)
where S =
Pd−1
j=0 ω
j and ω = e
2πiν/d. If ν ̸= 0 then ω ̸= 1 is a d’th root of unity (since
ω
d = 1). Multiplying both sides by ω − 1 yields (ω − 1)S =
Pd−1
j=0 ω
j+1 −
Pd−1
j=0 ω
j
. The
terms ω, ω2
, . . . , ωd−1
cancel out, leaving (ω − 1)S = ω
d − 1 = 0. Since ω ̸= 1 we can divide
both sides by ω − 1, giving S = 0. If ν = 0 then S = d.
Combining Eqs. (A.20) and (A.22), we finally have
H
′ = IS ⊗ HB, (A.24)
i.e., H′
SB = H′
S = 0, leaving only the pure-bath term. This proves that the HWG is a
universal decoupling group.
We numerically confirm this universality in Fig. A.1. For various dimensions 2 ≤ d ≤ 10,
we consider a system-bath Hamiltonian containing all the HW operators with randomized
coefficients (i.e., a classical bath). We then apply the corresponding universal sequence and
compare the fidelity of the resulting unitary to the identity operator I in each case. Given
that the errors in the unitary evolution under DD are suppressed to the first order [Eq. (A.1)],
i.e., leaving the leading order term O(T
2
) where T ∝ τ , we expect the fidelity to scale as
O(τ
4
). This is confirmed in Fig. A.1.
A.1 Application to single-axis, pure dephasing noise
Let us utilize the formalism we developed above to analyze the single-axis noise (i.e., pure
dephasing) problem as a special case. This is the basis for the transmon qutrit and ququart
130
0.01 0.05 0.10 0.50 1
10-6
0.001
Pulse interval τ
Infidelity 1
-
ℱ
τ4
d=2
d=3
d=4
d=5
d=6
d=7
d=8
d=9
d=10
τ4
d=2
d=3
d=4
d=5
d=6
d=7
d=8
d=9
d=10
Figure A.1: Numerical simulation of the the universality of the DD sequence generated by
cycling over the HWG for 2 ≤ d ≤ 10. Here we plot the infidelity of the resulting unitary
evolution as a function of the pulse interval time τ . Since the sequence is expected to cancel
the errors to the first order O(τ
2
), the infidelity should scale as O(τ
4
), as confirmed by our
simulations.
experiments we presented in the Chapter 3, where dephasing is the dominant source of decoherence. In this single-axis scenario, the system-bath interaction component of Eq. (A.19)
reduces to
H
Z
SB =
X
d−1
ν=1
Λ0ν ⊗ Bν, (A.25)
where Λ0ν = Z
ν
d
[Eq. (A.8)].
We could choose the full HWG as a decoupling group, but since the Zd-type HW operators
commute with HZ
SB, we need only consider the pure Xd-type decoupling operators Λα0 =
Xα
d
, which satisfy the commutation relations Eq. (A.15) non-trivially. I.e., for pure qudit
dephasing the relevant decoupling group is the order-d HWG subgroup G
X
d = {Λα0}
d−1
α=0 =
{I, Xd, X2
d
, · · · , Xd−1
s }.
To show that G
X
d decouples HZ
SB, we observe, using Eq. (A.18), that the effective Hamil131
tonian is
(H
Z
SB)
′ = PGX
d
(H
Z
SB) (A.26a)
=
1
d
X
d−1
α=0
Λ
†
α0
X
d−1
ν=1
Λ0νΛα0 ⊗ Bν (A.26b)
=
1
d
X
d−1
ν=1
fνΛ0ν ⊗ Bν, (A.26c)
where
fν =
X
d−1
α=0
γ
να
d = dδν0, (A.27)
and the last equality is again due to zero-sum property of the d’th root of unity. It follows
that (HZ
SB)
′ = 0, i.e., G
X
d
is a decoupling group for qudit dephasing.
A.2 Qutrit (d = 3) dynamical decoupling and 3X3
For concreteness, we now illustrate the results above by giving the explicit form of the HW
subgroup G
X
d
for the case of qutrit dephasing. This is the simplest non-trivial example going
beyond qubits.
