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Protecting Hamiltonian-based quantum computation using error suppression and error correction
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Protecting Hamiltonian-based quantum computation using error suppression and error correction
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PROTECTING HAMILTONIAN-BASED QUANTUM COMPUTATION USING ERROR SUPPRESSION AND ERROR CORRECTION by Milad Marvian Mashhad A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) May 2018 Copyright 2018 Milad Marvian Mashhad Acknowledgments I am sincerely grateful to my advisor, Daniel A. Lidar. He gave me the freedom to explore while providing the guidance I needed. I am incredibly thankful for his endless support in all stages of my Ph.D., especially during the events of last year. At USC I had the privilege to learn from and also collaborate with Tameem Albash, Todd A. Brun, Itay Hen, Daniel A. Lidar, Ben Reichardt, and Paolo Zanardi. Also, I have benefited a lot from the excellent courses taught by these great mentors, which all directly or indirectly have influenced my research. I am also very thankful to all the members of Lidar’s Research Group for lively discussions and also their friendship. In particular, I would like to thank Kristen PudenzandZhihuiWangwhohelpedmetosettledowntheveryfirstdaysIarrived at USC. Also, I would like to thank Siddharth Muthukrishnan for all the fruitful discussions and friendship. I have been blessed to have Sepideh on my side during all these years. I am grateful for her love and continued support. I am extremely thankful to my wonderfulfriendswhomIhaveenjoyedtheircompanionship. Thislonglistincludes Sahel, Kavoos, Andisheh, Babak, Keyvan, Nooshin, and MovieFa! I was fortunate that Iman, my brother, joined USC as a postdoc shortly after I started my Ph.D. I am thankful for his support during these years. ii Last but not least, my deepest thanks go to my parents who have always supported and encouraged me my entire life. iii Contents Acknowledgments ii List of Figures vii Abstract ix 1 Introduction 1 1.1 Hamiltonian-based quantum computation . . . . . . . . . . . . . . . 1 1.2 Quantum Error Correction . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Shor’s code . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Stabilizer formalism . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.3 Beyond stabilizer formalism: non-additive codes . . . . . . . 7 1.2.4 Beyond subspace codes: subsystem codes . . . . . . . . . . . 7 1.2.5 Generalized Bacon-Shor codes using Bravyi’s A-matrix con- struction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.6 General Subsystem codes . . . . . . . . . . . . . . . . . . . . 11 1.3 Protecting adiabatic quantum computation using error correction codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 ErrorSuppressionforHamiltonianQuantumComputinginMarko- vian Environments 13 2.1 Error suppression for general master equations in Lindblad form . . 14 2.1.1 General expression for the excitation rate after encoding and error suppression . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.2 The excitation rate scales only polynomially in the system size 20 2.1.3 Theexcitationrateisexponentiallysuppressedbytheenergy penalty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.4 Relation to the JFS work . . . . . . . . . . . . . . . . . . . . 23 2.2 Error suppression for non-Lindblad Markovian master equations . . 24 2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 iv 3 Error suppression for Hamiltonian-based quantum computation using subsystem codes 27 3.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Results in the infinite penalty limit . . . . . . . . . . . . . . . . . . 28 3.2.1 Proof of Theorem 1 and Theorem 2, and derivation of Eqs. (5a) and (5b) . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.2 Boundingtheerrorforfinitepenaltiesassumingblockencoding 33 3.3 Protection using stabilizer codes . . . . . . . . . . . . . . . . . . . . 35 3.3.1 Protection using general subsystem codes . . . . . . . . . . . 36 3.3.2 A simplified sufficient condition . . . . . . . . . . . . . . . . 37 3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4.1 Stabilizer penalty Hamiltonians . . . . . . . . . . . . . . . . 38 3.4.2 Gauge group penalty Hamiltonians . . . . . . . . . . . . . . 39 3.4.3 [[4, 1, 2]] code . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4.4 Encode and Protect the adiabatic swap gate using 2-local interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4.5 Ising chain in a transverse field . . . . . . . . . . . . . . . . 42 3.5 Non-additive codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.6 A semi-distance and its relation to distinguishability in logical sub- systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.6.1 General subsystem case . . . . . . . . . . . . . . . . . . . . . 50 3.6.2 Trace distance of states and operator norm of evolutions . . 52 3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4 Suppression of effective noise in Hamiltonian simulations 54 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.4 Simulation: Ideal case . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.4.1 Commutator method . . . . . . . . . . . . . . . . . . . . . . 59 4.4.2 Simulating one plaquette operator . . . . . . . . . . . . . . . 62 4.4.3 Simulating all the plaquette operators on the grid . . . . . . 64 4.4.4 Simulating all the vertex operators on the grid . . . . . . . . 66 4.4.5 Strength of the effective Hamiltonian . . . . . . . . . . . . . 67 4.4.6 Boundaries and the surface code . . . . . . . . . . . . . . . . 67 4.4.7 π/4-Conjugation Method . . . . . . . . . . . . . . . . . . . . 69 4.4.8 Alternative Connectivity . . . . . . . . . . . . . . . . . . . . 71 4.4.9 Applying the scheme to fault-tolerant holonomic quantum computation in surface codes . . . . . . . . . . . . . . . . . 72 4.5 Effect of noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.5.1 Magnus expansion for piecewise constant Hamiltonian . . . . 76 4.5.2 Protecting against noise at first and second order in δt . . . 77 v 4.5.3 Third order error terms . . . . . . . . . . . . . . . . . . . . . 78 4.5.4 Strength of the simulated Hamiltonian vs. effective noise . . 82 4.5.5 Locality and strength of noise . . . . . . . . . . . . . . . . . 85 4.5.6 DD pulses for a local bath . . . . . . . . . . . . . . . . . . . 89 4.6 Error suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 96 5 Conclusion 99 A Appendix 100 A.1 Proof of the subsystem error detection condition, Eq. (1.31) . . . . 100 Reference List 102 vi List of Figures 3.1 Numerically computed gap of the penalty Hamiltonian for the pro- tected Ising chain in a transverse field, as a function of the number of qubits, given the scaling relation gap∼ 1/(N + 1). . . . . . . . . 46 4.1 Interacting qubits on a grid. Blue lines represent XX interactions between qubits (circles). There is an interaction between nearest neighbor and next-nearest neighbor qubits. For the toric Hamilto- nian, the goal is to simulate X ⊗4 on the shaded areas and Z ⊗4 on the light areas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Depiction of H x = P 1≤i<j≤4 X i X j for four qubits in a plaquette. . . 61 4.3 Generating plaquette operators. (a) The location of Z pulses is shown in blue. (b) The remaining XX interactions after (a) are shown in red. Their sum defines H temp . (c) Labelling of qubits according to Fig. 4.2. (d) The result after applying pulses as is Sec. 4.4.2 to (c). This achieves the first goal of generating plaquette operators in the green areas. . . . . . . . . . . . . . . . . . . . . . 64 vii 4.4 Generating H a on the grid. (a) Evolving under H temp [Fig. 4.3(b)], we apply Z pulses on the blue qubits at times δt and 2δt. (b) The effectiveHamiltonianattime 2δt. (c)H a isgeneratedbyconjugating the Hamiltonian in Fig. 4.4(b) with Hadamard pulses. The effective Hamiltonian becomes patterns of Z 1 Z 2 +X 3 X 4 , depicted as green coupled pairs (ZZ) and black coupled pairs (XX). . . . . . . . . . 65 4.5 GeneratingH b andH c onthegrid. (a)EvolvingunderH temp [Fig.4.3(b)], we apply Z pulses on the blue qubits at times δt and 2δt. (b) The effective Hamiltonian at time 2δt. (c) H b is generated by conju- gating the Hamiltonian in Fig. 4.5(b) with S pulses on the orange qubits. TheeffectiveHamiltonianbecomespatternsofY 1 X 4 +X 2 X 3 , depicted as orange-black coupled pairs (YX) and black coupled pairs (XX). (d)H c is generated by conjugating the Hamiltonian in Fig. 4.5(b) with S pulses on the orange qubits. Now the effective Hamiltonian becomes patterns of X 1 X 4 +Y 2 X 3 , depicted as black coupled pairs (XX) and orange-black coupled pairs (YX). . . . . . 66 4.6 Labeling of qubits connected to a vertex . . . . . . . . . . . . . . . 71 4.7 Timing of the pulses: combining a DD scheme withN DD pulses and a simulation with N sim steps. . . . . . . . . . . . . . . . . . . . . . . 85 viii Abstract In this dissertation, we present general conditions for quantum error suppres- sion for Hamiltonian-based quantum computation using subsystem codes. This involves encoding the Hamiltonian performing the computation using an error detecting subsystem code and the addition of a penalty term that commutes with the encoded Hamiltonian. We illustrate the power of subsystem-based error sup- pression with several examples of two-local constructions for protection against local errors, which circumvent an earlier no-go theorem about two-local commut- ing Hamiltonians. We also discuss the generalization of the quantum error suppression results of Jordan, Farhi, and Shor to arbitrary Markovian dynamics. In this setting we show that it is possible to suppress the initial decay out of the encoded ground state with an energy penalty strength that grows only logarithmically in the system size, at a fixed temperature. Finally we discuss how to simulate high-weight Hamiltonians. Such a simu- lation can convert local noise on the original Hamiltonian into undesirable non- local noise on the simulated Hamiltonian. We show how starting from two-local Hamiltonian in the presence of non-Markovian noise, a desired computation can be simulated as well as protected using fast pulses, while maintaining an energy gap against the errors created in the process. ix Chapter 1 Introduction 1.1 Hamiltonian-based quantum computation Adiabatic quantum computing (AQC) [1] is a model that can achieve univer- sality [2, 3, 4, 5, 6, 7] and appears promising for near future large scale realization (for a review see Ref. [8]). In AQC, the computation is performed using a time- dependent HamiltonianH that evolves slowly from an initial HamiltonianH i with a known and easily preparable ground state, to a final Hamiltonian H f whose ground state is unknown and encodes the desired result. H(t) =a(t)H i +b(t)H f 0≤t≤T (1.1) where a(T ) =b(0) = 0 and a(0) =b(T ) = 1. Theadiabatictheoremguaranteesthatthefinalstatewillbeclosetotheground state of the final Hamiltonian if the evolution is sufficiently slow [9]. Unfortunately, AQC lacks a theory of fault tolerance, unlike all other universal models of quantum computation [10]. The passive schemes we consider in this thesis suppress errors during AQC, while not necessarily providing fault tolerance for arbitrary long computations. Holonomic quantum computing (HQC) is another universal model, wherein quantum gates are performed as holonomies (non-Abelian geometric phases) in the degenerate ground eigensubspace of the system Hamiltonian [11, 12, 13]. The 1 effects of decoherence and its mitigation in HQC have also been the subject of intensive study [14, 15, 16, 17, 18, 19, 20]. While unlike AQC, a theory of fault- tolerance has been developed for HQC [21, 22], it is of interest to develop less demanding alternatives, such as the error suppression strategy we consider in this thesis. AQC and HQC are examples of Hamiltonian-based quantum computation. More generally we are considering any quantum computation that can be per- formed by continuously changing a Hamiltonian. The problem of protecting and also simulating general Hamiltonian-based quantum computations is the main topic of this thesis. An essential ingredient of the proposed schemes is quantum error correction and detection. Here is a very brief introduction to quantum error correction. 1.2 Quantum Error Correction Here is a very short review of quantum error correction tools we use in this thesis. (See [23, 24, 10] for more detailed introductions.) 1.2.1 Shor’s code Shor’s code is one of the earliest and also simplest to explain quantum codes. It encodes one logical qubit into nine physical qubits and can correct any arbitrary single-qubit error. To understand how it works we can consider the following quantum state: |000i+|111i √ 2 . We note that this state is +1 eigenvector of both Z 1 Z 2 and Z 2 Z 3 (parity operators). Therefore if we measure this state using Z 1 Z 2 or Z 2 Z 3 , the state would not change. Similar to classical codes, the location of any single bit flip error on this state can be determined using the results of these two 2 measurements. This is also true for |000i−|111i √ 2 : it is +1 eigenvector of the parity operators, and measuring the parity operators would reveal the location of any single bit flip error but would not distinguish it from |000i+|111i √ 2 . Using both of these quantum states the following encoding α|0i +β|1i→α |000i +|111i √ 2 +β |000i−|111i √ 2 (1.2) can protect the information against bit-flip errors. By measuring Z 1 Z 2 and Z 2 Z 3 , we can detect the location of any bit flip error on the encoded state and then correct it by applying a bit flip gate on that location. But any single phase flip error (Pauli Z) can convert a code state to another codestate. The interesting observation is that we can use this building block, which can correct bit flip errors, repeatedly to also correct for any single phase flip error. To be more explicit we can use the following encoding: α|0i +β|1i→α| ¯ 0i +β| ¯ 1i (1.3) with | ¯ 0i = |000i +|111i √ 2 ⊗ |000i +|111i √ 2 ⊗ |000i +|111i √ 2 (1.4) | ¯ 1i = |000i−|111i √ 2 ⊗ |000i−|111i √ 2 ⊗ |000i−|111i √ 2 (1.5) As before, measuringZ i Z i+1 withi∈{1, 2, 4, 5, 7, 8} would specify the location of a bit flip error. In addition to this, measuringX 1 ...X 6 andX 4 ...X 9 would specify the building block on which the phase flip error has occurred. Any phase flip error can be corrected by applying a phase flip gate on any qubit of that building block. Correcting forX andZ error would automatically correct for a Pauli Y error, as we 3 have Y =iZX. This process not only corrects the discrete set of Pauli errors but also can correct any 1-local rotation. This is evident by noting that{I,X,Y,Z} span the space of any 1-local rotation. The described set of measurements would collapse any state with a general 1-local rotation error to a code state with one of the single Pauli operators as the error. As we have already discussed, these errors can be corrected. To sum up, the two-dimensional codespace would be the +1 eigensubspace of the following eight operators: Z i Z i+1 i∈{1, 2, 4, 5, 7, 8} (1.6) X 1 X 2 X 3 X 4 X 5 X 6 ,X 4 X 5 X 6 X 7 X 8 X 9 and with measuring these operators and applying appropriate gates any single error can be corrected. 1.2.2 Stabilizer formalism The idea of using a subspace stabilized by a set of Pauli operators as the codespacecanbegeneralized[25]. Ingeneral, astabilizercodecanbedefinedasthe subspace stabilized by an Abelian groupS =hS 1 ,...,S s i of Pauli operators, with −I / ∈S, where{S i } s i=1 are the group generators. The projector onto the codespace isP C = Q s i=1 I+S i 2 . Propertiesofthestabilizergeneratorswoulddeterminetheerror detecting and error correcting power of code. For example, a stabilizer code can detectasetofPaulioperators{E i }ifforeveryE i thereexistsatleastonegenerator of the code that anti-commutes with E i : E j is a detectable error ⇐⇒ ∃S i ∈S s.t.{E j ,S i } = 0 (1.7) 4 This condition guarantees that an erred state E j | ¯ φi is detectable by the measure- ment S i as it is its eigenvector with eigenvalue−1: S i (E j | ¯ φi) =−E j S i | ¯ φi =−(E j | ¯ φi) (1.8) where we have used S i | ¯ φi =| ¯ φi asS i is a stabilizer of the code. Similarly a set of Pauli operators{E i } can be distinguished from each other and therefore corrected if ∀E j ,E j 0∈{E j },∃S i ∈S s.t.{E † j E j 0,S i } = 0 (1.9) ⇒{E j } are correctable errors (1.10) To see this we consider E j | ¯ φ 1 i and E j 0| ¯ φ 2 i as two erred codestates. If the above condition holds then we would have: h ¯ φ 1 |E † j E j 0| ¯ φ 2 i =h ¯ φ 1 |E † j E j 0S| ¯ φ 2 i = −h ¯ φ 1 |SE † j E j 0| ¯ φ 2 i = −h ¯ φ 1 |E † j E j 0| ¯ φ 2 i = 0 (1.11) so the two errors are distinguishable and also correctable. This is a sufficient condition but not a necessary condition. For example, as we discussed for the Shor’s code the effect of errors Z 1 and Z 2 could not be distinguished but they could be corrected. It turns out that the only necessary condition is that Z 1 Z 2 (and all the other combinations of single errors) should have the same expected value for all the states in the codespace. For general subspace codes, not necessarily in the stabilizer formalism, one can ask what are the sufficient and necessary conditions that a quantum codeC, and the error channelE generated by set of errors{E j } should satisfy, so that the 5 information encoded can be recovered using a valid quantum operationR. In other words, what are the sufficient and necessary conditions to have (R◦E)(ρ)∝ ρ. The Knill-Laflamme subspace condition states that [23]: (R◦E)(ρ)∝ρ ⇐⇒ P C E † i E j P C =c ij P c (1.12) with c being a Hermitian matrix. Logical operators of the code are the operators that preserve the codespace but can affect it nontrivially. For example, in the Shor’s code, we can define ¯ X =Z 1 Z 4 Z 7 (1.13) ¯ Z =X 1 X 2 X 3 . (1.14) We can check the following properties: ¯ X| ¯ 0i =| ¯ 1i, ¯ X 2 =I, ¯ Z| ¯ 0i =| ¯ 0i, ¯ Z| ¯ 1i =−| ¯ 1i, ¯ Z 2 =I. (1.15) So these operators generate the same algebra on the codespace, as the algebra that single Pauli X and Z would generate on a single qubit. One can show that for n-qubit stabilizer codes with s generators, these logical operators always show up in conjugate pairs: ( ¯ X 1 , ¯ Z 1 ), ( ¯ X 2 , ¯ Z 2 ),.., ( ¯ X k , ¯ Z k ) with k =n−s. Operators with different indices all commute with each other. ¯ X and ¯ Z with the same indices anti-commute. 