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Error correction and cryptography using Majorana zero modes

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Error correction and cryptography using Majorana zero modes

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Content Error correction and cryptography using Majorana zero modes
by
Sourav Kundu
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 2024
Copyright 2024 Sourav Kundu

Acknowledgements
I would like to express my heartfelt gratitude to my advisor, Ben Reichardt, for his unwavering
support, expertise, and insightful feedback throughout my research journey.
I am grateful to my qualifying and defense committee members: Todd Brun, Rosa Di Felice,
Daniel Lidar and Eli Levenson-Falk for their valuable suggestions and thoughtful perspectives that
enhanced the quality of this work.
Special thanks go to Yi-Kai Liu and Atul Mantri for engaging discussions on one-time property
and cryptography. I am also thankful to my teammates - Rui Chao, Prithviraj Prabhu, Yuanjia
Wang, Edward Kim, and Ammar Babar - for numerous enlightening conversations.
My heartfelt debt of gratitude extends to my family and friends for their understanding,
encouragement, and unwavering support through all these years.
Finally, I acknowledge the contributions of all the researchers and scholars whose work has
paved the way for this study. Your collective efforts serve as a constant source of inspiration and
knowledge.
My research have been supported by the following grants:
NSF grant CCF-1254119
ARO grant W911NF-12-1-0541
MURI Grant FA9550-18-1-0161
U.S. Department of Energy, Office of Science, National Quantum Information Science
Research Centers, Quantum Systems Accelerator
ii

Table of Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Majorana zero modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Current architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.3 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.4 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 From braiding to measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 From perfect to imperfect MZMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5.1 Majorana operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5.2 Logical degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5.3 Code notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5.4 Majorana codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Motivation and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Chapter 2: Majorana qubit codes that also correct odd-weight errors . . . . . . . . . . . . 18
2.1 Error correction principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 Tetrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 Bosonic and fermionic errors in tetrons . . . . . . . . . . . . . . . . . . . . . 21
2.1.3 Bosonic error correction in tetron architecture . . . . . . . . . . . . . . . . . 21
2.1.4 Challenges of fermionic error correction . . . . . . . . . . . . . . . . . . . . 22
2.1.5 Proposed fermionic error correction principle . . . . . . . . . . . . . . . . . 23
2.2 Review of bosonic codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 B 7→ F codes: Fermionic codes from bosonic codes . . . . . . . . . . . . . . . . . . 24
2.3.1 Recipe for fermionic code construction . . . . . . . . . . . . . . . . . . . . . 24
2.3.2 Code distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.3 Decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.4 Error correction on small codes . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.5 Error correction on color codes . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.6 Error correction on surface codes . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
iii

2.4.1 Code capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.2 Fault tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Chapter 3: Majorana subsystem qubit codes that also correct odd-weight errors . . . . . . 38
3.1 BC 7→ FS codes: Fermionic subsystem codes from bosonic and classical codes . . . 39
3.1.1 Recipe for fermionic code construction . . . . . . . . . . . . . . . . . . . . . 39
3.1.2 Code distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.3 Decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.4 Example 1: J10, 1, 2, df = 3K subsystem code . . . . . . . . . . . . . . . . . 43
3.1.5 Example 2: J12, 1, 3, df = 3K subsystem code . . . . . . . . . . . . . . . . . 43
3.1.6 Example 3: J14, 1, 4, df = 3K subsystem code . . . . . . . . . . . . . . . . . 44
3.1.7 Code capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.8 Fault tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Alternate constructions for fermionic subsystem codes . . . . . . . . . . . . . . . . 51
Chapter 4: One-time memory from isolated Majorana islands . . . . . . . . . . . . . . . . . 57
4.1 Majorana operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Single octon as OTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Octon cluster as OTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.1 Bit availability without leakage . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.2 Bit availability with leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Error correction on cluster OTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.1 Data error rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.2 Error correcting codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.3 Bacon-Shor code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.4 Other codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5 Dishonest recipient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.6 1/n OTM and (n − 1)/n OTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.6.1 Example: 1
3 OTM on J7, 1, 3K code . . . . . . . . . . . . . . . . . . . . . . . 74
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Appendix A: Measurement architecture for B 7→ F codes with and without coherent links . 83
A.1 One-sided tetrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
A.2 Two-sided tetrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
A.3 Two-sided tetrons and coherent links . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Appendix B: B 7→ FS codes: Fermionic subsystem codes from bosonic codes . . . . . . . . 89
B.1 Recipe for fermionic code construction . . . . . . . . . . . . . . . . . . . . . . . . . 89
B.2 Code distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
B.3 Decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
B.4 Partitioning scheme for color codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
B.5 Example 1: J14, 1, 2, df = 3K subsystem code . . . . . . . . . . . . . . . . . . . . . . 92
B.6 Example 2: J38, 1, 1, df = 9K subsystem code . . . . . . . . . . . . . . . . . . . . . . 92
B.7 Example 3: J74, 1, 3, df = 9K subsystem code . . . . . . . . . . . . . . . . . . . . . . 93
B.8 Scalability of fermionic error correction in B 7→ FS codes . . . . . . . . . . . . . . 95
B.8.1 Starting from mixed error configuration . . . . . . . . . . . . . . . . . . . . 96
B.8.2 From mixed error to fermionic error configuration . . . . . . . . . . . . . . 96
B.8.3 Decoupling disconnected errors . . . . . . . . . . . . . . . . . . . . . . . . . 96
B.8.4 Even intersection between errors and plaquettes . . . . . . . . . . . . . . . 97
B.8.5 Fermionic error correction capacity of partitions . . . . . . . . . . . . . . . 97
B.9 Code capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
iv

B.10 Fault tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Appendix C: Arbitrary measurements on one octon . . . . . . . . . . . . . . . . . . . . . . 106
C.1 Basics of weak measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
C.2 Sequence of measurement rounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
C.3 Information about classical bits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
C.4 Parity knowledge after 1 round of measurement . . . . . . . . . . . . . . . . . . . . 109
C.5 Parity knowledge after n rounds of measurement . . . . . . . . . . . . . . . . . . . 112
C.6 Variation in state integrity due to one round of weak measurement . . . . . . . . . 112
C.7 Variation in state integrity due to multiple rounds of weak measurements . . . . . 114
C.8 Optimized sequence of measurement rounds . . . . . . . . . . . . . . . . . . . . . . 115
Appendix D: Arbitrary measurements on cluster . . . . . . . . . . . . . . . . . . . . . . . . 118
D.1 Individual bit availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
D.2 Combined bit availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Appendix E: Dishonest OTM recipient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
E.1 Malicious strategy on 1
2 OTM using Steane code . . . . . . . . . . . . . . . . . . . 121
E.2 Malicious strategy on 1
2 OTM using Golay code . . . . . . . . . . . . . . . . . . . . 123
E.3 Malicious strategy on 1
3 OTM using Steane code . . . . . . . . . . . . . . . . . . . 124
v

List of Tables
3.1 This table compares the code capacity and fault-tolerant pseudothreshold between
non-subsystem fermionic codes and currently proposed subsystem fermionic codes.
The codes are compared at various values of noise bias η, which is the ratio
between fermionic error probability and bosonic error probability. We observe
that the J10, 1, 2, df = 3K code has the shortest syndrome-measurement sequence
that is fault-tolerant against a single error, and thus has the best fault-tolerant
pseudothreshold among these codes. We observe that the threshold of subsystem
codes exceed their non-subsystem counterparts. For the above codes and bias values,
the threshold improvement percentage ranges from 10% to as much as 84%. The
fault-tolerant sequences for these codes are provided in Section 2.4.2 and Section 3.1.8. 39
C.1 There are eight ways in which pure states can be damaged by a pair of weak
measurements. Normalization constants are not shown in the final damaged states. 113
vi

List of Figures
1.1 The above figure illustrates a tetron architecture. A single tetron comprises two
topological superconducting nanowires that host four Majorana zero modes at their
ends, and they are connected by a superconductor. . . . . . . . . . . . . . . . . . . 1
1.2 (a) An electron can be considered to be equivalent to a hole when it is surrounded
by Cooper pairs in a superconductor. (b) The Bogoliubov - de Gennes formalism of
superconductivity necessitates electron-hole symmetry in energy. (c) Each Majorana
mode is a superposition of an electron and a hole, and hence they are bound to the
middle of the band, at the zero energy state. . . . . . . . . . . . . . . . . . . . . . 2
1.3 (a) This shows two equivalent braids performed on a pair of MZMs. They undergo
a unitary transformation when moved around each other, and hence they are
non-Abelian anyons. (In contrast, Abelian anyons merely pick up a phase factor
when looped around each other.) This unitary transformation is resilient to small
disturbances, as the transformation depends only on the braid topology. For the
same reason, nontrivial braids cannot be performed in 3-dimensional space, since
such a path can always be continuously deformed into a trivial path. Nontrivial
braids can be performed when constrained to 2 dimensional planes, or 1 dimensional
wire networks.
(b) This braiding sequence corresponds to CNOT gate between |ψ1⟩ and |ψ2⟩,
where time moves from left to right. Here, |ψ1⟩ is densely encoded into the MZMs
γ1, γ2, γ3, γ4 and |ψ2⟩ is densely encoded into the MZMs γ3, γ4, γ5, γ6. The dense
encoding scheme is provided in Ref. [DFN15]. . . . . . . . . . . . . . . . . . . . . 3
1.4 The figure illustrates measurements for the X, Y, Z operators. it also gives an
example of a joint measurement spanning multiple tetron islands. . . . . . . . . . . 9
1.5 (a) A CNOT gate from |ψ1⟩ to |ψ2⟩ is achieved using an ancilla, measurements, and
classical tracking. (b) A Clifford gate that utilizes a phase gate S, a Hadamard gate
H, and a CNOT gate, is implemented using an ancilla, measurements, and classical
tracking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
vii

1.6 The left figure illustrates a stabilizer supported over four Z operators on 4 tetrons.
Each yellow square corresponds to a tetron island which supports the stabilizer. A
tetron has 4 MZMs at 4 corners, similar to the inset of Fig. 1.1. The green line in a
yellow square denotes a Majorana operator in a tetron island, which is supported on
2 MZMs at the endpoints of the line. For example, the green line on the top edge
of square denotes the Majorana operator Z = γaγb. The middle figure illustrates a
modified stabilizer where one tetron operator is modified. The red line denotes a
modified Majorana operator, which is supported on 2 MZMs at the endpoints of the
line. For example, the red line on the bottom edge of square denotes the Majorana
operator γcγd. The right figure shows that the product of these two stabilizers
is equivalent to the tetron stabilizer γaγbγcγd. Thus, the 4-MZM tetron operator
belongs to the stabilizer group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 The top row A illustrates the 4 stabilizers of the 5-qubit Pauli code. Each yellow
square corresponds to a tetron island which supports the stabilizer. A green line in
a yellow square denotes a 2-MZM operator in a tetron island. The middle row B
shows a modified set of stabilizers, derived from the 3 stabilizers directly above them
in row A. Each red line denotes a modified Majorana operator, which is supported
on 2 MZMs at endpoints of the line. The bottom row A × B shows 3 members
of the stabilizer group, which are formed by the product of the stabilizer pair in
row A and row B directly above it. Here each stabilizer is a set of tetrons, and it
corresponds to the non-zero locations of a matrix row. This matrix is the parity
matrix of [5, 2, 3] classical code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.8 We use a CSS quantum code such as the Bacon-Shor code to encode two classical bits
in a one-time memory. The X stabilizers shown as yellow row stabilizers protect one
bit in the top layer, and the Z stabilizers shown as blue column stabilizers protect
one bit in the bottom layer. Each qubit in this code corresponds to multiple smaller
one-time memory devices, and each small one-time memory device corresponds to a
single octon island. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 (a) A tetron hosts four MZMs at the locations a, b, c, d. (b) A bosonic error affects
even number of MZMs on the tetron. (c) A fermionic error affects odd number
of MZMs on the tetron. (d) R and R
′ are two complementary sets of weight-2
operators in a tetron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 (a) The figure shows the stabilizer generators of the J14, 1, df = 6K fermionic code.
The three colored plaquettes correspond to three X stabilizers and three Z stabilizers,
similar to the Steane color code. In addition, the seven tetrons also belong to the
stabilizer generator group, and they are shown as yellow squares. Although the
tetrons belong to the stabilizer generator group, they are not directly measurable.
(b) The figure shows the measurable stabilizers of the J14, 1, df = 6K fermionic
code, derived from the J7, 1, 3K bosonic color code. Each stabilizer plaquette is
supported on four Majorana operators at its four vertices. These operators are
defined according to Fig. 2.1(d). (c) The figure demonstrates that the stabilizer
group contains one tetron operator. Similarly, all seven tetrons belong to the
stabilizer group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
viii

2.3 (a) The figure shows the stabilizer generators of the J10, 1, df = 6K fermionic code,
derived from the J6, 1, 3K Pauli stabilizer code. The set 0 stabilizers correspond to
the Pauli code, and the stabilizers in sets 1 to 5 lead to the inclusion of 5 tetrons
in the stabilizer group. (b) The figure shows the measurable stabilizers of the
J12, 1, df = 6K fermionic code, derived from the J6, 1, 3K Pauli stabilizer code. The
set 0 stabilizers correspond to the Pauli code, and the stabilizers in sets 1 to 6 lead
to the inclusion of 6 tetrons in the stabilizer group. . . . . . . . . . . . . . . . . . 27
2.4 The figure illustrates an optimized syndrome measurement sequence for stabilizers
in sets 1 to 19 of J38, 1, df = 10K code. Each pink striped plaquette corresponds to
a stabilizer supported on Z
′ at the vertex marked by a red circle, and supported on
Z at all other vertices of that plaquette. Similarly, each green checkered plaquette
corresponds to a stabilizer supported on X′ at the vertex marked by a red circle,
and supported on X at all other vertices of that plaquette. . . . . . . . . . . . . . 29
2.5 The figure illustrates an optimized syndrome measurement sequence for stabilizers
in sets 1 to 37 of J74, 1, df = 14K code. Each pink striped plaquette corresponds to
a stabilizer supported on Z
′ at the vertex marked by a red circle, and supported on
Z at all other vertices of that plaquette. Similarly, each green checkered plaquette
corresponds to a stabilizer supported on X′ at the vertex marked by a red circle,
and supported on X at all other vertices of that plaquette. . . . . . . . . . . . . . 29
2.6 (a) The figure shows the stabilizers of the J25, 1, 5K bosonic code, which are the
same as the set 0 stabilizers of the J50, 1, df = 10K code. The pink striped stabilizer
plaquettes are supported on Z operators at its vertices, while the green checkered
stabilizer plaquettes are supported on X operators at its vertices. The X stabilizers
of set 0 require 2 steps for syndrome extraction, similarly the Z stabilizers of set
0 require another 2 steps for syndrome extraction. (b) The figure illustrates an
optimized four-step syndrome measurement sequence for stabilizers in sets 1 to 25
of J50, 1, df = 10K code. Each pink striped plaquette corresponds to a stabilizer
supported on Z
′ at the vertex marked with a red circle, and supported on Z
at all other vertices of that plaquette. Similarly, each green checkered plaquette
corresponds to a stabilizer supported on X′ at the vertex marked with a red circle,
and supported on X at all other vertices of that plaquette. . . . . . . . . . . . . . 30
2.7 The figure shows the variation of pseudothreshold with noise bias for J10, 1, df = 6K,
J12, 1, df = 6K, J14, 1, df = 6K, J38, 1, df = 10K and J74, 1, df = 14K codes. . . . . . 32
2.8 Code capacity logical error plots for J10, 1, df = 6K and J12, 1, df = 6K fermionic
codes at bias values η = 0.1, 1, 10. The 95% confidence interval bars are smaller
than the marker size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.9 Code capacity logical error plots for J14, 1, df = 6K, J38, 1, df = 10K and J74, 1, df =
14K fermionic codes at bias values η = 0.1, 1, 10. The 95% confidence interval bars
are smaller than the marker size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.10 Figures (a) to (i) demonstrate a sequence of fault-tolerant measurements for the
J10, 1, df = 6K fermionic code. This sequence can tolerate one bosonic error or one
fermionic error. The tetron operators follow the same notation as Fig. 2.1(d). . . 35
ix

2.11 The logical error plots for the fault-tolerant implementation of the J10, 1, df = 6K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively.
The graph also shows the 95% confidence intervals. . . . . . . . . . . . . . . . . . 35
2.12 Figures (a) to (n) demonstrate a sequence of fault-tolerant measurements for the
J12, 1, df = 6K fermionic code. This sequence can tolerate one bosonic error or one
fermionic error. The tetron operators follow the same notation as Fig. 2.1(d). . . 36
2.13 The logical error plots for the fault-tolerant implementation of the J12, 1, df = 6K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively.
The graph also shows the 95% confidence intervals. . . . . . . . . . . . . . . . . . 36
2.14 Figures (a) to (p) demonstrate a sequence of fault-tolerant measurements for the
J14, 1, df = 6K fermionic code. This sequence can tolerate one bosonic error or one
fermionic error. The tetron operators follow the same notation as Fig. 2.1(d). . . 37
2.15 The logical error plots for the fault-tolerant implementation of the J14, 1, df = 6K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively.
The graph also shows the 95% confidence intervals. . . . . . . . . . . . . . . . . . 37
3.1 (a) This figure shows the 7 stabilizer generators of the J10, 1, 2, df = 3K Majorana
fermionic subsystem code in the rows A and B. The third row shows the stabilizer
formed by the product of corresponding generators in the first and second rows.
Each stabilizer in the third row is a set of tetrons, corresponding to the rows of
parity matrix H. (b) This figure shows the logical qubit and the gauge qubits of the
J10, 1, 2, df = 3K Majorana fermionic subsystem code. Each gauge qubit corresponds
to a row of the generator matrix G. (c) This figure shows the classical parity matrix
H and the generator matrix G of the [5, 2, 3] classical code. The tetron numbering
scheme shown in this figure is used for the matrix columns. . . . . . . . . . . . . . 45
3.2 (a) This figure shows the 8 stabilizer generators of the J12, 1, 3, df = 3K Majorana
fermionic subsystem code in the rows A and B. The third row shows the stabilizer
formed by the product of corresponding generators in the first and second rows.
Each stabilizer in the third row is a set of tetrons, corresponding to the rows of
parity matrix H. (b) This figure shows the logical qubit and the gauge qubits of the
J12, 1, 3, df = 3K Majorana fermionic subsystem code. Each gauge qubit corresponds
to a row of the generator matrix G. (c) This figure shows the classical parity matrix
H and the generator matrix G of the [6, 3, 3] classical code. The tetron numbering
scheme shown in this figure is used for the matrix columns. . . . . . . . . . . . . . 46
3.3 (a) This figure shows the 9 stabilizer generators of the J14, 1, 4, df = 3K Majorana
fermionic subsystem code in the rows A and B. The third row shows the stabilizer
formed by the product of corresponding generators in the first and second rows.
Each stabilizer in the third row is a set of tetrons, corresponding to the rows of
parity matrix H. (b) This figure shows the logical qubit and the gauge qubits of the
J14, 1, 4, df = 3K Majorana fermionic subsystem code. Each gauge qubit corresponds
to a row of the generator matrix G. (c) This figure shows the classical parity matrix
H and the generator matrix G of the [7, 4, 3] classical code. The tetron numbering
scheme shown in this figure is used for the matrix columns. . . . . . . . . . . . . . 47
x

3.4 The figure shows the variation of pseudothreshold with noise bias for J10, 1, 2, df = 3K,
J12, 1, 3, df = 3K, J12, 1, 1, df = 6K and J14, 1, 4, df = 3K subsystem codes. . . . . . 48
3.5 Code capacity logical error plots for currently proposed fermionic subsystem codes
as well as previously proposed non-subsystem fermionic codes. The top row shows
the plot for the 5 tetron codes, the middle row shows the plot for the 6 tetron
codes, and the bottom row shows the plot for the 7 tetron codes. For each code,
the logical error rates are evaluated at bias values η = 0.1, 1, 10. In these plots, the
95% confidence interval bars are smaller than the marker size. . . . . . . . . . . . 49
3.6 Figures (a) to (h) demonstrate a sequence of fault-tolerant measurements for the
J10, 1, 2, df = 3K fermionic subsystem code. This sequence can tolerate one bosonic
error or one fermionic error. The tetron operators follow the same notations as
Fig. 2.1(d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.7 The logical error plots for the fault-tolerant implementation of the J10, 1, 2, df = 3K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively.
The graphs also show the 95% confidence intervals. . . . . . . . . . . . . . . . . . 52
3.8 Figures (a) to (h) demonstrate a sequence of fault-tolerant measurements for the
J12, 1, 1, df = 6K fermionic subsystem code. This sequence can tolerate one bosonic
error or one fermionic error. The tetron operators follow the same notations as
Fig. 2.1(d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.9 The logical error plots for the fault-tolerant implementation of the J12, 1, 1, df = 6K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively.
The graphs also show the 95% confidence intervals. . . . . . . . . . . . . . . . . . 53
3.10 Figures (a) to (h) demonstrate a sequence of fault-tolerant measurements for the
J12, 1, 3, df = 3K fermionic subsystem code. This sequence can tolerate one bosonic
error or one fermionic error. The tetron operators follow the same notations as
Fig. 2.1(d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.11 The logical error plots for the fault-tolerant implementation of the J12, 1, 3, df = 3K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively.
The graphs also show the 95% confidence intervals. . . . . . . . . . . . . . . . . . 54
3.12 Figures (a) to (h) demonstrate a sequence of fault-tolerant measurements for the
J14, 1, 4, df = 3K fermionic subsystem code. This sequence can tolerate one bosonic
error or one fermionic error. The tetron operators follow the same notations as
Fig. 2.1(d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.13 The logical error plots for the fault-tolerant implementation of the J14, 1, 4, df = 3K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively.
The graphs also show the 95% confidence intervals. . . . . . . . . . . . . . . . . . 55
xi