The generators of the qutrit HWG are X3 and Z3, which can be seen as generalizations of
the Pauli matrices σx and σz, respectively. Setting ω ≡ γ3 = e
2πi/3
(the cube root of unity),
the shift and phase operators are
Λ10 = X3 =
0 0 1
1 0 0
0 1 0
, Λ01 = Z3 =
1 0 0
0 ω 0
0 0 ω
2
(A.28a)
Λ20 = X
2
3 =
0 1 0
0 0 1
1 0 0
, Λ02 = Z
2
3 =
1 0 0
0 ω
2 0
0 0 ω
. (A.28b)
132
Their action on the qutrit computational basis states |m⟩ (m = 0, 1, 2) is X3|m⟩ = |(m +
1) mod 3⟩ and Z3|m⟩ = ω
m|m⟩. Note that O−1 = O† = O2 and O3 = I for O = X3, Z3, X2
3
,
and Z
2
3
.
Their commutation properties follow from Eq. (A.12):
X3Z3 = ω
2Z3X3, X2
3Z3 = ωZ3X
2
3
, (A.29a)
X3Z
2
3 = ωZ2
3X3, X2
3Z
2
3 = ω
2Z
2
3X
2
3
. (A.29b)
Since X
†
3 = X2
3
, we can write
(X
2
3
)
†Z3X
2
3 = ω
2Z3, X†
3Z3X3 = ωZ3, (A.30a)
(X
2
3
)
†Z
2
3X
2
3 = ωZ2
3
, X†
3Z
2
3X3 = ω
2Z
2
3
, (A.30b)
which is a special case of Eq. (A.18). The decoupling group is G
X3
3 = {I, X3, X2
3 }, which
suppresses dephasing due to the system-bath interaction Hamiltonian HZ
SB with d = 3
[Eq. (A.25)]. For example, consider free evolution subject to a term of the form Z3 ⊗ B;
writing out the DD sequence Eq. (A.1) explicitly, we have, for T = 3τ :
U(T) =
Ifτ I
(X
2
3
)
†
fτX
2
3
X
†
3
fτX3
(A.31a)
= e
−iτZ3⊗B
e
−iτ(X2
3
)
†Z3X2
3 ⊗B
e
−iτX†
3Z3X3⊗B
(A.31b)
= e
−iτ(Z3⊗B)
e
−iτ(ω
2Z3⊗B)
e
−iτ(ωZ3⊗B)
(A.31c)
= e
−iτ(1+ω
2+ω)Z3⊗B + O(T
2
) (A.31d)
= I + O(T
2
), (A.31e)
i.e., suppression to second order of dephasing due to Z3 ⊗B. Replacing Z3 ⊗B with Z
2
3 ⊗B′
simply rearranges the order the roots of unity, yielding 1+ω+ω
2 = 0 instead of 1+ω
2+ω = 0
in Eq. (A.31d), with the same outcome.
Note that since (X2
3
)
† = X3 = X2
3X
†
3
, this DD sequence in fact reduces to three equidistant pulses of type X3:
U(T) = X3fτX3fτX3fτ (A.32)
which is the reason we called it 3X3 in the Chapter 3. We could have equivalently used the
sequence consisting of three X2
3 pulses.
A.3 Proof of first order suppression of cross-Kerr coupling by the CKDD sequence
The goal of the CKDD sequence is to suppress the cross-Kerr coupling Hamiltonian Eq. (3.3):
HCK =
X
d−1
i,j=1
αij |i⟩⟨i| ⊗ |j⟩⟨j|. (A.33)
Since the HWG is an operator basis, and in particular its phase elements {Z
α
d
}
d−1
α=0 are a
basis for diagonal operators, we can expand each diagonal term in HCK as
|i⟩⟨i| =
X
d−1
k=0
cikZ
k
d
, (A.34)
where
cik =
1
d
Tr[(Z
†
d
)
k
|i⟩⟨i|] = 1
d
γ
−ik
d
. (A.35)
This is a consequence of Eq. (A.6) and the zero-sum property: Tr[(Z
†
d
)
kZ
l
d
] = Tr(Z
l−k
d
) =
Pd−1
m=0 γ
m(l−k)
d = dδkl. Thus, we can rewrite the cross-Kerr Hamiltonian as
HCK =
X
d−1
k,l=1
ζklZ
k
d ⊗ Z
l
d
, (A.36)
134
where
ζkl =
X
d−1
i,j=1
cikcjlαij =
1
d
2
X
d−1
i,j=1
γ
−(ik+jl)
d αij . (A.37)
The CKDD sequence Ud
2τ
[Eq. (3.4)] consists of an inner Xd-type sequence applied to the
first qudit and an outer Xd-type sequence applied to the second qudit. Namely, Ud
2τ ≡
U
(2)
dτ ◦ U
(1)
dτ = (Id ⊗ Xd)U
(1)
dτ (Id ⊗ Xd)· · · U
(1)
dτ (Id ⊗ Xd)U
(1)
dτ , where U
(1)
dτ ≡ dXd ⊗ Id and
U
(2)
dτ ≡ Id ⊗ dXd.