6 1.2.3 Beyond stabilizer formalism: non-additive codes The stabilizer formalism is extremely intuitive and useful not only to design codes but also to connect them to different well studied areas of physics. But non- additive codes (also known as non-stabilizer codes) can achieve higher rates (ratio of the number of encoded to physical qubits) than stabilizer codes [26, 27, 28, 29]. For example, using 5 physical qubits to detect any single-qubit error stabilizer codes can encode at most 2 qubits, but using a non-additive code one can encode up to log 2 6 qubits [26]. 1.2.4 Beyond subspace codes: subsystem codes Stabilizer subsystem codes [30] are of particular interest. Intuitively, one can think of such codes as subspace stabilizer codes [25] where some logical qubits and the corresponding logical operators are ignored. To induce a subsystem structure we define logical operatorsL and gauge operatorsA as Pauli operators that leave the codespace invariant, and also demand that the three setsS,L, andA mutually commute. The generators ofL andA can be organized in canonical conjugate pairs: the set of bare logical operatorsL = {Z 1 ,X 1 ,...,Z k ,X k } that preserve the code space and act trivially on the gauge qubits [31], and the set of gauge operatorsA ={Z 0 1 ,X 0 1 ,...,Z 0 r ,X 0 r }, where for A,B∈{X,Z} or A,B∈{X 0 ,Z 0 } we have [A i ,B j ] = 0 if i6= j, and{X i ,Z i } = 0. The gauge group is defined asG = hS 1 ,...,S s ,Z 0 1 ,X 0 1 ,...,Z 0 r ,X 0 r i, and is non-Abelian. A Pauli error E j is detectable iff it anti-commutes with at least one of the stabilizer generators [30], or equivalently iff P C E j P C = 0 [since (I +S i )(I−S i ) = 0∀i]. 7 Bacon-Shor code Consider the following [[9, 5, 3]] subspace code (a [[n,k,d]] code is a quantum code that encodes k logical qubits into n physical qubits and can correct up to b d−1 2 c errors): ¯ X 1 =X ⊗9 , ¯ Z 1 =Z ⊗9 , (1.16) ¯ X 2 =X 1 X 2 , ¯ Z 2 =Z 1 Z 4 , (1.17) ¯ X 3 =X 2 X 3 , ¯ Z 3 =Z 3 Z 6 , (1.18) ¯ X 4 =X 7 X 8 , ¯ Z 4 =Z 4 Z 7 , (1.19) ¯ X 5 =X 8 X 9 , ¯ Z 5 =Z 6 Z 9 , (1.20) with the following stabilizer generators: S 1 =Z 1 ...Z 6 , (1.21) S 2 =Z 4 ...Z 9 , (1.22) S 3 =X 1 X 2 X 4 X 5 X 7 X 8 , (1.23) S 4 =X 2 X 3 X 5 X 6 X 8 X 9 (1.24) The dimension of the codespace is 2 9 /2 4 = 2 5 , and we have five pairs of logical operators. The Bacon-Shor code [32] is a [[9, 1, 3]] code that uses only the first pair of logi- cal operators of the [[9, 5, 3]] code and ignores the other 4 pairs. These pairs induce a tensor product structure in the code space: C =A⊗B. A, the gauge subsystem, is 2 4 dimensional andB, the information subsystem that we are interested in, is 2 dimensional. 8 The benefit of not using these logical operators is in the simplification of the decoding scheme. Rather than measuring the weight 6 stabilizers, one can combine the result of measurements using weight 2 gauge operators to detect errors. Simi- larly, having the gauge freedom makes the process of error correction simpler. This simplifications makes subsystem codes suitable for fault-tolerant schemes [33]. Under geometrical locality constraint for generators of the code, subsystem stabilizer codes are more powerful than subspace stabilizer codes. For example, in Ref. [34] it is shown that parameters of any [[n,k,d]] subspace stabilizer code with local stabilizer generators on a 2D lattice satisfy: kd 2 =O(n). (1.25) This bound indicates that: requiring the distance of the code to grow with the lattice size d = O( √ n) limits the number of logical qubits to a constant number (k =O(1)). In Ref. [31], Bravyi has shown that using subsystem stabilizer codes, with the same locality constraint on the gauge operators, more logical qubits (k = O( √ n)) can be encoded. The corresponding bound for subsystem codes would be kd =O(n). (1.26) which is saturated. For a 2D code, it is known that the distance of a stabilizer code, whether subspace or subsystem code, cannot grow faster than √ n [35]: d =O( √ n). (1.27) 9 Toprovetheseresults, BravyihasintroducedgeneralizedBacon-Shorcodesand has shown that it can outperform the subspace codes. A review of his construction follows. 1.2.5 Generalized Bacon-Shor codes using Bravyi’s A- matrix construction HerewebrieflyreviewtheconstructionandpropertiesofthegeneralizedBacon- Shor codes introduced in Ref. [31]. Let A be a square binary matrix. Following Bravyi, we associate a subsystem code to this matrix. Each non-zero element of A represents a physical qubit, so the number of physical qubits, n, is just the Hamming weight|A|. The gauge group of the code,G, is generated by: 1. Gauge generators X a X b , for each pair of qubits a,b in the same row, 2. Gauge generators Z a Z b , for each pair of qubits a,b in the same column. Note that, by definition, the generators can be overcomplete. It suffices to retain generators for consecutive pairs. Given the gauge group, in principle all properties of the code can be derived using: S =G∩C(G) (stabilizer group) , (1.28a) L =C(S)/G (dressed logical operators) , (1.28b) d = min P∈L |P| (code distance) . (1.28c) 10 But for this specific construction, all these properties can be directly related to properties of theA matrix over the binary fieldF 2 . As shown in Ref. [31] (Theorem 2 there) the properties of the code are: [[n,k,d]] = [[|A|, rank(A), min{d row ,d col }]] (1.29) The rank ofA overF 2 is equal to half the number of logical operators of the code, k. The code distance is the minimum weight of the non-zero vectors in the row space and the non-zero vectors in the column space. We present the A matrices and corresponding gauge groups for the examples given in the main text. We use the notation X i,j to represent the X operator acting on the qubit located in the i th row and j th column (in the main text we used one index to label the qubits for simplicity). Example: [[4, 1, 2]] code: A = 1 1 1 1 , so G =hX 1,1 X 1,2 ,X 2,1 X 2,2 ,Z 1,1 Z 2,1 ,Z 1,2 Z 2,2 i . (1.30) This is the code for encoding one qubit. 1.2.6 General Subsystem codes The idea of protecting a subsystem in the codespace could be generalized beyond the stabilizer formalism. Assume that the system’s Hilbert space can be decomposed asH S =C⊕C ⊥ , whereC =A⊗B. The channel (completely positive map)E ={E j } is detectable on the “information subsystem" B if: ∀E j ∃G j :P C E j P C =P C G j ⊗I B P C , (1.31) 11 where I B is the identity on B andP C denotes the projector onto C. Here A plays the role of a “gauge subsystem"; the G j operators are arbitrary and do not affect the information stored in subsystem B. For completeness, we provide a proof of the sufficiency of the error detection condition in the Appendix (see Ref. [36, 37, 38] for necessary and sufficient condi- tions and proofs for correctable errors on a subsystem.) 1.3 Protecting adiabatic quantum computation using error correction codes The first scheme to suppress the detrimental effect of the bath on AQC [39, 40, 41, 42, 43, 44, 45, 46, 47] was proposed by Jordan, Farhi, and Shor (JFS) [48]. Inthisscheme,astabilizersubspacecodethatcandetecttheerrorsintroduced by the system-bath interaction Hamiltonian is chosen, and the system Hamilto- nian is encoded using the logical operators of the same code. Adding a penalty Hamiltonian breaks the induced degeneracy and stabilizes the computation in the code-subspace, while any excitation out of this subspace is penalized. The short time performance of this scheme was investigated for a specific Markovian model in Ref. [48] and also for a general non-Markovian bath in [49], where numerical sim- ulations were used to extend the study beyond the short time limit. For a general but local non-Markovian bath in the regime of weak coupling to the bath, it was shown that, modulo a unitary rotation in the codespace due to the Lamb shift, the same scheme can result in an exponential suppression of decoherence [50]. Variants of the JFS scheme tailored to current experimental quantum annealing [51], where encoding of the initial Hamiltonian is not possible, have also been proposed and studied [52, 53, 54]. This method is the subject of the next two chapters. 12 Chapter 2 Error Suppression for Hamiltonian Quantum Computing in Markovian Environments Here we extend the original result of Jordan, Farhi, and Shor (JFS) to general Markovian environments, not necessarily in Lindblad form. We show that the main conclusion of the original JFS study holds under these general circumstances: assuming a physically reasonable bath model, it is possible to suppress the initial decay out of the encoded ground state with an energy penalty strength that grows only logarithmically in the system size, at a fixed temperature. The reason we are interested in Markovian models, despite the fact that general results of a similar nature have already been established for non-Markovian models [49, 50], is that Markovian models are special: not only are they widely used [55], decay in these models (e.g., of the purity) is always exponential [56]. This means that they preclude any use of ultra-short time recurrence effects that soften deco- herence. In particular, error suppression techniques such as dynamical decoupling [57, 58, 59] or the Zeno effect [60] (shown to be formally equivalent to the JFS scheme [61]) are ineffective for Markovian models. In this sense, error suppression 13 for Hamiltonian computation in the presence of a Markovian environment is more challenging than in the non-Markovian case. 2.1 Error suppression for general master equa- tions in Lindblad form Assuming a time-dependent system Hamiltonian H(t), a general bath Hamil- tonian H B , and an interaction Hamiltonian H SB = P α A α ⊗B α , an adiabatic Markovian master equation in Lindblad form [62, 63] can be derived [64]: ˙ ρ =−i[H(t) +H LS (t),ρ] +D(t)[ρ] , (2.1) whereH LS (t) is the Lamb shift, andD(t) denotes the dissipative, i.e., non-unitary part (see Ref. [47] for a concise summary and definitions), and we set ~ ≡ 1 throughout. Henceforth we mostly suppress the time-dependence of the various termsfornotationalsimplicity, butitimportanttorememberthatallourquantities are explicitly time-dependent. Consider the spectral decomposition H = X l≥0 l Π l , (2.2) 14 i.e., Π l denotes the projection onto the (possibly degenerate) H-eigensubspace with energy l . The eigenprojectors are orthogonal: Π l Π l 0 = δ ll 0Π l . Defining A α (ω) = P l 0− l =ω Π l A α Π l 0, the dissipator becomes: D[ρ] = X ω X αβ γ αβ (ω)[A β (ω)ρA † α (ω) − 1 2 {A † α (ω)A β (ω),ρ}] , (2.3) where the matrix of decay rates γ αβ (ω) = Z ∞ −∞ dt e iωt hB αβ (t)i =γ ∗ βα (ω) , (2.4) is the Fourier transform of the bath correlation function hB αβ (t)i = Tr[ρ B e −iH B t B α e iH B t B β ] , (2.5) where ρ B is the initial state of the bath. From now on we assume that the system-bath coupling exhibits a local struc- ture, in the sense that the system operatorsA α inH SB = P α A α ⊗B α arek-local, with k a constant that is independent of the number of qubits n. This guarantees that the interaction Hamiltonian can be expressed in terms of a number of terms that is polynomial in n. 15 2.1.1 General expression for the excitation rate after encoding and error suppression Assumethatthesystemisinitiallypreparedinthe(possiblydegenerate)ground subspace of the Hamiltonian H, with energy 0 , i.e., ρ(0) = Π 0 ρ(0)Π 0 . We are interested in the initial excitation rate out of the ground subspace: R≡∂ t Tr[Π 0 ρ]| t'0 , (2.6) First we note that this expression can be simplified. The excitation rate can be re-written as R = Tr[ ˙ Π 0 ρ(0)] + Tr[Π 0 ˙ ρ(0)] and we observe that the first term vanishes. Toseethis,weusethefactthattheinitialstatesatisfiesρ(0) = Π 0 ρ(0)Π 0 , so that Tr[ ˙ Π 0 ρ(0)] = Tr[Π 0 ˙ Π 0 Π 0 ρ(0)]. Now, differentiating the identity Π 2 0 = Π 0 yields Π 0 ˙ Π 0 + ˙ Π 0 Π 0 = ˙ Π 0 =⇒ ˙ Π 0 Π 0 = Π ⊥ 0 ˙ Π 0 =⇒ Π 0 ˙ Π 0 Π 0 = 0 , (2.7) where Π ⊥ 0 =I− Π 0 . So the initial excitation rate becomes: R = Tr[Π 0 ˙ ρ(0)] . (2.8) In AQC one is usually interested in the case that the ground state of the initial Hamiltonian is non-degenerate and the initial state is pure. In this case the excitation rate R is proportional to the initial purity decay, with purity defined as Trρ 2 . In HQC the initial state belongs to a degenerate subspace. In Eq. (2.8) 16 we do not assume that the initial state is pure, and later consider the special case when it is [see below Eq. (2.18)]. It is not hard to show that the dissipative part yields: Tr{Π 0 D[ρ(0)]} =− X αβ X l6=0 γ αβ ( 0 − l )Tr[ρ(0)A † α Π l A β ] . (2.9) Proof: We explicitly compute the terms that need to be summed. We will use the fact that the initial state is in Π 0 : ρ(0) = Π 0 ρ(0)Π 0 . First: X ω γ αβ (ω)Tr[Π 0 A β (ω)ρ(0)A † α (ω)] = X ω γ αβ (ω)Tr[Π 0 A β (ω)Π 0 ρ(0)Π 0 A † α (ω)] = X ω γ αβ (ω) X l 0− l =ω X l 000− l 00=ω Tr[Π 0 (Π l A β Π l 0)Π 0 ρ(0)Π 0 (Π l 000A † α Π l 00)] =γ αβ (0)Tr[Π 0 A β ρ(0)A † α ] =γ αβ (0)Tr[ρ(0)A † α Π 0 A β ] (2.10) 17 Second, similarly: X ω γ αβ (ω)Tr[Π 0 A † α (ω)A β (ω)ρ(0)] = X ω γ αβ (ω) X l 000− l 00=ω X l 0− l =ω Tr[Π 0 (Π l 000A † α Π l 00)(Π l A β Π l 0)Π 0 ρ(0)] = X l γ αβ ( 0 − l )Tr[Π 0 A † α Π l A β ρ(0)] = X l γ αβ ( 0 − l )Tr[ρ(0)A † α Π l A β ] . (2.11) We also note that Tr[Π 0 A † α (ω)A β (ω)ρ(0)] = Tr[Π 0 ρ(0)A † α (ω)A β (ω)], and so both terms of the anti-commutator produce the same result. Adding these terms accord- ing to the dissipator, Eq. (2.3), yields Eq. (2.9). We now choose a code C that can detect all the errors (system operators) A α in the system-bath Hamiltonian [65]: ∀α : P C A α P C = 0 , (2.12) where P C projects onto the code space. We encode the system Hamiltonian using the logical operators of this code, and add a penalty Hamiltonian that has the codespace as its ground-subspace. Such a Hamiltonian can be constructed by summing the stabilizer generators of the code [48]. Thus, H is the sum of an encoded computational Hamiltonian H S and a penalty Hamiltonian H p : H(t) =H S (t) +η p H p , (2.13) where the dimensionless quantity η p > 0 quantifies the strength of the energy penalty, and by construction [H S (t),H p ] = 0. This allows us to choose the Π l ’s as 18 the simultaneous eigenprojectors ofH S andH p , and write the eigenvalues ofH(t) as l (t) =ω l (t) +η p ξ l , (2.14) where ω l (t) and ξ l are, respectively, the eigenvalues of H S (t) and H p . Let us assume thatH p is chosen so that its ground subspace is the codespace, defined by the projection operator P C = X l∈C Π l , (2.15) and that the initial state belongs to the (now definitely degenerate) ground sub- space of H(t), i.e., again ρ(0) = Π 0 ρ(0)Π 0 . Since Tr(X[Y,X]) = 0 for any pair of operators X,Y, the unitary part−i[H +H LS ,ρ(t)] of the master equation (2.1) does not contribute to the initial excitation rate. 1 Moreover, because of the error detection properties of the code [Eq. (2.12)] we have ∀l∈C : Π 0 A α Π l = 0 . (2.16) Using the master equation (2.1) and Eq. (2.9) we thus have: R =− X αβ X l∈C ⊥ γ αβ ( 0 − l )Tr[ρ(0)A † α Π l A β ] . (2.17) Note that the matrix γ is positive semi-definite and can be diagonalized by a unitary U: P α 0 β 0(U † ) αα 0γ α 0 β 0U β 0 β = δ αβ γ α with positive γ α (the eigenvalues of γ). Introducing new Lindblad operators F α 0(ω) via A α (ω) = P α 0U αα 0F α 0(ω) into Eq. (2.17) we have the following general expression for the excitation rate: 1 The effect of the Lamb shift on the codespace is captured by other measures such as the fidelity (see, e.g., Ref. [50]). 19 R =− X α X l∈C ⊥ γ α ( 0 − l )Tr[ρ(0)F † α Π l F α ] . (2.18) Alternatively, when the initial state is a pure state|ψ 0 i we can define the excitation rate as R 0 ≡ Tr[|ψ 0 ihψ 0 | ˙ ρ(0)]|, but it is easy to check that as a result of the encoding we have R = R 0 . This means that the encoding also suppresses the errors induced by the system-bath interaction in the ground subspace, which are logical errors for HQC. 2.1.2 The excitation rate scales only polynomially in the system size Despite the fact that the sum over l∈C ⊥ involves exponentially many terms, the excitation rate scales only polynomially in the system size. To see this, we first define γ max ≡ max l∈C ⊥ ,α γ α ( 0 − l ) . (2.19) Using the spectral decomposition ρ(0) = P i λ i |iihi|, it is clear that Tr[ρ(0)F † α Π l F α ] = P i λ i kΠ l F α |iik 2 ≥ 0. Therefore, using Eq. (2.15), the exci- tation rate satisfies the bound |R|≤γ max X α X l∈C ⊥ Tr[ρ(0)F † α Π l F α ] =γ max X α Tr[ρ(0)F † α F α ]− X α Tr[ρ(0)F † α P C F α ] ! ≤γ max X α Tr[ρ(0)F † α F α ] (2.20) 20 None of the terms in the last sum depends on the number of qubitsn. The number of terms itself can increase at most polynomially inn, due to the sum overα [both the explicit one in Eq. (2.20) and also the implicit one in F α = P α 0(U † ) αα 0A α 0]. This proves that the excitation rate grows at most polynomially in n. 2.1.3 The excitation rate is exponentially suppressed by the energy penalty Next, let us show that for reasonable models of the bath the excitation rate is exponentially suppressed with increasing energy penalty η p . If the bath is in ther- mal equilibrium at inverse temperature β, then under rather general conditions (analyticity of the bath correlation function in a strip) the matrix of decay rates satisfies the quantum detailed balance, or Kubo-Martin-Schwinger (KMS) condi- tion [66]: γ αβ (−ω) = e −βω γ βα (ω). The diagonalization used above then implies that the eigenvalues of the γ matrix also satisfy the KMS condition, i.e., γ α (−ω) =e −βω γ α (ω) . (2.21) Let Π l denote an eigenprojector ofH with energy l = Tr[Π l H]. It follows from Eq. (2.13) and [H S ,H p ] = 0 that these are simultaneous eigenstates ofH S andH p aswell. LetusassumethatH p hasagroundstategapg = min l∈C ⊥ Tr[(Π l −Π 0 )H p ]. We have∀l∈C ⊥ : l − 0 = Tr[Π l (H S +η p H p )]− Tr[Π 0 (H S +η p H p )] = Tr[(Π l − Π 0 )H S ] +η p Tr[(Π l − Π 0 )H p ] ≥η p g . (2.22) 21 When H p is a sum of commuting terms, as is true for the stabilizer construction we consider here, the gap g is guaranteed to be a constant [67]. 