3.14 Fault-tolerance logical error plots for currently proposed fermionic subsystem codes
as well as previously proposed non-subsystem fermionic codes. The top row shows
the plot for the 5 tetron codes, the middle row shows the plot for the 6 tetron codes,
and the bottom row shows the plot for the 7 tetron codes. For each code, the logical
error rates are evaluated at bias values η = 0.1, 1, 10. The graphs also show the 95%
confidence intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.1 Figure (a) shows a one-sided octon. Figure (b) shows a two-sided octon. Figure
(c) illustrates the three qubits present in an octon. The first qubit stores the first
classical bit, the second qubit stores the second classical bit, and the third qubit
stores the measurement basis. Figure (d) shows an X measurement on the one-sided
octon. Figure (e) shows a Z measurement on the two-sided octon. Observe that a
coherent link (floating 1D topological superconductor) is necessary for measuring
MZMs on opposite sides. Figure (f) shows an X measurement on the two-sided octon. 60
4.2 Alice randomly chooses parity of the octon halves, and accordingly decides to store
the classical bits in the vertical operators or the horizontal operators. . . . . . . . . 61
4.3 Figure (a) shows the classical repetition code corresponding to the two X stabilizers
of a Bacon-Shor code. It also shows three copies of the logical X operator highlighted
in brown color. Figure (b) shows the classical repetition code corresponding to the
two Z stabilizers of a Bacon-Shor code. It also shows three copies of the logical
Z operator highlighted in blue. Figure (c) shows that each node of the 3 × 3
Bacon-Shor code is a cluster of k octons. . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 The soundness of the Bacon-Shor code reduces and approaches 0.5 if we increase
the cluster size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5 The above plots show the logical error rate of the chosen bit x for various codes and
various values of cluster size k. We consider the cases of (a) no code, (b) J7, 1, 3K
Steane code, (c) J9, 1, 3K Bacon-Shor code, (d) J17, 1, 5K color code, (e) J19, 1, 5K
color code, (f) J23, 1, 7K Golay code, and (g) J25, 1, 5K Bacon-Shor code. In the plots
(b) to (g), the 95% confidence interval is smaller than the size of plot marker. . . 68
4.6 The above plots show the soundness of the remaining bit y for various codes and
various values of cluster size k. We consider the cases of (a) no code, (b) J7, 1, 3K
Steane code, (c) J9, 1, 3K Bacon-Shor code, (d) J17, 1, 5K color code, (e) J19, 1, 5K color
code, (f) J23, 1, 7K Golay code, and (g) J25, 1, 5K Bacon-Shor code. The plots (b) to
(g) also display the 95% confidence intervals. The graph does not show variation of
soundness with physical error rate, because we had conservatively assumed that an
adversary does not face errors while obtaining the remaining bit y. . . . . . . . . 69
xii

4.7 In the top graph, we plot the codes with the least overhead which achieves 95%
availability of the chosen bit, for different regimes of soundness and physical error
rate. Each sector of the graph is labeled in the form n×k, where n is the number of
physical qubits in CSS code and k is the cluster size. In the middle graph and the
bottom graph, we similarly plot the codes with the least overhead which achieves
99% and 99.9% availability of the chosen bit, respectively. In all three graphs, an
octon cluster is better than codes when error rate approaches zero. The legend
shows the colors for various CSS codes, including the J7, 1, 3K Steane code, the
J9, 1, 3K Bacon-Shor code, the J17, 1, 5K color code, the J19, 1, 5K color code, the
J23, 1, 7K Golay code, and the J25, 1, 5K Bacon-Shor code. . . . . . . . . . . . . . . 70
4.8 A dishonest recipient can perform this malicious sequence of measurements to
maximize his chance of cheating and obtaining both bits stored in the Bacon-Shor
OTM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.9 (a) This illustrates the basic unit of a 1/n OTM. It corresponds to an island
with 4n MZMs, and it has n parts with 4 MZMs in each part. The X operators
and Z operators in each part are shown as red and green lines respectively. The
measurement basis of a part depends on its own parity.
(b) This illustrates the basic unit of an (n−1)/n OTM. Here, the measurement basis
of any part is determined by parity of the preceding part. For example, the basis of
part 2 is determined by parity of part 1, and the basis of part 1 is determined by
parity of part n, and so on. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.10 This shows error correction on the 1
3 OTM. The X stabilizers protect the logical X
operator in the top layer, and they are shown in yellow polygons and brown line
respectively. The Y stabilizers protect the logical Y operator in the middle layer,
and they are shown as light green polygons and a dark green line respectively. The
Z stabilizers protect the logical X operator in the bottom layer, and they are shown
as light blue polygons, and a dark blue line respectively. . . . . . . . . . . . . . . 75
A.1 Figure (a) shows a one-sided tetron with MZMs at locations a, b, c, d. Its Pauli
operators are X = γbγc, Z = γaγb. Figure (b) highlights one stabilizer of the
J14, 1, df = 6K code which is being measured. Figure (c) shows the tetron configuration for this stabilizer code. We use doubled black lines to denote quantum dot
mediated coupling, which is used to create the measurement loop corresponding to
the X′
1X2X3X4 stabilizer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
A.2 The figure illustrates a two-sided tetron configuration that can implement the
J38, 1, df = 10K code. Figure (a) shows two examples of syndrome measurement,
which are highlighted in green. Figure (b) illustrates the tetron configuration for
this stabilizer code. We use doubled black lines to denote quantum dot mediated
coupling, which is used to create the measurement loops corresponding to the
stabilizers Z2Z3Z7Z9Z10Z11 and X4X5X′
13X15. . . . . . . . . . . . . . . . . . . . 85
xiii

A.3 Figure (a) illustrates 7 tetrons and 10 measurement links (using the legend of
Fig. 1.1), and labels them for convenience. Figure (b) shows the parity matrix for
three classical repetition codes for the measurement links. In subsequent figures, we
use doubled black lines to denote quantum dot mediated coupling, which is used to
create measurement loops for Majorana operators. Figure (c) shows measurement of
X1X2X3X4, L5L8, L6L9 and L7L10. Figure (d) shows measurement of X2X4X6X7,
L3L5 and L1L4. Figure (e) shows measurement of X3X4X5X6 and L2L7. Figure (f)
shows measurement of X′
1X2X3X4. Figure (g) shows measurement of X′
2X4X6X7.
Figure (h) shows measurement of X′
3X4X5X6. . . . . . . . . . . . . . . . . . . . . 87
A.4 This is a continuation of Fig. A.3, which illustrates the syndrome measurement for the
J14, 1, df = 6K quantum code over 7 tetrons, in parallel with syndrome measurement
for the classical code over 10 measurement links. Figure (a) shows measurement
of Z1Z2Z3Z4. Figure (b) shows measurement of Z3Z4Z5Z6 and L3L4L8L9 (which
yields L4L6). Figure (c) shows measurement of Z2Z4Z6Z7. Figure (d) shows
measurement of Z1Z2Z3Z
′
4
. Figure (e) shows measurement of Z3Z4Z
′
5Z6. Figure
(f) shows measurement of Z2Z4Z
′
6Z7. Figure (g) shows measurement of Z2Z4Z6Z
′
7
. 88
B.1 (a) The top panel shows two partitions of the 7-tetron color code. The bottom
panel shows the stabilizers in sets 0, 1, 2 of J14, 1, 2, df = 3K code. The X and
Z stabilizers in set 0 are supported on X and Z operators, respectively, at their
plaquette vertices. The X stabilizer plaquettes in set 1 are supported on X′ at the
plaquette vertices marked by red circles, and X on other vertices of that plaquette.
The Z stabilizer plaquettes in set 2 are supported on Z
′ at the plaquette vertices
marked by red circles, and Z on other vertices of that plaquette. This group has 11
independent stabilizers. (b) The figure shows the logical qubit and the gauge qubits
of J14, 1, 2, df = 3K code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
B.2 The 19-tetron code is divided into two partitions, where partition 1 comprises n1 = 9
tetrons at odd distance from the center, and partition 2 comprises n2 = 10 tetrons
at even distance from the center. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
B.3 (a) The top panel shows two partitions of the 19-tetron color code. The bottom
panel shows the stabilizers in sets 0, 1, 2 of J38, 1, 1, df = 9K code. The X and
Z stabilizers in set 0 are supported on X and Z operators, respectively, at their
plaquette vertices. The X stabilizer plaquettes in set 1 are supported on X′ at the
plaquette vertices marked by red circles, and X on other vertices of that plaquette.
The Z stabilizer plaquettes in set 2 are supported on Z
′ at the plaquette vertices
marked by red circles, and Z on other vertices of that plaquette. This group has 36
independent stabilizers. (b) Logical qubit and gauge qubit of J38, 1, 1, df = 9K code. 93
B.4 (a) The top panel shows two partitions of the 37-tetron color code. The bottom
panel shows the stabilizers in sets 0, 1, 2 of J74, 1, 3, df = 9K code. The X and
Z stabilizers in set 0 are supported on X and Z operators, respectively, at their
plaquette vertices. The X stabilizer plaquettes in set 1 are supported on X′ at the
plaquette vertices marked by red circles, and X on other vertices of that plaquette.
The Z stabilizer plaquettes in set 2 are supported on Z
′ at the plaquette vertices
marked by red circles, and Z on other vertices of that plaquette. This group has 70
independent stabilizers. (b) Logical qubit and gauge qubits of J74, 1, 3, df = 9K code. 94
xiv

B.5 Figures (a) and (b) illustrate the stabilizer sets 2 and 1 respectively. In both figures,
the white plaquettes yield zero syndrome as they are unaffected by any error. The
blue striped plaquettes yield zero syndrome as they are affected by an even number
of errors. However, the red checkered plaquettes yield non-zero syndrome as they
are affected by an odd number of errors. Figure (a) illustrates that if a convex
corner of the hull lies in the interior of the triangle, then we obtain a non-zero
syndrome. Figure (b) illustrates that if the hull exterior contains an obtuse angle
between a triangle edge and a hull edge, then the syndrome is non-zero. . . . . . . 98
B.6 The seven geometry classes of error configurations. . . . . . . . . . . . . . . . . . . 99
B.7 If an error configuration with one of the first four geometries produce zero syndrome,
then the same configuration would continue to yield zero syndrome in larger codes,
without requiring the error configuration to be scaled up with code size. . . . . . . 100
B.8 The figure shows the variation of pseudothreshold with noise bias for J14, 1, 2, df = 3K,
J38, 1, 1, df = 9K and J74, 1, 3, df = 9K subsystem codes. . . . . . . . . . . . . . . . 102
B.9 Code capacity logical error plots for fermionic subsystem codes derived from color
codes. The top row shows the plot for the 7 tetron code, the middle row shows the
plot for the 19 tetron code, and the bottom row shows the plot for the 37 tetron
code. In these plots, the 95% confidence interval bars are smaller than the marker
size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
B.10 Figures (a) to (o) demonstrate a sequence of fault-tolerant measurements for the
J14, 1, 2, df = 3K fermionic subsystem code. This sequence can tolerate one bosonic
error or one fermionic error. The stabilizer being measured is highlighted in color,
and it is supported on four operators at four vertices. The tetron operators follow
the same notations as Fig. 2.1(d). . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
B.11 The logical error plots for the fault-tolerant implementation of the J14, 1, 2, df = 3K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively.
The graphs also show the 95% confidence intervals. . . . . . . . . . . . . . . . . . 105
C.1 A Monte Carlo simulation of variation in state integrity over multiple rounds of
weak measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
C.2 According to the first sequence, we perform weak measurement on part i followed
by strong measurement on part i and finally strong measurement on part (3 − i).
Here p denotes probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
C.3 According to second sequence, we perform weak measurement on part i followed by
strong measurement on part (3 − i) and finally strong measurement on part i. Here
p is probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
xv

E.1 Optimal malicious strategy used by a dishonest recipient of a 1/2 OTM encoded
in the Steane code. The recipient sequentially measures the clusters numbered 1
through 7. He reads the top bits from the brown clusters, he reads the bottom bits
from the blue clusters, and he can choose to read any bit from the purple cluster.
(a) The clusters are shown on the stabilizer diagram of the Steane code. Both X
and Z stabilizers have the same structure. (b)The clusters are numbered on the
stabilizer matrix of the Steane code. Both X and Z stabilizers have the same matrix. 122
E.2 Optimal malicious strategy used by a dishonest recipient of a 1/2 OTM encoded
in the Golay code. The recipient sequentially measures the clusters numbered 1
through 23. He reads the top bits from the brown clusters, and he reads the bottom
bits from the blue clusters. He can choose either strategy for the purple cluster.
The clusters are numbered and colored on the stabilizer matrix of the Golay code.
Both X and Z stabilizers have the same matrix. . . . . . . . . . . . . . . . . . . . 123
E.3 Optimal malicious strategy used by a dishonest recipient of a 1/3 OTM encoded
in the Steane code. The recipient sequentially measures the clusters numbered 1
through 7. He reads the top bits from the brown clusters, and the bottom bits
from the blue clusters. (a) The clusters are shown on the stabilizer diagram of the
Steane code. Both X and Z stabilizers have the same structure. (b) The clusters
are numbered on the stabilizer matrix of the Steane code. Both X and Z stabilizers
have the same matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
xvi

Abstract
Majorana zero modes (MZMs) are promising candidates for topological quantum computation.
Karzig et al. have proposed a tetron architecture for measurement-based quantum computation
using Majorana zero modes. In this architecture, each tetron island hosts four MZMs, and possible
measurements are constrained to span zero or two MZMs per island. Such measurements are
known to be sufficient for correcting “bosonic errors”, which infect an even number of MZMs
per tetron. We demonstrate that such measurements are also sufficient for correcting “fermionic
errors”, which infect an odd number of MZMs in a tetron. We propose several strategies for
constructing “fermionic codes” that can correct fermionic errors in addition to bosonic errors.
Then we extrapolate the nanowire architecture and consider octon islands, where each island
contains eight MZMs. We show that octons can be used to construct a cryptographic primitive
called one-time memory (OTM). One-time memories have been shown to allow for secure classical
and quantum computations. An OTM is a memory device that stores two classical bits, such
that either bit can be retrieved, but not both. We prove that a malicious recipient performing an
arbitrary sequence of strong and weak measurements can not obtain more information than an
honest recipient performing only strong measurements. We show that errors in the two stored bits
can be corrected by a pair of classical codes derived from a quantum CSS code. Finally, we show
that the one bit out of two bit OTM construction generalizes to 1-out-of-n and (n − 1)-out-of-n
OTMs.
xvii

Chapter 1
Introduction
Topological quantum computation has intrigued researchers in the recent years as it can protect
quantum information from local environmental perturbations [Kit01, Kit03, Kit06, NSS+08,
SLTD10, BTL10, Ali12, LF12, MCA+14, DFN15, AHM+16, LBK+18, FvS21, AAA+23]. One of
the promising candidates for this is the Majorana-based nanowire architecture [KKL+17, Lv17,
Lv18b, KBPK18, KORF19], shown in Fig. 1.1. This architecture provides a convenient means
for manipulating quantum information using measurement-only protocols. The basic unit of this
architecture is a superconducting island, which hosts an even number of Majorana zero modes
(MZMs). We explore techniques for error correction and cryptography in this architecture.
Topological superconductor Superconductor
Majorana zero mode Semiconductor
훾
훾
훾
훾
Figure 1.1: The above figure illustrates a tetron architecture. A single tetron comprises two
topological superconducting nanowires that host four Majorana zero modes at their ends, and they
are connected by a superconductor.
1

1.1 Majorana zero modes
We begin with a brief introduction to general Majorana operators and general Majorana error
correcting codes. Additional background information will be provided in each chapter. We refer the
reader to [BTL10] for further exposition on Majorana operators, as well as [KKL+17, KBPK18]
for a comprehensive overview of the hardware architecture.
In 1928, Paul Dirac proposed the relativistic wave equation that described electrons and other
massive spin 1
2
particles [Dir28]. The symmetric nature of his equation also predicted the existence
of antiparticles that could interact with the particles and annihilate each other. Almost a decade
later, Ettore Majorana proposed a variation of the Dirac equation that predicted a new class of
neutral massless particles which could be their own antiparticles - the Majorana fermions [Maj37].
These solutions take the form of purely imaginary matrices that satisfy Clifford algebra. No
fundamental particles have been conclusively known to be Majorana fermions yet. However,
they can emerge as quasiparticles or particle-like excitations from the collective behavior of other
fundamental particles. For example, superposition of electrons and holes in a superconductor can
exhibit the characteristics of Majorana fermions under specific circumstances. This is explained by
the fact that superconductors have a plethora of Cooper pairs, which can be added or removed from
the superconducting sea without any variation in its properties. So, we can consider an electron to
be equivalent to a hole (along with a trivial Cooper pair addition), as illustrated in Fig. 1.2(a).
≈ +
(a)
Δ
−Δ
0
Energy

† = −
Trivial superconductor
(b)
Δ
−Δ
0
Energy
Topological superconductor
0
† = 0
(c)
Figure 1.2: (a) An electron can be considered to be equivalent to a hole when it is surrounded by
Cooper pairs in a superconductor. (b) The Bogoliubov - de Gennes formalism of superconductivity
necessitates electron-hole symmetry in energy. (c) Each Majorana mode is a superposition of an
electron and a hole, and hence they are bound to the middle of the band, at the zero energy state.
2

Furthermore, the superconducting nature of the substrate screens the electromagnetic properties
of electrons and holes, so that they behave as neutral particles [Wil09]. The Bogoliubov-de Gennes
equation ensures that electrons and holes in a superconductor must have equal and opposite
energy, as shown in Fig. 1.2(b). Consequently, a superposition of electrons and holes must have
zero energy [Bro12], as shown in Fig. 1.2(c). The zero modes corresponding to the charge-neutral
superpositions of electrons and holes are Majoranas. Henceforth, we shall refer to them as Majorana
zero modes (or MZMs). Each MZM can be thought of as a half-fermion [Ali10], and two MZMs
can fuse together to yield either an electron or a hole. Thus, the total number of MZMs (NMaj ) in
a system must always be even in order to ensure an integral number of fermions. We can partition
these MZMs into NMaj/2 pairs, and each partitioning choice corresponds to a basis. Measurement
of any MZM pair yields either even or odd parity. There are 2NMaj /2 possibilities corresponding
to 2NMaj /2 degenerate parity eigenstates. We can encode information in this subspace by braiding
Majorana modes around one another, which causes nontrivial unitary transformation on the
degenerate subspace [NSS+08]. Figure 1.3(a) shows that moving MZMs around each other creates
a braid along timeline, and Fig. 1.3(b) illustrates the braiding sequence for a CNOT gate. Although
Time
(a) 1
2
3
4
5
6
(b)
Figure 1.3: (a) This shows two equivalent braids performed on a pair of MZMs. They undergo a
unitary transformation when moved around each other, and hence they are non-Abelian anyons.
(In contrast, Abelian anyons merely pick up a phase factor when looped around each other.) This
unitary transformation is resilient to small disturbances, as the transformation depends only on
the braid topology. For the same reason, nontrivial braids cannot be performed in 3-dimensional
space, since such a path can always be continuously deformed into a trivial path. Nontrivial braids
can be performed when constrained to 2 dimensional planes, or 1 dimensional wire networks.
(b) This braiding sequence corresponds to CNOT gate between |ψ1⟩ and |ψ2⟩, where time moves
from left to right. Here, |ψ1⟩ is densely encoded into the MZMs γ1, γ2, γ3, γ4 and |ψ2⟩ is densely
encoded into the MZMs γ3, γ4, γ5, γ6. The dense encoding scheme is provided in Ref. [DFN15].
3

braiding operations can be experimentally difficult, equivalent transformations can be achieved
by measurement-based quantum computation [BFN08], which is experimentally easier. These
transformations only depend on the braid topology, and are unaffected by local perturbations
as long as the perturbations obey the symmetries of the system [DKC15]. However, tunneling
of an unpaired electron (quasiparticle) from the environment is one kind of perturbation that
breaks the particle hole symmetry, and can cause undesirable transitions between the degenerate
parity eigenstates. These events are called quasiparticle poisoning, and they lead to decoherence in
Majorana-based quantum computation. Although it is theoretically predicted that the probability
of these events should reduce exponentially at low temperatures, experimental evidence reveals
that the quasiparticle density saturates to a constant value instead of dropping to zero at low
temperatures [RL12]. This necessitates error correction schemes to ensure the integrity of any
information encoded in the above-mentioned subspace.
1.2 Historical background
The use of Majorana modes for quantum computation was first envisioned by Bravyi and
Kitaev in their seminal work of 2000 [BK02]. In the same year, Kitaev demonstrated that these
Majorana zero modes can emerge at the end of a 1-dimensional p-wave superconducting wire,
or at the junctions between topological superconducting (TSC) wire and non-TSC wire [Kit01].
Similarly, there were early proposals for Majorana modes in the vortices of 2-dimensional px +
ipy superconductors [RG00, Iva01]. However, such superconductors do not seem to exist in
nature, which motivated new ideas for creating them artificially from heterostructures with more
conventional materials.
The next milestone came in 2007, when Fu and Kane showed that the interface between an
s-wave superconductor and a strong topological insulator resembles a px + ipy superconductor,
which can host Majorana zero modes at its vortex cores [FK08, FK09]. In 2009, Sau et al. proposed
4

that Majorana zero modes can appear in the vortices of a thin semiconductor layer sandwiched
between an s-wave superconductor and a magnetic insulator [SLTD10]. In this proposal, the
chief ingredients for generation of Majorana modes are spin-orbit coupling of the semiconductor,
proximity-induced Zeeman splitting, and s-wave superconductivity. These continued to be used as
common ingredients in future device designs. In the same year, Alicea proposed a simplified device
design, where one side of the semiconductor is directly coupled to an s-wave superconductor, which
allowed an easier probe for Majorana modes from the opposite side [Ali10].
However, the most appealing proposals for experimentalists were the 1-dimensional device
designs that emerged after that [LSD10, ORv10, STL+10]. These designs involve a semiconducting
nanowire placed on an s-wave superconductor under a weak magnetic field. The magnetic field
induces a Zeeman splitting that can drive the nanowire into a topological phase, which can be
controlled by a gate voltage along the wire. Alicea et al. [AOR+11] showed that networks of
these nanowires can enable braiding Majorana modes, which is a key operation for quantum
computation [Kit03].
This sparked a flurry of experiments to detect signatures of Majorana zero modes. The
signature that most groups were searching for was quantized conductance peaks at zero sourcedrain voltage bias. MZMs are theoretically predicted to exhibit zero-bias conductance peak (ZBCP)
quantized at 2e
2/h height. Since 2012, several groups claimed to have observed signatures of
Majorana zero modes in their nanowire devices [MZF+12, DRM+12, DYH+12]. However, later
simulations revealed that while MZMs are resilient to weak disorders, strong disorders could
mimic their signatures like stable ZBCPs [PD20]. Subsequently, some of the previous results were
retracted [GCZ+17, HPS+17, ZLG+18], and the experimental focus shifted to reducing sample
disorder, and developing better methods for distinguishing MZMs from disorders.
We will specifically focus on the semiconductor-superconductor heterostructure architecture
that is being spearheaded by the Microsoft group [KKL+17]. Additional details about this
architecture are provided in Section 1.3. The Microsoft group has proposed a topological gap
5

protocol to identify MZMs and distinguish them from disorders, by utilizing a combination of local
and non-local conductance measurements performed over a large parameter space [PvK+21].
Ref. [DP23] classified sample disorders according to their strength. Weak disorders have
strength < 0.3 meV and MZMs are immune to disorders in this regime, while strong disorders with
strength > 10 meV are harmful for experiments. A recent Microsoft experiment [AAA+23] utilized
InAs samples with a disorder strength of 0.6 − 1.2 meV [LSD24], which falls in the intermediate
disorder regime. This is a considerable improvement over previous experiments, which were in the
strong disorder regime. In this intermediate disorder regime, topological phase can only persist
within narrow parameter ranges, and further material advancements are needed to obtain stable
MZMs necessary for topological qubits.
The low temperature mobility of InAs electrons in this recent experiment [AAA+23] was
5 × 104
cm2/Vs [DSS23], corresponding to a significant level of disorder. Multiple research groups
are working to improve the sample mobility which would also reduce disorders, since disorders can
potentially mimic topological signatures in non-topological regime. For example, Ref. [ZLG+23]
has demonstrated InAs samples with mobility surpassing 105
cm2/Vs. Ref. [LSD24] has proposed
that Germanium hole nanowires can be a better substitute for InAs nanowires since the low
temperature mobility of Germanium holes is as high as 1.2 × 106
cm2/Vs. Hence Germanium
nanowires on Aluminum superconductor can be a viable platform for future MZM experiments.
Apart from material advances for reducing disorder, there is also ongoing research on algorithmic
methods to overcome disorder. For example, Ref. [TR23] and Ref. [TR24] have recently suggested
machine learning techniques to tune the gate arrays so that they can compensate even strong
disorders. MZMs affected by strong disorders can be restored by tuning the voltages of only 20
gates. So we are optimistic, and we look forward to the realization of topological qubits soon.
In the following section, we will give a brief overview of the Majorana architecture being
developed by the Microsoft group, which constitutes the main component of this thesis.
6