Generalizing from the single qudit dephasing case, the effect of the inner U
(1)
dτ sequence
is to project HCK into the commutant of G
X
d ⊗ Id, i.e.,
H
′
CK = PGX
d ⊗Id
(HCK) (A.38a)
=
1
d
X
d−1
α=0
X
α†
d ⊗ Id
X
d−1
k,l=0
ζklZ
k
d X
α
d ⊗ Z
l
d
(A.38b)
=
1
d
X
d−1
k,l=0
fkζklZ
k
d ⊗ Z
l
d
, (A.38c)
where fk =
Pd−1
α=0 γ
kα
d = dδk0, just as in Eq. (A.27). Thus, the effect of the inner sequence is
to leave just the identity term on the first qudit:
H
′
CK =
X
d−1
l=0
ζ0lId ⊗ Z
l
d
. (A.39)
The outer sequence then removes the remaining dephasing terms:
H
′′
CK = PId⊗GX
d
(H
′
CK) (A.40a)
=
1
d
X
d−1
α=0
Id ⊗ X
α†
d
X
d−1
l=0
ζ0lId ⊗ Z
l
dX
α
d
(A.40b)
=
1
d
X
d−1
l=0
flζ0lId ⊗ Z
l
d = ζ00Id
2 . (A.40c)
Consequently, it follows from Eq. (A.1) that Ud
2τ = e
−iζ00T
Id ⊗ Id + O(T
2
), where T = d
2
τ
135
and ζ00 =
1
d
2
Pd−1
i,j=1 αij . This proves that the CKDD sequence achieves first-order decoupling
of the cross-Kerr interaction.
Note that applying simultaneous Xd-type sequences to both qudits does not work. I.e.,
using G
X
d ⊗ GX
d
as the decoupling group results instead of Eq. (A.38) in the projection
H
′
CK = PGX
d ⊗GX
d
(HCK) (A.41a)
=
1
d
2
X
d−1
α=0
X
d−1
k,l=0
ζklX
α†
d Z
k
d X
α
d ⊗ X
α†
d Z
l
dX
α†
d
(A.41b)
=
1
d
2
X
d−1
k,l=0
ζklZ
k
d ⊗ Z
l
d
X
d−1
α=0
γ
(k+l)α
d
, (A.41c)
and Pd−1
α=0 γ
(k+l)α
d = dδk+l,d, i.e., terms of the form Z
k
d ⊗ Z
d−k
d
are not suppressed. This is a
generalization of the qubit case, where it is well known that simultaneous X-type sequences
do not cancel crosstalk as shown in Chapter 2.
136
Appendix B
Fidelities of Y Y and XX in
open quantum system
We assume a Markovian environment which we model via the Lindblad master equation
with time-dependent driving:
ρ˙ = −i[H(t), ρ] + LD(ρ) (B.1a)
LD(ρ) = X
α
γα(LαρL†
α −
1
2
{L
†
αLα, ρ}). (B.1b)
We recognize that this is a phenomenological model and that rigorously derived master
equations for time-dependent driving have a different structure, where the Lindblad operators
Lα become time-dependent [270, 271, 272, 188, 273, 274, 275]. However, this simplified
model is analytically solvable and suffices for our purposes.
B.1 Background
For analysis purposes, we work in the Bloch vector picture, which substitutes ρ =
1
2
(I+vxσx+
vyσy + vzσz), H =
1
2
(hxσx + hyσy + hzσz), where v = (vx, vy, vz) ∈ R
3
, h = (hx, hy, hz) ∈ R
3
.