2 Now, using the KMS condition (2.21), we have: γ α ( 0 − l ) =e −β( l − 0 ) γ α ( l − 0 ) ≤e −βgηp γ α ( l − 0 ) . (2.23) It follows that γ max ≤e −βgηp max l∈C ⊥ ,α γ α ( l − 0 ) , (2.24) and thus, the bound on|R| depends on max l∈C ⊥ ,α γ α ( l − 0 ) = max l∈C ⊥ ,α γ α (ω l + η p ξ l − 0 ), where we used Eq. (2.14). To ensure a non-trivial bound on|R| this quantity has to be finite, which is a natural assumption. For example, for a bath satisfying an Ohmic-like relation of the form γ(ω) = μω k e −ω/ωc for ω > 0, where ω c is a finite cutoff frequency, the maximum value of γ(ω) is μ(kω c ) k e −k . Even if this is not the case (e.g., in the quantum optical master equation γ(ω)∝ ω 3 for sufficiently large ω [55]) it is reasonable to assume that the system itself imposes a high-frequency cutoff, i.e., that max l∈C ⊥{ω l ,ξ l }<∞. 3 We also assume that γ(ω) is a polynomial function (or any subexponential function in ω) for ω > 0; this too is an assumption that is compatible with all commonly used bath models [55]. 2 For H p that is a sum of non-commuting terms, e.g., when it is chosen as a sum of gauge group elements [68, 69], g may decrease with increasing system size. Even this case remains interesting if the gap of H S decreases faster in the system size than g [69]. 3 This is certainly reasonable for condensed matter systems, where the finite number density naturally imposes a high-frequency cutoff, such as a Debye frequency. 22 Combining this with Eq. (2.20), we have: |R|≤ exp(−βgη p )poly(η p )poly(n) . (2.25) It follows that the excitation rate is exponentially suppressed as the penalty strength η p is increased. In other words, by using stabilizer error detecting codes (constant g), for Markovian models with a thermal bath that satisfy our assump- tionsabove, tokeeptheinitialexcitationrate(orpuritydecay)outofthecodespace constant while the system size n increases, one only needs to increase the strength of energy penalty, η p , logarithmically in n, at any fixed inverse temperature β. The flatter the initial purity decay, the longer the adiabatic or holonomic quantum computation will proceed in the ground state. 2.1.4 Relation to the JFS work In the pioneering JFS work [48], a very similar result to Eq. (2.25) was already established, under somewhat less general conditions. Rather than dealing with a general Markovian master equation, they assumed a particular system of spins weakly coupled to a photon bath and a pure initial state. They then provided the lowest-weight possible subspace stabilizer codes for detecting 1-local and 2-local noise compatible with the error suppression scheme. Here, following and generaliz- ing the JFS proof technique and providing all the necessary details, we generalized the suppression result to arbitrary Markovian master equation in Lindblad form and arbitrary stabilizer subspace error detection codes, while allowing for a degen- erate initial state. We now proceed to establish the result even more generally, for Markovian master equations derived without the rotating wave approximation. 23 2.2 Error suppression for non-Lindblad Marko- vian master equations The derivation from first principles of a master equation with a dissipator in Lindblad form [Eq. (2.3)] requires several approximations [55]. Prominent among these is the rotating wave approximation (RWA), whose validity has often been questioned [70, 71, 72, 73, 74, 75, 76, 77]. Ref. [64] presented a derivation not only of the Lindblad-form adiabatic master equation (2.1), which required the use of the RWA and guarantees complete positivity, but also a so-called double sided adiabatic master equation (DSAME), derived without the RWA (the Lindblad form follows from the latter master equation after using the RWA). In this section we show that the DSAME also exhibits the same suppression of purity decay or the excitation rate out of the ground space. Thus, the suppression effect does not depend on the RWA. The DSAME has the following form: ˙ ρ =−i[H(t),ρ] + ˜ D[ρ] (2.26a) ˜ D[ρ] = X αβ X ll 0 Γ αβ (ω ll 0)[Π l A β Π l 0ρ,A α ] +h.c. , (2.26b) where h.c. denotes the Hermitian conjugate. Note the non-Lindblad form of the dissipator ˜ D and the absence of an explicit Lamb shift term (such a term, i.e., a Hermitian part, can nevertheless be separated from ˜ D). Here the frequencies are 24 the time-dependent Bohr frequencies of the system: ω ll 0(t) = l 0(t)− l (t), where H(t)| l (t)i = l (t)| l (t)i, and Γ αβ (ω) = Z ∞ 0 dt e iωt hB αβ (t)i (2.27) is the one-sided Fourier transform of the bath correlation function. The double- sided and one-sided Fourier transforms are related via Γ αβ (ω) = 1 2 γ αβ (ω) +iS αβ (ω) , (2.28) where S αβ (ω) = S ∗ βα (ω) is the remaining Cauchy principal value (see, e.g., Ref. [64]). We again calculate the excitation rate Tr[Π 0 ˙ ρ(0)] for a state initialized in the ground subspace of the Hamiltonian H, with energy 0 , i.e., ρ(0) = Π 0 ρ(0)Π 0 . First: Tr{Π 0 X ll 0 Γ αβ (ω ll 0)[Π l A β Π l 0ρ(0),A α ]} = X ll 0 Γ αβ (ω ll 0) (Tr{Π 0 Π l A β Π l 0[Π 0 ρ(0)Π 0 ]A α } (2.29) −Tr{Π 0 A α Π l A β Π l 0[Π 0 ρ(0)Π 0 ]}) = Γ αβ (0)Tr[ρ(0)A α Π 0 A β ]− X l Γ αβ (ω 0l )Tr[ρ(0)A α Π l A β ] . Next, after subtracting the l = 0 term, we are left just with the sum over l6= 0. Thus, using Eq. (2.28): Tr{Π 0 ˜ D[ρ(0)]} = (2.30) − X αβ X l6=0 [ 1 2 γ αβ (ω 0l ) +iS αβ (ω 0l )]Tr[ρ(0)A α Π l A β ] +h.c. 25 Accounting for the fact that without loss of generality we can always choose the system operators A α to be Hermitian, and that the sum over all α and β allows us to interchange the order of summation, the imaginary part vanishes after sum- mation with the Hermitian conjugate, and we are left exactly with Eq. (2.9) for the Lindblad form. The unitary part has no effect in the DSAME either (i.e., Tr{ρ(0)[H(0),ρ(0)]} = 0). Therefore, the same conclusions as reported in the pre- vious section for master equations in Lindblad form, follow for the DSAME about the excitation rate out of the ground subspace of the Hamiltonian. 2.3 Conclusion We have extended the JFS result [48], that it suffices to increase the energy penalty logarithmically with system size in order to protect AQC against excita- tions out of the ground state, to general Markovian dynamics and mixed states. We have also pointed out that these results apply to HQC, and shown that the same results continue to hold even if the master equation is not in Lindblad form, i.e., without assuming the rotating wave approximation. These results only con- cern the initial excitation rate. A natural next generalization of these results is to subsystem codes and longer evolutions. 26 Chapter 3 Error suppression for Hamiltonian-based quantum computation using subsystem codes By construction, JFS encoding necessitates greater than two-body interactions, which can make its implementation challenging. An important open question is whether there exist quantum error suppression schemes that involve only two- body interactions. However, even for the special case of quantum memory, invok- ing penalty terms but no encoding, two-body commuting Hamiltonians cannot in general provide suppression [78]. This no-go result left open the possibility that non-commuting two-local Hamiltonians might nevertheless suffice for quan- tum error suppression. Examples based on (generalized) Bacon-Shor codes [32] were recently given in Ref. [68] to show that this is the case for penalty terms and encoded single-qubit operations, and for some encoded two-qubit interactions, but without general conditions or performance bounds. Here we show how general subsystem codes can be used for quantum error suppression. Using an exact, non-perturbative approach, we find conditions that penalty Hamiltonians should satisfy to guarantee complete error suppression in the infinite energy penalty limit. We derive performance bounds for finite energy 27 penalties. Our formulation accounts for stabilizer subspace and subsystem codes as special cases, including the examples of Refs. [48, 49, 68]. We provide sev- eral examples where our approach results in encoded Hamiltonians and penalty terms that involve purely two-body interactions [79]. These examples include the swap gate used in adiabatic gate teleportation [80], and the Ising chain in a trans- verse field frequently encountered in adiabatic quantum computation and quantum annealing. 3.1 Setting We wish to protect a quantum computation performed by a system with Hamil- tonianH S (t) against the system-bath interaction V = P j E j ⊗B j , to a bath with HamiltonianH B . WeconstructtheencodedsystemHamiltonian,H S (t), byreplac- ing every operator in H S (t) by the corresponding logical operators of a subsystem code [36, 37, 38]. The strategy for protecting the computation performed byH S (t) is to add a penalty Hamiltonian E p H p , chosen so that [H S (t),H p ] = 0 in order to prevent interference with the computation [48]. As the energy penalty E p is increased, errors should become more suppressed. 3.2 Results in the infinite penalty limit We now state our main results, in the form of two related theorems that give sufficient conditions for complete error suppression in the large E p limit. These results incorporate both those for general stabilizer penalty Hamiltonians intro- duced in [48, 49] and the subsystem penalty Hamiltonian examples introduced in [68]. They are also related to a dynamical decoupling approach for protecting adiabatic quantum computation [57] via a formal equivalence found in Ref. [61]. 28 LetU 0 ,U p ,U V , andU W be the unitary evolutions generated byH 0 =H S +H B , E p H p , H V = H 0 +E p H p +V, and H W = H 0 +E p H p +W, respectively. As will become clear later,W will play the role of the suppressed version ofV. We assume thatkVk,kWk<∞, wherek·kdenotesanyunitarilyinvariantnorm[81]. LetP be an arbitrary projection operator and letH p = P a λ a Π a be the eigendecomposition of the penalty term. Theorem 1. Set W =cI (c∈R, I is the identity operator) and assume that [H S ,P] = [H S ,H p ] = 0 , (3.1a) X a Π a V Π a P =cP . (3.1b) Then lim Ep→∞ kU V (T )P−U W (T )Pk = 0 , (3.2) where U W (T ) =e −icT U 0 (T )U p (T ). Theorem1statesthatintheinfinitepenaltylimitandoverthesupportofP,the evolution generated by the total system-bath HamiltonianH V is indistinguishable (up to a global phase) from the decoupled evolution generated byH 0 +E p H p . The conditions in Eq. (3.1a) ensure compatibility of the subspace defined by P and of the type of penalty HamiltonianH p with the given encoded HamiltonianH S . The condition in Eq. (3.1b) ensures the absence of a term that cannot be removed by the penalty. 29 Theorem 2. Set W = P a∈I Π a V Π a , whereI is some index set. Assume that in addition to Eq. (3.1a) also P = P a∈I Π a . Then Eq. (3.2) holds again, with U W (T ) =T exp Z T 0 (H 0 (t) +E p H p +W )dt (3.3) (T denotes time-ordering). Theorem 2 is similar to Theorem 1, except that it allows for a more general target evolution operator U W (T ). As discussed below, Theorem 1 is suitable for stabilizer subsystem codes, while Theorem 2 is suitable for general subsystem codes. Proof sketch.—Both Theorems 1 and 2 establish the desired decoupling result, andshowthatinprincipleitispossibletocompletelyprotectHamiltonianquantum computation against coupling to the bath. To prove them we define K(t) = Z t 0 U † p (τ)(V−W )U p (τ)dτP , (3.4) and derive the following bounds in the next section: kU V (T )P−U W (T )Pk≤kK(T )k (3.5a) +T sup t k[K(t),H 0 (t)]k +T (kVk +kWk) sup t kK(t)k kK(t)k≤ 2 E p X a6=a 0 kV−Wk |λ a −λ a 0| . (3.5b) Theorems 1 and 2 follow in the large E p limit, since in this limitkK(T )k→ 0, andk[K(t),H 0 (t)]k≤ 2kK(t)kkH 0 k. An error bound for finite E p follows directly from Eq. (3.5b) (for related results see Refs. [49, 82]). While a tighter bound may not be possible without introducing additional assumptions, we note that for a Markovian bath in a thermal state, it is possible to show that the excitation rate out of the code space is exponentially suppressed as a function ofE p , andE p need 30 only grow logarithmically in the system size to achieve a constant excitation rate, assuming the gap of H p is constant [48, 83]. 3.2.1 Proof of Theorem 1 and Theorem 2, and derivation of Eqs. (5a) and (5b) Proof. Let H V = H 0 + E p H p + V and H W = H 0 + E p H p + W, with corre- sponding unitary evolutions U V and U W , respectively. We would like to bound kU V (T )P−U W (T )Pk, where T is the final time. Below we suppress the time dependence for notational simplicity unless it is essential. OnlyV,W, andH p are time-independent. Going to the interaction picture defined byH 0 +E p H p , we let ˜ V =U † 0 U † p VU p U 0 and ˜ W = U † 0 U † p WU p U 0 . We denote the corresponding unitary evolutions by ˜ U V and ˜ U W . Now, for any unitarily invariant norm [81], in particular the operator norm: kU V P−U W Pk = U 0 U p ˜ U V P−U 0 U p ˜ U W P = ˜ U V P− ˜ U W P = ( ˜ U † V ˜ U W −I)P = Z T 0 d[ ˜ U † V (τ) ˜ U W (τ)] dτ dτP = Z T 0 ˜ U † V [ ˜ V− ˜ W ] ˜ U W dτP = Z T 0 ˜ U † V U † 0 U † p (V−W )U p U 0 ˜ U W dτP . (3.6a) We assume that [P, ˜ W ] = 0 and verify this condition below. Then: kU V P−U W Pk = Z T 0 ˜ U † V U † 0 U † p (V−W )U p PU 0 ˜ U W dτP . (3.7) 31 Recall that K(t) = R t 0 U † p (τ)(V−W )U p (τ)dτP [Eq. (4)]. Using integration by parts, 1 we have: U V P−U W P = Z T 0 ˜ U † V U † 0 dK dt U 0 ˜ U W dtP (3.8a) = ˜ U † V (T )U † 0 (T )K(T )U 0 (T ) ˜ U W (T )P (3.8b) − Z T 0 (i ˜ U † V ˜ V )U † 0 KU 0 ˜ U W dtP (3.8c) − Z T 0 ˜ U † V U † 0 KU 0 (−i ˜ W ˜ U W )dtP (3.8d) − i Z T 0 ˜ U † V U † 0 [H 0 ,K]U 0 ˜ U W dtP . (3.8e) Using the triangle inequality, unitary invariance, and submultiplicativity we thus have the upper bound: kU V (T )P−U W (T )Pk ≤ kK(T )k + (3.9) T (kVk +kWk) sup t kK(t)k +T sup t k[K(t),H 0 (t)]k , which is Eq. (5a). Next, let us substitute the eigendecomposition H p = P a λ a Π a into K(t): K(t) = Z t 0 X a6=a 0 e i(λa−λ a 0)Epτ Π a (V−W )Π a 0dτP + Z t 0 X a Π a (V−W )Π a dτP .(3.10) The last term vanishes under the assumptions of Theorems 1 and 2. To see this, note that Theorem 1 corresponds to the case where W =cI, so also ˜ W =cI and [P, ˜ W ] = 0 is satisfied. Additionally, P a Π a (V−W )Π a P becomes P a Π a V Π a P = cP, which is satisfied by assumption in Theorem 1. 1 The version we need here is the following: let A, B and C be operators. Then (ABC) 0 = aBC +AbC +ABc where a = A 0 etc. By the fundamental theorem of calculus R s r (ABC) 0 = ABC| s r = R s r (aBC +AbC +ABc). We set A = ˜ U † V U † 0 , b = dK dt , and C =U 0 ˜ U W . 32 Similarly, Theorem2correspondstothecasewhereW = P a∈I Π a V Π a andP = P a∈I Π a , so again [P, ˜ W ] = 0 is satisfied. In this case too, P a Π a (V−W )Π a P = 0. Carrying out the remaining integral in Eq. (3.10), we thus obtain: kK(t)k = X a6=a 0 e i(λa−λ a 0)Ept − 1 (λ a −λ a 0)E p Π a (V−W )Π a 0P (3.11a) ≤ 1 E p X a6=a 0 2 |λ a −λ a 0| kΠ a (V−W )Π a 0Pk (3.11b) ≤ 2 E p X a6=a 0 kV−Wk |λ a −λ a 0| Ep→∞ −→ 0 , (3.11c) which confirms Eq. (5b). 3.2.2 Bounding the error for finite penalties assuming block encoding In this section we show that the bound resulting from Eq. (5b) for the finite penalty case can be tightened, and in particular does not depend extensively on the bath size viakV−Wk if the bath couples locally to the system. For simplicity we assume a block encoding with one logical qubit per block. The same method can be used when several logical qubits are encoded in each block. Let us expand V and W as V = P n i=1 P j∈i v i j and W = P n i=1 P j∈i w i j . The first sum is over the logical qubits, the second is over terms with support on a logical qubit. 2 Accordingly we define: K i j (t) = Z t 0 U † p (τ)(v i j −w i j )U p (τ)Pdτ . (3.12) 2 E.g., assuming local terms act on system sites, and encoding each logical qubit into n c physical qubits, j in the second sum runs from 1 to 3n c for each i. 33 With the penalty Hamiltonian represented as a sum over logical qubits, H p = P n i=1 h i p , we have: K i j (t) = Z t 0 e ih i p τ (v i j −w i j )e −ih i p τ Pdτ . (3.13) Thus, using the general bound (3.10): kU V (T )P−U W (T )Pk ≤ X i X j∈i K i j (T ) + (3.14) T (kVk +kWk) sup t K i j (t) +T sup t [K i j (t),H 0 ] . The only part of H 0 (t) = H S (t) +H B that appears in the bound is that which does not commute with K i j : [K i j (t),H 0 (t)] = Z t 0 e ih i p τ [v j −w j ,H 0 (t)]e −ih i p τ Pdτ = Z t 0 e ih i p τ [v i j −w i j ,h i S,j (t) +h i B,j ]e −ih i p τ Pdτ = [K i j (t),h i S,j (t) +h i B,j ] , (3.15a) where H S (t) = P i P j∈i h i S,j (t) and H B = P i P j∈i h i B,j , with h i S,j (t) representing the part of the system that has nontrivial support on sitej in blocki of the system and h i B,j representing the bath Hamiltonian part that has nontrivial support on the bath part of the interaction Hamiltonian, here corresponding tov i j −w i j . Using the triangle inequality and unitary invariance of the norm we have: kU V (T )P−U W (T )Pk ≤ X i X j∈i kK i j (T )k +T (kVk +kWk) sup t kK i j (t)k + 2T sup t kK i j (t)k(kh i S,j (τ)k +kh i B,j k) . (3.16) 34 This involves the local bath component h i B,j , as opposed to depending extensively onkH B k. The only remaining ingredient is: K i j (t) = Z t 0 e +ih i p τ (v i j −w i j )e −ih i p τ p i dτ , (3.17) and we repeat the steps leading from Eq. (3.10) to Eq. (3.11). Namely, using the eigendecomposition h i p = P a e i a π i a and assuming that π i a (v i j −w i j )π i a p i = 0 ∀a,i,j (3.18) holds [a generalization of Eq. (8) allowing forW6= 0] so that the term correspond- ing to the second integral in Eq. (3.10) vanishes, the bound becomes: K i j (t) ≤ X a6=a 0 π i a (v i j −w i j )π i a 0p i |e +iEp(e i a −e i a 0 )t − 1| |E p (e i a −e i a 0)| ≤ 2 E p X a6=a 0 kv i j −w i j k |e i a −e i a 0| .