1.3 Current architecture
1.3.1 Structure
The basic unit of this architecture is a Coulomb-blockaded superconducting island with several
one-dimensional topological superconductors, for example, the inset of Fig. 1.1 shows a tetron
island with 2 topological superconductor nanowires. A topological superconductor is formed by a
semiconductor nanowire with strong spin-orbit coupling (e.g. InAs nanowire) proximity-coupled
with an s-wave superconductor (e.g. Al in the form of a half-shell around the nanowire), which
is tuned to the topological phase by means of an external magnetic field in the direction of the
nanowire. These topological superconductor nanowires are connected by a non-topological s-wave
superconducting bridge, sometimes referred to as the “backbone”, and shown as a checkered
rectangle in Fig. 1.1.
1.3.2 Dimension
The separation between adjacent topological nanowires is chosen to be smaller than the
superconducting coherence length ξ, so that the MZMs near the backbone strongly hybridize,
and we are only left with MZMs at the non-backbone endpoints of the nanowire. If the wire has
length L, then the hybridization of MZMs at the wire endpoints would be suppressed by a factor
of exp(−2L/ξ) [KKL+17]. In [AAA+23], the coherence length of a disorder-free InAs-Al device is
given to be 100-250 nm, and the optimal nanowire length for weak to moderate disorders is given
to be 3 µm. This corresponds to the base temperature of dilution refrigerator being 20 mK.
1.3.3 Errors
The values of L and ξ affect the dephasing time T2. For example, increasing L/ξ from 5 to 30
can improve T2 from 200 nanoseconds to 7 minutes, for the parameters given in Ref. [KKLN18].
The dephasing time also depends on the quasiparticle density. For a non-equilibrium quasiparticle
7

density of ∼ 1 µm−3 and L/ξ = 3, the corresponding quasiparticle poisoning timescale is ∼
2 ms [ARA+24].
There is a significant variation in the error rates estimated by various literature. For example,
Ref. [KCP21] estimates that the rate of quasiparticle creation, as well as the rate of quasiparticle
poisoning, may vary from once in 0.1 seconds to once per 10 days. Ref. [BMT+22] estimates that
the realistic and optimistic Clifford error rates are 10−4 and 10−6
respectively. The architecture is
in active development, and the error rates will be more precise with improvements in device and
material parameters.
If a tetron island is affected by even-weight errors, then such errors can be classically tracked
without requiring active correction. However, if a tetron is affected by odd-weight errors, then
we can perform active error correction by intentionally injecting quasiparticles into the device
[ARA+24, BG20].
1.3.4 Measurement
Parity measurement of a topological superconductor can be performed by a quantum dot
interferometer, as outlined in Ref. [ARA+24]. To measure the parity of a 2-MZM operator on a
tetron island, we create an interferometric loop that passes through the 2 MZMs of our interest,
and the loop is completed by quantum dots. The loop encloses a magnetic flux which can be
externally tuned. Probing the quantum capacitance of the quantum dot yields the fermion parity
for the 2 MZMs of our choice.
Figure 1.4 illustrates some parity measurement examples. Consider the Pauli operators
X = γbγc, Y = γaγc, Z = γaγb, where γa, γb, γc, γd are located at four endpoints of the tetron
island, as shown in Fig. 1.1. Parity measurement for the Y operator simply requires a loop through
the semiconducting channels its side. However, parity measurements for the X and Z operators
require an additional floating topological superconducting link, sometimes known as “coherent
link” [KKL+17]. The figure also illustrates a joint XXXX measurement spanning four tetrons.
8

Topological superconductor
Non-topological superconductor
Semiconductor
Quantum dot mediated coupling
Majorana zero mode (MZM)
Y measurement
XXXX measurement
X measurement
Z measurement
Figure 1.4: The figure illustrates measurements for the X, Y, Z operators. it also gives an example
of a joint measurement spanning multiple tetron islands.
Interestingly, no coherent links are needed for this joint measurement. In fact, coherent links can
be avoided for many conventional stabilizer codes such as color codes, surface codes, and so on.
Note that a single measurement loop can only pass through 2 MZMs per island. Odd-weight
Majorana operators are not measurable. Furthermore, if a Majorana operator is supported on a
larger number of MZMs on one or more islands, then the measurement must be decomposed into
smaller measurements that span only 2 MZMs per island.
1.4 From braiding to measurements
Braiding was the original technique for quantum computation using non-Abelian anyons [Kit03].
It was known that braiding sequences are sufficient for universal quantum computation. However,
braiding anyons requires fine experimental control and seems challenging to accomplish. Therefore
researchers shifted to a measurement-based approach for quantum computation using non-Abelian
anyons. It was proven that measurements are equivalent to braid sequences [BFN08], and universal
quantum gates can be implemented by using measurements and classical tracking [Lv17].
In particular, the logical phase gate and Hadamard gates can be achieved by edge tracking [Lv18a]. For example, the Hadamard gate can be achieved by classically swapping the
9

S
Z
Z X
X
m
Z m
X Z
Z
X m +m
m
m
(a)
S
H
S
H
Z
Z X
X
X
m
Z m
X Z
Z
X m +m
X m +m
m
m
(b)
Figure 1.5: (a) A CNOT gate from |ψ1⟩ to |ψ2⟩ is achieved using an ancilla, measurements, and
classical tracking. (b) A Clifford gate that utilizes a phase gate S, a Hadamard gate H, and a
CNOT gate, is implemented using an ancilla, measurements, and classical tracking.
definitions of X and Z operators, and changing the sign of Y operator. Similarly, the phase
gate changes the X operator to the Y operator, the Y operator to the -X operator, and the Z
operator remains unchanged. CNOT gates can be decomposed into measurements and Pauli
corrections [LKEv17], as shown in Fig. 1.5(a). Pauli gates can be classically tracked through
Clifford gates [DA07]. So, Clifford operations can be reduced to measurements. Figure 1.5(b) shows
an example of a Clifford gate that is implemented using measurements and classical operations.
Furthermore, it has also been shown that the magic gate operation can be achieved using
measurements alone [KORF19]. Thus, universality can be achieved by a combination of Clifford
gates and magic gates.
1.5 From perfect to imperfect MZMs
In the initial formulation of quantum computation using non-Abelian anyons, it was assumed
that the corresponding quantum memory would be stable by virtue of its physical properties
instead of requiring active error correction [Kit03]. However, later works on Majorana zero mode
platforms revealed that while error rates are expected to be low, it is not zero. Symmetry-breaking
errors continue to be present even at low temperatures, and hence error correction is necessary
for a reliable quantum memory. Here, we set the background for the error correction proposals
10

in the thesis. We describe Majorana operators, their commutation relations, notations used for
Majorana codes, and a review of some existing Majorana codes.
1.5.1 Majorana operators
Consider a system with NMaj MZMs γ1, γ2, . . . γNMaj , where NMaj is even. They satisfy:
γj = γ
†
j
, γ2
j = I, {γj , γk} = 2δj,k .
In this system, we can define a Majorana operator γS = i
|S|/2 Q
j∈S
γj , which is supported on
an MZM set S ⊆ {1, 2, . . . , NMaj}. This Majorana operator may also be alternately represented
as |S⟩ =
P
j∈S
|j⟩, where the MZM |j⟩ is a basis vector in F
NMaj
2
. The commutation relation
between two Majorana operators |A⟩ and |B⟩ is given by |A⟩ × |B⟩ = (−1)|A|·|B|+|A∩B| × |B⟩ × |A⟩.
However, parity measurement is only possible for Majorana operators [BTL10] which are supported
on an even number of MZMs. Two even-weight Majorana operators commute only if they overlap
on an even number of Majorana modes. The parity measurement of an even-weight Majorana
operator |M⟩ yields either even parity (0) or odd parity (1). However, if it is infected with an
error |E⟩ that anticommutes with |M⟩, then the parity measurement of |M⟩ is toggled. On the
other hand, if |E⟩ commutes with |M⟩, then the parity measurement of |M⟩ is unaffected.
1.5.2 Logical degrees of freedom
The total number of MZMs in a system must be an even integer, say NMaj . They can be
partitioned into NMaj/2 MZM pairs, wherein each partitioning choice corresponds to a basis. Each
MZM pair exists in a superposition of two degenerate parity eigenstates. Thus, the Majorana
system has NMaj/2 degrees of freedom and 2NMaj /2 degenerate parity eigenstates in total. For
example, if a system has n tetrons with 4n MZMs, then it would have 2n degrees of freedom.
Among these 2n degrees of freedom, we restrict ns = 2n − k degrees of freedom by requiring that
11

the allowed wavefunctions be eigenstates of ns stabilizer operators. This leaves us with k degrees
of freedom, wherein k logical qubits can be encoded.
1.5.3 Code notations
Let us define some common notations that we will use for classical codes, Pauli stabilizer codes
and tetron stabilizer codes. A classical linear binary code with parameters [n, k, dc] denotes that
k bits of information are encoded into an n bit code space, such that the minimum Hamming
distance between its codewords is dc. A Pauli code with parameters Jn, k, dbK represents a code
over n physical qubits with k logical qubits, where the logical operators have a least Pauli weight
of db. A tetron code with parameters J2n, k1, k2, df K represents a code over n tetrons with k1
logical qubits and k2 gauge qubits, where the dressed logical operators have a least Majorana
weight of df . If a tetron code has no gauge qubits, then it is simply denoted by 3 parameters -
J2n, k1, df K.
1.5.4 Majorana codes
We shall review some of the recent advances in Majorana error correcting codes.
1.5.4.1 Static stabilizer codes
It was known that Majorana codes can be directly obtained from Pauli quantum codes [Kit06,
BTL10], and they can also be adapted from weakly self-dual classical codes [VF17]. Numerical
search was also used to obtain Majorana codes with small overhead and small distance, some of
which were also found to have an optimal number of logical qubits [Has17]. Litinski et al. [Lv18b]
demonstrated that such small codes can be concatenated with Majorana surface codes to obtain a
family of Majorana color codes. They also proposed lattice surgery protocols for logical operations
on Majorana surface codes and Majorana color codes.
12

Several researchers have also worked on optimizing error correction procedures to use low-weight
measurements. For example, Chao et al. [CBDH20] optimized the surface code error correction
scheme to use only 1-qubit and 2-qubit measurements, which are easier to implement experimentally.
Tran et al. [TBBB20] have shown how surface code syndrome measurements can be compiled
into low-weight measurements, and optimized them for a tetron-hexon checkerboard layout.
Ref. [GSMA+23] further optimized the surface code syndrome measurements for implementation
in an array of tetron qubits, accounting for layout constraints.
1.5.4.2 Dynamic Floquet codes
Recently a new class of Floquet codes has emerged in the pursuit of codes suitable for the
Majorana architecture. They utilize a periodic sequence of non-commuting measurements which
generate an instantaneous stabilizer group and encode a dynamic logical qubit. The first example
of such a code was the honeycomb code, which used a hexagonal lattice of qubits to encode two
dynamic logical qubits [HH21]. The first version of this code required a toric geometry, which was
experimentally challenging. Subsequently, two groups independently proposed two different planar
boundary conditions for the honeycomb code, thereby resolving the experimental difficulties of the
original toric code proposal [HH22, GNM22]. Paetznick et al. [PKD+23] showed that the Floquet
honeycomb code as well as a new Floquet 4.8.8 code was especially well-suited for implementation
on the Majorana-based tetron array architecture.
However, all of these error correction efforts have focused only on correcting even-weight errors
on Majorana islands. We will demonstrate how an arbitrary stabilizer code can be extended to
correct both even-weight errors as well as odd-weight errors on Majorana islands.
13

1.6 Motivation and results
Quasiparticle poisoning events that change the fermion parity of a Majorana island are
considered to remove it from the computational subspace [RL12, Ov20]. Hence, tetron islands
are Coulomb-blockaded with high charging energy and maintained in even parity state, which
suppresses odd-weight quasiparticle poisoning events.
But in general, a tetron island can be affected by both odd-weight errors or “fermionic errors”,
and even-weight errors or “bosonic errors” [VVF19]. The even-weight errors correspond to Pauli
errors which are easily correctable by Pauli stabilizer codes, but these codes cannot correct the
non-Pauli errors of odd weight. Such uncorrected fermionic errors can eventually corrupt the
logical information [KBPK18].
Explicit correction of fermionic errors has previously required experimental changes and
challenges [KBPK18, BG20]. Conventionally, it is assumed that fermionic errors can be suppressed
by high charging energy, and only bosonic errors are corrected. However, we extend conventional
stabilizer codes to enable explicit correction of both fermionic errors and bosonic errors without
any experimental modification, and without affecting the logical operators or possible logical
operations.
We propose several different methods for this. But the basic principle can be stated as follows:
In the absence of fermionic errors, the tetron has two equivalent representations for the same Pauli
operator. However, when a fermionic error affects the tetron, this equivalence breaks down, and
their parity differs. This change in parity can be observed by appropriate syndrome measurements.
In the second chapter, we describe how each tetron can be included in the stabilizer group. We
choose one stabilizer code, and consider one of its stabilizers. For example, Fig. 1.6 shows one
stabilizer supported on 4 Z operators on 4 tetrons. Then we add a modified stabilizer where one
tetron operator is switched to another operator. We observe that product of these two stabilizers
yields a tetron stabilizer. The tetron stabilizer is particularly helpful for fermionic error correction,
14

× =
Figure 1.6: The left figure illustrates a stabilizer supported over four Z operators on 4 tetrons.
Each yellow square corresponds to a tetron island which supports the stabilizer. A tetron has 4
MZMs at 4 corners, similar to the inset of Fig. 1.1. The green line in a yellow square denotes a
Majorana operator in a tetron island, which is supported on 2 MZMs at the endpoints of the line.
For example, the green line on the top edge of square denotes the Majorana operator Z = γaγb.
The middle figure illustrates a modified stabilizer where one tetron operator is modified. The red
line denotes a modified Majorana operator, which is supported on 2 MZMs at the endpoints of
the line. For example, the red line on the bottom edge of square denotes the Majorana operator
γcγd. The right figure shows that the product of these two stabilizers is equivalent to the tetron
stabilizer γaγbγcγd. Thus, the 4-MZM tetron operator belongs to the stabilizer group.
because each fermionic error anticommutes with some tetron stabilizer, and thus can be detected
and corrected. We use this principle to extend several “bosonic stabilizer codes” into “fermionic
stabilizer codes”, which can correct both bosonic and fermionic errors.
In the third chapter, we provide a construction for subsystem fermionic codes from bosonic
codes and classical codes. For a stabilizer code over n tetrons, we choose an n-bit classical code.
We incorporate additional tetron stabilizers into this code, where each tetron stabilizer corresponds
to a row in the parity matrix of this classical code. For example, in Fig. 1.7, we use a 5-qubit Pauli
code and a [5, 2, 3] classical code. This parity matrix of this classical code has 3 rows, and these 3
rows are used to define 3 additional stabilizers, such that the nonzero locations of a parity matrix
row correspond to the support of a multi-tetron stabilizer. This produces our desired subsystem
fermionic code.
If the classical code encodes k bits of logical information, then the resulting subsystem fermionic
code will have k gauge qubits. The gauge group generators can be mapped to the k rows of the
generator matrix of this classical code. The k gauge qubits imply that the stabilizer group is now
generated by k fewer generators. This requires less syndrome measurements, and the corresponding
fault-tolerant sequence can also be made shorter. This directly translates to higher fault-tolerant
15

A
B A × B
2
3
5 5
4
2
1
5
11001
01101
00011
 
       
1 2 3 4 5
Figure 1.7: The top row A illustrates the 4 stabilizers of the 5-qubit Pauli code. Each yellow
square corresponds to a tetron island which supports the stabilizer. A green line in a yellow
square denotes a 2-MZM operator in a tetron island. The middle row B shows a modified set of
stabilizers, derived from the 3 stabilizers directly above them in row A. Each red line denotes a
modified Majorana operator, which is supported on 2 MZMs at endpoints of the line. The bottom
row A × B shows 3 members of the stabilizer group, which are formed by the product of the
stabilizer pair in row A and row B directly above it. Here each stabilizer is a set of tetrons, and it
corresponds to the non-zero locations of a matrix row. This matrix is the parity matrix of [5, 2, 3]
classical code.
pseudothreshold as compared to the non-subsystem fermionic code counterpart, and the observed
pseudothreshold improvement can be as high as 84%. This is interesting because we accomplish
two goals at once: reducing the number of syndrome measurements, and improving the logical
error rate and fault-tolerant pseudothreshold.
In the fourth chapter, we consider an extension of the nanowire architecture that hosts eight
MZMs per island, which are called octon islands. We show that the octon islands can be used to
construct a cryptographic primitive called one-time memory (OTM). It is a powerful primitive
16

…
Figure 1.8: We use a CSS quantum code such as the Bacon-Shor code to encode two classical bits
in a one-time memory. The X stabilizers shown as yellow row stabilizers protect one bit in the top
layer, and the Z stabilizers shown as blue column stabilizers protect one bit in the bottom layer.
Each qubit in this code corresponds to multiple smaller one-time memory devices, and each small
one-time memory device corresponds to a single octon island.
which is sufficient to construct quantum one-time programs, with applications in software protection,
electronic tokens, electronic cash, and more.
A single octon island corresponds to a simple but imperfect one-time memory device, where two
classical bits can be stored. The recipient of this memory device can obtain any one bit perfectly,
and the second bit would be obtained with 75% probability. We can further reduce the availability
of the second bit from 75% to nearly 50%, by using a cluster of octons. This approaches a perfect
OTM, since this allows only a single bit to be obtained from the device, and the second bit is
almost certainly lost, and the recipient can only guess it with nearly 50% probability of success.
In addition, we can also protect the bits from errors by utilizing a quantum CSS code, such
that the X stabilizers protect one bit encoded in the logical X operator of the top layer, and the Z
stabilizers protect one bit encoded in the logical Z operator of the bottom layer. This is illustrated
in Fig. 1.8 with the example of a 9-qubit Bacon-Shor code.
Similarly, we can generalize this to a 1-out-of-n OTM device and also (n − 1)-out-of-n OTM
device by utilizing Majorana islands that host 4n Majorana zero modes. We also show that
our OTM devices perform well against an arbitrary sequence of strong and weak measurements
performed by a malicious recipient who wishes to obtain more information than is permitted by
the OTM device.
17