The dissipator is replaced by (R, c) with Rij ≡
1
2Tr(σiLD(σj )) and ci ≡
1
2Tr(σiLD(I)). The
Lindblad master equation then is v˙ = h × v + Rv + c (see, e.g., Ref. [276] for derivations of
137
all these results). The corresponding fidelity with the |+⟩ state is given by
F(t) = 1
2
(1 + vx(t)). (B.2)
We model our system as a qubit with energy gap ω01 coupled to a thermal environment
at inverse temperature β along with dephasing noise. The combined noise model translates
to R = diag(−γ2, −γ2, −γ1) and c = (0, 0, η γ1) where
γ2 =
1
2
γ1 + γϕ, (B.3)
and
η =
1 − exp(−βω01)
1 + exp(−βω01)
∈ [0, 1]. (B.4)
B.2 Y Y sequence
For a square Y Y pulse, h(t) = (0, εtot, ∆err) where εtot = ε + εerr. Then the Bloch vector
equation becomes v˙ = Gv + c, where
G =
−γ2 −∆err εtot
∆err −γ2 0
−εtot 0 −γ1
. (B.5)
We first solve the case where ∆err = 0 (δϕ = 0). Define
δG =
0 −∆err 0
∆err 0 0
0 0 0
. (B.6)
138
B.2.1 Exact solution for ∆err = 0
The Bloch vector equation for ∆err = 0 therefore becomes
v˙ = (G − δG)v + c. (B.7)
In this equation, vy is decoupled from the other two components and to solve for vx(t) and
vz(t), the equation requires exponentiation of a 2 × 2 matrix. Given that v(0) = (1, 0, 0),
the relevant solutions for the Bloch vector components take the following form:
vy(t) = 0 (B.8a)
vx(t) = v∞ + (1 − v∞)e
−γ∗t
h
cos (ω∗t) (B.8b)
+
(γ1 − γ2) − (γ1 + γ2)(1 − v∞)
2ω∗(1 − v∞)
sin (ω∗t)
where
v∞ = η
εtotγ1
ε
2
tot + γ1γ2
(B.9a)
γ∗ =
1
2
(γ1 + γ2) (B.9b)
ω∗ =
r
ε
2
tot −
1
4
(γ1 − γ2)
2
. (B.9c)
Along with Eq. (B.2), Eq. (B.8b) is close to, but not quite Eq. (5.3) from Chapter 5.
B.2.2 Perturbation theory for ∆err = 0
In the regime where ε ≫ εerr, γ1, γ2, the quantities in Eq. (B.9) simplify to
v∞ ≈ η
γ1
ε
, ω∗ ≈ εtot = ε + εerr, (B.10)
139
to first order. Moreover, since we are interested in the fidelities after each DD sequence,
i.e., at t = 2ntg (n ∈ N), sin (2ω∗ntg) ≈ sin (2nδθ) and cos (2ω∗ntg) ≈ cos (2nδθ). The
term corresponding to sin (2ω∗ntg) vanishes to first order initially because it is multiplied by
another first-order term and later because it is exponentially suppressed. This results in a
simplified expression for vx(2ntg):
vx(2ntg) = v∞ + (1 − v∞)e
−2ntgγ∗
cos (2nδθ) (B.11)
The fidelity of the |+⟩ state as a function of Y Y repetitions n, Fδϕ=0(n), is therefore given
by
Fδϕ=0(tn) = 1
2
(1 + v∞) + 1
2
(1 − v∞)e
−γ∗tn
cos(2nδθ). (B.12)
Eq. (B.12) is identical in form to Eq. (5.3), with a = v∞ ≈ γ1/ε, TD = 1/γ∗ and 2ωtg ≈ δθ.
Note that the oscillation period of 2nδθ matches that of the fidelity in the closed system case
as discussed below Eq. (5.4a) and Eq. (5.4b).
B.2.2.1 Perturbation theory for ∆err ̸= 0
If the solution to the Bloch vector equation given in Eq. (B.7) is denoted by v0(t), then in
the case where ∆err ̸= 0, let the deviation of the actual Bloch vector trajectory v(t) from
v0(t) be given by ξ(t) ∈ R
3
, i.e., v(t) = v0(t) + ξ(t). Substituting this form into the Bloch
vector equation v˙ = Gv + c gives
˙ξ = Gξ + δGv0. (B.13)
The solution for ξ(t) is then given by
ξ(t) = Z t
0
e
G(t−s)
δGv0(s) ds. (B.14)
140
Taking the norm on both sides and moving the norm inside the integral, we obtain
∥ξ(t)∥ ≤ Z t
0
∥e
G(t−s)
∥∥δGv0(s)∥ ds. (B.15)
Using Eq. (B.6), ∥δGv0(s)∥ simplifies to |∆errv0,x(s)| since v0,y(s) = 0. From Eq. (B.8b),
∥v0,x(s)∥ in turn can be strictly upper bounded by |v∞| + ke−γ∗s
for some constant k. A
smaller value of k will result in a tighter bound. In the approximation regime discussed
above, k can be conveniently chosen to be 2.