(3.19) Thus, the bound for the finite penalty case only depends locally on the coupling to the bath, viakv i j −w i j k. As expected, the bound remains extensive in the system size, via the sum over i in Eq. (3.16). 3.3 Protection using stabilizer codes Tosatisfythecondition [H S ,H p ] = 0inEq.(3.1a)wemaychooseH p asalinear combination of elements of the gauge groupG (not necessarily the generators) [68, 84], H p = X i α i g i , g i ∈G , |α i |≤ 1 ,∀i . (3.20) 35 To satisfy the condition [H S ,P] = 0 we may chooseP = P a∈I Π a . Equation (3.1b) then becomes Π a V Π a = cΠ a ∀a∈I, a condition that is already satisfied with c = 0 for a stabilizer error detecting code (for which P C VP C = 0) if the support of P is in the codespace (i.e., PP C = P C P = P). This is true, in particular, ifI contains just the ground subspace of H p . We may thus state the following corollary of Theorem 1: For H p chosen as in Eq. (3.20), the joint system-bath evolution completely decouples in the large penalty limit for initial states in the ground subspace of H p , with this subspace itself being a subspace of the codespace. Note that the difference between the subspace and subsystem case manifests itself in the appearance ofU p (T ) in Eq. (3.2). If the penalty Hamiltonian consisted of only stabilizer terms [i.e., g i ∈S ∀i in Eq. (3.20)], the penalty Hamiltonian would at most change the overall phase of states in the codespace. But here, as the elements of penalty Hamiltonian can be any element of the gauge group, U p can have a nontrivial effect on states in C. Nevertheless, as the gauge operators commute with the logical operators of the code, this unitary does not change the result of a measurement of the logical subsystem. 3.3.1 Protection using general subsystem codes Choose a code C with projector P C such that the error-detection condi- tion (1.31) is satisfied for all the error operators{E j } inV = P j E j ⊗B j . Assume that the penalty is chosen so that [H S ,H p ] = 0 in Eq. (3.1a) holds, and set P = P C in Theorem 2 (thus also the condition [H S ,P] = 0 holds). Then Π a V Π a = P j (Π a G j ⊗I B Π a )⊗B j ∀a∈I, sothatW = P a∈I P j (Π a G j ⊗I B Π a )⊗B j , with trivial action (I B ) on the information subsystem B. The unitary U W (T ) [Eq. (3.3)] appearing in Theorem 2 thus has a non-trivial effect on B only via the H 0 (t) term, as desired. 36 Block encoding.—A useful simplification results when the logical qubits can be partitioned into n separate blocks. In this case the total penalty Hamiltonian becomes H p = P n i=1 h i p , where h i p = P j α i j g i j denotes the penalty Hamiltonian on logical qubit i, with g i j ∈G, and [h i p ,h j p ] = 0 for i6= j. The code space projector becomes P C =⊗ n i=1 p i , where p i is the projector onto the code space of the ith logical qubit. We may also partition the system-bath interaction according to the logicalqubitsitactson: V = P n i=1 v i (notethatwedonotassumethat [v i ,v j ] = 0). Clearly,K(t) can also be expressed as a sum over blocks, as can inequality (3.5a). Using the eigendecompositionh i p = P a e i a π i a , condition (3.1b) can then be replaced by π i a v i π i a p i =c i p i ∀a,i . (3.21) Using the block encoding structure, later we tighten the error bound resulting from Eq. (3.5b). We show, in particular, that the bound is extensive in the system size and depends only on the bath degrees of freedom that couple locally to the system, so that the bound is not extensive in the bath size. 3.3.2 A simplified sufficient condition To check whether Theorem 1 applies one can simply find the eigendecomposi- tion of h i p and check if Eq. (3.21) holds for a given system-bath interaction and choice of code space. Instead, we next identify conditions that are less general but are easier to check. We assume that the interaction Hamiltonian has the 1-local form V = P i v i , where v i = P j σ i j ⊗B i j and σ i j is an arbitrary non-identity Pauli operator acting on qubit j in block i. From now on we drop the block superscript 37 for notational simplicity. Furthermore, we choose a penalty term that satisfies [h p ,p] = 0 given a code block projector p, which implies [π a ,p] = 0∀a. A sufficient condition for Eq. (3.21), and hence for Theorem 1, is then the following: Condition1. h p pandσ j h p σ j pdonotshareaneigenvalueforanyσ j inthesupport of p. To see that this is a sufficient condition, we note that π a p and σ j π a σ j p are both projectors, corresponding to the same eigenvalue e a of h p and σ j h p σ j . If both projectors are nonzero then there exists at least one (nonzero) eigenvector for each ofh p p andσ j h p σ j p with eigenvaluee a , in contradiction to our condition. So, the stated condition guarantees that for any eigenvalue e a we have either π a p = 0 or σ j π a σ j p = 0. Thus,∀a: 0 = (σ j π a σ j p)(π a p) = σ j π a σ j ppπ a = σ j π a σ j pπ a = σ j (π a σ j π a p), so that,∀a: π a σ j π a p = 0, which implies Eq. (3.21) (with c i = 0∀i). 3.4 Examples We now consider a number of interesting cases, and show that Condition 1 holds, thus guaranteeing error suppression via Theorem 1. 3.4.1 Stabilizer penalty Hamiltonians As in Ref. [48], let h p = X i α i S i (3.22) with S i ∈S, α i 6= 0 and p = p c . Clearly [h p ,p] = 0. Let us define a ij = 0 or 1 if [S i ,σ j ] = 0 or{S i ,σ j } = 0, respectively. In the support of p (i.e., in the 38 code space) h p p = ( P α i )p, so the eigenvalue of h p there equals P i α i , while the eigenvalue of σ j h p σ j p there equals P i α i (−1) a ij . Condition 1 thus requires∀j: P i α i 6= P i α i (−1) a ij . When all α i have the same sign this becomes the familiar error detection condition, that every σ j anticommutes with at least one of the terms in the sum of stabilizers. The penalty Hamiltonian considered in Ref. [49] corresponds to h p =I−p, so that [h p ,p] = 0 holds. Condition 1 is also satisfied in this case since sinceh p p = 0, while σ j h p σ j p = p−σ j pσ j p = p (where we used the error detection condition pσ j p = 0), so in the support of p the eigenvalues are, respectively, 0 and 1. 3.4.2 Gauge group penalty Hamiltonians AfamilyofgeneralizedBacon-Shorcodescanbeidentifiedwithabinarymatrix A, which fully characterizes all the code properties [31]. E.g., each nonzero element of A corresponding to a qubit on a planar grid, and two ones in a row (column) of the matrix correspond to an XX (ZZ) generator acting on the corresponding qubits. As pointed out in Ref. [68], because of the locality of the generators of these codes, they are promising candidates for use in error suppression schemes. We present several examples for suppressing local errors that originate from this construction. 3.4.3 [[4, 1, 2]] code The [[4, 1, 2]] code was proposed in Ref. [68] to overcome the aforementioned no-go theorem for error suppression using 2-local commuting Hamiltonians [78]. Each qubit is encoded into four qubits using this code (block encoding), so the 39 entire code corresponds to a block diagonalA matrix, with 2×2 blocks of all ones, A = 1 1 1 1 . The stabilizer, gauge and bare logical generators are: S =hS 1 =X ⊗4 ,S 2 =Z ⊗4 i (3.23a) A ={X 0 =X 1 X 2 ,Z 0 =Z 1 Z 3 } (3.23b) L ={X =X 1 X 3 ,Z =Z 1 Z 2 } . (3.23c) ThusG =hS 1 ,S 2 ,X 0 ,Z 0 i =hS 1 X 0 ,S 2 Z 0 ,X 0 ,Z 0 i =hX 3 X 4 ,Z 2 Z 4 ,X 1 X 2 ,Z 1 Z 3 i≡ h{g i } 4 i=1 i, i.e., the generators are 2-local. The penalty Hamiltonian is h p = E p P 4 i=1 g i and again, clearly [h p ,p] = 0. One may check that the eigenvalues of h p p and σ j h p σ j p are 0,±2E p and±2 √ 2E p , respectively. Thus Condition 1 is satisfied. While the penalty Hamiltonian is 2-local, unfortunately the encoding of a 2-local interaction (which is necessary for universal quantum computation), still requires 4-local interactions. Checking Condition 1 for the [[4, 1, 2]] code Recall that S = hX 1 X 2 X 3 X 4 ,Z 1 Z 2 Z 3 Z 4 i (3.24a) G = hX 3 X 4 ,Z 2 Z 4 ,X 1 X 2 ,Z 1 Z 3 i≡h{g i } 4 i=1 i (3.24b) h p = E p 4 X i=1 g i . (3.24c) Firstnotethath p P C = 2E p (g 1 +g 2 )P C . Theeigenvaluesofthismatrixare±2E p √ 2 (note that the spectrum is equivalent to the spectrum of XIII +ZIII, as there exists a unitary transformation between the terms). 40 Next, observe that a single PauliX at any of the four locations commutes with both g 1 ,g 3 and it commutes with one of g 2 ,g 4 and anticommutes with the other one. So, for σ j being X at any location we have σ j h p σ j P C = 2E p g 1 P C . Similarly, for σ j being Z at any location we have σ j h p σ j P C = 2E p g 2 P C . However, if σ j =Y then at any location it commutes with one of g 1 ,g 3 and anti-commutes with the other one, and also commutes with one ofg 2 ,g 4 and anti-commutes with the other one; thus σ j h p σ j P C = 0. Therefore the eigenvalues are either±2E p or 0. 3.4.4 Encode and Protect the adiabatic swap gate using 2-local interactions We show how to encode and protect the adiabatic swap gate introduced in [80] using purely 2-local interactions. This Hamiltonian is one of the key building blocks of a proposal for universal quantum computation using adiabatic gate tele- portation. The Hamiltonian is: H(s) = (1−s)(X b X c +Z b Z c ) +s(X a X b +Z a Z b ). By slowly increasing s from 0 to 1 any state initially prepared on qubit a trans- fers onto qubit c. To encode and protect this Hamiltonian, we use the following [[8, 3, 2]] subsystem code: A = 1 1 0 0 0 1 1 0 0 0 1 1 1 0 0 1 S =hS 1 =X ⊗8 ,S 2 =Z ⊗8 i (3.25a) L ={X 1 =X 1 X 8 ,X 2 =X 1 X 2 X 3 X 8 ,X 3 =X 4 X 5 , Z 1 =Z 1 Z 2 ,Z 2 =Z 3 Z 4 Z 5 Z 6 ,Z 3 =Z 5 Z 6 } (3.25b) G =hX 1 X 2 ,X 3 X 4 ,X 5 X 6 ,X 7 X 8 ,Z 2 Z 3 ,Z 4 Z 5 ,Z 6 Z 7 ,Z 8 Z 1 i 41 The penalty Hamiltonian is the sum of all the gauge group generators g i ∈G, which is manifestly 2-local. One can check that Condition 1 is satisfied for this Hamiltonian, and so we obtain the desired protection. The encoded Hamiltonian becomes: H(s) = (1−s)(X 2 X 3 +Z 2 Z 3 ) +s(X 1 X 2 +Z 1 Z 2 ) = (1−s)(X 6 X 7 +Z 3 Z 4 ) +s(X 2 X 3 +Z 7 Z 8 ), (3.26) where in the second line we used the fact that X 2 X 3 = S 1 X 6 X 7 and Z 1 Z 2 = S 2 Z 7 Z 8 are equivalent logical operators. Thus, the encoded Hamiltonian remains 2-local. 3.4.5 Ising chain in a transverse field Our next example, an open Ising chain in a transverse field, does not involve block encoding: H S (s) = (1−s) N X i=1 X i +s N−1 X i=1 J i Z i Z i+1 . (3.27) This Hamiltonian appears frequently in adiabatic quantum optimization. The goal is again to provide encoding and error suppression using only 2-local Hamiltonians. 42 We use the following A-matrix: A = 1 1 0 0... 0 0 0 1 1 0... 0 0 0 0 1 1... 0 0 . . . . . . 0 0 0 0... 1 1 1 0 0 0... 0 1 a×a . (3.28) This corresponds to a [[|A|, rank(A), min(d row ,d col )]] = [[2a,a− 1, 2]] code, a = N + 1, and the generators ofG are: 1≤∀i≤a− 1 : X i,i X i,i+1 (3.29a) X a,1 X a,a (3.29b) 1≤∀i≤a− 1 : Z i,i+1 Z i+1,i+1 (3.29c) Z 1,1 Z a,1 . (3.29d) The logical operators can be chosen to be 1≤∀i≤a− 1 : X i =X i,i+1 X i+1,i+1 (3.30a) 1≤∀i≤a− 1 : Z i = i Y j=1 Z j,j Z j,j+1 . (3.30b) Thus 2≤∀i≤a− 1 :Z i−1 Z i =Z i,i Z i,i+1 , a 2-local physical interaction. Substituting for logical operators we have: H S (s) = (1− s) P N i=1 X i + s P N−1 i=1 J i Z i Z i+1 = (1−s) P N i=1 X 2i X 2i+1 +s P N−1 i=1 J i Z 2i+1 Z 2i+2 . 43 In summary: H p =− N+1 X i=1 X 2i−1 X 2i + N X i=1 Z 2i Z 2i+1 +Z 1 Z 2N+2 H S (s) = (1−s) N X i=1 X 2i X 2i+1 +s N−1 X i=1 J i Z 2i+1 Z 2i+2 . (3.31) We have verified numerically that the ground subspace of H p is a subspace of the codespace, which as we showed above is sufficient for error suppression in the stabilizer case. We also find numerically that the minimum gap ofH p decreases as 1/(N + 1), so thatE p should grow withN to maintain the protection obtained in this case as the system size increases, since this gap separates the logical ground subspace from the undecodable excited states. While in general this is undesirable, itiscompatiblewithexampleswhereH S (andhencealsoH S )exhibitsmorerapidly closing gaps for certain choices of the couplings{J i } (e.g., an exponentially small gap [85]). Details for the protected Ising chain in a transverse field The spectrum of the penalty Hamiltonian given in the main text, H p = −( P N+1 i=1 X 2i−1 X 2i + P N i=1 Z 2i Z 2i+1 +Z 2N+2 Z 1 ) [acting on 2(N + 1) qubits] can be found by considering the spectrum of the following Hamiltonian, with s x and s z set to±1: −( N X i=1 X i +s x N Y i=1 X i +Z 1 + N−1 X i=1 Z i Z i+1 +s z Z N ) . (3.32) The minimum energy of this Hamiltonian appears in the s x =s z = +1 sector and so the ground subspace is in the codespace, as required. Figure 3.1 shows how 44 the gap of the penalty Hamiltonian changes as a function of the number of qubits. This gap is proportional to (N + 1) −1 . To arrive at Eq. (3.32) we define 1≤∀i≤N : ˆ X i =X 2i−1 X 2i (3.33a) ˆ Z 1 =Z 2N+2 Z 1 (3.33b) 2≤∀i≤N : ˆ Z i =Z 2N+2 Z 1 Π i−1 j=1 Z 2j Z 2j+1 . (3.33c) Note that these operators are similar in form but different in indexing to the logical operators in Eq. (3.29). With the stabilizers S x (S z ) being the product of all X (Z) Pauli operators on all 2(N + 1) qubits, the original penalty Hamiltonian can be rewritten as −( N X i=1 ˆ X i +S x N Y i=1 ˆ X i + ˆ Z 1 + N−1 X i=1 ˆ Z i ˆ Z i+1 +S Z ˆ Z N ) (3.34) Now, as these two stabilizers commute with all the terms, we can diagonalize the Hamiltonian in the corresponding±1 sectors separately. As these hatted operators satisfy the same algebra as single Pauli operators the spectrum will be the same. 3.5 Non-additive codes Theorems1and2allowustogobeyondtheframeworkofRef.[49]andexamples of Ref. [68], and employ non-additive codes (also known as non-stabilizer codes) to encode and protect evolutions 3 . Non-additive codes can achieve higher rates (ratio of the number of encoded to physical qubits) than stabilizer codes [26, 27, 28, 29]. 3 Ref.[49]provedalessgeneralversionofTheorem1, whereEq.(3.1b)isreplacedbyP C VP C = 0; this excludes non-additive codes. Ref. [68] used certain stabilizer subsystem codes but did not consider non-additive codes. 45 1 1.5 2 2.5 3 ln(N+1) -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 ln(gap) Figure3.1: NumericallycomputedgapofthepenaltyHamiltonianfortheprotected Ising chain in a transverse field, as a function of the number of qubits, given the scaling relation gap∼ 1/(N + 1). For example, using 5 physical qubits to detect any single-qubit error stabilizer codes can encode at most 2 qubits, but using a non-additive code one can encode up to log 2 6 qubits [26]. The encoding procedure is straightforward. Choosing a subspace code C, one can expand the system Hamiltonian in a basis{|ii} and then replace each basis vector in the expansion with the corresponding code state {| ¯ ii}. One possible choice of a penalty Hamiltonian is E p H p , where H p =−P C and P C = P i∈C | ¯ iih ¯ i|. Theorem 1 guarantees that with this choice, starting from an initial state in the codespace, leakage out of the codespace is suppressed in the large E p limit, and the desired system Hamiltonian is implemented in the codespace with a higher rate than what could be achieved using stabilizer codes. Moreover, Theorem 2 allows using non-additive subsystem codes such as the codes introduced in Ref. [29]. 46 3.6 A semi-distance and its relation to distin- guishability in logical subsystems We argued that in the infinite penalty case measurement outcomes do not change despite the fact that the state evolves under U p (T )U 0 (T ) rather than U 0 (T ). To see this, consider replacing a generalized measurement M with measurement operators {M m } after evolving the initial state ρ(0) subject to H S (t), by the encoded version M = {M m } after evolving the encoded ini- tial state ρ(0) using the encoded Hamiltonian H S (t). Using Theorem 1, U V (T )ρ(0)U † V (T ) Ep→∞ −→ U † p (T )U † 0 (T )ρ(0)U 0 (T )U p (T )≡ ρ(T ), so the probability of outcome m is Tr[ρ(T )M † m M m ] = Tr[U † 0 (T )ρ(0)U 0 (T )M † m M m ], where we used [U p ,M m ] = 0 since the gauge operators commute with the logical operators. Thus, measurement outcomes do not change despite the fact that the state evolves under U p (T )U 0 (T ) rather than U 0 (T ). To make this more precise and to relate it to the bounds derived for the finite penalty case, one can define a semi-distance that quantifies state distinguishability using measurements restricted to the logical subsystem. First we limit our discussion to the stabilizer subsystem setting; later we show how this can be generalized to general subsystem codes. Let us denote the unitary that implements the encoding in the stabilizer for- malism by U enc . This unitary maps an initial state, consisting of s ancillas in the |0i state, some arbitrary state on r gauge qubits|φi, and a k-qubit information- carrying state |ψi, to an encoded state over n = s + r + k physical qubits: |ψi = U enc (|0i ⊗s |φi|ψi). This unitary also converts the single Pauli operators onn qubits to the correspondings generators of the stabilizer,r gauge generators, and 2k logical operators of the code [86]. 47 We are interested in bounding the distance between the following two states: ρ full =U V (T )ρU † V (T ) , ρ ideal =U 0 (T )ρU † 0 (T ) , (3.35) where ρ =|ψihψ| is an initial state prepared in the support of P. The state ρ full represents the evolution of this initial state under the effect of the total system- bath Hamiltonian, whileρ ideal is the state resulting from the ideal, fully decoupled evolution subject purely to H 0 . However, the gauge degrees of freedom need to be removed before a meaningful distance can be computed, since the state of the gauge subsystem is completely arbitrary. To account for this we need to define an appropriate distance measure, namely: d(ρ,σ) = 1 2 Tr gauge U † enc ρU enc − Tr gauge U † enc σU enc 1 , (3.36) which is the trace distance between states after tracing out the r gauge qubits, and so it quantifies state distinguishability after a measurement of the logical subsystem. The unencoding transformation U † enc is inserted in this definition in order to ensure a tensor-product structure between the gauge qubits and the rest. We are thus interested in bounding d(ρ full ,ρ ideal ), and proceed to do so. For our purposes it suffices to consider unitary operators that only act on the gauge degrees of freedom. Thus, we define a unitary family U G ={u g :U † enc u g U enc =u gauge g 0 } , (3.37) i.e., all the unitary operatorsu g whose effect on the codespace after unencoding is a unitary that has support only on the gauge qubits. 