Chapter 2
Majorana qubit codes that also correct odd-weight errors
A tetron is a superconducting island, hosting four localized Majorana zero modes (MZMs),
schematically shown in Fig. 2.1(a). One can measure the fermion parity of any operator that spans
zero or two MZMs per tetron. Using such measurements, we can define stabilizer codes in a system
of several tetrons. The possible errors in a tetron architecture include bosonic and fermionic errors.
Bosonic errors are those which affect two MZMs per tetron as shown in Fig. 2.1(b), while fermionic
errors affect an odd number of MZMs per tetron as shown in Fig. 2.1(c). Bosonic errors can be
mapped to Pauli errors, and can be corrected using conventional stabilizer codes. This is not the
case for fermionic errors.
Fermionic errors are typically suppressed by high charging energy. Short-lived fermionic
errors relax to bosonic errors. However, a sufficiently long-lived fermionic error can disrupt
measurement outcomes, and even spread to adjacent tetrons during connected measurements.
Hence Ref. [KBPK18] suggests the use of fermionic codes that can correct fermionic errors, so that
they do not spread and cause a logical error.
Although several Majorana fermionic codes have been proposed [Has17, VF17], there exist
implementation challenges specific to the tetron architecture. For example, one of the chief
experimental hurdles in employing a fermionic code is the inability to directly measure the fourMZM parity of a tetron [KBPK18]. If the tetron parity were measurable, then a fermionic error
18

a b
c d
(a) (b) (c)
ℛ X= Y= Z=
ℛ′ X′= Y′= Z′=
(d)
Figure 2.1: (a) A tetron hosts four MZMs at the locations a, b, c, d. (b) A bosonic error affects
even number of MZMs on the tetron. (c) A fermionic error affects odd number of MZMs on the
tetron. (d) R and R
′ are two complementary sets of weight-2 operators in a tetron.
could be detected. However, a measurement can overlap only two MZMs per tetron. We will show
that such measurements suffice for fermionic error correction.
In Sec. 2.3, we derive fermionic error correcting codes from conventional bosonic codes such as
color codes and surface codes. We introduce additional stabilizers in these codes, in which one
qubit has a different Pauli to MZM operator mapping. This places the tetron in the stabilizer
group, enabling fermionic error correction. Figure 2.2 shows an example.
Finally, in Sec. 2.4, we study the code capacity of some small fermionic codes, and provide
some examples of a fault-tolerant measurement schemes.
2.1 Error correction principle
We begin with a brief introduction to the tetron architecture, bosonic and fermionic errors on
tetrons, and the motivation behind fermionic error correction.
2.1.1 Tetrons
A tetron is a superconducting island with four Majorana zero modes (MZMs), maintained
in an overall even parity state by high charging energy [KKL+17]. The MZMs of a tetron are
denoted as γm, where the character m denotes the MZM location according to Fig. 2.1(a).
A system with 4n Majorana modes would have 2n degrees of freedom. But if they are
implemented on a tetron architecture, then n degrees are restricted by the requirement of even
19

X stabilizer Z stabilizer
Set 0 Sets 1
→
7

(a)
1
2
4
5 6 7
3
(b)
× =
(c)
Figure 2.2: (a) The figure shows the stabilizer generators of the J14, 1, df = 6K fermionic code.
The three colored plaquettes correspond to three X stabilizers and three Z stabilizers, similar to
the Steane color code. In addition, the seven tetrons also belong to the stabilizer generator group,
and they are shown as yellow squares. Although the tetrons belong to the stabilizer generator
group, they are not directly measurable. (b) The figure shows the measurable stabilizers of
the J14, 1, df = 6K fermionic code, derived from the J7, 1, 3K bosonic color code. Each stabilizer
plaquette is supported on four Majorana operators at its four vertices. These operators are defined
according to Fig. 2.1(d). (c) The figure demonstrates that the stabilizer group contains one tetron
operator. Similarly, all seven tetrons belong to the stabilizer group.
20

parity on all n tetrons. Thus, each tetron is left with a single degree of freedom, wherein one
qubit information can be stored [Kit06]. Each Pauli operator of this qubit has two possible
representations, as shown in Fig. 2.1(d). For example, the Pauli operator X can be represented as
either γbγc or γaγd. Both of them yield the same parity measurement since the tetron has overall
even parity.
While addressing Majorana codes described on multiple tetrons, we would often use a numerical
subscript q to denote that Majorana operators such as γaq or Xq correspond to the qth tetron.
2.1.2 Bosonic and fermionic errors in tetrons
A tetron might be affected by an error that affects 1, 2, 3 or all 4 of its MZMs. An odd-weight
error is said to be a fermionic error while an even-weight error is said to be a bosonic error [VVF19].
Among these errors, the weight-4 error is trivial since it does not affect the parity of any Pauli
operator. Also, a weight-3 error is equivalent to a weight-1 error, for example γa ≡ γbγcγd. So,
error correction is only required for weight-2 Pauli errors and weight-1 fermionic errors.
2.1.3 Bosonic error correction in tetron architecture
A tetron is maintained in an overall even parity state by a high charging energy, which suppresses
fermionic errors [KKL+17, KBPK18]. As long as the charging energy preserves even parity on
tetrons, only Pauli errors can occur on a tetron, and a conventional bosonic error correcting code
will be sufficient to correct it.
In a system with n tetrons maintained in even parity, we have n degrees of freedom. We can
define ns stabilizer operators which restrict ns degrees of freedom and leaves k = n − ns degrees
for encoding k logical qubits. This forms an Jn, k, dbK bosonic error correcting code, where db
denotes the bosonic distance corresponding to the minimum qubit weight of all logical operators.
21

Note that each measurable Majorana operator should span zero or two MZMs per tetron, so as
to preserve the even parity subspace of each tetron qubit. Parity measurement of all four MZMs
of a tetron are not considered because of experimental challenges.
2.1.4 Challenges of fermionic error correction
If the charging energy is insufficient to prevent long-lived fermionic errors on tetrons, then
we cannot assume that n degrees of freedom are restricted by high charging energy. Instead, we
would need to manually restrict these degrees of freedom by a fermionic error correcting code.
In a system with n tetrons, where the tetrons are susceptible to fermionic errors, there are 2n
degrees of freedom. Thus, a fermionic error correcting code on n tetrons would have J2n, k, df K
parameters, where df denotes the fermionic distance corresponding to the minimum Majorana
weight of all logical operators.
One possible approach is to restrict n degrees of freedom by introducing n stabilizers, where the
qth stabilizer is supported on the 4 MZMs of the qth tetron, such as in Ref. [Lv18b]. The syndrome
of these n stabilizers would allow us to detect fermionic errors on any tetron. When coupled with
syndromes of other stabilizers, it would enable us to correct both fermionic and bosonic errors
affecting the Majorana code. Unfortunately, as we have previously discussed, parity measurement
of all four MZMs of a tetron has turned out to be experimentally challenging [KBPK18]. A second
approach involves dynamically varying the number of MZMs on an island, but that has significant
experimental challenges as well [KBPK18]. Ref. [BG20] proposed a fermionic error correction
scheme for a quantum wire, where system parameters are tuned to generate additional Majorana
modes at the wire endpoints, such that stabilizers defined on these Majorana modes can correct
fermionic errors. However, this can also be experimentally challenging.
22

2.1.5 Proposed fermionic error correction principle
We describe the basic principle of fermionic error correction on a single tetron. We define two
complementary sets of weight-2 operators, shown in Fig. 2.1(d).
R = {X, Y, Z}, where X = γbγc, Y = γaγc, Z = γaγb
R
′= {X′
, Y ′
, Z′
},where X′= γaγd, Y ′= γdγb, Z′= γcγd
If a weight-2 error affects the tetron, then both sets of operators are similarly affected. For
example, a weight-2 error γbγc toggles the parity of operators Y, Y ′
, Z, Z′
, while leaving the
X, X′ operators unaffected. This forms the building block of all bosonic error correction schemes.
However, if a weight-1 error affects the tetron, then the two sets of operators are oppositely affected.
For example, a fermionic error γa toggles the parity of Y and Z, while leaving X unaffected.
Conversely, the same γa error leaves the parity of Y
′ and Z
′ unaffected, while toggling the parity
of X′
. Thus, we can identify and correct a fermionic error on a tetron. This is the building block
of our fermionic error correction proposal.
2.2 Review of bosonic codes
Before moving on to fermionic error correcting codes, we shall discuss how bosonic codes may
be used to deal with long-lived fermionic errors, as well as its shortcomings. Consider an Jn, k, dbK
bosonic code implemented on a tetron architecture, where each Pauli operator maps to a Majorana
operator in R [BTL10]. This code is capable of correcting up to tb = ⌊(db − 1)/2⌋ number of
bosonic errors if no fermionic errors occur. Now suppose that the Majorana code is not only
affected by bosonic errors on the tetron set Tb, but also by fermionic errors of types γa, γb, γc, γd in
the tetron sets Tfa, Tf b, Tf c, Tfd respectively. Observe that a fermionic γx error on a tetron yields
the same syndromes as a bosonic error γxγd (or its equivalent) on that tetron. Thus, fermionic
23

errors of types γa, γb, γc are identified as X, Y, Z errors respectively, while γd errors are invisible.
Thus, a γx error corresponds to a γxγd correction. Such a correction can be uniquely performed
only if |Tb| + |Tfa| + |Tf b| + |Tf c| ≤ tb. If we correct a γx error by a γxγd correction, then we
essentially shift the error from γx to γd. If a fermionic error occurs at γd, then it would stay there
undetected. Thus, the dth MZM would act as a reservoir for fermionic errors at all locations of
the tetron.
It might appear that the remnant γd errors do not affect either stabilizers or logical operators,
and hence they can be left uncorrected. However, Ref. [KBPK18] notes that such uncorrected
excitations may propagate from one tetron to other neighboring tetrons via connecting measurements, and finally culminate in errors of high weight. To efficiently mitigate the spread of such
excitations, it recommends the usage of fermionic codes.
2.3 B 7→ F codes: Fermionic codes from bosonic codes
We show that a bosonic code with parameters Jn, k, dbK can be translated into a fermionic
Majorana code with parameters J2n, k, df K where df = 2db.
2.3.1 Recipe for fermionic code construction
We choose a bosonic stabilizer code Jn, k, dbK from any scalable code family. We can derive a
new Majorana fermion code from the ns stabilizers of the bosonic code. The Majorana fermion
code contains several stabilizers, which we group into overlapping sets for convenience.
Set 0 contains all the ns stabilizers of the bosonic code, wherein every Pauli operator of the
bosonic code maps to Majorana operators in R.
Set i contains all the ns stabilizers of the bosonic code, wherein Pauli operators of the ith
qubit map to Majorana operators in R
′
, and all other Pauli operators map to Majorana
operators in R. Here, i varies from 1 to n.
24

Thus, we have n + 1 overlapping stabilizer sets, which form a stabilizer group of rank 2n − k. This
stabilizer group characterizes the new Majorana fermion code with J2n, k, df K parameters.
We provide an example of this scheme in Fig. 2.2(a), which shows the J14, 1, df = 6K fermionic
code derived from a J7, 1, 3bK bosonic code. Its logical qubit is characterized by X¯ = X1X3X5, Y¯ =
Y1Y2Y7,Z¯ = Z5Z6Z7.
Claim 1. Any combination of fermionic error yields a non-zero syndrome on a B 7→ F Majorana
fermion code.
Proof. Suppose a Majorana fermion code is affected by some fermionic errors and optionally some
more bosonic errors. The fermionic error would anticommute with one or more tetron operators, as
they have odd intersection. The tetron operator belongs to the stabilizer group, as the ith tetron
operator can be obtained by multiplying the independent stabilizer in set i with the corresponding
stabilizer in set 0, as shown in Fig. 2.2(c). Thus, any fermionic error would yield non-zero syndrome
on a B 7→ F code. If the ith tetron is affected by fermionic error, the corresponding operators
in R and R
′ would yield opposite parities, and so we would measure opposite parities on the
independent stabilizer of set i and the corresponding set 0 stabilizer.
2.3.2 Code distance
In this section, we show that the derived fermionic codes have the same logical operator as
the bosonic code. Hence, the fermionic code distance, or the least Majorana weight of its logical
operators, is given by twice the Pauli distance of the bosonic code.
Claim 2. If the bosonic code has Jn, k, dbK parameters, then the resultant fermionic code has
J2n, k, df K parameters, where df = 2db.
Proof. The bosonic code has n degrees of freedom and n − k stabilizer generators, so it has k
logical operators. The fermionic code has 2n degrees of freedom, and 2n − k stabilizer generators.
25

The fermionic code has n additional tetron stabilizers, as compared to the bosonic code. Thus,
the fermionic code also has k logical operators.
Next, we note that the k logical operators in the bosonic code are also valid logical operators
for the fermionic code. This is true because the set 0 stabilizers of the fermionic code are the same
as the stabilizers of the bosonic code. Furthermore, each logical operator of the bosonic code can
be mapped to a Majorana operator that spans 2 MZMs per tetron, and hence it commutes with
each tetron stabilizer. So, the k logical operators of the bosonic code satisfy all stabilizers of the
fermionic code, and are the same as the k logical operators of the fermionic code.
Finally, the code distance of the bosonic code is db, which means that the logical operators
have a least weight of db Pauli operators. Since the same set of logical operators is shared between
both codes, and each Pauli operator can be mapped to 2 MZMs, so the fermionic code has code
distance df = 2db, where df is the least Majorana weight of its logical operators.
2.3.3 Decoder
We use the BPOSD decoder, proposed by Roffe et al. [RWBC20, Rof22] for error correction.
A code with distance df can correct any error that has Majorana weight < df /2. As the code size
increases, so does the code distance df and the error correction capacity.
2.3.4 Error correction on small codes
In this section, we shall discuss Majorana fermion codes derived from two small codes. The
error correction latency of a tetron code is limited by the fact that only operators with disjoint
tetron support can be measured in parallel. Although it is theoretically possible to measure two
copies of the same Pauli operator on a tetron, it has been recommended that such measurements
be avoided to prevent the spread of correlated errors [KBPK18].
26

The J10, 1, df = 6K fermionic code is derived from the J5, 1, 3K Pauli stabilizer code. The
fermionic code has 9 independent stabilizers, each of which overlaps with the other, as shown
in Fig. 2.3(a). Thus, one round of syndrome extraction requires 9 steps, and the error
correction latency is 9.
The J12, 1, df = 6K fermionic code is derived from the J6, 1, 3K Pauli stabilizer code. The
fermionic code has 11 independent stabilizers, as shown in Fig. 2.3(b). These stabilizers can
be divided into 5 disjoint sets, so the error correction latency is 9.
Set 0 Sets 1 → 5
(a)
Set 0 Sets 1 → 6
(b)
Figure 2.3: (a) The figure shows the stabilizer generators of the J10, 1, df = 6K fermionic code,
derived from the J6, 1, 3K Pauli stabilizer code. The set 0 stabilizers correspond to the Pauli code,
and the stabilizers in sets 1 to 5 lead to the inclusion of 5 tetrons in the stabilizer group. (b) The
figure shows the measurable stabilizers of the J12, 1, df = 6K fermionic code, derived from the
J6, 1, 3K Pauli stabilizer code. The set 0 stabilizers correspond to the Pauli code, and the stabilizers
in sets 1 to 6 lead to the inclusion of 6 tetrons in the stabilizer group.
27

2.3.5 Error correction on color codes
In this section, we shall discuss Majorana fermion codes derived from CSS color codes. The
CSS color code based on the 6.6.6 tessellation is 3-colorable, so the corresponding set 0 stabilizers
require 3 steps for measuring all X stabilizers and 3 steps for measuring all Z stabilizers.
If the Majorana fermion code is described on n tetrons, then it would have n independent
stabilizers in addition to the set 0 stabilizers. In each of these n additional stabilizers, one of the
tetron operators is switched from R to R
′
. These additional stabilizer syndromes can be extracted
in seven steps. So, the fermionic error correction latency is 13.
The J14, 1, df = 6K code has 13 independent stabilizers, each of which overlaps with the
other, as shown in Fig. 2.2. Thus, one round of syndrome extraction requires 13 steps, and
the error correction latency is 13.
The J38, 1, df = 10K code has 37 independent stabilizers. Set 0 contains 9 X stabilizers, 9 Z
stabilizers, and it needs 3 + 3 steps to be measured. Sets 1 to 19 contain one independent
stabilizer each, and they need 7 steps to be parallelly measured, as shown in Fig. 2.4. Thus,
the error correction latency is 13.
The J74, 1, df = 14K code has 73 independent stabilizers. Set 0 contains 18 X stabilizers
and 18 Z stabilizers, and it needs 3 + 3 steps to be measured. Sets 1 to 37 contain one
independent stabilizer each, and they need 7 steps to be parallelly measured, as shown in
Fig. 2.5. Thus, the error correction latency is 13.
We provide the physical nanowire structure for some color codes in Appendix A, where we show
how these codes can be implemented both with additional coherent links and without such links.
The code capacity and fault-tolerance analysis provided later assumes a link-free architecture,
since the error rate would be lower and the Majorana overhead would be lower as well.
Before proceeding to code capacity and fault-tolerance analysis, let us see another example of
this fermionic code construction recipe.
28

Figure 2.4: The figure illustrates an optimized syndrome measurement sequence for stabilizers
in sets 1 to 19 of J38, 1, df = 10K code. Each pink striped plaquette corresponds to a stabilizer
supported on Z
′ at the vertex marked by a red circle, and supported on Z at all other vertices of
that plaquette. Similarly, each green checkered plaquette corresponds to a stabilizer supported on
X′ at the vertex marked by a red circle, and supported on X at all other vertices of that plaquette.
Figure 2.5: The figure illustrates an optimized syndrome measurement sequence for stabilizers
in sets 1 to 37 of J74, 1, df = 14K code. Each pink striped plaquette corresponds to a stabilizer
supported on Z
′ at the vertex marked by a red circle, and supported on Z at all other vertices of
that plaquette. Similarly, each green checkered plaquette corresponds to a stabilizer supported on
X′ at the vertex marked by a red circle, and supported on X at all other vertices of that plaquette.
29

(a) (b)
Figure 2.6: (a) The figure shows the stabilizers of the J25, 1, 5K bosonic code, which are the
same as the set 0 stabilizers of the J50, 1, df = 10K code. The pink striped stabilizer plaquettes
are supported on Z operators at its vertices, while the green checkered stabilizer plaquettes are
supported on X operators at its vertices. The X stabilizers of set 0 require 2 steps for syndrome
extraction, similarly the Z stabilizers of set 0 require another 2 steps for syndrome extraction. (b)
The figure illustrates an optimized four-step syndrome measurement sequence for stabilizers in sets
1 to 25 of J50, 1, df = 10K code. Each pink striped plaquette corresponds to a stabilizer supported
on Z
′ at the vertex marked with a red circle, and supported on Z at all other vertices of that
plaquette. Similarly, each green checkered plaquette corresponds to a stabilizer supported on X′
at the vertex marked with a red circle, and supported on X at all other vertices of that plaquette.
2.3.6 Error correction on surface codes
In this section, we consider Majorana fermion codes derived from rotated surface codes. Its
latency for fermionic error correction is 8.
If the Majorana fermion code is described on n tetrons, then it would have additional n
independent stabilizers on top of the set 0 stabilizers. In each of these n additional stabilizers,
one of the tetron operators is switched from R to R
′
. This defines the stabilizers for the fermionic
version of the rotated surface code.
For example, let us consider the J25, 1, 5K bosonic rotated surface code, shown in Fig. 2.6(a).
From this, we derive the J50, 1, df = 10K code, containing 49 independent stabilizers. Set 0 contains
12 X stabilizers and 12 Z stabilizers, and it needs 2 + 2 steps to be measured. Sets 1 to 25 contain
one independent stabilizer each, and they need 4 steps to be parallelly measured, as shown in
30

Fig. 2.6(b). Thus, one round of fermionic syndrome extraction requires 8 syndrome measurement
steps, and so the fermionic error correction latency is 8.
Thus, we have analyzed the B 7→ F codes, wherein the stabilizer group is generated by toggling
the tetron operators at one tetron in one measurement operator at a time. The tetrons are included
in the resultant stabilizer group.
2.4 Error analysis
2.4.1 Code capacity
We consider a simple noise model where each tetron is subjected to bosonic errors as well as
fermionic errors. If the total error rate is p and the noise bias is η, then the bosonic error rate
is pb =
p
η+1 and the fermionic error rate is pf =
pη
η+1 . As the bosonic errors X, Y, Z are equally
likely, so pX = pY = pZ =
p
3(η+1) =
pb
3
. Similarly, the fermionic errors γa, γb, γc, γd are equally
likely, so pγa = pγb = pγc = pγd =
pη
4(η+1) =
pf
4
. The physical error rate is given by pb +
3pf
4
since
all bosonic errors affect a physical qubit, but only 3 out 4 fermionic errors affect a physical qubit
on a tetron. The logical error rate for the fermionic codes are evaluated by using the BPOSD
decoder, and the variation of their pseudothreshold with noise bias is provided in Fig. 2.7. The
code capacity logical error plots for bias values η = 0.1, 1, 10 are provided in Fig. 2.8 and Fig. 2.9.
31

10, 1, df =6 code
12, 1, df = 6 code
14, 1, df = 6 code
38, 1, df = 10 code
74, 1, df = 14 code
Pseudothreshold
0
0.1
0.2
0.3
0.4
0.5
Bias
103 0.01 0.1 1 10 100 1000
Figure 2.7: The figure shows the variation of pseudothreshold with noise bias for J10, 1, df = 6K,
J12, 1, df = 6K, J14, 1, df = 6K, J38, 1, df = 10K and J74, 1, df = 14K codes.
10, 1, df = 6, bias = 0.1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(a)
10, 1, df = 6, bias = 1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(b)
10, 1, df = 6, bias = 10
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(c)
12, 1, df = 6, bias = 0.1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(d)
12, 1, df = 6, bias = 1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(e)
12, 1, df = 6, bias = 10
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(f)
Figure 2.8: Code capacity logical error plots for J10, 1, df = 6K and J12, 1, df = 6K fermionic codes
at bias values η = 0.1, 1, 10. The 95% confidence interval bars are smaller than the marker size.
32

14, 1, df = 6, bias = 0.1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(a)
14, 1, df = 6, bias = 1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(b)
14, 1, df = 6, bias = 10
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(c)
38, 1, df = 10, bias = 0.1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(d)
38, 1, df = 10, bias = 1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(e)
38, 1, df = 10, bias = 10
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(f)
74, 1, df = 14, bias = 0.1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(g)
74, 1, df = 14, bias = 1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(h)
74, 1, df = 14, bias = 10
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(i)
Figure 2.9: Code capacity logical error plots for J14, 1, df = 6K, J38, 1, df = 10K and J74, 1, df = 14K
fermionic codes at bias values η = 0.1, 1, 10. The 95% confidence interval bars are smaller than
the marker size.
33