If G = SDS−1 where D is the diagonal matrix of G’s eigenvalues λi and S is the similarity
matrix with the corresponding eigenvectors of G as its columns,
∥e
G(t−s)
∥ ≤ ∥S∥∥e
D(t−s)
∥∥S
−1
∥ ≤ κe−λ(t−s)
(B.16)
where κ = ∥S∥∥S
−1∥ and λ = − maxi ℜ(λi) [277]. This gives the inequality
∥ξ(t)∥ <
κ|∆err|
1−e−λt
λ
|v∞| + k
e−γ∗t−e−λt
λ−γ∗
if λ ̸= γ∗
κ|∆err|
1−e−λt
λ
|v∞| + kte−λt
otherwise
(B.17)
Note that the first term is O(∆errv∞/λ).
From first-order eigenvalue perturbation theory, we have
δλi =
wi
· δGvi
wi
· vi
, (B.18)
where wi and vi are the left and right eigenvectors of G in Eq. (B.5) with eigenvalues
{λi} = {γ∗ ± iω∗, −γ2}. Therefore, using Eq. (B.6):
λ1,2 = (−γ∗ ± iω∗) + O(∆err) (B.19a)
λ3 = −γ2 + O(∆err). (B.19b)
141
Under the additional assumption ∆err/γ1,2 = O(1), the first summand in Eq. (B.17) simplifies
to O(v∞) = O(γ1/ε). Combining this with the fact that the second summand decays exponentially with time, ξ(t) is a small enough quantity such that v(t) can be well approximated
by v0(t). Therefore the resulting fidelity expression FY Y (n) in a ∆err ̸= 0 system is still well
captured by Eq. (B.12). It matches Eq. (5.3) with an error |FY Y (n)−Fδϕ=0(n)| <
1
2
∥ξ(2ntg)∥,
with the factor 1/2 due to Eq. (B.2). In big-O notation,
|FY Y (n) − Fδϕ=0(n)| = O(γ1/ε) = O(γ1tg). (B.20)
B.3 XX sequence
We solve for a simplified model in which Tϕ = 2T1, i.e., γ1 = γ2: all the Bloch vector components decay uniformly to the steady state (0, 0, η). The resulting Bloch vector equations
are:
v˙(t) =
G+v(t) + c, if (2k)tg < t ≤ (2k + 1)tg, k ∈ Z
G−v(t) + c, if (2k − 1)tg < t ≤ (2k)tg, k ∈ Z
(B.21)
where c = (0, 0, η γ), γ = γ1 = γ2, and the matrices G± are
G± =
−γ −∆err 0
∆err −γ ∓εtot
0 ±εtot −γ
. (B.22)
The solutions for the Bloch vector after each XX sequence (which corresponds to a duration
of 2tg) are then given recursively by
v(2(n + 1)tg) = Av(2ntg) − ζ (B.23)
142
where ζ ∈ R
3
, and
A = e
G−tg e
G+tg
(B.24a)
ζ = (I − e
G−tg
)(G−)
−1
c + e
G−tg
(I − e
G+tg
)(G+)
−1
c. (B.24b)
Unraveling the recursion yields
v(2ntg) = A
nv(0) −
Xn−1
k=0
A
k
ζ. (B.25)
We are interested in calculating only vx(2ntg):
vx(2ntg) = (1, 0, 0) · A
nv(0) −
Xn−1
k=0
(1, 0, 0) · A
k
ζ (B.26)
The required elements of the matrix An
can be written as:
(A
n
)11 =
1
2
(p
2n + q
2n
) (B.27a)
(A
n
)12 =
−iω′
(1 + e
iω′
tg )
2χ
(p
2n − q
2n
) (B.27b)
(A
n
)13 =
−iεtot(1 − e
iω′
tg )
2χ
(p
2n − q
2n
) (B.27c)
143
where
ω
′ ≡
q
ε
2
tot + ∆2
err (B.28a)
χ ≡
p
4e
iω′tgω′2 + (1 − e
iω′tg )
2∆2
err (B.28b)
p ≡
e
−γtg−iω′
tg
2ω′2
(1 + e
2iω′
tg
)∆2
err + 2e
iω′
tg ε
2
tot+
(1 − e
iω′
tg
)∆errχ
(B.28c)
q ≡
e
−γtg−iω′
tg
2ω′2
(1 + e
2iω′
tg
)∆2
err + 2e
iω′
tg ε
2
tot−
(1 − e
iω′
tg
)∆errχ
. (B.28d)
Substituting these into Eq. (B.26) gives
vx(2ntg) = 1
2
(p
2n + q
2n
) + 1 − p
2n
2(1 − p)
× (B.29)
ζx −
i(1 + e
iω′
tg )ω
′
χ
ζy −
(1 − e
iω′
tg )εtot
χ
ζz
+
1 − q
2n
2(1 − q)
ζx +
i(1 + e
iω′
tg )ω
′
χ
ζy +
(1 − e
iω′
tg )εtot
χ
ζz
,
where ζ carries the temperature dependence. Under first order approximations, χ ≈ 2e
iω′
tg/2ω
′
,
ω
′ ≈ εtot, and
p ≈ e
−γtg−iϑ (B.30a)
q ≈ e
−γtg+iϑ (B.30b)
ϑ ≡ tan−1
2 sin
ω
′
tg
2
∆err
ω′
. (B.30c)
The values of ζ’s components are lengthy expressions and are therefore not included here.