48 For elements of U G we have Tr gauge U † enc u g σu † g U enc = Tr gauge U † enc u g U enc U † enc σU † enc U enc u † g U enc = Tr gauge u gauge g 0 U † enc σU † enc u gauge† g 0 = Tr gauge U † enc σU enc , (3.38) and so ∀u g ∈U G : d(ρ,σ) =d(ρ,u g σu † g ) . (3.39) From this we conclude that: d(ρ,σ) = 1 2 min ug∈U G Tr gauge U † enc ρU enc − Tr gauge U † enc u g σu † g U enc 1 (3.40) ≤ 1 2 min ug∈U G U † enc ρU enc −U † enc u g σu † g U enc 1 = 1 2 min ug∈U G ρ−u g σu † g 1 . (3.41) Now note thatU p (T )∈U G , as it is a unitary generated by a linear combination of elements of the gauge group. Therefore we have d(ρ full ,ρ ideal ) ≤ 1 2 min ug∈U G ρ full −u g ρ ideal u † g 1 (3.42a) ≤ 1 2 min ug∈U G U V ρU † V −u g U 0 ρU † 0 u † g 1 (3.42b) ≤ 1 2 U V ρU † V −U 0 U p ρU † 0 U † p 1 , (3.42c) 49 Theorem 1 involves the operator normkU V P−U 0 U p Pk, so another step is required in order to connect the bound ond(ρ full ,ρ ideal ) with Theorem 1. This can be easily done using 1 2 U V ρU † V −U W ρU † W 1 ≤kU V P−U W Pk , (3.43) where it is assumed that the initial stateρ is in the support ofP (for completeness the bound is derived in the next section). Thus, we have obtained the desired bound as: d(ρ full ,ρ ideal )≤ U V P−e −ic U 0 U p P . (3.44) Of course this distance also goes to zero in the large penalty limit. 3.6.1 General subsystem case To generalize the distance we have defined to general subsystem codes we note that, unlike in the case of stabilizer codes, U † enc only produces the tensor structure in the codespace (H =C⊕C ⊥ , whereC =A⊗B.) In this case we can modify the defined distance to trace out the gauge degrees of freedom only in the codespace: d(ρ,σ) = 1 2 Tr gauge U † enc P C ρP C U enc − Tr gauge U † enc P C σP C U enc 1 . (3.45) As in Theorem 2, here we also assume that [P,H p ] = 0. All the steps described for the stabilizer case above can be repeated with small modifications. We define the unitary family U G ={u g : [u g ,P C ] = 0, U † enc u g U enc =u gauge g 0 } , (3.46) 50 i.e., all the unitary operators u g that both commute with P C and whose effect on the codespace after unencoding is a unitary that has support only on the gauge subsystem. Using this one can easily show that ∀u g ∈U G : d(ρ,σ) =d(ρ,u g σu † g ) . (3.47) From this we conclude that: d(ρ,σ) ≤ 1 2 min ug∈U G P C ρP C −u g P C σP C u † g 1 . (3.48) Thus, we have: d(ρ full ,ρ ideal ) = 1 2 Tr gauge U † enc P C U V ρU † V P C U enc − Tr gauge U † enc P C U 0 ρU † 0 P C U enc 1 ≤ 1 2 min ug∈U G P C U V ρU † V P C −u g P C U 0 ρU † 0 P C u † g 1 (3.49a) ≤ 1 2 P C U V ρU † V P C −P C U 0 U p ρU † 0 U † p P C 1 (3.49b) ≤ 1 2 U V ρU † V −U 0 U p ρU † 0 U † p 1 (3.49c) ≤ U V P−e −ic U 0 U p P , (3.49d) where in the last step we assumed that the initial state ρ is in the support of P. Weremarkthataswearejustcomparingthepartofthestatesinthecodespace, the distanced defined in Eq. (3.45) can vanish for two different states that are both outside the codespace. However, when one of the states ρ C is in the codespace, while the other state σ is not necessarily in the codespace, having d(ρ C ,σ)≤ guarantees that Tr(P C σ) ≥ 1− . Thus, if the distance d between a general state and a state in the codespace is small, it is guaranteed that σ is close to the codespace. 51 We can relate this to the problem of optimally distinguishing states, with general prior probabilities: For two general states with nonzero support in the codespace, we can rewrite the distance definition as d(ρ,σ) = 1 2 kpρ 0 −qσ 0 k 1 , (3.50) where we have defined p = TrP C ρ and ρ 0 = Tr gauge U † enc P C ρP C U enc /p, and likewise, q = TrP C σ and σ 0 = Tr gauge U † enc P C σP C U enc /q. This quantity has an operational meaning connected to the minimum error of distinguishing states ρ 0 and σ 0 with prior probabilities p and q: p error = 1 2 (1−kpρ 0 −qσ 0 k 1 ) [87]. 3.6.2 Trace distance of states and operator norm of evolu- tions For pure states, there exists a tight relation between fidelity and trace distance of states [23]: D(|ψ 1 ihψ 1 |,|ψ 2 ihψ 2 |) = q 1−|hψ 1 |ψ 2 i| 2 ≤k|ψ 1 i−|ψ 2 ik . (3.51) Thus, if|ψ 1 i =U V |ψ 0 i and|ψ 2 i =U W |ψ 0 i for some pure state|ψ 0 i, andP|ψ 0 i = |ψ 0 i we have: D(|ψ 1 ihψ 1 |,|ψ 2 ihψ 2 |) ≤ k(U V −U W )|ψ 0 ik = k(U V −U W )P|ψ 0 ik≤kU V P−U W Pk , (3.52) where we used the definition of the operator norm in the last inequality. 52 The same method extends to the case in which the initial state is a mixed state, by decomposing the initial state into an ensemble of pure states: ρ = P i p i |ψ i ihψ i |, where{p i } is a probability distribution. If this state is in the support of P, then: D(U V ρU † V ,U W ρU † W ) = 1 2 X i p i (U V |ψ i ihψ i |U † V −U W |ψ i ihψ i |U † W ) 1 (3.53a) ≤ 1 2 X i p i (U V |ψ i ihψ i |U † V −U W |ψ i ihψ i |U † W ) 1 (3.53b) = X i p i D(U V |ψ i ihψ i |U † V ,U W |ψ i ihψ i |U † W ) (3.53c) ≤ sup |ψ i i D(U V |ψ i ihψ i |U † V ,U W |ψ i ihψ i |U † W ) (3.53d) ≤kU V P−U W Pk , (3.53e) where we used Eq. (3.52) in Eq. (3.53e). 3.7 Conclusions We have presented conditions guaranteeing error suppression for Hamiltonian quantum computation using general subsystem error detecting codes, along with conditions that the corresponding penalty Hamiltonians should satisfy, and perfor- mance bounds that improve monotonically with increasing energy penalty. Stabi- lizer subsystem codes are more flexible than stabilizer subspace codes when there are constraints on the spatial locality of the generators of the code [31]. This allowed us to use these codes to present examples of fully 2-local encoded Hamil- tonian quantum information processing with error suppression. This should hope- fully pave the way towards a similar result for protected universal Hamiltonian quantum computation. 53 Chapter 4 Suppression of effective noise in Hamiltonian simulations 4.1 Introduction The theory of quantum fault tolerance ensures that quantum computers can operate reliably in the presence of decoherence and noise [88, 10]. The quantum accuracy threshold theorem, in various incarnations [89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99], guarantees that arbitrary long, reliable quantum computation is achievable if the error rate is below a threshold. Although in principle the exis- tence of this threshold means that scalable quantum computation is possible, in practice the value of this threshold and the required overhead are very important, as achieving them in experiments remains extremely challenging. One approach to reduce these requirements is to use active quantum error correction in combination with other methods that provide additional robustness against instability or noise. Among such methods are holonomic quantum com- putation [11] and topological quantum computation [100]. In [101, 102] schemes have been proposed based on encoding the information in the ground subspace of a Hamiltonian with a constant energy gap and topological properties, such as the surface code Hamiltonian, and performing the computation by adiabatically deforming the Hamiltonian. The energy gap of the Hamiltonian suppresses the 54 thermal excitations induced by the environment. Compatible active error cor- rection is performed frequently enough to prevent logical errors. As the energy gap protection from a Hamiltonian stabilizer code on a two-dimensional lattice cannot increase with system size [35], to sustain arbitrary long universal quantum computation on such a lattice with stabilizer codes requires active error correction. Although this construction is appealing, the surface code Hamiltonian and also the time-dependent Hamiltonian implementing the deformation consist of up to 4-local interactions [101]. Implementing such interactions experimentally is difficult (though not impossible and various proposals exist [103, 104, 105]). One way to circumvent this is to simulate these Hamiltonians using other available resources. But the noise of the original system may be dramatically transformed by the simulation procedure. It is important to investigate how the simulation process converts realistic noise on the original resources to new effective noise on the simulated Hamiltonian. For example, local noise on the original Hamiltonian can be converted into nonlocal noise on the simulated Hamiltonian, and this could reduce the effectiveness of the simulated Hamiltonian in suppressing errors. This is a general concern for various simulation methods, including stroboscopic methods (see, e.g., [106, 107]) and perturbative gadgets [108, 109, 110]. The question of the effect of a Markovian noise on simulation has been partially studied [111], but despite its importance [112], only a limited amount of attention has been devoted to the effects of general noise on the simulation of high-weight Hamiltonians. Inthisworkweshowhow—startingfromanentanglingHamiltonian in the presence of a general local non-Markovian environment—one can generate the desired nonlocal Hamiltonian by the application of one-local unitary operators. By combining simulation with schemes for dynamical decoupling (DD) [113, 114, 10], we construct a sequence of pulses that suppresses errors while simulating 55 the desired interactions. The relationship between the strength of the simulated Hamiltonian and the strength of effective noise is investigated. We also consider how far the transformed errors spread, based on the locality (both geometric and algebraic) of the bath Hamiltonian and the system-bath interaction. We illustrate our construction with the Hamiltonians used in Ref. [101] for fault-tolerant quantum computation in the surface code, but it can easily be mod- ified to simulate other, similar Hamiltonians, including the Hamiltonians used for protection ofadiabatic quantumcomputation in Ref.[48]. In the latter, a stabilizer code that can detect the effect of the environment on the system is chosen, the system Hamiltonian is then encoded using the logical operators, while a penalty Hamiltonian is added to break the degeneracy. Again, the energy gap of the con- structed Hamiltonian suppresses the rate of excitation out of the code space [83]. This scheme also requires 4-body interactions. 4.2 Summary of main results Our main contributions in this work can be summarized as follows: • We provide a method to simulate Hamiltonians with high-weight interac- tions on a grid, such as surface code Hamiltonians, in a number of steps that is independent of the size of the grid. We demonstrate this with detailed construction steps for surface code Hamiltonians, using two different strobo- scopic simulation techniques. • We provide a method to design new pulse sequences that can simulate a desiredHamiltonianwhilepushingerrors(causedbythepresenceofthebath) to higher orders, by combining DD and Hamiltonian simulation techniques. To do so, we use two application of the symmetrization procedure [114, 115]. 56 The first application shows how to protect a Hamiltonian while averaging out all the errors. The second application reduces the number of pulses needed to average out local errors. • For local, but otherwise general, non-Markovian noise on the original Hamil- tonianweshowhowtheeffectiveerroronthesimulatedHamiltonianbecomes non-local. We demonstrate that this effective noise is mostly suppressed dur- ing the simulation of a surface code Hamiltonian because of an effective gap. This chapter is organized as follows. In Sec. 4.3 we introduce the resources and the goal of the simulation. In Sec. 4.4, we describe two different methods to simulate the desired high-weight Hamiltonian on a grid in the noise-free setting. We also discuss how to simulate a time-dependent Hamiltonian that, with the help of active error correction, can perform universal quantum computation fault- tolerantly. In Sec. 4.5, we discuss the effects of general non-Markovian noise on the simulation. We then show how using DD techniques in the simulation procedure can reduce the strength of the effective noise. We also discuss how the locality of the bath and system-bath interaction Hamiltonian changes the spread of the noise and also the resources required for a successful simulation. In Sec. 4.6, we explicitly show that during the simulation an effective gap against the strongest errors created in the process is maintained. Section 4.7 is devoted to conclusion and discussion. 4.3 Setup Assume N qubits are placed on a two-dimensional square lattice as depicted in Fig. 4.1. Let d(i,j) denote the Euclidean distance between qubits i and j; we 57 choose units such that d(i,j) = 1 if qubits i and j are nearest neighbors. A two- body entangling Hamiltonian acts on nearest neighbor and next-nearest neighbor qubits (with d(i,j) = √ 2): H X = X hhi,jii X i X j , (4.1) where X i denotes the Pauli σ x matrix acting on qubit i. (An example of an alternative connectivity is provided in Sec. 4.4.8.) Let us start with the toric code Hamiltonian; later we will convert it to the surface code Hamiltonian with cuts. In this case the Hamiltonian is: H p = X v A v + X p B p , (4.2) with each vertex operator A v corresponding to an X ⊗4 term acting on the spins connected to the vertex v and each plaquette operator B p corresponding to a Z ⊗4 term acting on the spins in the plaquette p; see Fig. 4.1 (Z denotes the Pauli σ z matrix). This Hamiltonian is 4-local, and as the first step we show how to simulate it in the ideal case (no noise) using the 2-local Hamiltonian H X . Having stroboscopic simulation in mind, for now we assume that we can apply instantaneous, ideal unitary pulses on each qubit. For simplicity we assume that the time interval between pulses is fixed (more general schemes can improve the results). 58 Figure 4.1: Interacting qubits on a grid. Blue lines represent XX interactions between qubits (circles). There is an interaction between nearest neighbor and next-nearest neighbor qubits. For the toric Hamiltonian, the goal is to simulate X ⊗4 on the shaded areas and Z ⊗4 on the light areas. 4.4 Simulation: Ideal case Assuming there is no noise, here we show how to use the commutator method to simulate the HamiltonianH p of Eq. (4.2) using the HamiltonianH X of Eq. (4.1). We set~≡ 1. 4.4.1 Commutator method One way to build a 3-local Hamiltonian using 2-local Hamiltonians is to use the identity: Ω τ (A,B) ≡ e iBτ e iAτ e −iBτ e −iAτ = e −iτ 2 (i[A,B]) +O(τ 3 ) , (4.3) 59 valid for any pair of operators A and B. This allows us to generate an effective 3-local Hamiltonianiδt[H a ,H b ] using appropriate 2-local HamiltoniansH a andH b : Ω δt (H a ,H b ) =e −iδt 2 (i[Ha,H b ]) +O(δt 3 ) . (4.4) Note that in oder to neglect the higher order terms, obviouslykH a kδt and kH b kδt must be small. Also, the effective Hamiltonian containsδt, and so is much weaker than the Hamiltonians we started with. The same method can be used iteratively to increase the locality of the effective Hamiltonian. To generate a 4-local Hamiltonian we notice that by negating H b or H a , the term [H a ,H b ] is also negated. So we repeat the procedure with H a ,−H b , and a new Hamiltonian H c : U = e iHcδt Ω δt (H a ,−H b )e −iHcδt Ω δt (H a ,H b ) = e −iδt 3 ([[Ha,H b ],Hc]+[[Ha,H b ],H b ]]) +O(δt 4 ) . (4.5) The term δt 2 [[H a ,H b ],H c ] is the desired 4-local Hamiltonian. The extra term, δt 2 [[H a ,H b ],H b ], is the result of the third-order error from Eq. (4.4) which becomes relevant now: Ω δt (H a ,H b ) =e −iδt 2 (i[Ha,H b ])−iδt 3 [[Ha,H b ],Ha+H b ] +O(δt 4 ) , (4.6) 60 1 2 3 4 Figure 4.2: Depiction of H x = P 1≤i<j≤4 X i X j for four qubits in a plaquette. where the third-order error is not negated by the replacement H b 7→−H b ; the term [[H a ,H b ],H b ] remains. Using the operator identity ue A u † =e uAu † = e A [u,A] = 0 e −A {u,A} = 0 (4.7) valid for unitaryu and arbitraryA, one can apply a pulseu, itself generated by a 2- local Hamiltonian, that commutes with the desired Hamiltonian and anticommutes with the extra term to eliminate it: UuUu † =e −2iδt 3 [[Ha,H b ],Hc] +O(δt 4 ) . (4.8) This process takes time 20δt. This type of simulation is stroboscopic, so we obtain the desired effective Hamiltonian at a specific time (here at t = 20δt). In the above discussion we used three different Hamiltonians H a,b,c . For the actual simulation, we will assume that H X is always on, and then apply pulses (not generated by H X ) to generate the effective Hamiltonians H {a,b,c} . 61 4.4.2 Simulating one plaquette operator Let us now show how to use the Hamiltonian H x = X 1≤i<j≤4 X i X j (4.9) (not to be confused with H X ) to simulate the Hamiltonian X 1 X 2 X 3 X 4 on the qubits of a plaquette (see Fig. 4.2). We will repeatedly use some basic identities, which are listed next for conve- nience, as they will be used throughout the remainder of this work: WXW =Z, WYW =−Y, WZW =X, SXS † =Y, SYS † =−X, SZS † =Z, (4.10) where W is the Hadamard gate and S is the phase gate. Let the system evolve with the Hamiltonian H x for a time 2δt, and apply W 1 W 2 ,u =Z 1 Z 2 andW 1 W 2 u pulses at times t = 0, δt and 2δt respectively, where W i is a Hadamard pulse on the i-th qubit: W 1 W 2 (ue −iHxδt u)e −iHxδt W 1 W 2 =e −i2(Z 1 Z 2 +X 3 X 4 )δt . (4.11) The resulting Hamiltonian is H a = 2(Z 1 Z 2 +X 3 X 4 ). We generate H b = 2(Y 1 X 4 +X 2 X 3 ) by first applying the inverse phase gate S † 1 to the first qubit, then applying u =Z 1 Z 4 at time δt, and finally S 1 u at time 2δt: S 1 (ue −iHxδt u)e −iHxδt S † 1 =e −i2(Y 1 X 4 +X 2 X 3 )δt . (4.12) 62 We also generate H c = 2(Y 2 X 3 +X 1 X 4 ) by applying pulses S † 2 , u =Z 1 Z 4 , and S 2 u at times 0,δt and 2δt: S 2 (ue −iHxδt u)e −iHxδt S † 2 =e −i2(X 1 X 4 +Y 2 X 3 )δt . (4.13) Generating any of−H a,b,c is almost the same as generating H a,b,c , but with a few additional pulses. For example, for H a we notice that: (X 1 Z 3 )e −i2(Z 1 Z 2 +X 3 X 4 )δt (X 1 Z 3 ) =e i2(Z 1 Z 2 +X 3 X 4 )δt . (4.14) In the same way one can generate−H b by conjugating H b with X 1 Z 2 pulses, and−H c by conjugating H c with Z 1 Z 2 pulses. Having all the ingredients needed for Eq. (4.5), the effective Hamiltonian at orderδt 3 contains the desired 4-body Hamiltonian with some extra terms that can be removed with only one extra step. To see this, note that: [[H a ,H b ],H c ] = 32(X 1 X 2 X 3 X 4 +Y 1 Y 2 X 3 X 4 ) , (4.15a) [[H a ,H b ],H b ]] = 64(Z 1 Z 2 +X 1 Y 2 X 3 X 4 ) . (4.15b) One then follows Eq. (4.8) with an extra pulse u = Z 1 Y 2 , which commutes with the desired term (X 1 X 2 X 3 X 4 ) and anticommutes with all the other terms. The effective Hamiltonian becomes 64δt 2 X 1 X 2 X 3 X 4 , as desired. Therefore, in N X = 40 steps we can convert the always-on Hamiltonian H x (after a timeN X δt) to 64δt 2 X 1 X 2 X 3 X 4 , with errors of orderδt 4 , which is the same as saying that the original Hamiltonian is stroboscopically converted to a 4-local Hamiltonian at the price of making it 64δt 2 /N X times weaker. 