2.4.2 Fault tolerance
Fault-tolerance can be achieved by careful ordering of stabilizer measurements, and might
require some redundant syndrome measurements. We provide the fault-tolerance sequences of
some small codes, which can tolerate 1 input error or 1 intermediate error, either bosonic or
fermionic. We analyze these sequences against a noise with η bias, a bosonic error rate of pb =
p
η+1 ,
a fermionic error rate of pf =
pη
η+1 , and a measurement error rate of p.
Figure 2.10 shows the fault-tolerant sequence for the J10, 1, df = 6K fermionic code, and Fig. 2.11
shows the existence of a fault-tolerant threshold for noise bias η = 0.1, 1, 10. Figure 2.12 shows the
fault-tolerant sequence for the J12, 1, df = 6K fermionic code, and Fig. 2.13 shows the existence
of a fault-tolerant threshold for noise bias η = 0.1, 1, 10. Figure 2.14 shows the fault-tolerant
sequence for the J14, 1, df = 6K fermionic code, and Fig. 2.15 shows the existence of a fault-tolerant
threshold for noise bias η = 0.1, 1, 10.
34

(e) Z1 Z2 X3 I4 X5
′
(a) X1
′ Z2 Z3 X4 I5 (b) I1 X2
′ Z3 Z4 X5 (c) X1 I2 X3
′ Z4 Z5
(d) Z1 X2 I3 X4
′ Z5 (f) X1 Z2 Z3 X4 I5
(g) I1 X2 Z3 Z4 X5 (h) X1 I2 X3 Z4 Z5 (i) Z1 X2 I3 X4 Z5
Figure 2.10: Figures (a) to (i) demonstrate a sequence of fault-tolerant measurements for the
J10, 1, df = 6K fermionic code. This sequence can tolerate one bosonic error or one fermionic error.
The tetron operators follow the same notation as Fig. 2.1(d).
× 10-4
10, 1, df = 6, bias = 0.1
Physical error rate
Logical error rate
0
103
2×103
3×103
4×103
5×103
6×103
7×103
Bosonic error rate + fermionic error rate (p)
0 5 10 15 20 25
(a)
× 10-4
10, 1, df = 6, bias = 1
Physical error rate
Logical error rate
0
103
2×103
3×103
4×103
5×103
6×103
7×103
Bosonic error rate + fermionic error rate (p)
0 5 10 15 20 25
(b)
× 10-4
10, 1, df = 6, bias = 10
Physical error rate
Logical error rate
0
103
2×103
3×103
4×103
5×103
6×103
7×103
Bosonic error rate + fermionic error rate (p)
0 5 10 15 20 25
(c)
Figure 2.11: The logical error plots for the fault-tolerant implementation of the J10, 1, df = 6K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively. The graph also
shows the 95% confidence intervals.
35

(a) Z1 X2 X3 X4 X5 I6 (b) Z1 Z2 I3 Z4 I5 Z6 (c) I1 Z2 X3 X4 Y5 Y6
(e) I1 I2 Z3 I4 Z5 I6 (f) Z1
′ X2 X3 X4 X5 I6 (g) Z1 Z2
′ I3 Z4 I5 Z6 (h) I1 Z2 X3
′ X4 Y5 Y6
(i) X1 I2 I3 X4
′ Z5 Z6 (j) I1 I2 Z3 I4 Z5
′ I6 (k) Z1 Z2 I3 Z4 I5 Z6
′
(m) Z1 X2 X3 X4
′ X5 I6 (n) I1 Z2 X3 X4 Y5
′
Y6
(d) X1 I2 I3 X4 Z5 6
(l) Z1 Z2 I3 Z4 I5 Z6
Figure 2.12: Figures (a) to (n) demonstrate a sequence of fault-tolerant measurements for the
J12, 1, df = 6K fermionic code. This sequence can tolerate one bosonic error or one fermionic error.
The tetron operators follow the same notation as Fig. 2.1(d).
12, 1, df = 6, bias = 0.1
Physical error rate
× 10-4
Logical error rate
0
2×104
4×104
6×104
8×104
103
1.2×103
1.4×103
1.6×103
Bosonic error rate + fermionic error rate (p)
01234567
(a)
12, 1, df = 6, bias = 1
Physical error rate
× 10-4
Logical error rate
0
2×104
4×104
6×104
8×104
103
1.2×103
1.4×103
1.6×103
Bosonic error rate + fermionic error rate (p)
01234567
(b)
12, 1, df = 6, bias = 10
Physical error rate
× 10-4
Logical error rate
0
2×104
4×104
6×104
8×104
103
1.2×103
1.4×103
1.6×103
Bosonic error rate + fermionic error rate (p)
01234567
(c)
Figure 2.13: The logical error plots for the fault-tolerant implementation of the J12, 1, df = 6K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively. The graph also
shows the 95% confidence intervals.
36

(a) X1 X2 X3 X4 (b) X2 X4 X6 X7 (c) X3 X4 X5 X6 (d) Z1 Z2 Z3 Z4
(e) Z2 Z4 Z6 Z7 (f) Z3 Z4 Z5 Z6 (g) X1
′ X2 X3 X4 (h) X3
′ X4 X5 X6
(i) X2
′ X4 X6 X7 (j) Z1 Z2 Z3 Z4
′ (k) Z3 Z4 Z5
′ Z6 (l) Z2 Z4 Z6
′ Z7
(m) Z2 Z4 Z6 Z7
′ (n) Y1
′
Y2
′
Y3
′
Y4
′ (o) Y2
′
Y4
′
Y6
′
Y7
′ (p) Y3
′
Y4
′
Y5
′
Y6
′

Figure 2.14: Figures (a) to (p) demonstrate a sequence of fault-tolerant measurements for the
J14, 1, df = 6K fermionic code. This sequence can tolerate one bosonic error or one fermionic error.
The tetron operators follow the same notation as Fig. 2.1(d).
× 10-4
14, 1, df = 6, bias = 0.1
Physical error rate
Logical error rate
0
4×104
8×104
1.2×103
1.6×103
2×103
2.4×103
Bosonic error rate + fermionic error rate (p)
01234567
(a)
× 10-4
14, 1, df = 6, bias = 1
Physical error rate
Logical error rate
0
4×104
8×104
1.2×103
1.6×103
2×103
2.4×103
Bosonic error rate + fermionic error rate (p)
01234567
(b)
× 10-4
14, 1, df = 6, bias = 10
Physical error rate
Logical error rate
0
4×104
8×104
1.2×103
1.6×103
2×103
2.4×103
Bosonic error rate + fermionic error rate (p)
01234567
(c)
Figure 2.15: The logical error plots for the fault-tolerant implementation of the J14, 1, df = 6K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively. The graph also
shows the 95% confidence intervals.
37

Chapter 3
Majorana subsystem qubit codes that also correct
odd-weight errors
In the previous chapter, we have proposed Majorana codes that can correct fermionic errors by
incorporating each tetron in the stabilizer group. In this chapter, we show that we can construct
subsystem codes capable of fermionic error correction by using a classical code over the tetrons
such that tetrons belong to the gauge group. The proposed subsystem codes require less stabilizer
generators compared to the previous approach, and can potentially lead to smaller fault-tolerant
sequences yielding higher pseudothreshold. This is depicted in Table 3.1, which compares codes
constructed using the previous approach, with currently proposed codes.
We describe a general recipe for the construction of fermionic subsystem codes using Pauli
stabilizer codes and classical codes. We choose a Pauli stabilizer code with parameters Jn, k1, dbK,
and introduce additional stabilizers supported on one or more tetrons in accordance with an
[n, k2, dc] classical code. This results in a J2n, k1, df K subsystem code with k2 gauge qubits, where
the fermionic code distance df is bounded by db ≤ df ≤ 2db. For convenience, we will denote this
subsystem code as J2n, k1, k2, df K, and will continue to use this 4-parameter notation later. We
provide error analysis on some small codes constructed with this recipe, as well as examples of
some fault-tolerant schemes.
38

Code parameter Code capacity Fault-tolerant
sequence length
Fault-tolerance pseudothreshold
η = 0.1 η = 1 η = 10 η = 0.1 η = 1 η = 10
J10, 1, df = 6K 0.137 0.196 0.423 9 0.00134 0.00145 0.00146
J10, 1, 2, df = 3K 0.135 0.169 0.255 8 0.00164 0.00170 0.00161
J12, 1, df = 6K 0.103 0.177 0.391 14 0.00043 0.00045 0.00045
J12, 1, 1, df = 6K 0.103 0.176 0.346 12 0.00060 0.00062 0.00062
J12, 1, 3, df = 3K 0.097 0.122 0.197 12 0.00059 0.00060 0.00059
J14, 1, df = 6K 0.072 0.128 0.418 16 0.00027 0.00028 0.00029
J14, 1, 4, df = 3K 0.067 0.077 0.155 12 0.00047 0.00051 0.00051
Table 3.1: This table compares the code capacity and fault-tolerant pseudothreshold between
non-subsystem fermionic codes and currently proposed subsystem fermionic codes. The codes are
compared at various values of noise bias η, which is the ratio between fermionic error probability
and bosonic error probability. We observe that the J10, 1, 2, df = 3K code has the shortest
syndrome-measurement sequence that is fault-tolerant against a single error, and thus has the best
fault-tolerant pseudothreshold among these codes. We observe that the threshold of subsystem
codes exceed their non-subsystem counterparts. For the above codes and bias values, the threshold
improvement percentage ranges from 10% to as much as 84%. The fault-tolerant sequences for
these codes are provided in Section 2.4.2 and Section 3.1.8.
3.1 BC 7→ FS codes: Fermionic subsystem codes from
bosonic and classical codes
We show that a non-subsystem bosonic code on n qubits and a classical code over n bits can
be used to construct a subsystem fermionic code over n tetrons. We apply this recipe to obtain
several small subsystem fermionic codes.
3.1.1 Recipe for fermionic code construction
We use an Jn, kb, dbK non-subsystem bosonic code and an [n, k, dc] classical code to derive a
J2n, kb, k, df K subsystem fermionic code.
We start with the stabilizers of the Jn, kb, dbK code and convert its Pauli operators X, Y, Z to
the Majorana operators X = γbγc, Y = γaγc, Z = γaγb, according to Fig. 2.1(d). For example, we
can choose a J5, 1, 3K Pauli stabilizer code with stabilizer generators IZXZX, XXIZZ, ZIXXZ,
and XZZXI. These generators are illustrated in row A of Fig. 3.1(a).
39

Then we choose the classical code [n, k, dc] with parity matrix H and generator matrix G.
We choose G to be in the systematic form [Ik|P] where Ik is a k × k identity matrix and P is a
k × (n − k) matrix. Then we add tetron stabilizers according to the rows of its parity matrix.
For example, we can choose the [5, 2, 3] classical code with parity matrix
H =








1 1 0 0 1
0 1 1 0 1
0 0 0 1 1








and a systematic generator matrix
G =




1 0 1 1 1
0 1 0 1 1




Since the first row of H is 11001, then we need to add the stabilizer T1T2T5 comprising the first,
second and fifth tetrons. Now, we cannot directly measure the parity of one or more tetrons. So in
order to add such a stabilizer, we would choose an existing stabilizer, for example S1 = XXIZZ,
that is supported on at least the first, second and fifth tetrons. Then we add a modified stabilizer
where the operators on the first, second and fifth tetrons are switched from R to R
′
, for example
S1m = X′X′
IZZ′
. This ensures that S1 × S1m = T1T2T5 belongs to the stabilizer group.
In this step, it is essential that the stabilizer code and the classical code are compatible with
each other, such that there exists a stabilizer supported on the non-zero tetrons in each row of the
parity matrix.
The resulting subsystem fermionic code has n − kb stabilizers from the bosonic code and n − k
stabilizers from the classical code. This code has the same kb logical qubits as the bosonic code.
This code also has k gauge qubits, corresponding to the k rows of the generator matrix of
the classical code. Each row of the generator matrix can be mapped to two gauge operators, the
40

first being a tetron operator at the first non-zero entry of the generator matrix row. The second
operator is a fermionic operator corresponding to γd on all non-zero entries of that generator
matrix row. For example, if the generator matrix row is 11001, then the gauge operators are T1
(the first tetron) and γd1γd2γd5 (supported on γd operator of tetrons 1, 2, and 5). This provides us
with 2k gauge operators which generate a group of 22k
size, and contains k gauge qubits.
Claim 3. The above code construction utilizing an Jn, kb, dbK non-subsystem bosonic code and an
[n, k, dc] classical code results in a subsystem code with k gauge qubits.
Proof. We will show that the chosen gauge operators form the generators of a gauge group.
Suppose the classical code has parity matrix H and generator matrix G. If the group formed
by rows of H contains any single tetron at index D, then the entire column D of the generator
matrix must be zero so that G remains orthogonal with H. As we have chosen our tetron
gauges from non-zero locations of the generator matrix rows, so these chosen tetrons do not
belong to the stabilizer group.
As logical operators are only supported on γa, γb, γc but not on the γd operators, so the
tetrons cannot belong to the group of logical operators.
The k chosen fermionic operators do not belong to stabilizer group or logical operator group.
As the stabilizers and the logical operators are supported on 2 MZMs per tetron, so their
group cannot have any fermionic operator.
The k chosen tetrons are independent of each other as they have no overlap.
The k chosen fermionic operators are independent of each other because the rows of generator
matrix are independent of each other.
The tetrons and the fermionic operators are independent of each other, since the fermionic
operators are supported only on γd operators and none of the γa, γb, γc operators.
41

The k chosen fermionic operators commute with all the tetron stabilizers formed by the rows
of parity matrix H, since G and H are orthogonal.
The k chosen fermionic operators are entirely supported on γd operators, whereas the original
Pauli stabilizers of the bosonic code as well as the logical operators are supported only on
γa, γb, γc operators. So the k chosen fermionic operators have no intersection with them, and
so commute with them.
The tetrons commute with all stabilizers and logical operators.
Thus we have k tetrons and k fermionic operators which are independent of each other, as well
as independent of stabilizers and logical operators. They commute with both stabilizers and logical
operators, and hence these 2k operators form a gauge group of size 22k
, which can accommodate
k gauge qubits.
Note that the least weight of all fermionic gauges, or the least weight of the group formed by
generator matrix rows is the same as the classical code distance dc.
3.1.2 Code distance
Code distance of subsystem codes are given by the least weight of all nontrivial dressed logical
operators. If a logical operator of minimum weight 2db and a fermionic gauge operator are
supported on the same tetrons, then the dressed logical operator formed by their product has the
least Majorana weight db. If their support is different, then the dressed logical operator would have
a higher weight. Thus, the code distance df of the subsystem code is bounded by db ≤ df ≤ 2db.
The code distance for the proposed codes are evaluated by exhaustive search.
3.1.3 Decoder
We use the BPOSD decoder, proposed by Roffe et al. [RWBC20, Rof22] for correcting errors in
the subsequent examples. We use the “product sum” method for belief propagation, the “osd cs”
42

method for the ordered statistics decoder, a a maximum of 5 iterations, and a search depth of
2N+1 where N is the number of tetrons.
3.1.4 Example 1: J10, 1, 2, df = 3K subsystem code
We derive the J10, 1, 2, df = 3K subsystem Majorana fermionic code from the J5, 1, 3K Pauli
stabilizer code and the [5, 2, 3] classical code. The stabilizer generators of this code are shown in
rows A and B of Fig. 3.1(a), and its logical and gauge qubits are illustrated in Fig. 3.1(b). The
third row of Fig. 3.1(a) shows that the product of corresponding stabilizer generators in rows A
and B yields tetron sets in the stabilizer group. These tetron stabilizers correspond to the rows
of a parity matrix H shown in Fig. 3.1(c). The two gauge qubits correspond to the two rows of
the generator matrix G shown in Fig. 3.1(c). These two matrices characterize the [5, 2, 3] classical
code. The smallest error which affects the logical qubit but yields zero syndrome is γb3γc4γb5,
so the Majorana code has distance df = 3. As each stabilizer generator overlaps with another
generator, so we require 7 time steps for syndrome measurement.
Figure 3.1 illustrates that the rows of the parity matrix correspond to the tetron stabilizers,
and the rows of the generator matrix correspond to the gauge operators. Note that for each gauge
qubit, one gauge operator corresponds to the first non-zero entry of the generator matrix row, and
another gauge operator is a fermionic operator corresponding to γd on all non-zero entries of that
generator matrix row. For example, if the generator matrix row is 11001, then the gauge operators
are T1 (the first tetron) and γd1γd2γd5 (supported on γd operator of tetrons 1, 2, and 5). We can
verify the commutation relation between these operators by inspection.
3.1.5 Example 2: J12, 1, 3, df = 3K subsystem code
We derive the J12, 1, 3, df = 3K subsystem Majorana fermionic code from the J6, 1, 3K Pauli
stabilizer code and the [6, 3, 3] classical code. The stabilizer generators of this code are shown in
rows A and B of Fig. 3.2(a), and its logical and gauge qubits are illustrated in Fig. 3.2(b). The
43

third row of Fig. 3.2(a) shows that the product of corresponding stabilizer generators in rows A
and B yields tetron sets in the stabilizer group. These tetron stabilizers correspond to the rows of
a parity matrix H shown in Fig. 3.2(c). The three gauge qubits correspond to the three rows of
the generator matrix G shown in Fig. 3.2(c). These two matrices characterize the [6, 3, 3] classical
code. The smallest error which affects the logical qubit but yields zero syndrome is γc2γc3γc6, so
the Majorana code has distance df = 3. This code has 8 stabilizer generators, but 2 of them can
be parallelly measured, so we need 7 time steps for syndrome measurement.
Figure 3.2 illustrates that the rows of the parity matrix correspond to the tetron stabilizers,
and the rows of the generator matrix correspond to the gauge operators.
3.1.6 Example 3: J14, 1, 4, df = 3K subsystem code
We derive the J14, 1, 4, df = 3K subsystem Majorana fermionic code from the J7, 1, 3K Pauli
Steane code and the [7, 4, 3] classical Hamming code. The stabilizer generators of this code are
shown in rows A and B of Fig. 3.3(a), and its logical and gauge qubits are illustrated in Fig. 3.3(b).
The third row of Fig. 3.3(a) shows that the product of corresponding stabilizer generators in rows
A and B yields tetron sets in the stabilizer group. These tetron stabilizers correspond to the rows
of a parity matrix H shown in Fig. 3.3(c). The four gauge qubits correspond to the four rows of
the generator matrix G shown in Fig. 3.3(c). These two matrices characterize the [7, 4, 3] classical
Hamming code. The smallest error which affects the logical qubit but yields zero syndrome is
γa1γa6γa7, so the Majorana code has distance df = 3. This code has 9 stabilizer generators, all of
which overlap with one another. Hence, we require 9 time steps for syndrome measurement.
Figure 3.3 illustrates that the rows of the parity matrix correspond to the tetron stabilizers,
and the rows of the generator matrix correspond to the gauge operators.
44

A
B A × B
2
3
5 5
4
2
1
5
(a)
Operator 1 Operator 2
Logical qubit
Gauge qubit 1
― γ d
― γ aγ bγ cγ d
Gauge qubit 2
(b)
11001
H 01101
00011
 
 
=      
1 2 3 4 5
10111
G
01011
 
=    
2
1
3
5
4
(c)
Figure 3.1: (a) This figure shows the 7 stabilizer generators of the J10, 1, 2, df = 3K Majorana
fermionic subsystem code in the rows A and B. The third row shows the stabilizer formed by the
product of corresponding generators in the first and second rows. Each stabilizer in the third row
is a set of tetrons, corresponding to the rows of parity matrix H. (b) This figure shows the logical
qubit and the gauge qubits of the J10, 1, 2, df = 3K Majorana fermionic subsystem code. Each
gauge qubit corresponds to a row of the generator matrix G. (c) This figure shows the classical
parity matrix H and the generator matrix G of the [5, 2, 3] classical code. The tetron numbering
scheme shown in this figure is used for the matrix columns.
45

A
B A × B
1
4
5
4
6 2
4
5 3
6
(a)
Operator 1 Operator 2
Logical qubit
Gauge qubit 1
― γ d
― γ aγ bγ cγ d
Gauge qubit 2
Gauge qubit 3
(b)
100110
H 010101
001111
 
 
=
 
100101
G 010110
001111
 
 
=  
 
1 2 3 4 5 6
1
4
5 3
6 2
(c)
Figure 3.2: (a) This figure shows the 8 stabilizer generators of the J12, 1, 3, df = 3K Majorana
fermionic subsystem code in the rows A and B. The third row shows the stabilizer formed by the
product of corresponding generators in the first and second rows. Each stabilizer in the third row
is a set of tetrons, corresponding to the rows of parity matrix H. (b) This figure shows the logical
qubit and the gauge qubits of the J12, 1, 3, df = 3K Majorana fermionic subsystem code. Each
gauge qubit corresponds to a row of the generator matrix G. (c) This figure shows the classical
parity matrix H and the generator matrix G of the [6, 3, 3] classical code. The tetron numbering
scheme shown in this figure is used for the matrix columns.
46

A
B A × B

3 7
1
5
7
6
3
2
5
7
6 4

(a)
Operator 1 Operator 2
Logical qubit
Gauge qubit 1
― γ d
― γ aγ bγ cγ d
Gauge qubit 2
Gauge qubit 3
Gauge qubit 4

(b)
1010101
H 0110011
0001111
 
 
=
 
1000011
0100101
G
0010110
0001111
 
 
=
 

2 6 4
3 7
1
5
1 2 3 4 5 6 7
(c)
Figure 3.3: (a) This figure shows the 9 stabilizer generators of the J14, 1, 4, df = 3K Majorana
fermionic subsystem code in the rows A and B. The third row shows the stabilizer formed by the
product of corresponding generators in the first and second rows. Each stabilizer in the third row
is a set of tetrons, corresponding to the rows of parity matrix H. (b) This figure shows the logical
qubit and the gauge qubits of the J14, 1, 4, df = 3K Majorana fermionic subsystem code. Each
gauge qubit corresponds to a row of the generator matrix G. (c) This figure shows the classical
parity matrix H and the generator matrix G of the [7, 4, 3] classical code. The tetron numbering
scheme shown in this figure is used for the matrix columns.
47