Substituting these values and removing second and higher-order terms gives the following
144
final expression:
vx(2ntg) ≈
1
2
1 + η
∆err
εtot
(1 − e
−γtg
)
(p
2n + q
2n
)
− η
∆err
εtot
(1 − e
−γtg
) (B.31a)
=
1 + η
∆err
εtot
(1 − e
−γtg
)
e
−2nγtg
cos 2nϑ
− η
∆err
εtot
(1 − e
−γtg
). (B.31b)
The expression for the fidelity with the |+⟩ state is, therefore
F(tn) = 1
2
1 − η
∆err
εtot
(1 − e
−γtg
)
(B.32)
+
1
2
1 + η
∆err
εtot
(1 − e
−γtg
)
e
−γtn
cos (2nϑ).
This expression matches Eq. (5.3) with 2ωtg = ϑ, TD = 1/γ and a = −η∆err(1 − e
−γtg )/εtot.
Moreover, the oscillation frequency matches the closed system fidelity oscillation frequency
[Eq. (5.5)] up to the first order (noting that θerr = ω
′
tg).
Solving the general case where γ1 ̸= γ2 is difficult: we cannot apply perturbative techniques directly to bound the deviation of the Bloch vector trajectory in a {γ1 ̸= γ2, ∆err ̸= 0}
environment (where vx exhibits oscillating exponential decay) from a Bloch vector trajectory
in a {γ1 ̸= γ2, ∆err = 0} environment (where vx exhibits exponential decay). We rely on
numerical simulations of the Lindblad equation instead.
145
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Abstract (if available)
Abstract
Superconducting quantum systems, particularly those based on transmon qubits, have become one of the leading platforms for quantum computing research. Essential to the advancement of these systems are the challenges of crosstalk errors, accurate noise modeling, reliable benchmarking of quantum gates, and the development of efficient multi-level quantum processors, or qudits. This dissertation explores these critical areas to enhance the performance and scalability of superconducting quantum processors.
We begin by addressing crosstalk errors, one of the major challenges in superconducting qubits. By exploring the underlying physics and developing strategies using dynamical decoupling, we demonstrate effective methods to suppress qubit-qubit $ZZ$ crosstalk errors. Both experimental results and theoretical analyses, showcasing significant reductions in crosstalk, and thus, enhanced state protection and gate performance in coupled qubit systems. Extending the study to transmon qudits, we introduce a general framework for utilizing dynamical decoupling to suppress low-frequency noise and qudit-qudit crosstalk errors. This work generalizes the $ZZ$ suppression technique for qubits and is supported by experimental results. We then focus on characterization of both low- and high-frequency noise components in transmon qubits. Using hybrid Redfield model, we learn the noise parameters and elucidate the effects of environmental noise on qubit performance. Furthermore, we develop deterministic benchmarking techniques for single qubit gates providing strategies to quantify gate infidelities in terms of coherent and incoherent contributions. Finally, we investigate the role of virtual $Z$ gates in gate compilation strategies within open quantum systems, offering explanations for several previously observed phenomena in dynamical decoupling studies.
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Creator
Tripathi, Vinay
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Core Title
Characterization and suppression of noise in superconducting quantum systems
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Physics
Degree Conferral Date
2024-08
Publication Date
07/30/2024
Defense Date
07/22/2024
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dynamical decoupling,noise characterization,noise suppression,OAI-PMH Harvest,superconducting qubits,transmon
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Lidar, Daniel (
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), Brun, Todd (
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), Levenson-Falk, Eli (
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), Takahashi, Susumu (
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), Zhuang, Quntao (
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tripathi4781@gmail.com,vinaytri@usc.edu
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Tags
dynamical decoupling
noise characterization
noise suppression
superconducting qubits
transmon