63 (a) (b) 1 2 3 4 1 2 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 2 1 2 3 4 1 3 4 1 2 4 1 2 4 1 4 1 2 3 4 1 2 3 4 2 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 3 4 2 3 4 2 3 4 2 3 4 2 (c) (d) Figure 4.3: Generating plaquette operators. (a) The location of Z pulses is shown in blue. (b) The remainingXX interactions after (a) are shown in red. Their sum defines H temp . (c) Labelling of qubits according to Fig. 4.2. (d) The result after applying pulses as is Sec. 4.4.2 to (c). This achieves the first goal of generating plaquette operators in the green areas. 4.4.3 Simulating all the plaquette operators on the grid We assume that simultaneous pulses on all the qubits of the grid can be applied in parallel. Using this we can simulate all the plaquette operators on the grid. To do so, we start with the Hamiltonian H X and repeat the steps provided for one plaquette operator, while ensuring that no unwanted terms are created in the process. We start by creating patterns similar to H x by letting H X evolve and apply Z pulses on the qubits that are colored in blue in Fig. 4.3(a) at timesδt and 2δt, but leave alone the qubits that are colored in black. (More precisely, we conjugate byZ pulsesallthequbitsofaplaquetteiftheplaquetteisinbothanoddrowandanodd column of the grid, counting from the top left corner.) Every pair of neighboring blue and black qubits is thus decoupled, since Z anticommutes with XX, while every pair of neighboring blue qubits is unaffected, sinceZZ commutes withXX. This results in an effective Hamiltonian, called H temp , shown in Fig. 4.3(b) by red lines as the surviving XX interactions (the plaquettes in an odd row and an odd column, or plaquettes in an even row and an even column of the grid.) Now we use H temp and apply pulses similar to Sec. 4.4.2 on the labelled qubits in 64 (a) (b) 1 2 3 4 (c) Figure 4.4: Generating H a on the grid. (a) Evolving under H temp [Fig. 4.3(b)], we applyZ pulsesonthebluequbitsattimesδtand 2δt. (b)TheeffectiveHamiltonian at time 2δt. (c)H a is generated by conjugating the Hamiltonian in Fig. 4.4(b) with Hadamard pulses. The effective Hamiltonian becomes patterns of Z 1 Z 2 +X 3 X 4 , depicted as green coupled pairs (ZZ) and black coupled pairs (XX). Fig. 4.3(c) in parallel to generate plaquette operators in half of the grid, as shown in Fig. 4.3(d). The steps of this procedure, and the corresponding components H {a,b,c} are presented in Figs. 4.4 and 4.5. As discussed in Sec. 4.4.2, by applying Z 1 Y 2 (but now in parallel on the whole grid) one can remove all the extra unwanted terms, and all that remains are the plaquette operators shown in Fig. 4.3(d). Up to now, the effective Hamiltonian only contains half of the plaquette opera- tors needed. The other half can be generated by repeating the same procedure, but shifting the locations of the pulses by one qubit. In this case plaquette operators in the shaded area of Fig. 4.3(d) are generated. The combined process gives all the plaquette operators needed: e −ihδt 3 P v Av +O(δt 4 ) , (4.16) where hδt 2 is strength of the effective Hamiltonian, and h accounts for various numerical factors (such as 64/N X in Sec. 4.4.2). 65 (a) (b) 1 2 3 4 (c) 1 2 3 4 (d) Figure 4.5: Generating H b and H c on the grid. (a) Evolving under H temp [Fig. 4.3(b)], we apply Z pulses on the blue qubits at times δt and 2δt. (b) The effective Hamiltonian at time 2δt. (c)H b is generated by conjugating the Hamilto- nian in Fig. 4.5(b) with S pulses on the orange qubits. The effective Hamiltonian becomes patterns of Y 1 X 4 +X 2 X 3 , depicted as orange-black coupled pairs (YX) andblackcoupledpairs(XX). (d)H c isgeneratedbyconjugatingtheHamiltonian in Fig. 4.5(b) with S pulses on the orange qubits. Now the effective Hamiltonian becomes patterns of X 1 X 4 +Y 2 X 3 , depicted as black coupled pairs (XX) and orange-black coupled pairs (YX). 4.4.4 Simulating all the vertex operators on the grid Noticing, using Eq. (4.7), that W 1 W 2 W 3 W 4 e −iX 1 X 2 X 3 X 4 δt W 1 W 2 W 3 W 4 =e −iZ 1 Z 2 Z 3 Z 4 δt , (4.17) the process to create the vertex operators on the grid is similar to the process for plaquette operators. One can just shift the location of pulses and then conjugate the whole effective Hamiltonian with Hadamard pulses on all the qubits of the grid. 66 4.4.5 Strength of the effective Hamiltonian By simulating the plaquette operators and the vertex operators consecutively, we can effectively simulate the toric code Hamiltonian in N sim = 320 steps, 1 and so in time N sim δt, independent of size of the grid. The generated 4-body operator on the grid is 2 9 δt 3 H p . Taking the strength of H X as 1, the strength of each H {a,b,c} is 4. Inserting these into the commutators, [[H a ,H b ],H c ]] has strength 4 4 . Another factor 2 is added when we remove the error term. Thus 4 2 ∗ 2δt 3 = 2 9 δt 3 . That is, we use H X to generate the toric code Hamiltonian while making it 2 9 δt 3 /320δt = 1.6δt 2 times weaker. As expected, decreasing δt makes the ratio smaller. 4.4.6 Boundaries and the surface code The toric code is defined on a torus, without boundaries. We are also interested in the surface code, which has boundaries [116]. To simulate the boundaries for the surface code Hamiltonian, whether outer boundaries or the inner cuts representing qubits, two other procedures are also needed. The first is generating holes (either Z-cut or X-cut), which is straightforward. For example, wherever it is necessary to create a hole one can choose H 0 c such that it commutes with [H a ,H b ]. To accomplish this, rather than applying S 2 in Eq. (4.13), if we apply the Hadamard pulse W 2 , then H 0 c = Z 2 X 3 +X 1 X 4 , which commutes with [H a ,H b ], and so the effective Hamiltonian is zero to orderδt 3 . Thus, simply by using a Hadamard pulse rather than phase gate on a specific qubit we can generate a hole. 1 10∗ 2 comes from the commutator method, a factor of 4 comes from generating each of H {a,b,c} from the original Hamiltonian H X . A factor of 2 is needed to generate the other half of the plaquettes, and another factor of 2 is needed to generate the vertex operators. Thus N sim = 10∗2∗4∗2∗2 = 320. 67 The second procedure needed is the simulation of 3-body interactions. One approach is to simulate them using Eq. (4.6) at order δt 2 , while eliminating errors of order δt 3 . Using the same pattern as Sec. 4.4.3, starting from H x in Fig. 4.2 we can choose H a = Z 1 X 2 +X 3 X 4 and H b = Y 1 X 4 +X 2 X 3 . This results in an effective Hamiltonian X 1 X 2 X 4 δt 2 + (Y 1 X 4 +Z 1 X 2 )δt 3 +O(δt 4 ). By decoupling using one extra pulseX 1 , we can remove the terms proportional to δt 3 . Also, if in Fig. 4.2 the third qubit and the terms it involves are missing (these are edges of the Hamiltonian, corresponding to outer boundaries), then again we can generate H a =Z 1 X 2 and H b =Y 1 X 4 , and then apply X 1 with the same final result. To balance the different strength of the generated terms (boundaries are pro- portional to δt 2 while the rest of the construction is proportional to δt 3 ), the proportion of time slices implemented from each of these terms (in Trotterization of the evolutions) can be chosen to beδt. Another approach is to implement these boundaries together with the other terms, but with flipping the signs of some of the Hamiltonians so that all but a fraction δt of the 3-body terms cancel out. (In our example, we could use Z 1 pulses: Z 1 e −iX 1 X 2 X 4 δt 2 Z 1 =e iX 1 X 2 X 4 δt 2 .) An alternative approach to avoid the complicated scheduling described above is to apply pulses such as e −iYδt to implement single-body Hamiltonians. For example, starting from H x in Fig. 4.2 we can repeat the procedure in Sec. 4.4.2 to generate H a =Z 1 X 2 +X 3 X 4 and H b =Y 1 X 4 +X 2 X 3 , but now generate H c = X 1 X 4 +X 3 X 4 +X 1 X 3 +Y 2 . If so, the effective Hamiltonian to order δt 3 (after applyingZ 1 Y 2 pulses to remove the extra terms, as mentioned in Sec. 4.4.2 ) would becomeX 1 X 2 X 4 as desired. To generate such anH c we can follow this procedure: Z 2 e −iHxδt Z 2 e −iHxδt e −i2Y 2 δt =e −i2(X 1 X 4 +X 3 X 4 +X 1 X 3 +Y 2 )δt . (4.18) 68 4.4.7 π/4-Conjugation Method An alternative method for simulating high-weight Hamiltonians is based on the π/4-conjugation identity (see, e.g., [106]). This identity for Pauli operators A,B with{A,B} = 0 is e −iπ/4A e iθB e iπ/4A =e iθ(iAB) , (4.19) which is exact. Obviously if [A,B] = 0 then we have e −iπ/4A e iθB e iπ/4A =e iθB . (4.20) Assume the entangling Hamiltonian is cH x =c P i<j X i X j with c being a con- stant representing the strength of the entangling Hamiltonian: e −i π 4c (cHx) e −iθY 1 e i π 4c (cHx) =e iθZ 1 X 2 X 3 X 4 . (4.21) Using this we can show that: W 1 [e −i π 4c (cHe) e iΔtY 1 (Y 1 e −i π 4c (cHe) Y 1 )]W 1 =ie −iΔtX 1 X 2 X 3 X 4 . (4.22) So we apply three fast pulses while having H x on for a length of time 2π/4c. It is important to notice that in this method the simulation time, here 2π/4c, is independent of Δt, the time over which we wish to simulate the evolution. 69 To simulate plaquette operators we again can conjugate this with Hadamard pulses: W 1 W 2 W 3 W 4 e −iΔtX 1 X 2 X 3 X 4 W 1 W 2 W 3 W 4 =e −iΔtZ 1 Z 2 Z 3 Z 4 . (4.23) Implementation of the Hamiltonian on the grid can be done similarly to Sec. 4.4.3, with slight modifications. Using this procedure there is no error in simulation (as the identity is exact) and it takes a time proportional to π/c independent of Δt. In contrast to the commutator method, where using the entangling Hamilto- nian for a short time produces a weak effective Hamiltonian, here the entangling Hamiltonian has to evolve for the fixed time π/4c to produce the desired effective Hamiltonian. For an experimental setup that can only generate weak entangling Hamiltonian (small c), this method requires a large time which may make the implementation more prone to noise. For experiments with access to strong entan- gling Hamiltonians compared to the single-body gates and measurements, this method is beneficial as it is exact and the simulation time is independent of the time over which we wish to simulate a Hamiltonian. See [111] for a comparison of the π/4-conjugation method to the commutator method in a Markovian environ- ment. 70 1 2 3 4 Figure 4.6: Labeling of qubits connected to a vertex 4.4.8 Alternative Connectivity AnalternativeconnectivityfortheHamiltonianonthesquaregridiswhenthere is only nearest-neighbor coupling. In this case the Hamiltonian would become: H X = X hi,ji X i X j . (4.24) Here we show how to use the HamiltonianH x = (X 1 +X 3 )(X 2 +X 4 ) to simulate the HamiltonianX 1 X 2 X 3 X 4 on qubits connected to a vertex, for an effective time Δt. (See Figure 4.6.) We denote the Hadamard gate by W, and the Phase gate by S. Now, (S 1 S 2 )e −iH X δt (S 1 S 2 ) =e −i(Y 1 +X 3 )(Y 2 +X 4 )δt (4.25) generates H a = (Y 1 +X 3 )(Y 2 +X 4 ). Likewise, (W 1 S 2 )e −iH X δt (W 1 S 2 ) =e −i(Z 1 +X 3 )(Y 2 +X 4 )δt (4.26) 71 generates H b = (Z 1 +X 3 )(Y 2 +X 4 ), and (W 2 )e −iH X δt (W 2 ) =e −i(X 1 +X 3 )(Z 2 +X 4 )δt (4.27) generates H c = (X 1 +X 3 )(Z 2 +X 4 ). All of these Hamiltonians can be negated using X 2 Z 4 pulses. Using the com- mutator method given in Eq. (4.5), we will have the the desired Hamiltonian 8X 1 X 2 X 3 X 4 plus some extra terms. Again, by doubling the number of pulses and conjugating the new ones by Z 1 Y 2 , we can get rid of all the extra terms, and so the desired Hamiltonian becomes e −i16X 1 X 2 X 3 X 4 δt 3 +O(δt 4 ). So using N X = 20 pulses, and applying the Hamiltonian H x for a time N X δt we can simulate 16X 1 X 2 X 3 X 4 at order δt 3 with errors of order δt 4 . Generating this Hamiltonian on a grid can be done using the methods from Sec. 4.4 with slight modifications. 4.4.9 Applying the scheme to fault-tolerant holonomic quantum computation in surface codes Using adiabatic deformation of the surface code Hamiltonian in combination with active error correction it is possible to achieve fault-tolerant universal quan- tum computation [101]. In such a scheme, a gapped Hamiltonian protects the groundspacefromthedeleteriouseffectsofthermalnoise, andthegatesareapplied by slowly deforming the Hamiltonian, such that the desired gate is applied to the ground state of the new Hamiltonian (holonomic quantum computation [11]). 72 Explicit constructions to implement all the gates needed for universal quantum computation were proposed in [101] for such deformations. The Hamiltonians needed for these schemes can be made geometrically local [101], but in practice it is challenging to implement them as they consist of up to 4-local interactions. Using the methods we proposed above, it is straightforward to simulate these Hamiltonians as well. For the purpose of illustration we choose a few examples from [101] and explain the needed procedure, but the constructions for the remaining needed interactions are similar. As the first example, we note that the creation of a|+i state for an X-cut double qubit (and of a|0i state for a Z-cut double qubit) is done by just turning off two stabilizer terms (Sec IV.A in [101]). As explained in Sec. 4.4.6, this can be done by changing a pulse such that the O(δt 3 ) effective Hamiltonian vanishes. Wealsoneedtheabilitytoenlargeahole. Theproposedtime-dependentHamil- tonian implementing the deformation is of the form: H(t) =−J (1− t t 1 )B 2 + t t 1 X 1 + X p6=1,2 B p + X v A v , (4.28) fort∈ [0,t 1 ](seeEq.(37)in[101], withaspecificchoiceofthemonotonicfunction). Tosimulatethetime-dependentHamiltonian, wecanuseTrotter-Suzukiexpan- sions (see, e.g., [117]), and simulate each piece by applying the appropriate pulses. To do so, we define Δt = t 1 Ntr and approximate the evolution generated by the time-dependent Hamiltonian with ordered evolutions consisting of several time independent Hamiltonians: Ntr Y m=0 e −i((1− m N tr )B 2 + P p6=1,2 Bp)JΔt e −i m N tr X 1 JΔt e −iJ P v Av Δt (4.29) 73 To simulate these terms, one can generate the weakest 4-body interaction in this expansion using the building blocks we constructed in Sec. 4.4.3, setting JΔt Ntr = hδt 3 : Ntr Y m=0 (e −iB 2 hδt 3 ) Ntr−m (e −i P p6=1,2 Bphδt 3 ) Ntr (e −iX 1 hδt 3 ) m (e −ih P v Avδt 3 ) Ntr . (4.30) Each 4-body term in this expansion can be simulated as described in Sec. 4.4.3. We notice that B 2 can be constructed in parallel with the rest of the plaquette operators. Also, e −iX 1 hδt 3 is just a single qubit rotation applied using a pulse. The error in the Trotterization procedure scales as Δt 2 , and is independent of the system size (the only non-commuting term is X 1 ; see [117]). The same procedure works for the other Hamiltonians needed, such as another Hamiltonianneededtoenlargeahole(Eq.(39)in[101]), ortheHamiltonianneeded for moving logical qubits (Eq. (47) in [101]). The ¯ Z and ¯ X logical operators of the surface code are just strings of Z and X operators that connect appropriate boundaries (or holes). Thus any product thereof can be formed by applying parallel Z and X pulses (in practice one may avoid applying them and instead keep track of updates in software). 4.5 Effect of noise So far we have considered only the noise-free case. In this section we study the effect of noise on the simulation, and propose methods to suppress the effective noise of this process. 74 Again we use the entangling Hamiltonian H X [Eq. (4.1)], but now we assume the presence of a general non-Markovian bath with Hamiltonian H B , interacting with the system via the HamiltonianH SB . For now we assume that this interaction Hamiltonian is 1-local: H B = I⊗B 0 , (4.31) H SB = X i X α∈{x,y,z} σ α i ⊗B α i , (4.32) where i labels the the system qubit (when it is clear from the context, we will combine this index and the Pauli operator index into a single index: H SB = P β σ β ⊗B β .) The total Hamiltonian is: H =H X +H B +λH SB , (4.33) withthedimensionlessparameterλbeingthestrengthofthesystem-bathcoupling. As the effective 4-body interactions constructed in previous sections appear in order δt 3 , our strategy is to choose a series of DD pulses that commute with the Hamiltonian H X but suppress the errors. This use of DD pulses that commute with the Hamiltonian is an example of the general idea of using the stabilizer generatorsofaquantumerrorcorrection(ordetection)codetoperformadecouple- while-compute operation, which is possible since the Hamiltonian is a sum of the logical elements of the same code [118, 119, 120]. We will show how to protect the Hamiltonian H X against the noise at least up to order δt 3 , so that the simulated Hamiltonian is not overwhelmed by the noise. To simulate the desired interactions, we combine the pulses designed for simulation with these DD pulses. Note that DD schemes are effective when the timescale of the bath is long compared to the timescales of the pulses, and in fact DD can be shown to fail in 75 the Markovian limit [121]. Therefore the method proposed here is only effective for non-Markovian environments. We begin by first finding pulses that can suppress the errors in a general non- Markovian environment, and then discuss how further assumptions on the locality of the bath and the system-bath interaction allow for more efficient schemes. To analyze the effects of DD pulses, and the corresponding effective Hamilto- nian, it is handy to use the Magnus expansion. 