10, 1, 2, 3f code
12, 1, 3, 3f code
12, 1, 1, 6f code
14, 1, 4, 3f code
Pseudothreshold
0
0.1
0.2
0.3
0.4
0.5
Bias
103 0.01 0.1 1 10 100 1000
Figure 3.4: The figure shows the variation of pseudothreshold with noise bias for J10, 1, 2, df = 3K,
J12, 1, 3, df = 3K, J12, 1, 1, df = 6K and J14, 1, 4, df = 3K subsystem codes.
3.1.7 Code capacity
We analyze four fermionic subsystem codes, and plot their variation of pseudothreshold with
noise bias in Fig. 3.4. Their logical error plots at various noise bias are provided in Fig. 3.5. The
bosonic codes and classical codes from which these four fermionic subsystem codes are obtained
are listed below:
J5, 1, 3K + [5, 2, 3] → J10, 1, 2, df = 3K
J6, 1, 3K + [6, 1, 6] → J12, 1, 1, df = 6K
J6, 1, 3K + [6, 3, 3] → J12, 1, 3, df = 3K
J7, 1, 3K + [7, 4, 3] → J14, 1, 4, df = 3K
For the above Majorana fermionic subsystem codes, the stabilizers of the J5, 1, 3K, J6, 1, 3K, and
J7, 1, 3K codes are illustrated in Fig. 3.1(a), Fig. 3.2(a), and Fig. 3.3(a) respectively. The classical
48

10, 1, df = 6, bias = 0.1
10, 1, 2, df = 3, bias = 0.1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(a)
10, 1, df = 6, bias = 1
10, 1, 2, df = 3, bias = 1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(b)
10, 1, df = 6, bias = 10
10, 1, 2, df = 3, bias = 10
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(c)
12, 1, df = 6, bias = 0.1
12, 1, 1, df = 6, bias = 0.1
12, 1, 3, df = 3, bias = 0.1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(d)
12, 1, df = 6, bias = 1
12, 1, 1, df = 6, bias = 1
12, 1, 3, df = 3, bias = 1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(e)
12, 1, df = 6, bias = 10
12, 1, 1, df = 6, bias = 10
12, 1, 3, df = 3, bias = 10
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(f)
14, 1, df = 6, bias = 0.1
14, 1, 4, df = 3, bias = 0.1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(g)
14, 1, df = 6, bias = 1
14, 1, 4, df = 3, bias = 1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(h)
14, 1, df = 6, bias = 10
14, 1, 4, df = 3, bias = 10
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5
(i)
Figure 3.5: Code capacity logical error plots for currently proposed fermionic subsystem codes as
well as previously proposed non-subsystem fermionic codes. The top row shows the plot for the 5
tetron codes, the middle row shows the plot for the 6 tetron codes, and the bottom row shows
the plot for the 7 tetron codes. For each code, the logical error rates are evaluated at bias values
η = 0.1, 1, 10. In these plots, the 95% confidence interval bars are smaller than the marker size.
49

codes [5, 2, 3], [6, 3, 3], [7, 4, 3] are defined in Fig. 3.1(c), Fig. 3.2(c), and Fig. 3.3(c) respectively.
The [6, 1, 6] code is a classical repetition codes.
In this analysis, we utilize a biased noise model. We consider that fermionic errors are η times
more likely to occur than bosonic errors. There are 3 possible modes for bosonic errors - X, Y,
Z errors, each of which occurs with equal probability pX = pY = pZ =
p
3(η+1) =
pb
3
. There are
4 possible modes for fermionic errors - γa, γb, γc, γd, each of which occurs with equal probability
pγa = pγb = pγc = pγd =
pη
4(η+1) =
pf
4
. We use the BPOSD decoder to obtain the logical error rates
for bias values η = 0.1, 1, 10. We evaluate the pseudothreshold as the p value below which the
logical error rate is lower than the physical error rate. Note that the physical error rate is given by
pb +
3pf
4
since a tetron qubit is affected by all bosonic errors but only 3 out of 4 fermionic errors.
3.1.8 Fault tolerance
Fault-tolerance can be achieved by careful ordering of stabilizer measurements, and might
require some redundant syndrome measurements. We provide the fault-tolerance sequences of
some small codes, which can tolerate 1 input error or 1 intermediate error, either bosonic or
fermionic. We analyze these sequences against a noise with η bias, a bosonic error rate of pb =
p
η+1 ,
a fermionic error rate of pf =
pη
η+1 , and a measurement error rate of p.
Figure 3.6 shows the fault-tolerant sequence for the J10, 1, 2, df = 3K fermionic code, and
Fig. 3.7 shows the existence of a fault-tolerant threshold for noise bias η = 0.1, 1, 10. Figure 3.8
shows the fault-tolerant sequence for the J12, 1, 1, df = 6K fermionic code, and Fig. 3.9 shows
the existence of a fault-tolerant threshold for noise bias η = 0.1, 1, 10. Figure 3.10 shows the
fault-tolerant sequence for the J12, 1, 3, df = 3K fermionic code, and Fig. 3.11 shows the existence of
a fault-tolerant threshold for noise bias η = 0.1, 1, 10. Figure 3.12 shows the fault-tolerant sequence
for the J14, 1, 4, df = 3K fermionic code, and Fig. 3.13 shows the existence of a fault-tolerant
threshold for noise bias η = 0.1, 1, 10.
50

A comparison of the fault-tolerant pseudothresholds of the small subsystem fermionic codes
and non-subsystem fermionic codes is provided in Fig. 3.14. We observe that subsystem fermionic
codes with smaller fault-tolerant sequences have better fault-tolerant pseudothresholds.
3.2 Alternate constructions for fermionic subsystem codes
We propose another construction for fermionic subsystem codes in Appendix B. This construction is suitable for color codes such as the 6.6.6 CSS color code family.
51

(a) Z1 I2 X3 X4 Z5 (b) X1 Z2 Z3 X4 I5 (c) I1 Z2
′ X3
′ Z4 X5
′ (d) X1
′ X2
′ I3 Z4 Z5
′
(e) Z1 I2 X3 X4
′ Z5
′ (f) I1 Z2 X3 Z4 X5 (g) X1 X2 I3 Z4 Z5 (h) X1 Z2
′ Z3
′ X4
′ I5
Figure 3.6: Figures (a) to (h) demonstrate a sequence of fault-tolerant measurements for the
J10, 1, 2, df = 3K fermionic subsystem code. This sequence can tolerate one bosonic error or one
fermionic error. The tetron operators follow the same notations as Fig. 2.1(d).
× 10-4
10, 1, 2, df = 3, bias = 0.1
Physical error rate
Logical error rate
0
103
2×103
3×103
4×103
5×103
6×103
Bosonic error rate + fermionic error rate (p)
0 5 10 15 20 25
(a)
× 10-4
10, 1, 2, df = 3, bias = 1
Physical error rate
Logical error rate
0
103
2×103
3×103
4×103
5×103
6×103
7×103
Bosonic error rate + fermionic error rate (p)
0 5 10 15 20 25
(b)
× 10-4
10, 1, 2, df = 3, bias = 10
Physical error rate
Logical error rate
0
103
2×103
3×103
4×103
5×103
6×103
7×103
Bosonic error rate + fermionic error rate (p)
0 5 10 15 20 25
(c)
Figure 3.7: The logical error plots for the fault-tolerant implementation of the J10, 1, 2, df = 3K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively. The graphs also
show the 95% confidence intervals.
52

(a) Z1 X2 X3 X4 X5 I6 (b) Z1 Z2 I3 Z4 I5 Z6 (c) I1 Z2 X3 X4 Y5 Y6
(e) I1 I2 Z3 I4 Z5 I6 (f) Z1
′ X2
′ X3 X4 X5 I6 (g) Z1
′ Z2 I3 Z4 I5 Z6
′ (h) I1 Z2 X3 X4 Y5
′
Y6
′
(i) X1 I2 I3 X4
′ Z5
′ Z6 (j) I1 I2 Z3
′ I4 Z5
′ I6 (k) I1 Z2 X3 X4 Y5 Y6
(d) X1 I2 I3 X4 Z5 6
(l) I1 Z2 X3
′ X4
′ Y5 Y6
Figure 3.8: Figures (a) to (h) demonstrate a sequence of fault-tolerant measurements for the
J12, 1, 1, df = 6K fermionic subsystem code. This sequence can tolerate one bosonic error or one
fermionic error. The tetron operators follow the same notations as Fig. 2.1(d).
12, 1, 1, df = 6, bias = 0.1
Physical error rate
× 10-4
Logical error rate
0
2×104
4×104
6×104
8×104
103
1.2×103
Bosonic error rate + fermionic error rate (p)
01234567
(a)
12, 1, 1, df = 6, bias = 1
Physical error rate
× 10-4
Logical error rate
0
2×104
4×104
6×104
8×104
103
1.2×103
Bosonic error rate + fermionic error rate (p)
01234567
(b)
12, 1, 1, df = 6, bias = 10
Physical error rate
× 10-4
Logical error rate
0
2×104
4×104
6×104
8×104
103
1.2×103
Bosonic error rate + fermionic error rate (p)
01234567
(c)
Figure 3.9: The logical error plots for the fault-tolerant implementation of the J12, 1, 1, df = 6K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively. The graphs also
show the 95% confidence intervals.
53

(a) Z1 X2 X3 X4 X5 I6 (b) Z1 Z2 I3 Z4 I5 Z6 (c) I1 Z2 X3 X4 Y5 Y6
(e) I1 I2 Z3 I4 Z5 I6 (f) Z1
′ X2 X3 X4
′ X5
′ I6 (g) Z1 Z2
′ I3 Z4
′ I5 Z6
′ (h) I1 Z2 X3
′ X4
′ Y5
′
Y6
′
(i) Z1 X2 X3 X4 X5 I6 (j) Z1 Z2 I3 Z4 I5 Z6 (k) Z1
′ X2 X3 X4
′ X5
′ I6
(d) X1 I2 I3 X4 Z5 6
(l) I1 Z2
′ X3 X4
′ Y5 Y6
′
Figure 3.10: Figures (a) to (h) demonstrate a sequence of fault-tolerant measurements for the
J12, 1, 3, df = 3K fermionic subsystem code. This sequence can tolerate one bosonic error or one
fermionic error. The tetron operators follow the same notations as Fig. 2.1(d).
12, 1, 3, df = 3, bias = 0.1
Physical error rate
× 10-4
Logical error rate
0
2×104
4×104
6×104
8×104
103
1.2×103
Bosonic error rate + fermionic error rate (p)
01234567
(a)
12, 1, 3, df = 3, bias = 1
Physical error rate
× 10-4
Logical error rate
0
2×104
4×104
6×104
8×104
103
1.2×103
Bosonic error rate + fermionic error rate (p)
01234567
(b)
12, 1, 3, df = 3, bias = 10
Physical error rate
× 10-4
Logical error rate
0
2×104
4×104
6×104
8×104
103
1.2×103
Bosonic error rate + fermionic error rate (p)
01234567
(c)
Figure 3.11: The logical error plots for the fault-tolerant implementation of the J12, 1, 3, df = 3K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively. The graphs also
show the 95% confidence intervals.
54

(a) X1 X2 X3 X4 (b) X2 X4 X6 X7 (c) X3 X4 X5 X6 (d) Z1 Z2 Z3 Z4
(e) Z2 Z4 Z6 Z7 (f) Z3 Z4 Z5 Z6 (g) X1
′ X2
′ X3
′ X4
′ (h) X2
′ X4
′ X6
′ X7
′
(i) X3
′ X4
′ X5
′ X6
′ (j) Z1
′ Z2
′ Z3
′ Z4
′ (k) Z2
′ Z4
′ Z6
′ Z7
′ (l) Z3
′ Z4
′ Z5
′ Z6
′

Figure 3.12: Figures (a) to (h) demonstrate a sequence of fault-tolerant measurements for the
J14, 1, 4, df = 3K fermionic subsystem code. This sequence can tolerate one bosonic error or one
fermionic error. The tetron operators follow the same notations as Fig. 2.1(d).
× 10-4
14, 1, 4, df = 3, bias = 0.1
Physical error rate
Logical error rate
0
2×104
4×104
6×104
8×104
103
1.2×103
1.4×103
1.6×103
Bosonic error rate + fermionic error rate (p)
01234567
(a)
× 10-4
14, 1, 4, df = 3, bias = 1
Physical error rate
Logical error rate
0
2×104
4×104
6×104
8×104
103
1.2×103
1.4×103
1.6×103
Bosonic error rate + fermionic error rate (p)
01234567
(b)
× 10-4
14, 1, 4, df = 3, bias = 10
Physical error rate
Logical error rate
0
2×104
4×104
6×104
8×104
103
1.2×103
1.4×103
1.6×103
Bosonic error rate + fermionic error rate (p)
01234567
(c)
Figure 3.13: The logical error plots for the fault-tolerant implementation of the J14, 1, 4, df = 3K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively. The graphs also
show the 95% confidence intervals.
55

× 10-4
10, 1, df = 6, bias = 0.1
10, 1, 2, df = 3, bias = 0.1
Physical error rate
Logical error rate
0
103
2×103
3×103
4×103
5×103
6×103
7×103
Bosonic error rate + fermionic error rate (p)
0 5 10 15 20 25
(a)
× 10-4
10, 1, df = 6, bias = 1
10, 1, 2, df = 3, bias = 1
Physical error rate
Logical error rate
0
103
2×103
3×103
4×103
5×103
6×103
7×103
Bosonic error rate + fermionic error rate (p)
0 5 10 15 20 25
(b)
× 10-4
10, 1, df = 6, bias = 10
10, 1, 2, df = 3, bias = 10
Physical error rate
Logical error rate
0
103
2×103
3×103
4×103
5×103
6×103
7×103
Bosonic error rate + fermionic error rate (p)
0 5 10 15 20 25
(c)
12, 1, df = 6, bias = 0.1
12, 1, 1, df = 6, bias = 0.1
12, 1, 3, df = 3, bias = 0.1
Physical error rate
× 10-4
Logical error rate
0
2×104
4×104
6×104
8×104
103
1.2×103
1.4×103
1.6×103
1.8×103
Bosonic error rate + fermionic error rate (p)
01234567
(d)
12, 1, df = 6, bias = 1
12, 1, 1, df = 6, bias = 1
12, 1, 3, df = 3, bias = 1
Physical error rate
× 10-4
Logical error rate
0
2×104
4×104
6×104
8×104
103
1.2×103
1.4×103
1.6×103
1.8×103
Bosonic error rate + fermionic error rate (p)
01234567
(e)
12, 1, df = 6, bias = 10
12, 1, 1, df = 6, bias = 10
12, 1, 3, df = 3, bias = 10
Physical error rate
× 10-4
Logical error rate
0
2×104
4×104
6×104
8×104
103
1.2×103
1.4×103
1.6×103
1.8×103
Bosonic error rate + fermionic error rate (p)
01234567
(f)
× 10-4
14, 1, df = 6, bias = 0.1
14, 1, 4, df = 3, bias = 0.1
Physical error rate
Logical error rate
0
4×104
8×104
1.2×103
1.6×103
2×103
2.4×103
Bosonic error rate + fermionic error rate (p)
01234567
(g)
× 10-4
14, 1, df = 6, bias = 1
14, 1, 4, df = 3, bias = 1
Physical error rate
Logical error rate
0
4×104
8×104
1.2×103
1.6×103
2×103
2.4×103
Bosonic error rate + fermionic error rate (p)
01234567
(h)
× 10-4
14, 1, df = 6, bias = 10
14, 1, 4, df = 3, bias = 10
Physical error rate
Logical error rate
0
4×104
8×104
1.2×103
1.6×103
2×103
2.4×103
Bosonic error rate + fermionic error rate (p)
01234567
(i)
Figure 3.14: Fault-tolerance logical error plots for currently proposed fermionic subsystem codes
as well as previously proposed non-subsystem fermionic codes. The top row shows the plot for the
5 tetron codes, the middle row shows the plot for the 6 tetron codes, and the bottom row shows
the plot for the 7 tetron codes. For each code, the logical error rates are evaluated at bias values
η = 0.1, 1, 10. The graphs also show the 95% confidence intervals.
56

Chapter 4
One-time memory from isolated Majorana islands
A one-time memory (OTM) is a hypothetical device that stores two bits such that either can
be read, but not both. Once the first bit is read, the other is lost. Perfect OTMs are impossible
classically, because classical information can always be copied. In 1983, Bennett et al. [BBBW83]
introduced OTM, and proposed a quantum implementation. Two classical bits can be stored
in a qubit, such that with appropriate measurements any one bit can be obtained with > 85%
probability, while the other bit is lost. Bennett et al. suggested that by encoding the classical bits
into two classical codes derived from a Calderbank-Steane-Shor (CSS) code, 15% errors on the
chosen bit can be corrected. More specifically, we would require classical codes derived from CSS
codes with threshold greater than 15%. Unfortunately, now we know that CSS codes can have a
maximum threshold of 11% [DKLP02, Pou06, RWH+12, AWK15] and hence this OTM scheme is
not feasible. It is now known that one-time memory cannot exist in an unrestricted quantum world
because it would enable secure two-party computation, which is impossible [Lo97, BCS12, Liu14a].
A secure OTM can exist only if the adversary is restricted in some manner.
In 2008, Goldwasser formalized the notion of one-time memory (OTM) as a cryptographic
primitive and showed that classical one-time programs can be constructed from classical OTMs
[GKR08]. In 2013, Broadbent et al. showed that ideal classical OTMs are sufficient for noninteractive unconditionally secure quantum computation [BGS13]. Broadbent et al. also proposed
57

an OTM construction using a tamper-proof classical hardware token along with a quantum key
[BGZ21].
Liu has proposed an OTM using isolated qubits [Liu14a, Liu14b, Liu15]. He encodes two
messages as two codewords of an error correcting code that approaches the channel capacity.
The code is divided into multiple qubit blocks, and each block is randomly initialized in either
the standard basis or the Hadamard basis. The first message can be obtained by measuring all
qubits in the standard basis, and the second message can be obtained by measuring all qubits in
Hadamard basis.
We propose an alternate construction for a simple OTM. We show that a single Majorana
island with 8 modes is equivalent to an OTM storing 2 bits, wherein the chosen bit can be perfectly
obtained, and the remaining bit can be obtained with 75% probability. If the 2 bits are stored
in a cluster of k Majorana islands, then any one chosen bit can be perfectly obtained, and the
remaining bit can be obtained with a probability of 1
2 +
1
2
k+1 . This approximates an ideal OTM,
where choosing any one bit destroys the remaining bit, and hence the destroyed bit can only be
guessed with 50% probability. But we can increase k such that the proposed OTM approaches
an ideal OTM. In this scheme, the adversary is restricted from performing joint measurements
spanning multiple islands, or the Majorana islands are isolated from each other.
4.1 Majorana operators
We briefly review the properties of Majorana operators. Consider a system with one or more
Majorana islands, wherein each island contains an even number of MZMs. We can define Majorana
operators supported on some MZMs, with the following characteristics:
All physically measurable operators should be supported on an even number of MZMs per
island.
Operator measurement gives either even parity (0) or odd parity (1).
58

Two even-weight Majorana operators commute only if they intersect on an even number of
MZMs.
Two even-weight Majorana operators anticommute if they intersect on an odd number of
MZMs.
If an error infects an even number of MZMs in an operator support, then the operator parity
remains unchanged.
If an error infects an odd number of MZMs in an operator support, it toggles the operator
parity.
4.2 Single octon as OTM
We adopt the nanowire architecture proposed by Karzig et al. [KKL+17], which uses semiconducting nanowires with strong spin-orbit coupling and Zeeman fields in the presence of the s-wave
superconducting proximity effect. An even number of Majorana zero modes (MZMs) can be hosted
in a superconducting island maintained in an overall even parity state by high charging energy.
In this work, we use Majorana islands with 8 MZMs, also called as octons. Figure 4.1 shows the
hardware of an octon, which serves as an OTM unit. The octon can have either a one-sided design
or a two-sided design. We shall use the qubit structure of the two-sided design for representational
convenience in subsequent schematic diagrams such as Fig. 4.2. Note that in both designs, we can
measure the parity of only two MZMs at a time [KBPK18].
Now let us show that each octon is equivalent to an OTM storing 2 bits, where the chosen bit
can be perfectly obtained, and the remaining bit can be obtained with 75% probability.
Note that the octon has an overall even parity. So if we divide the octon into two parts - with
the top part having 4 MZMs and the bottom part having 4 MZMs, then both the top part and the
bottom part should have equal parity. When Alice wants to store two bits in the OTM, she would
59

1
2
3
4
5
6
7
8
(a)
1
3
5
7
2
4
6
8
(b)
ܼଶ
ܼଵ
ܼଷ
ܺଷ
ܺଵ
ܺଶ
(c)
(d) (e) (f)
Topological
superconductor
Non-topological
superconductor Semiconductor Quantum dot
mediated coupling
Majorana
zero mode
Figure 4.1: Figure (a) shows a one-sided octon. Figure (b) shows a two-sided octon. Figure (c)
illustrates the three qubits present in an octon. The first qubit stores the first classical bit, the
second qubit stores the second classical bit, and the third qubit stores the measurement basis.
Figure (d) shows an X measurement on the one-sided octon. Figure (e) shows a Z measurement
on the two-sided octon. Observe that a coherent link (floating 1D topological superconductor)
is necessary for measuring MZMs on opposite sides. Figure (f) shows an X measurement on the
two-sided octon.
at first randomly choose bit c ∈R {0, 1} and then use forced measurement [BFN08] of operator
X3 to initialize Z3 with parity c. Then she attempts to store two classical bits a1 and a2 in the
top part and the bottom part respectively. If c = 0, Alice stores the classical bits in operators
Z1 and Z2. If c = 1, Alice stores the two classical bits in X1 and X2. Note that 2-MZM forced
measurements are sufficient for OTM initialization. A brief summary of the OTM initialization
protocol is shown in Fig. 4.2.
Suppose Alice stores bits a1 and a2 in two parts of the octon, and gives it to Bob. Bob knows
that the two bits are stored in part 1 and part 2, but he does not know the correct measurement
basis. So, if he wants bit ax, he obtains parity of part y by measuring two vertical operators or
60

Odd
Odd
Even
Even ܽଵ
ܽଶ
1 െ ܽଵ
1 െ ܽଶ
ܽଵ
ܽଶ
ܽଵ
ܽଶ
50% 50%
Figure 4.2: Alice randomly chooses parity of the octon halves, and accordingly decides to store
the classical bits in the vertical operators or the horizontal operators.
two horizontal operators. Then he can find the basis, and check whether his choice was correct or
wrong.
With 1
2
probability, his chosen basis was correct, and he obtains bit ay.
With 1
2
probability, he chooses the wrong basis. After getting the parity, he realizes that bit
ay has been destroyed due to his measurements in wrong basis. Then he guesses the value of
bit ay with 1
2
probability of success.
So Bob obtains bit ay with probability 1
2 +

1
2 ×
1
2

=
3
4
. After finding the parity of part y, Bob
knows the correct measurement basis, and can always perfectly obtain his chosen bit ax. Thus,
the octon functions as an imperfect OTM where the chosen bit can be correctly obtained, and the
remaining bit is obtained with 75% probability.
In this section we have considered only strong measurements. Appendix C extends the result for
an arbitrary sequence of weak and strong measurements on one octon, and shows that (probability
of finding bit a1)+(probability of finding bit a2) ≤ 1.75.
6

4.3 Octon cluster as OTM
4.3.1 Bit availability without leakage
In this section we show that a cluster of octons approximate an ideal OTM, where any one bit
can be perfectly obtained, and the remaining bit can be obtained with nearly 50% probability.
Consider a cluster of k octons, each isolated from the other. The i
th octon stores the classical
bits a1i and a2i
in the top part and bottom part respectively. The octon cluster stores the classical
bits (a1 = a11 ⊕ a12 ⊕ . . . a1k) and (a2 = a21 ⊕ a22 ⊕ a23 . . . a2k).
If we wish to obtain bit a1 correctly, then we would need to correctly obtain the top bits of all
k octons. So, we would at first obtain the parity of bottom part of all octons, and then use the
correct basis to read all top bits. This yields bit a1 correctly, but probably damages bit a2.
If we correctly choose the basis for parity measurement of all k octons, then we can obtain
all the bottom bits correctly. This has a probability of 1
2
k .
If the basis choice for any parity measurement is wrong, then the bottom bit would be
destroyed. Then a2 can only be guessed with 1
2
probability.
Thus, the probability of obtaining the bit a2 is given by 1
2
k +

1 −
1
2
k

1
2 =
1
2 +
1
2
k+1 . This is
close to an ideal OTM where the remaining bit gets destroyed, and can only be guessed with 1
2
probability.
Alternately, we can obtain bit a2 perfectly, while bit a1 would be nearly destroyed.
Appendix D extends the result for an arbitrary sequence of weak and strong measurements on
the octon cluster, and shows that (probability of finding bit a1)+(probability of finding bit a2)
≲ 1.5.
6

4.3.2 Bit availability with leakage
Now let us discuss how leakage of mutual information about the two classical bits affects the
bit availability in the ideal OTM and the octon cluster OTM.
Suppose Bob has an ideal OTM, which outputs bit ax and destroys bit ay. However, Bob has
access to some leaked mutual information about the two bits, and so he knows the value of the
destroyed bit ay with py probability.
Now suppose that Bob has an OTM cluster of k octons. He wants to correctly output bit ax,
so he measures the parity of part y of all octons, and then uses the obtained basis to output bit ax.
He has 1
2
k probability to choose the correct basis for all k octons and obtain bit ay, and he has

1 −
1
2
k

probability to destroy bit ay by measurements in wrong basis. If the bit ay gets destroyed,
he can use the leaked mutual information to guess it with py probability. Thus, probability to
obtain bit ay is
1
2
k
+

1 −
1
2
k

py = py +
1 − py
2
k
= py + ϵ.
Thus, we have shown that the octon cluster OTM and the ideal OTM have approximately the
same bit availability in the case of information leakage.
4.4 Error correction on cluster OTM
4.4.1 Data error rate
Consider a cluster of k octons, where each octon has p probability of being infected by error.
So there is ≈ kp probability for an infected octon to exist in the cluster. We assume that errors
are bosonic in nature and find the corresponding data error rates. Likewise we can extend this for
fermionic errors or a combination of both errors.
6

Suppose we wish to obtain bit ax stored in part x of all octons in the cluster. To obtain ax, we
would perform parity measurements on part y of all octons in the cluster, and then obtain the
correct basis for measuring ax.
There are 8C2 = 28 possible configurations of 2-MZM errors on an octon, each with kp
28
probability. We analyze these error configurations below:
1. There are 4C2 = 6 ways in which the error occurs in part x :
(a) With 2kp
28 probability, the error in part x has a good configuration, meaning that it does
not damage the bit ax. Hence, we can obtain the bit ax with probability px = 1 and we
can obtain the bit ay with py =
1
2 +
1
2
k+1 .
(b) With 4kp
28 probability, the error in part x has a bad configuration, meaning that it
damages the bit ax. Then bit ax is obtained with px = 0 and bit ay is obtained with
py =
1
2 +
1
2
k+1 .
2. There are 4C2 = 6 ways in which the error occurs in part y :
(a) With 2kp
28 probability, the error in part y has a good configuration, meaning that it does
not damage the bit ay. Hence, we can obtain the bit ax with probability px = 1 and we
can obtain the bit ay with py =
1
2 +
1
2
k+1 .
(b) With 4kp
28 probability, the error in part y has a bad configuration, meaning that it
damages the bit ay. Then bit ax is obtained with px = 1 and the bit ay is obtained
with probability py =

1 −
1
2
k

·
1
2 =
1
2 −
1
2
k+1 .
3. There are 4C1 × 4C1 = 16 ways in which the error is spread out in both part x and part y.
Thus, with probability 16kp
28 the parity is corrupted, and the obtained basis is also wrong. In
this case, px =
1
2
and py =
1
2
.
Thus, the expected value of px = 1 −
3kp
7
.
The expected value of py =
1
2
k ·

1 −
3kp
7

+

1 −
1
2
k

·
1
2
.

We can compare this with the general form of probability pi = (probability that part i is
measured in correct basis × probability that part i has no error) + (probability that part i is
measured in wrong basis × probability of correctly guessing bit ai).
By comparison, we can see that an octon cluster error rate of kp is equivalent to a data error
rate of 3kp
7
for bit a1 as well as for bit a2.
However, we will make a conservative assumption that only the chosen bit has a data error
rate of 3kp
7
, and the adversary does not face any errors if it wishes to obtain the remaining bit.
4.4.2 Error correcting codes
We can use classical codes to correct data errors on the top bit and the bottom bit. We would
use any CSS code and map the X stabilizers to a classical code for the top layer, and map the Z
stabilizers to another classical code for the bottom layer. If the CSS code requires n qubits, then
we would use n clusters, each containing k octons. For each qubit in the CSS code, we would place
an octon cluster in that node.
For each X stabilizer, the top bits of all octons in all clusters in the support of that X stabilizer
would have overall even parity. Similarly, for each Z stabilizer, the bottom bits of all octons in all
clusters in the support of that Z stabilizer would have overall even parity.
This CSS code stores two classical bits. The first classical bit a1 is stored in the parity of the
logical operator X. This means that the classical bit a1 is equal to the parity of all the top bits in
all octons in all clusters that belong to the support of the logical operator X. Similarly, the second
classical bit a2 is stored in the parity of the logical operator Z.
4.4.3 Bacon-Shor code
The Bacon-Shor code is a good example of a CSS code. Figure 4.3 shows how the Bacon-Shor
code maps to two classical repetition codes. The top layer code spans the top bits of all octons,
and the bottom layer code spans the bottom bits of all octons. The logical operator X in the top
65

Top layer
Logical ܺ
(a)
Bottom layer
Logical ܼ
(b)
݇ octons
(c)
Figure 4.3: Figure (a) shows the classical repetition code corresponding to the two X stabilizers
of a Bacon-Shor code. It also shows three copies of the logical X operator highlighted in brown
color. Figure (b) shows the classical repetition code corresponding to the two Z stabilizers of a
Bacon-Shor code. It also shows three copies of the logical Z operator highlighted in blue. Figure
(c) shows that each node of the 3 × 3 Bacon-Shor code is a cluster of k octons.
layer intersects the logical operator Z in the bottom layer, and all stabilizer equivalents of these
logical operators must also intersect each other. Hence, obtaining the parity of any one logical
operator reduces the probability of obtaining the parity of the remaining logical operator.
These codes can correct data errors on the chosen bit. Without loss of generality, we can
assume that the chosen bit was a1, and it was encoded in the logical X operator of the top layer
code. We obtained the chosen bit a1 and performed error correction on it. Now we shall analyze
the soundness or availability of the remaining bit a2.
For the purpose of error correction, all the octons needed to be measured. Hence, each octon
cluster now has a
1 −
1
2
k

probability to have a bottom bit damaged by measurements in wrong
basis.
The logical Z operator has three copies in three columns. The probability that all three clusters
in one column have been damaged is given by
1 −
1
2
3k

. Hence, the probability that all three
copies of the logical Z operator are damaged is given by
1 −
1
2
3k
3
. If all three copies are damaged,
then bit a2 can only be guessed with 1
2
probability. Otherwise, if at least one copy of the logical

Cluster size ݇
Soundness
Figure 4.4: The soundness of the Bacon-Shor code reduces and approaches 0.5 if we increase the
cluster size.
Z operator is undamaged, with a probability of 1 −

1 −
1
2
3k
3
, then the bit a2 can be correctly
obtained. Thus, the soundness or availability of bit a2 is given by
1 −

1 −
1
2
3k
3
+
1
2

1 −
1
2
3k
3
= 1 −
1
2

1 −
1
2
3k
3
.
Figure 4.4 shows that soundness decreases when we increase the cluster size.
4.4.4 Other codes
We consider various CSS codes such as the J7, 1, 3K Steane code, the J17, 1, 5K and J19, 1, 5K
triangular color codes, the J9, 1, 3K and J25, 1, 5K Bacon-Shor codes, and the J23, 1, 7K Golay code.
For each code, we plot the logical error rate of chosen bit in Fig. 4.5. The availability of chosen bit
is given by (1 - logical error rate). Figure 4.6 shows the soundness or availability of the remaining
bit. In Fig. 4.7, we plot the best code for different regimes of error, and availability of the chosen
bit, and soundness of the remaining bit.
6

No code
k = 10
k = 9
k = 8
k = 7
k = 6
k = 5
k = 4
k = 3
k = 2
k = 1
Logical error rate of bit
x
0
0.1
0.2
0.3
0.4
Physical error rate (p)
0 0.02 0.04 0.06 0.08 0.1
(a)
⟦7,1,3⟧ code
k = 4
k = 3
k = 2
k = 1
Logical error rate of bit
x
0
0.1
0.2
0.3
0.4
Physical error rate (p)
0 0.02 0.04 0.06 0.08 0.1
(b)
⟦9,1,3⟧ code
k = 4
k = 3
k = 2
k = 1
Logical error rate of bit
x
0
0.1
0.2
0.3
0.4
Physical error rate (p)
0 0.02 0.04 0.06 0.08 0.1
(c)
⟦17,1,5⟧ code
k = 3
k = 2
k = 1
Logical error rate of bit
x
0
0.1
0.2
0.3
0.4
Physical error rate (p)
0 0.02 0.04 0.06 0.08 0.1
(d)
⟦19,1,5⟧ code
k = 3
k = 2
k = 1
Logical error rate of bit
x
0
0.1
0.2
0.3
0.4
Physical error rate (p)
0 0.02 0.04 0.06 0.08 0.1
(e)
⟦23,1,7⟧ code
k = 3
k = 2
k = 1
Logical error rate of bit
x
0
0.1
0.2
0.3
0.4
Physical error rate (p)
0 0.02 0.04 0.06 0.08 0.1
(f)
⟦25,1,5⟧ code
k = 3
k = 2
k = 1
Logical error rate of bit
x
0
0.1
0.2
0.3
0.4
Physical error rate (p)
0 0.02 0.04 0.06 0.08 0.1
(g)
Figure 4.5: The above plots show the logical error rate of the chosen bit x for various codes and
various values of cluster size k. We consider the cases of (a) no code, (b) J7, 1, 3K Steane code,
(c) J9, 1, 3K Bacon-Shor code, (d) J17, 1, 5K color code, (e) J19, 1, 5K color code, (f) J23, 1, 7K Golay
code, and (g) J25, 1, 5K Bacon-Shor code. In the plots (b) to (g), the 95% confidence interval is
smaller than the size of plot marker.
68

No code
Soundness of bit
y
0.50001
0.5001
0.501
0.51
0.55
0.75
1
Cluster size (k)
1 2 3 4 5 6 7 8 9 10
(a)
⟦7,1,3⟧ code
Soundness of bit
y
0.50001
0.5001
0.501
0.51
0.55
0.75
1
Cluster size (k)
1 2 3 4
(b)
⟦9,1,3⟧ code
Soundness of bit
y
0.50001
0.5001
0.501
0.51
0.55
0.75
1
Cluster size (k)
1 2 3 4
(c)
⟦17,1,5⟧ code
Soundness of bit
y
0.50001
0.5001
0.501
0.51
0.55
0.75
1
Cluster size (k)
1 2 3
(d)
⟦19,1,5⟧ code
Soundness of bit
y
0.50001
0.5001
0.501
0.51
0.55
0.75
1
Cluster size (k)
1 2 3
(e)
⟦23,1,7⟧ code
Soundness of bit
y
0.50001
0.5001
0.501
0.51
0.55
0.75
1
Cluster size (k)
1 2 3
(f)
⟦25,1,5⟧ code
Soundness of bit
y
0.50001
0.5001
0.501
0.51
0.55
0.75
1
Cluster size (k)
1 2 3
(g)
Figure 4.6: The above plots show the soundness of the remaining bit y for various codes and
various values of cluster size k. We consider the cases of (a) no code, (b) J7, 1, 3K Steane code,
(c) J9, 1, 3K Bacon-Shor code, (d) J17, 1, 5K color code, (e) J19, 1, 5K color code, (f) J23, 1, 7K Golay
code, and (g) J25, 1, 5K Bacon-Shor code. The plots (b) to (g) also display the 95% confidence
intervals. The graph does not show variation of soundness with physical error rate, because we had
conservatively assumed that an adversary does not face errors while obtaining the remaining bit y.
69

0.001
0
Maximum availability of bit

Physical error rate (p)
Minimum availability of bit ! = 0.95
1 × 1
1 × 2
1 × 3
1 × 4
1 × 5
1 × 6
1 × 7
1 × 8
1 × 9
1 × 1
1 × 2
1 × 3
1 × 4
1 × 5
1 × 6
1 × 7
1 × 8
1 × 9
7 × 2
9 × 2
9 × 1
25 × 1
25 × 2
7 × 3
9 ×3
7 × 4
17 × 2
23 × 2
17 × 3
19 × 2
0
0.001
1 × 1
1 × 2
7 × 1 17 × 1
23 × 3
19 × 2
9 × 1
25 × 1
25 × 2
17 × 2
17 × 3
23 × 2
23
× 1
19
× 1
7 × 2
9 × 2
7 × 3
9 × 3
7 × 4
1 × 1
1 × 2
1 × 3
1 × 4
1 × 5
1 × 6
1 × 7
1 × 8
1 × 9
Maximum availability of bit

Physical error rate (p)
Minimum availability of bit ! = 0.99
Minimum availability of bit ! = 0.999
0.001 0.01 0.02 0.03 0.04 0.05
0
1 × 1
1 × 2
1 × 3
1 × 4
1 × 5
1 × 6
1 × 7
1 × 8
1 × 9
7 × 1
9 × 1
17 × 1 23 × 1
25 × 1
7 × 2
9 × 2 17 × 2
25 × 2
19 × 2
23 × 2
19 × 3
17 × 3
7 × 3
9 × 3
7
×
4
23
× 3
Maximum availability of bit

Physical error rate (p)
Minimum availability of bit ! = 0.999
Legend No code 7, 1, 3 9, 1, 3 17, 1, 5 19, 1, 5 23, 1, 7 25, 1, 5
Figure 4.7: In the top graph, we plot the codes with the least overhead which achieves 95%
availability of the chosen bit, for different regimes of soundness and physical error rate. Each sector
of the graph is labeled in the form n × k, where n is the number of physical qubits in CSS code
and k is the cluster size. In the middle graph and the bottom graph, we similarly plot the codes
with the least overhead which achieves 99% and 99.9% availability of the chosen bit, respectively.
In all three graphs, an octon cluster is better than codes when error rate approaches zero. The
legend shows the colors for various CSS codes, including the J7, 1, 3K Steane code, the J9, 1, 3K
Bacon-Shor code, the J17, 1, 5K color code, the J19, 1, 5K color code, the J23, 1, 7K Golay code, and
the J25, 1, 5K Bacon-Shor code.
70

3
4
1 2 5
Figure 4.8: A dishonest recipient can perform this malicious sequence of measurements to maximize
his chance of cheating and obtaining both bits stored in the Bacon-Shor OTM.
4.5 Dishonest recipient
Suppose the recipient of an OTM is dishonest. The OTM has an error correction code intended
to correct errors on the chosen bit. However, the dishonest recipient chooses to forgo error
correction, and obtains the chosen bit without correction, and then he uses the code to obtain the
remaining bit with a higher probability. Let us see how the codes can be maliciously used.
Consider again the Bacon-Shor code shown in Fig. 4.3. It has two X stabilizers on the top layer,
which protect the logical X operator. The bottom layer also has two Z stabilizers that protect the
logical Z operator. The dishonest recipient can use the malicious measurement sequence shown in
Fig. 4.8.
1. At first, he measures the octon cluster labeled as 1 and highlighted in brown color, located in
the bottom left of the grid. He measures the parity from the bottom part of all octons in this
cluster, and then obtains the correct basis to measure all the top bits stored in this cluster.
He correctly obtains the top bits, and he also has 1
2
k probability to obtain the bottom bits
correctly. If the bottom bits are correctly obtained, then he can find the parity of logical X
operator passing through cluster 1, as well as the parity of logical Z operator passing through
cluster 1. So this step gives him 1
2
k probability to cheat and obtain both the classical bits.
71

2. In the second step, he measures the parity from the bottom part of all octons in cluster 2,
highlighted in brown. Then he obtains the correct measurement basis, and gets all the top
bits of cluster 2 correctly. This step gives him an additional 1
2
k probability to cheat and
measure both logical X operator and logical Z operator passing through cluster 2.
3. If the previous steps did not allow him to cheat, then he proceeds to cluster 3, where he
measures the parity of top parts of all octons, and uses the basis to correctly measure all
bottom bits. This step gives him 1
2
k probability to cheat and measure both logical operators
passing through cluster 3.
4. Then he proceeds to cluster 4, and measures the parity of top parts of all octons, and uses
the basis to correctly measure all bottom bits. This step gives him 1
2
k probability to cheat
and measure both logical operators passing through cluster 4.
5. Finally, if he has been unable to cheat in the previous steps, then he proceeds to cluster 5.
In this step, he can choose to prioritize any one of the two classical bits. If he prioritizes
the bit stored in logical X operator, then he can measure the parity of bottom part of all
octons in cluster 5, and use the obtained basis to correctly measure all the top bits in cluster
5. Then he uses the previously measured top bits of clusters 1 and 2 to obtain the logical X
operator. This gives him another 1
2
k probability to measure the logical Z operator passing
through cluster 5. Alternately, he could prioritize the bit stored in logical Z operator.
Thus the dishonest recipient has 5 chances to cheat, with a probability of 1 −

1 −
1
2
k
5
≈
5
2
k .
In a general CSS code with n qubits, the dishonest recipient can have as much as n chances to
cheat, with a probability of ≈
n
2
k . In Appendix E.1 and Appendix E.2 , we provide two examples
of CSS codes where a dishonest recipient has n chances to cheat.
7

4.6 1/n OTM and (n − 1)/n OTM
Until now, we have described 1-out-of-2 OTMs, where two classical bits are stored, and only
one bit can be retrieved. In this section we will show that the 1
2 OTM can be generalized into 1/n
OTM and (n − 1)/n OTM. Previous works [BCR86, BCS96, NP99] have showed that multiple
copies of 1
2 OTM can be used to construct arbitrary OTMs. For example, two copies of 1
2 OTM
can be used to create a 1
3 OTM. However, we show that a simple hardware extension can be used
to create a 1
3 OTM with only 50% additional overhead. We describe the principles of the hardware
extension in this section.
We shall use a Majorana island with 4n MZMs, and divide it into n parts with 4 MZMs each.
The i
th part has parity ci
, and it has two operators Xi and Zi as shown in Fig. 4.9(a). The overall
island is in even parity, however, the individual parts can have arbitrary parity. We shall store n
classical bits a1, a2, . . . an in these n parts.
To use this island as a 1/n OTM, we shall store the classical bits ai
in the operators HciZi for
all i ∈ [n], where [n] = {1, 2, 3, . . . n}. If the recipient wishes to obtain bit ai then he can measure
the parity of all other parts except part i. If all other parts have overall even parity, then the bit
ai
is stored in the Zi operator. Else if all other parts have overall odd parity, then the bit ai
is
stored in the Xi operator. Thus, bit ai can be obtained perfectly, whereas the other bits would be
damaged with some probability. We can use a cluster of such islands to reduce the soundness of
all other bits, in a similar manner to octon clusters.
To use this island as an (n−1)/n OTM, we shall adopt a different strategy. We define j = (i−2)
(mod n) + 1. We shall store the classical bits ai
in the operators HcjZi for all i ∈ [n]. Thus, the
measurement basis for part i is defined by parity of the previous part, as shown in Fig. 4.9(b).
We begin with parity measurement on some part x, which can potentially damage the stored bit
ax. However, its parity measurement would reveal the correct basis for the next part, and that
would again reveal the correct basis for the next part, and so on. Thus, apart from bit ax all other
73