4.5.1 Magnus expansion for piecewise constant Hamilto- nian A good reference introducing the Magnus expansion is [98]. Here are a few low-order terms for piecewise constant Hamiltonians [122]: e −iHnδt ···e −iH 2 δt e −iH 1 δt =e −iTnH eff (Tn) . (4.34) Here T n =nδt and H eff = P ∞ k=0 H (k) eff , with H (0) eff (T n ) = δt T n n X k=1 H k , (4.35) H (1) eff (T n ) = − i(δt) 2 2T n n X l=2 l−1 X k=1 [H l ,H k ], (4.36) H (2) eff (T n ) = (4.37) − (δt) 3 6T n ( n X m=3 m−1 X l=2 l−1 X k=1 [H m , [H l ,H k ] + [H m ,H l ],H k ] + 1 2 n X l=2 l−1 X k=1 [H l , [H l ,H k ] + [H l ,H k ],H k ]] ) . 76 Here we choose the simplest DD schemes to illustrate the main ideas, but there is much room to use more sophisticated DD schemes (see, e.g., Ref. [10] for a review). 4.5.2 Protecting against noise at first and second order in δt To remove the first order noise terms, we conjugate the Hamiltonian H X with pulsesI ⊗N ,X ⊗N ,Y ⊗N ,Z ⊗N in 4 steps. These pulses all commute with the Hamil- tonian but remove the noise from the effective Hamiltonian to first order, via symmetrization [115, 123]. Then, using the fact that all the even-order terms of the Magnus expansion vanish for time-symmetric Hamiltonians [98], we simply append another 4 steps, conjugating with the same 4 pulses but in reverse order. This removes the first and second order terms of the noise from the effective Hamiltonian: U sec = (I ⊗N e −iHδt I ⊗N )(X ⊗N e −iHδt X ⊗N ) (Y ⊗N e −iHδt Y ⊗N )(Z ⊗N e −iHδt Z ⊗N ) (Z ⊗N e −iHδt Z ⊗N )(Y ⊗N e −iHδt Y ⊗N ) (X ⊗N e −iHδt X ⊗N )(I ⊗N e −iHδt I ⊗N ) = e −i(H X +H B )8δt+O(δt 3 ) . (4.38) 77 Note that the middle Z ⊗N terms cancel, and that we can simplify this sequence to: U sec = e −iHδt X ⊗N e −iHδt Z ⊗N e −iHδt X ⊗N e −2iHδt X ⊗N e −iHδt X ⊗N e −iHδt X ⊗N e −iHδt = e −i(H X +H B )8δt+O(δt 3 ) . (4.39) At this point, having evolved for 8δt (independent of the grid size), we have protected theH X Hamiltonian from the effect of noise to first and second order in δt. 4.5.3 Third order error terms First using a lemma we describe a general approach to protect a Hamiltonian while averaging out all other errors at a fixed order of time. It can be understood as an application of the symmetrization schemes proposed in [114, 115]. Lemma 1. (Symmetrization lemma: Protecting interaction) Let P 1 ,...,P n be commuting Pauli operators, and letP be the group generated by these Pauli opera- tors. Denote the set of all Pauli operators that commute with all the P i s byN (P) (the normalizer ofP). Then for any Pauli operator s ∀s / ∈P : X g∈N (P) gsg † = 0, (4.40) ∀s∈P : X g∈N (P) gsg † =s|N (P)|. (4.41) As a special case, ifP only includes the identity operator we recover the usual Pauli twirling lemma (see, e.g., [87].) 78 Proof. The s∈P case is trivial. Assume s / ∈P and s / ∈N (P), which means that s anticommutes with at least one element ofP and ofN (P). Denoting a set of independent generators ofP by{P i } m i=1 , there exists a (non-identity) P ∗ in this set such that{P ∗ ,s} = 0. Obviously∀g∈N (P)⇒ P ∗ g∈N (P). From this we conclude that g and P ∗ g are two distinct elements ofN (P) such that one commutes withs, and the other one anticommutes with it. 2 Going through all the elements ofN (P) (and using P 2 ∗ = I), half the elements commute with s while the other half anticommute with s, and so the result holds in this case. For the case that s / ∈P and s∈N (P): elements ofN (P) can be generated using{P i } and n−m extra pairs of conjugate generators, where n is the number of qubits. Let us call these pairs ( ˆ X j , ˆ Z j ), where 1≤ j ≤ n−m. Note that s / ∈P and s∈N (P) means that s contains at least one element from these pairs, let us say an element from the j ∗ -th pair. Now we notice that∀g∈N (P)⇒ {g,g ˆ X j∗ ,g ˆ Z j∗ ,g ˆ X j∗ ˆ Z j∗ }∈N (P). All these elements are distinct, and two of them commute with s while two anticommute. Therefore, again the elements ofN (P) can be partitioned into half commuting and half anticommuting, which completes the proof. Using this lemma for the particular plaquette operator discussed in section 4.4.2, choosing P 1 = X 1 X 2 ,P 2 = X 2 X 3 ,P 3 = X 3 X 4 we need 4 4 /2 3 = 32 pulses to cancel out all the third-order errors. (Again, by symmetrization and doubling the number of pulses we get can get rid of fourth-order error as well.) But to protect the whole grid, without additional assumptions about the local- ity of the bath part of the interaction Hamiltonians (the{B α } operators), the number of pulses needed to cancel the terms at third order may grow with the grid 2 There are two cases depending on whether s commutes or anticommutes with g. If{s,g} = 0,{s,P ∗ } = 0 then P ∗ gs =−P ∗ sg = sP ∗ g, i.e., [s,P ∗ g] = 0. Likewise, [s,g] = 0,{s,P ∗ } = 0⇒ {s,P ∗ g} = 0. 79 size. The reason is that at higher orders, multiqubit error terms begin to appear without a locality assumption.ă To average out these terms by DD methods, the number of pulses must grow with the number of qubits. A simple example showing that the number of pulses to average out multi-qubit errors grows with the number of qubits is the following set of two-qubit errors: {Y i Y j : 1≤i<j≤N} (4.42) A simple lower bound on the number of pulses needed to average out all of these errors can be derived by noticing that each Y i has to have a distinct pattern of commutation/anticommutationwiththepulses: ifY i andY j havethesamepattern, then all the pulses will commute withY i Y j , so applying pulses will leave this term unchanged. From this we conclude that number of distinct patterns has to be greater than N, which means that the number of pulses needed to average out all the error terms has to increase with the number of qubits. If|P| is the number of pulses, this means that 2 |P| ≥ N, and so|P| ≥ log 2 N. For combinatorial approaches usingO(N) pulses to achieve first-order decoupling see Refs. [124, 125] and chapter 15 of Ref. [10]. Later we will consider the case where the terms{B α } are geometrically local, but here we show that even without this assumption it is possible to protect the Hamiltonian against the dominant noise terms appearing at third order inδt using a few pulses. To see this, we note that at order δt 3 , the terms that are first order 80 in the coupling strength λ have a specific form: they are nested commutators of σ α i ⊗B α i with two H X terms: [[H X ,σ α i ⊗B α i ],H X ] = [[X i ⊗ X hhi,i 0 ii X i 0,σ α i ⊗B α i ],H X ] = [[X i ,σ α i ]⊗ X hhi,i 0 ii X i 0⊗B α i ,H X ] = [[X i ,σ α i ],X i ]⊗ ( X hhi,i 0 ii X i 0) 2 ⊗B α i . (4.43) These terms are either Y i ,Z i ,Y i X i 0X i 00 or Z i X i 0X i 00, and are thus decoupled away by conjugating the evolution in Eq. (4.39) with I ⊗N and X ⊗N pulses: (I ⊗N U sec I ⊗N )(X ⊗N U sec X ⊗N ) =e −i(H X +H B )16δt+O(λ 2 δt 3 )+O(δt 4 ) (4.44) In fact, these two pulses not only remove theλ 1 terms at orderδt 3 , but at any order of δt. This is expected, as the first-order terms in λ are the result of the system Hamiltonian spreading the noise across more qubits. As the system Hamiltonian terms are all local and commuting, the noise to first order in λ can only have the form of eitherY i X...X orZ i X...X, and so conjugating with the proposed pulses can remove them. Assuming that the system is weakly coupled to the environment (small λ), the dominant terms in the expansions are the lower powers of λ, so using just two additional pulses can cancel out all errors to first order in λ. To sum up, we obtain this DD protection for H X on the grid by evolving for time 16δt, without any assumptions about geometric locality of the interaction Hamiltonian. With the noise removed by this DD procedure to order δt 3 , we can 81 use our earlier constructions to produce the effective Hamiltonian without its being overwhelmed by noise. 4.5.4 Strength of the simulated Hamiltonian vs. effective noise The effective noise process on the system depends on the particular schemes used for simulation and for DD.ă The ratio between this effective noise strength and the strength of the simulated Hamiltonian is important. If this ratio is small enough, the energy gap protection of the simulated Hamiltonian can be effective against the effective noise. Here we give estimates of the strength of each of these terms, but we also expect that for specific systems with more knowledge about the form of the noise, better bounds can be achieved. Weassumeδtisthetimeintervalbetweenpulses. Thisismostlikelydetermined by the physical limitations of the experiment. As described earlier, we choose N DD N sim pulses to simulate a building block of the desired Hamiltonian at order δt 3 , with the effective noise at second order in λ and third order in δt (Fig. 4.7). We first find a bound on the effective noise strength in each DD interval, and then add the effect of simulation on it. Following [98], we define the effective noise strength as: η =kU(N DD δt)−U ideal (δt)k, (4.45) withU being the unitary generated byH X +H B +H SB with the DD pulses applied in between. The operator U ideal (t)≡e −i(H B +H X )N DD t (4.46) 82 is the ideal case, with H SB absent. We denote the cumulative unitary generated by all pulses at each time (not including the Hamiltonian terms) by U pulse (t). We assume that the pulses all commute with the system HamiltonianH X and also assume that the product of all pulses in one cycle gives identity: U pulse (N DD δt) =I. Moving to the interaction picture defined by the pulses we have: η = ˜ U(N DD δt)−U ideal (δt) , (4.47) with ˜ U(N DD δt) =Texp " −i Z N DD δt 0 H X +H B + ˜ H SB (t)dt # , (4.48) and ˜ H SB (t) =U † pulse (t)H SB U pulse (t). (4.49) The Magnus expansion can be used to approximate ˜ U(N DD δt) with an effective HamiltonianH eff up to arbitrary order. Denoting ˜ U(N DD δt) = exp(−iH eff N DD δt), the Magnus expansion gives each term in H eff (see Sec. 4.5.1): H eff = X k=0 H (k) eff . (4.50) Using this we have an upper bound on η as η≤N DD δt X k=0 H (k) eff − (H B +H X ) . (4.51) 83 Assuming that the pulses achieve complete decoupling for first and second order, we have H (0) eff =H B +H X and H (1) eff = 0. So we have η≤N DD δt H (2) eff + X k≥3 H (k) eff (4.52) Now we can use the bounds on Magnus terms derived in [98], but a slight modification is needed. First is the addition of theH X Hamiltonian, and the other is taking into account the effect of the two extra pulses we used to remove the first order terms in λ. For the first term in Eq. (4.52), adding the terms corresponding to λ 2 ,λ 3 (adding Eqs. (112) and (113) in [98]: changing β tokH B +H X k and replacing withkH B +H X +λH SB k), we have: H (2) eff ≤ (4.53) (Nδt) 2 λ 2 kH SB k 2 (c 0 λkH SB k +c 1 kH B +H X k), where c 0 and c 1 are some small constants. For the second term we have (using Eqs. (119) and (125) in [98]): X k≥3 H (k) eff ≤ (4.54) (Nδt) 3 λkH SB k (λkH SB k +kH B +H X k) 3 ×(c 2 +c 3 (λkH SB k +kH B +H X k)N DD δt)), where c 2 and c 3 are some small constants. Denoting this upper bound by η 0 (the right hand side of Eq. (4.52)), the upper bound on the effective error strength of the combined DD and simulation procedure becomes N sim η 0 plus the error we get 84 t N DD t t N DD t N Sim (N DD t) Figure 4.7: Timing of the pulses: combining a DD scheme with N DD pulses and a simulation with N sim steps. from the simulation. The strongest of these errors is at order δt 4 , and a bound on these terms is c(N sim N DD δt) 4 kH X k. (This comes from counting the number of terms in the Magnus expansion and considering the locality of H X , so rather than havingkH X k 4 , we have a constant timeskH X k.) Similarly, higher order error terms of the simulation can also be included up to any desired accuracy. The strength of the Hamiltonian itself, after the pulses, becomes N DD H X , and so after the simulation the desired Hamiltonian shows up at order (N DD δt) 3 . IncreasingN DD makesthesimulatedHamiltonianstronger, buttheerrorassociated with DD can become worse, as can be seen from the bound. 4.5.5 Locality and strength of noise LocalnoiseontheoriginalHamiltoniancanactaseffectivenonlocalnoiseonthe simulated Hamiltonian. This is true for the DD process, and also for the simulation process. Here, starting from the Hamiltonian in Eq. (4.33), we investigate how the locality and strength of the effective noise changes after applying pulses. For generality, we assume that the system Hamiltonian is k-local and the system bath interaction, H SB , consists of l-local terms on the system (for our construction k = 2,l = 1). 85 Using the Magnus expansion it is evident that the terms inδt m are in the form of m− 1 nested commutators (basically, products of m Hamiltonians). Growth of nonlocality of noise in higher order terms First, let us assume that the bath part of the interaction Hamiltonian can be arbitrarilynonlocal:∀i6=j : [B i α ,B j β ]6= 0. Conjugatingasegmentoftheevolution with u a pulses, is equivalent to having a evolution with an effective Hamiltonian H i =H a,X +H B +λH a,SB , where: u a e −iHδt u † a =e −i(H a,X +H B +λH a,SB ) . (4.55) The pulses do not change the locality of the system Hamiltonian or the inter- action Hamiltonian (i.e., the locality of H a,X and H a,SB is the same as that H X and H SB ). The effective Hamiltonian at order δt m would be the result of m− 1 nested commutators of these H a segments. At this order in δt, for any q, the terms of orderλ q are at most [ql + (m−q)(k− 1)]-local. The most nonlocal terms are ml-local, and they appear at order λ m (which is quite weak). We can have a [m(k− 1) + 1]-local term at order λ 0 , which we use for the simulation. In the case of DD, if we choose the{u a } pulses to commute with the system Hamiltonian we get H a = H X +H B +λH a,SB . In this case, all the H B and H X terms commute with each other and the λ 0 term vanishes. The case of our construction For the case where the system Hamiltonian is 2-local and the interaction Hamiltonian is 1-local (k = 2,l = 1), errors of order λ q for ∀q ≥ 1 are [ql + (m−q)(k− 1)] = m-local, and at order λ 0 the simulated Hamiltonian is 86 (m + 1)-local. So at order δt 3 (m = 3) we have 3-local noise and the 4-body simulated Hamiltonian. Estimating the Number of Error Terms We can provide a worst-case estimate of the number of error terms that occur at order δt m while having the simulated Hamiltonian appear at order δt m+1 . Using our construction, we first apply N DD pulses to protect the system Hamiltonian and push the errors to order δt m . At this order there will be roughly N m DD terms of m− 1 nested commutators, with each term of the form H i =H X +H B +λH i,SB . At orderλ 0 there is no error, as all terms ofH X andH B commute with each other. At order λ 1 there are 3N terms (N being the number of qubits, and 3N being the number of terms), multiplied by a constant depending on the connectivity degree of H X , as the entangling Hamiltonian can only expand an existing string of errors locally to neighboring qubits. (The same argument is true for higher powers of λ.) Thus, at order λ 1 the total number of terms is of orderN m DD × 3N. Notice the growth with the number of qubits. At order λ q there are N m DD (3N) q terms. (Recall that we can remove all the errors of order λ 1 by doubling the number of pulses.) Therefore, to simulate the desired nonlocal Hamiltonian at orderδt m 0 , we apply N sim of these sequences of pulses consecutively. The number of error terms gen- erated by the simulation process at order δt m 0 +1 is N m 0 sim , which as we saw earlier can be [m 0 (k− 1) + 1]-local at orderλ 0 . Also, the number of error terms resulting from DD will be multiplied by N sim . In our construction, we use N sim N DD pulses to simulate hH p at order δt 3 with errors also at order δt 3 , with the errors of leading order λ 2 . (These terms are at most 3-local, and there are at mostN sim ×N 3 DD × (3N) 2 such terms.) The leading 87 order terms in the simulation error are of order δt 4 and of order λ 0 . (These terms are at most 5-local, and there are O(N 4 sim ) such terms.) Geometrically local bath It is reasonable to assume that the dynamics of the bath and the way it inter- acts with the system can be characterized by some local Hamiltonians. Assuming locality for the bath part of the interaction Hamiltonian and also the bath Hamil- tonian, one can show that the stronger noise terms (lower powers of λ) are more geometrically local. For example this means that the 3-local errors showing up in δt 3 are more concentrated in a geometrically local region of the lattice, rather than showing up in any three locations. The surface code Hamiltonian (and topological codesingeneral)areexpectedtoperformbetteriftheerrorsaremoregeometrically correlated. Now we assume locality for the bath part of the interaction Hamiltonian. We assume that the terms B α i and B β j from the interaction Hamiltonian commute with each other if the corresponding system qubits are at least distance r from each other. ∀i,j d(i,j)≥r⇒ [B i α ,B j β ] = 0. (4.56) Again, looking at the nested commutators (we will add the effect of the bath Hamiltonian later), we see that errors of order λ consist of a single term from H SB commuted with one or more terms from the system Hamiltonian. This error term therefore can be [l + (m− 1)(k− 1)]-local, but all the qubits it acts on are neighbors. In general, [ql + (m−q)(k− 1)]-local errors in λ q can appear with 88 the qubits separated by a distance of at most (q− 1)r, and the geometrically local terms coming from the system Hamiltonian can only grow from this base of qubits. We need to also include the effects of the bath Hamiltonian.ăH B does not itself increase the non-locality of the interaction Hamiltonian terms, but it can connect terms in different local regions and so affect the spread of error. One can separately assumegeometric localityforthe bathHamiltonian andrepeat theargumentabove to bound the spread of errors. Namely, we may assume that H B = P h B , where each term can have a nontrivial effect on the bath part of H SB corresponding to qubits separated by a distance of at most r 0 : ∀h B , d(i,j) ≥ r 0 : ⇒ [B i α ,h B ] = 0 or [B j α ,h B ] = 0. (4.57) If there are b copies of H B in the m nested commutators, at order λ d this results results in at most [dl + (m−d−b)ĹŮ(k− 1)]-local term that can spread up to a distance of br 0 + (d− 2)r with d≥ 2. 4.5.6 DD pulses for a local bath Suppose we want to protect N qubits on a line from any possible multiqubit error. Theusualtwirlinglemmawouldsuggestapplyingall 4 N possiblepulses. One can ask: what if each error term is supported on at most l neighboring qubits? Here we discuss another symmetrization lemma to lower the number of pulses to just 4 l multiqubit pulses, independent of N (benefiting from the parallelism in applying pulses). The construction of the pulses is as follows: list all 4 l possible Pauli operators on the first l qubits. Now extend each pulse by periodically repeating the Pauli 89 operator on each successive set of l qubits such that each of the pulses has the same Pauli on qubits i and i +l. A special case is 1-local noise on N qubits (l = 1, so the period is one). As we saw we can average out all the errors by considering just four pulses: I ⊗N ,X ⊗N ,Y ⊗N ,Z ⊗N . Clearly we can similarly generalize this construction to higher dimensional lat- tices: Lemma 2. (Symmetrization lemma: Local noise) Assume that the support of each error term on a D-dimensional regular lattice is contained in a hypercube of size l D qubits. Then with 4 l D pulses, independent of the size of the lattice, all the error terms can be averaged out. Applying this result to our 2-dimensional square lattice, we only need to con- struct the sequences by considering all possible Pauli operators on the qubits in squares of length l (there are 4 l 2 corresponding pulses), and then cover the lattice with parallel use of these patterns. For example, for r = 1 (bath operators from the interaction Hamiltonian, B α i and B β j , commute unless qubits i and j are nearest neighbors) all the errors of order δt 3 are strings of 3 Pauli operators. We can define a square of 3× 3 qubits, and apply all Pauli operators that commute with all the XX interactions in that square (8 independent interaction of such type). At most 4 9 /2 8 steps are needed (all the possible Pauli’s operators on 9 qubits that commute with 8 independent Pauli terms). Therefore, assuming locality for the bath, all error terms of order δt 3 can be removed with a number of pulses independent of the size of the grid. Of course this is a worst case analysis (assuming that any kind of three-Pauli error can be 90 generated), but when needed one can work out the details of the generated errors and find the pulses to remove them, resulting in much shorter sequences. 4.6 Error suppression Nowweanalyzetheeffectofthesimulatedenergygaponthenoise. Forsimplic- ity, rather than adding the simulation of the desired computation to this picture, we just repeat the process of simulating hH p δt 3 a total of k times (a quantum memory). Each repetition of the simulation represents a time step Δt =δt 3 . We recall that the combination of DD and simulation described earlier generates the following effective Hamiltonian in one cycle: e −i[hHp+H B /δt 2 −gV (δt)]Δt , (4.58) where h is the strength of the simulated Hamiltonian and V (δt) = P ∞ a=0 V a δt a is the effective error Hamiltonian. Each termV a can be decomposed according to the location of the system part: V a = P i,a V a i , and g is the overall scale [normalizing V (δt)]. The ratio between h and g quantifies the energy penalty as we will see later. We can think of the effective Hamiltonian as a constant Hamiltonian hH p + H B /δt 2 −gV (δt), that is on for time kΔt. The goal is to see how much error sup- pression we get from this process, assuming that we start in the ground subspace. To do so we bound the quantity: kU(kΔt)P−U ideal (Δt)Pk, (4.59) 91 where P is the projector onto the codespace, and U(t) is the evolution gener- ated by H(t) = hH P (t) +H B /δt 2 +V (t). This difference bounds how much the noisy evolution can make the state deviate from the ideal evolution represented by U ideal (t)≡e −i(hHp+H B /δt 2 )kt . To bound this difference, we start by moving to the interaction picture defined by the base Hamiltonian hH P +H B /δt 2 . We denote the evolution corresponding to this base Hamiltonian byU P (t). Moving to the interaction picture with respect to the base Hamiltonian we have: V I (t) = U † P (t)V (t)U P (t) , (4.60) U I (t) = U † P (t)U(t) , (4.61) where i ˙ U I (t) =V I (t)U I (t) . (4.62) Integrating this we have U † I (kΔt) =I +i Z kΔt 0 U † I (t)V I (t)dt . (4.63) We note that kU(kΔt)P−U P (kΔt)Pk = (I−U † I (kΔt))P = Z kΔt 0 dU † I (t) dt dtP = Z kΔt 0 U † I (t)V I (t)dtP . (4.64) 92 Following [49], we can define: F (t) = Z t 0 V I (τ)dτP . (4.65) Integrating by parts, Eq. (4.64) becomes: kU(kΔt)P−U P (kΔt)Pk = U † I (kΔt)F (kΔt)−i Z kΔt 0 U † I (τ)V I (τ)F (τ)dτ ≤ kF (kΔt)k + Z kΔt 0 U † (τ)V (τ)U P (τ)F (τ)dτ ≤ kF (kΔt)k + Z kΔt 0 kV (τ)kkF (τ)kdτ , (4.66) where we used the triangle inequality and the unitary invariance of the operator norm. F (t)quantifiestheaveragingoutoftheinteractionHamiltonianbytherotations induced by the penalty Hamiltonian (in the interaction picture), and the reduction of this term implies that the total evolution becomes closer to the ideal evolution (the difference between the actual and ideal evolutions goes to zero ifkFk goes to zero [49, 69]). To bound this term, we can evaluate: F (t 0 ) =g Z t 0 0 U † P (τ) X i,a V a i δt a U P (τ)Pdτ . (4.67) 93 We denote by c a i the number of stabilizer generators in the simulated Hamil- tonian H p that anticommute with the error V a i . Using this, and also the fact that H p P = 0 P, we have: F (t 0 ) =g Z t 0 0 e +i(H B /δt 2 )τ (4.68) × X i,a V a i δt a e −i(H B /δt 2 )τ e −i2h 0 c a i τ Pdτ . Breaking this sum into two parts depending on whether c j i is zero or nonzero, and then integrating by parts, we get: kF (kΔt)k≤ g h X i,a c a i 6=0 1 c a i (2 0 ) (2kV a i δt a k +kΔt [V a i ,H B ]δt a−2 ) +gkΔt|| X i,a c a i =0 V a i δt a || . (4.69) It is important to note that the first term, with the detectable errors (with c a i ≥ 1), is suppressed by a factor of g/h. So for large values of h, we get error suppression, and the total evolution becomes closer to the one with no error. This error suppression is similar to the error suppression we get from an ideal implemen- tation ofH p (rather than its simulation). This becomes clearer when we perform a similar calculation for the HamiltonianH p with energy penaltyE p , in the presence of a local system-bath interaction Hamiltonian λH SB for a duration of T 0 . Similar to the simulation case, using Eq. (4.66), we can bound e −i(EpHp+H B +λH SB )T 0 P−e −i(EpHp+H B )T 0 P ≤kF ideal (T 0 )k + Z T 0 0 kλH SB kkF ideal (τ)kdτ , (4.70) 94 where we have defined F ideal (t) as: F ideal (t) = Z t 0 e +i(EpHp+H B )τ ×(λH SB )e −i(EpHp+H B )τ Pdτ . (4.71) Decomposing the interaction Hamiltonian according to the location of the sys- tem part, H SB = P i h i SB , we get: kF ideal (t)k≤ λ E p X i 1 e i (2 0 ) (2 h i SB +t [h i SB ,H B ] ) , (4.72) where e i denotes the number of stabilizer generators in the ideal H p that anti- commute with the error term h i SB . In this case, by construction, we always have e i 6= 0. This shows that the bound in Eq. (4.70) actually contains a suppression factor of λ/E p . Clearly, in the simulated case, g/h plays the role of λ/E p . Incontrasttotheidealcasewhere∀i :e i 6= 0, inthesimulationnotalltheterms of the effective error are suppressed by the effective Hamiltonian. For the chosen H p , the only errors that commute with all the stabilizers (and so have c a i = 0) are (I) loops of X around A v operators and loops of Z around B p operators; and (II) logical operators, i.e., chains connecting boundaries. These errors are not suppressed by this mechanism, as can be seen from the second term in the bound. Fortunately, the type I errors are not (too) destructive, as they are just product of the stabilizers of the code. So, these terms in the effective error do not cause logical errors. The only effect they can have is changing the strength of each simulated stabilizer slightly (changing h to h±δh). The first error of this type is 95 4-local and so occurs at orderO(δt 4 ). Comparing to the Hamiltonian itself, which occurs at orderO(δt 3 ), it is at least a factor of order δt weaker. Also, the effect of these errors is expected to average out, as the sign of the errors changes, and so the effective δh should be small. The situation is different for the logical errors. If the distance of the code is d, these errors happen in O(δt d ) or higher, which is small for large d. While this error is small, to have an arbitrary long computation it is necessary to correct possible errors before they accumulate into logical errors. This is done by active error correction. 4.7 Summary and Conclusions High-weight Hamiltonians are frequently used in designing quantum algo- rithms, especially when the goal is to provide protection against noise. Implement- ing such high-weight Hamiltonians is experimentally challenging. One approach is to simulate such interactions using resources that are easier to implement. How- ever, this simulation procedure itself can spread the noise and convert it to some effective geometrically correlated noise on the simulated system. In this work we proposed combining techniques from dynamical decoupling and quantum simulation to simulate high-weight Hamiltonians such that the sim- ulated Hamiltonians are stronger than the new effective noise. The ratio of the strength between the simulated Hamiltonian and the effective noise depends on the strength of the original noise on the resources used and also on the specific type of DD and simulation techniques applied. The spread of the effective noise on the simulated system depends on the locality of the bath and the system-bath interaction Hamiltonians. The reasonable assumption that these Hamiltonians 96 are geometrically local, guarantees that the dominant terms in the effective noise Hamiltonian spread in a geometrically local region. Topological codes are expected to perform well in the presence of these types of geometrically correlated errors. For this reason we chose the surface code Hamiltonian and showed how to simulate the time dependent deformation of this Hamiltonian to perform universal quantum computation. Our analysis provided the details specific to this Hamiltonian, but the method is general. Similar to any other scheme performing universal quantum computation on a 2D grid, active error correction is necessary to guarantee fault tolerance. But as we showed explicitly, an energy gap is maintained during the simulation against the strongest errors generated in the process. The presence of this energy gap reduces the number of cycles of active error correction necessary during the simulation of the computation. We expect that the methods proposed in this work can also be used to simulate the Hamiltonians that use subsystem codes to reduce the non-locality, either for surface code Hamiltonians [126] or general Hamiltonian-based quantum computa- tion [69]. An interesting problem for future work is to consider other types of resources and simulation methods. One example is to consider simulation using perturba- tive gadgets and investigating methods to reduce the effect that local noise on a Hamiltonian has on the effective Hamiltonian in the low energy spectrum. In all the constructions in this work, we only used the simplest form of DD and simulation techniques. One can expect to gain performance improvements by using more complex DD pulses and simulation techniques. It then becomes more important to consider the effect of the imperfection and noise on the pulse sequences and their timing. 97 More generally, it is interesting to design methods that are natively optimized to generate the largest ratio between the strength of the simulated system Hamil- tonian and the strength of the effective noise. 98 Chapter 5 Conclusion In this dissertation, we extended the JFS result [48] to general Markovian dynamics. We have also shown that the same results hold even if the master equation is not in Lindblad form. We also have presented general conditions for quantum error suppression for Hamiltonian-based quantum computation using subsystem codes. This involves encoding the Hamiltonian performing the computation using an error detecting subsystemcodeandtheadditionofapenaltytermthatcommuteswiththeencoded Hamiltonian. The scheme is general and includes the stabilizer formalism of both subspace and subsystem codes as special cases. We have also derived performance boundsandhaveshownthatcompleteerrorsuppressionresultsinthelargepenalty limit. To illustrate the power of subsystem-based error suppression, we have intro- duced fully 2-local constructions for protection against local errors of the swap gate of adiabatic gate teleportation and the Ising chain in a transverse field. To simulate high-weight Hamiltonians, we have proposed combining techniques from dynamical decoupling and quantum simulation so that the simulated Hamil- tonians are stronger than the effective noise. We have shown that during the simulation an effective energy gap suppresses the strongest errors generated in the process. 99 Appendix A Appendix A.1 Proof of the subsystem error detection con- dition, Eq. (1.31) The channelE ={E i } is detectable by a code C if there exists a measurement that unambiguously reveals whether or not an error took place afterE acts on a state| ¯ ψ α i∈C,∀α. For subsystem codes, states in C are allowed to change by a gauge transformation. To show that Eq. (1.31) is sufficient for error detection we rewrite it as E i | ¯ ψ α i =G i ⊗I B | ¯ ψ α i +|φ ⊥ α,i i∀i,α , (A.1) for some (unnormalized) state|φ ⊥ α,i i∈C ⊥ . The action of the channel is then E(| ¯ ψ α ih ¯ ψ α |) = X i E i | ¯ ψ α ih ¯ ψ α |E † i = X i (G i ⊗I B )| ¯ ψ α ih ¯ ψ α |(G † i ⊗I B ) +|φ ⊥ α,i ih ¯ ψ α | (G † i ⊗I B ) + h.c. +|φ ⊥ α,i ihφ ⊥ α,i | . (A.2a) Since| ¯ ψ α i∈ C and C = A⊗B (where A is the gauge subsystem and B is the information subsystem), we have| ¯ ψ α i =| ¯ ψ A,α i⊗| ¯ ψ B,α i, where| ¯ ψ A,α i∈ A and | ¯ ψ B,α i∈B. Thus the first term in Eq. (A.2a) becomes P i (G i ⊗I B )| ¯ ψ α ih ¯ ψ α |(G † i ⊗ I B ) =ρ A,α ⊗| ¯ ψ B,α ih ¯ ψ B,α |, where ρ A,α = P i G i | ¯ ψ A,α ih ¯ ψ A,α |G † i . 100 NowconsidertheobservableM≡P C −P C ⊥; ithaseigenvalue +1(−1)forstates in (orthogonal to) the codespace. Thus measuring M is equivalent to detecting whether the measured state is in C or in C ⊥ . Clearly, measuring M annihilates the off-diagonal term|φ ⊥ α,i ih ¯ ψ α | (G † i ⊗I B ) and its Hermitian conjugate. The post- measurement states are E(| ¯ ψ α ih ¯ ψ α |) M 7−→ 1 p + P C E(| ¯ ψ α ih ¯ ψ α |)P C = 1 p + ρ A,α ⊗| ¯ ψ B,α ih ¯ ψ B,α | with prob. p + = Tr(ρ A,α )h ¯ ψ B,α | ¯ ψ B,α i 1 p − P C ⊥E(| ¯ ψ α ih ¯ ψ α |)P C ⊥ = 1 p − |φ ⊥ α,i ihφ ⊥ α,i | with prob. p − =hφ ⊥ α,i |φ ⊥ α,i i . (A.3) Thus, if after measuring M we obtain the outcome +1 corresponding to the projector P C , the state is projected to the original information subsystem state | ¯ ψ B,α ih ¯ ψ B,α | up to an irrelevant transformation on the gauge subsystem. In this case no error took place. On the other hand, if the outcome−1 corresponding to the projector P C ⊥ is obtained, then we know that an error has happened. This shows that Eq. (1.31) is sufficient for error detection using subsystem codes. 101 Reference List [1] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, “Quantum Computation by Adiabatic Evolution,” arXiv:quant-ph/0001106, Jan. 2000. [Online]. Available: http://arxiv.org/abs/quant-ph/0001106 [2] D. 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Abstract (if available)
Abstract
In this dissertation, we present general conditions for quantum error suppression for Hamiltonian-based quantum computation using subsystem codes. This involves encoding the Hamiltonian performing the computation using an error detecting subsystem code and the addition of a penalty term that commutes with the encoded Hamiltonian. We illustrate the power of subsystem-based error suppression with several examples of two-local constructions for protection against local errors, which circumvent an earlier no-go theorem about two-local commuting Hamiltonians. ❧ We also discuss the generalization of the quantum error suppression results of Jordan, Farhi, and Shor to arbitrary Markovian dynamics. In this setting, we show that it is possible to suppress the initial decay out of the encoded ground state with an energy penalty strength that grows only logarithmically in the system size, at a fixed temperature. ❧ Finally, we discuss how to simulate high-weight Hamiltonians. Such a simulation can convert local noise on the original Hamiltonian into undesirable non-local noise on the simulated Hamiltonian. We show how starting from two-local Hamiltonian in the presence of non-Markovian noise, a desired computation can be simulated as well as protected using fast pulses, while maintaining an energy gap against the errors created in the process.
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Marvian Mashhad, Milad
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Core Title
Protecting Hamiltonian-based quantum computation using error suppression and error correction
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Viterbi School of Engineering
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Doctor of Philosophy
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Electrical Engineering
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01/19/2018
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10/20/2017
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Lidar, Daniel A. (
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Tags
quantum computing
quantum error correction
quantum simulation