2
݅
ڭ
݊
1
ܼଵ
ܺଵ
ܺଶ
ܼଶ
ܺ௡
ܼ௡
ܺ௜
ܼ௜
(a)
ܿଶ = 0
ڭ
ܿ௡ = 1
ܿଵ = 1
ܽଵ
ܽଶ
ܽଷ
(b)
Figure 4.9: (a) This illustrates the basic unit of a 1/n OTM. It corresponds to an island with 4n
MZMs, and it has n parts with 4 MZMs in each part. The X operators and Z operators in each
part are shown as red and green lines respectively. The measurement basis of a part depends on
its own parity.
(b) This illustrates the basic unit of an (n − 1)/n OTM. Here, the measurement basis of any part
is determined by parity of the preceding part. For example, the basis of part 2 is determined by
parity of part 1, and the basis of part 1 is determined by parity of part n, and so on.
bits can be correctly obtained. We can reduce the soundness for bit ax by using a cluster of such
islands, in a similar manner as the octon cluster.
4.6.1 Example: 1
3 OTM on J7, 1, 3K code
We shall discuss how error correction can be performed on a 1
3 OTM by using the J7, 1, 3K
Steane code. Here the basic units are Majorana islands with 12 MZMs each, also called as dodecons.
74

Logical X
Logical Y
Logical Z
݇ dodecons
Figure 4.10: This shows error correction on the 1
3 OTM. The X stabilizers protect the logical X
operator in the top layer, and they are shown in yellow polygons and brown line respectively. The
Y stabilizers protect the logical Y operator in the middle layer, and they are shown as light green
polygons and a dark green line respectively. The Z stabilizers protect the logical X operator in the
bottom layer, and they are shown as light blue polygons, and a dark blue line respectively.
We use 7 clusters of k dodecons each, arranged in the form of a three layered code, as shown in
Fig. 4.10.
We encode three classical bits in three layers using the X, Y and Z stabilizers, where the Y
stabilizers are obtained by multiplying the X and Z stabilizers.
On the top layer, the classical analogue of X stabilizers encode a logical X operator, whose
parity is equal to the first classical bit.
On the middle layer, the classical analogue of Y stabilizers encode a logical Y operator,
whose parity is equal to the second classical bit.
On the bottom layer, the classical analogue of Z stabilizers encode a logical Z operator,
whose parity is equal to the third classical bit.
The three logical operators must intersect each other at one or more nodes, even if we consider
all stabilizer equivalents of these operators. Hence, obtaining the parity of one logical operator
75

significantly reduces the probability of obtaining parity of two other logical operators. Let us find
the probability of obtaining the remaining bits.
Suppose that the recipient obtains the first bit from the first layer by honestly using error
correction. So he had to correctly read the 7 bits stored in the 7 clusters of the first layer. This
damages all the bits stored in the second layer and the third layer with a high probability.
Let us focus on the second layer. Each cluster in the second layer can be either damaged or
undamaged. Thus, damage on the 7 clusters can lead to 27
configurations in total.
If a k-cluster remains undamaged with a probability of g =
1
2
k , then a configuration with x
damaged clusters, and (7 − x) undamaged clusters would have probability = x
1−g × (7 − x)
g
. We
consider the eight stabilizer equivalents of the logical operator, and find the probability for at least
one copy of the logical operator to remain undamaged, by programmatically enumerating all the 27
configurations in the second layer. We find that the logical operator is undamaged with probability
gL = 7g
3 − 21g
5 + 21g
6 − 6g
7
. If the logical operator is damaged, then it can be guessed with 1
2
probability. Hence, the availability of the second bit is given by
gL · 1 + (1 − gL) ·
1
2
=
1
2
+
gL
2
=
1
2
+
1
2
(7g
3 − 21g
5 + 21g
6 − 6g
7
)
=
1
2
+
7
2
3k+1 + O

1
2
5k

Similarly, we can show that the third bit stored in the third layer has the same availability. In
Appendix E.3 we show how a malicious recipient can sacrifice error correction and increase the
availability of the second remaining bit to O

1
2
k

, and he can increase the availability of the third
remaining bit to O

1
2
2k

.

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82

Appendix A
Measurement architecture for B 7→ F codes with and
without coherent links
We provide examples of stabilizer codes implemented using one-sided tetrons and two-sided
tetrons [KKL+17]. A large number of stabilizer codes can be implemented on the tetron architecture
without requiring additional measurement links. However, an arbitrary stabilizer code may require
additional floating topological superconducting links, sometimes known as “coherent links,” for
measurement purpose [KKL+17]. These additional coherent links increase the Majorana overhead
of a code, as well as increases error-prone locations in the code. So, we show how error correction
can be performed in such cases.
A.1 One-sided tetrons
For small error-correcting codes, stabilizer measurements can be performed using one-sided
tetrons. For example, Fig. A.1(a) shows a one-sided tetron with MZMs at locations a, b, c, d, and
its Pauli operators are X = γbγc, Z = γaγb. Figure A.1(b) shows the schematic of a 7-tetron
fermionic color code, where the highlighted stabilizer is being measured. Figure A.1(c) shows the
nanowire diagram corresponding to this stabilizer measurement. This tetron configuration can be
used for the fault-tolerant measurement sequence shown in Fig. 2.14.
83

T1
T3
T5
T7
T2
T4
T6
a
b
c
d
(a)

1
′
2
6 7
3 4
5
(b)
(c)
Figure A.1: Figure (a) shows a one-sided tetron with MZMs at locations a, b, c, d. Its Pauli
operators are X = γbγc, Z = γaγb. Figure (b) highlights one stabilizer of the J14, 1, df = 6K code
which is being measured. Figure (c) shows the tetron configuration for this stabilizer code. We
use doubled black lines to denote quantum dot mediated coupling, which is used to create the
measurement loop corresponding to the X′
1X2X3X4 stabilizer.
A.2 Two-sided tetrons
One-sided tetrons may not suffice for stabilizer codes with large number of qubits. We can use
two-sided tetrons to implement a variety of large stabilizer codes such as the 6.6.6 color code, the
4.8.8 color code, the CSS version of the rotated surface code, the XZZX version of the rotated
surface code, the subsystem surface code, among other codes.
For example, Fig. A.2 shows that the J38, 1, df = 10K code can be implemented using two-sided
tetrons. Figure A.2(a) shows the schematic of a 19-tetron fermionic color code, where the highlighted
stabilizers are being measured. Figure A.2(b) shows the nanowire diagram corresponding to the
measurement of the highlighted stabilizers.
Note that the tetrons are arranged in columns, such that any stabilizer is supported on an even
number of tetrons in adjacent columns. This allows the measurement of any stabilizer which is
84

1
7
9
10
11
17
16
19
13
′ 14
15
5
12 18
8
6
2
3
4
(a)
T16
T17
T18
T19
T1
T7
T8
T9
T10
T11
T12
T13
T14
T15
T6
T5
T4
T3
T2
(b)
Figure A.2: The figure illustrates a two-sided tetron configuration that can implement the
J38, 1, df = 10K code. Figure (a) shows two examples of syndrome measurement, which are
highlighted in green. Figure (b) illustrates the tetron configuration for this stabilizer code. We
use doubled black lines to denote quantum dot mediated coupling, which is used to create the
measurement loops corresponding to the stabilizers Z2Z3Z7Z9Z10Z11 and X4X5X′
13X15.
composed of X, Z, X′
, Z′ Pauli operators. If such stabilizers form the generators of a code, then
such stabilizers are also sufficient for fault-tolerant syndrome measurement.
A.3 Two-sided tetrons and coherent links
An arbitrary stabilizer code may require additional coherent links for measurement purpose.
Figure 1.1 shows a network of tetrons interleaved with coherent links. Each coherent link has 2
MZMs, and is maintained in even parity by high charging energy. As the links have only 2 MZMs,
so they are immune to bosonic errors. However, the links can be affected by fermionic errors,
which can be a source of measurement errors. Such errors can be corrected by classical codes.
To give an example of classical error correction on coherent links, we again use the J14, 1, df = 6K
code. But now we use the tetron configuration of Fig. A.3(a), such that stabilizers X1X2X3X4
and X′
1X2X3X4 cannot be measured without additional coherent links in the left column and
85

the right column. Furthermore, if we wish to measure stabilizers Y
′
1Y
′
2Y
′
3Y
′
4 and Y
′
3Y
′
4Y
′
5Y
′
6
for
fault-tolerant measurement, then coherent links are also necessary in the middle column.
Here, errors on the 7 tetrons are corrected according to the stabilizers of J14, 1, df = 6K
code, and errors on the coherent links are corrected according to the classical repetition code.
Figure A.3(b) shows the parity matrix for three copies of the classical repetition codes, used on
the coherent links. We illustrate one round of syndrome measurement for the stabilizer generators
and the coherent links in Fig. A.3(c) to Fig. A.3(h) and Fig. A.4(a) to Fig. A.4(g). Note that the
classical code on the links can be interleaved with the quantum code on the tetrons, such that
they can be parallelly executed without requiring any additional time steps.
86

L3
L5
L8
L1
L4
L6
L9
L2
L7
L10
T3
T5
T2
T7
T1
T4
T6
(a)
1100000000
0110000000
0001100000
0000110000
0000011000
0000000110
0000000011
 
 
 
L3 L5 L8 L1 L4 L6 L9 L2 L7 L10
(b)
(c) (d)
(e) (f)
(g) (h)
Figure A.3: Figure (a) illustrates 7 tetrons and 10 measurement links (using the legend of
Fig. 1.1), and labels them for convenience. Figure (b) shows the parity matrix for three classical
repetition codes for the measurement links. In subsequent figures, we use doubled black lines to
denote quantum dot mediated coupling, which is used to create measurement loops for Majorana
operators. Figure (c) shows measurement of X1X2X3X4, L5L8, L6L9 and L7L10. Figure (d)
shows measurement of X2X4X6X7, L3L5 and L1L4. Figure (e) shows measurement of X3X4X5X6
and L2L7. Figure (f) shows measurement of X′
1X2X3X4. Figure (g) shows measurement of
X′
2X4X6X7. Figure (h) shows measurement of X′
3X4X5X6.
87

(a) (b)
(c) (d)
(e) (f)
(g)
Figure A.4: This is a continuation of Fig. A.3, which illustrates the syndrome measurement for the
J14, 1, df = 6K quantum code over 7 tetrons, in parallel with syndrome measurement for the classical
code over 10 measurement links. Figure (a) shows measurement of Z1Z2Z3Z4. Figure (b) shows
measurement of Z3Z4Z5Z6 and L3L4L8L9 (which yields L4L6). Figure (c) shows measurement
of Z2Z4Z6Z7. Figure (d) shows measurement of Z1Z2Z3Z
′
4
. Figure (e) shows measurement
of Z3Z4Z
′
5Z6. Figure (f) shows measurement of Z2Z4Z
′
6Z7. Figure (g) shows measurement of
Z2Z4Z6Z
′
7
.
88

Appendix B
B 7→ FS codes: Fermionic subsystem codes from bosonic
codes
We show that a bosonic color code family can be translated into a fermionic subsystem code
family, wherein the fermionic code distance scales with the code size. We apply this recipe to
triangular 6.6.6 color codes, and provide three examples for illustration.
B.1 Recipe for fermionic code construction
We use an Jn, k, dbK non-subsystem bosonic code to derive a J2n, k′
, g, df K subsystem Majorana
fermion code. The tetrons of the Majorana fermion code are divided into multiple partitions, as
shown in the first row of Fig. B.1(a). From these partitions, we can derive the stabilizers of the
Majorana fermion code, which we group into several sets for convenience.
Set 0 contains all the stabilizers of the bosonic code, wherein every Pauli operator of the
bosonic code maps to Majorana operators in R.
Set i contains all the stabilizers of the bosonic code, wherein Pauli operators of the ith
partition map to Majorana operators in R
′
, and all other Pauli operators map to Majorana
operators in R. Here, i varies from 1 to the total number of partitions.
89

We provide an example of this scheme in Fig. B.1(a), which shows how to derive the stabilizers
of the J14, 1, 2, df = 3K fermionic code from a J7, 1, 3K bosonic code. Later, we will discuss various
parameters of this fermion code such as code distance, logical operators, etc.
ℛ
 
(a)
d
a b c d
(b)
Figure B.1: (a) The top panel shows two partitions of the 7-tetron color code. The bottom panel
shows the stabilizers in sets 0, 1, 2 of J14, 1, 2, df = 3K code. The X and Z stabilizers in set 0
are supported on X and Z operators, respectively, at their plaquette vertices. The X stabilizer
plaquettes in set 1 are supported on X′ at the plaquette vertices marked by red circles, and X on
other vertices of that plaquette. The Z stabilizer plaquettes in set 2 are supported on Z
′ at the
plaquette vertices marked by red circles, and Z on other vertices of that plaquette. This group
has 11 independent stabilizers. (b) The figure shows the logical qubit and the gauge qubits of
J14, 1, 2, df = 3K code.
B.2 Code distance
The code distance of the fermionic subsystem code is given by the least weight of the dressed
logical operators, which is equivalent to the least weight of an error configuration that can cause a
90

logical error without producing any syndrome. The code distance is obtained by an exhaustive
search. We later show that the code distance scales with the code size.
B.3 Decoder
We use the BPOSD decoder, proposed by Roffe et al. [RWBC20, Rof22] for correcting errors in
the subsequent examples. We use the “product sum” method for belief propagation, the “osd cs”
method for the ordered statistics decoder, a a maximum of 5 iterations, and a search depth of
2N+1 where N is the number of tetrons.
B.4 Partitioning scheme for color codes
A triangular n-qubit color code can be partitioned into:
Partition 1 with n1 = ⌊n/2⌋ tetrons at odd distance from the center, and
Partition 2 with n2 = ⌈n/2⌉ tetrons at even distance from the center.
We provide an example of this partitioning scheme in Fig. B.2, with reference to the J19, 1, 5K
Pauli CSS color code.
Figure B.2: The 19-tetron code is divided into two partitions, where partition 1 comprises n1 = 9
tetrons at odd distance from the center, and partition 2 comprises n2 = 10 tetrons at even distance
from the center.
91

We illustrate the B 7→ F S recipe with three examples, corresponding to three family members
of the triangular 6.6.6 color code.
B.5 Example 1: J14, 1, 2, df = 3K subsystem code
We derive the J14, 1, 2, df = 3K subsystem Majorana fermion code from the J7, 1, 3K bosonic
code according to the scheme given above. The stabilizers of this fermion code are illustrated in
Fig. B.1(a). This code has one logical qubit and two gauge qubits as shown in Fig. B.1(b).
The fermionic code distance is defined as the least weight of a syndromeless error configuration
that causes a logical error. In this example, consider the three tetrons that support the first gauge
operator of gauge qubit 1. If these tetrons were affected by three γa errors, then the error would
be both syndromeless and harmful for the logical qubit. Hence, this code has a fermionic code
distance of df = 3.
B.6 Example 2: J38, 1, 1, df = 9K subsystem code
We derive the J38, 1, 1, df = 9K subsystem Majorana fermion code from the J19, 1, 5K bosonic
code. The stabilizers of the fermion code are illustrated in Fig. B.3(a), and its logical qubit and
gauge qubit are shown in Fig. B.3(b).
Although the illustrated logical operators have a minimum weight of 10 Majorana modes, the
fermionic code distance is df = 9. To see why, consider the nine tetrons that support the first
gauge operator. If these tetrons were affected by nine γa errors, then the error would be both
syndromeless and harmful for the logical qubit. Hence, this code has a fermionic code distance of
df = 9.
92

 
(a)
d
a b c d
(b)
Figure B.3: (a) The top panel shows two partitions of the 19-tetron color code. The bottom
panel shows the stabilizers in sets 0, 1, 2 of J38, 1, 1, df = 9K code. The X and Z stabilizers in set
0 are supported on X and Z operators, respectively, at their plaquette vertices. The X stabilizer
plaquettes in set 1 are supported on X′ at the plaquette vertices marked by red circles, and X on
other vertices of that plaquette. The Z stabilizer plaquettes in set 2 are supported on Z
′ at the
plaquette vertices marked by red circles, and Z on other vertices of that plaquette. This group
has 36 independent stabilizers. (b) Logical qubit and gauge qubit of J38, 1, 1, df = 9K code.
B.7 Example 3: J74, 1, 3, df = 9K subsystem code
We derive the J74, 1, 3, df = 9K subsystem fermionic code from the J37, 1, 7K bosonic code.
Stabilizers of the fermion code are shown in Fig. B.4(a), and its logical qubit and gauge qubits are
shown in Fig. B.4(b).
Consider the nine tetrons supporting the first gauge operator of gauge qubit 2. If these tetrons
were affected by nine γa errors, then the error would be both syndromeless and harmful for the
logical qubit. Hence, the fermionic code distance is df = 9.
93

 
(a)
d
a b c d
(b)
Figure B.4: (a) The top panel shows two partitions of the 37-tetron color code. The bottom
panel shows the stabilizers in sets 0, 1, 2 of J74, 1, 3, df = 9K code. The X and Z stabilizers in set
0 are supported on X and Z operators, respectively, at their plaquette vertices. The X stabilizer
plaquettes in set 1 are supported on X′ at the plaquette vertices marked by red circles, and X on
other vertices of that plaquette. The Z stabilizer plaquettes in set 2 are supported on Z
′ at the
plaquette vertices marked by red circles, and Z on other vertices of that plaquette. This group
has 70 independent stabilizers. (b) Logical qubit and gauge qubits of J74, 1, 3, df = 9K code.
94

B.8 Scalability of fermionic error correction in B 7→ FS
codes
There are two main limits in the fermionic error correction scheme. The first limit is the
bosonic error correction distance, and we know that it scales with code size. The second limit is
the fermionic error correction capacity of the partitions, and we will prove that this too scales
with code size.
Additionally, we would also like to prevent fermionic errors that commute with the stabilizer
group as well as the logical operators. The motivation is that such uncorrected excitations may
propagate to nearby tetrons via connected measurements and culminate in errors of higher weight.
For example, suppose that an error has the same support as one of the three gauge operators
shown in the central column of Fig. B.4(b). Such errors would remain undetected by stabilizer
syndromes, but they may propagate to nearby tetrons and affect them in future. So, we shall
prove that the least weight of such syndromeless errors must scale with the code size.
Both of these are covered in the following proof.
Claim 4. For a B 7→ F S subsystem Majorana fermion code derived from a triangular 6.6.6 color
code, the fermionic distance of the subsystem code scales with the code size.
Proof. The fermionic code distance may be defined as the least weight of an error configuration
which anticommutes with any logical operator but commutes with the stabilizer group.
We can prove the scalability of fermionic code distance by proving a stronger version of the
claim: the least weight of a zero-syndrome error configuration must scale with the code size. The
proof is divided into several conceptual blocks for ease of readability.
95

B.8.1 Starting from mixed error configuration
For proof by contradiction, let us assume that a mixed error configuration of weight O(1) yields
zero syndrome. We choose the least weight configuration for purpose of our proof. This error
comprises bosonic errors on tetron set Tb and fermionic errors of types γa, γb, γc, γd on tetrons sets
Tfa, Tf b, Tf c, Tfd respectively.
B.8.2 From mixed error to fermionic error configuration
We already know that if |Tb| + |Tfa| + |Tf b| + |Tf c| < db, then the set 0 stabilizers must yield
a non-zero syndrome, regardless of the presence of any γd errors. Thus, any O(1) collection of
bosonic or fermionic errors of γa, γb, γc types cannot produce zero syndrome. An error of O(1)
weight must be solely comprised of γd errors so as to yield zero syndrome. So, our hypothetical
zero-syndrome error configuration of O(1) weight has been reduced to a simpler fermionic error
configuration, solely composed of γd errors.
B.8.3 Decoupling disconnected errors
A set of two γd errors are said to be connected if they affect a common stabilizer. We say
that a connecting path exists between these two errors. A set of multiple γd errors are said to be
connected if a connecting path exists between any two members of this error set.
Our hypothetical fermionic error set yields zero syndrome on the set of all stabilizers. If
our error set comprises multiple disconnected error subsets, then each error subset should affect
a different stabilizer subset. Hence, each disconnected error subset should be a zero-syndrome
configuration.
For example, errors in partition 1 and partition 2 are disconnected, since the γd MZMs of
tetron-1-partitions solely belong to the support of stabilizers in set 1, while the γd MZMs of
96

tetron-2-partitions solely belong to the support of stabilizers in set 2. Thus, the least-weight
fermionic error configuration that yields zero syndrome must be confined to a single partition.
So, our hypothetical zero-syndrome error configuration of O(1) weight has been reduced to a
simpler connected error configuration, solely composed of γd errors in a single partition.
B.8.4 Even intersection between errors and plaquettes
Suppose the zero-syndrome fermionic error configuration of O(1) weight is supported on γd
MZMs in tetrons of partition 1. All γd MZMs in the partition-1-tetrons belong to the support of
stabilizers in set 1. As all stabilizers in set 1 yield zero syndrome, so each plaquette must have an
even number of affected tetrons in partition 1.
Similarly, if we obtain zero syndrome due to an O(1) fermionic error configuration in partition
2, then each plaquette must have an even number of affected tetrons in partition 2.
A similar reasoning explains the geometry of gauge operators as shown in Figs. B.1(b), B.3(b)
and B.4(b). Among the two gauge operators of a gauge qubit, one operator is trivially supported
on the four MZMs of a tetron. The other gauge operator is supported on the γd MZMs of a single
partition, such that it has even intersection with each plaquette.
Now we shall analyze whether the least weight zero-syndrome fermionic error configuration
confined to a single partition can possibly have O(1) weight. This is equivalent to analyzing the
scalability of the fermionic error correction capacity of the partitions.
B.8.5 Fermionic error correction capacity of partitions
Consider our hypothetical zero-syndrome error configuration of O(1) weight, comprising
connected γd errors in a single partition. We construct a convex hull enclosing each affected tetron.
This hull has two properties:
97

d
(a)
d
(b)
Figure B.5: Figures (a) and (b) illustrate the stabilizer sets 2 and 1 respectively. In both figures,
the white plaquettes yield zero syndrome as they are unaffected by any error. The blue striped
plaquettes yield zero syndrome as they are affected by an even number of errors. However, the red
checkered plaquettes yield non-zero syndrome as they are affected by an odd number of errors.
Figure (a) illustrates that if a convex corner of the hull lies in the interior of the triangle, then
we obtain a non-zero syndrome. Figure (b) illustrates that if the hull exterior contains an obtuse
angle between a triangle edge and a hull edge, then the syndrome is non-zero.
1. The hull has several convex corners, which may lie on vertices, edges, or interior of the
triangle. If a convex corner lies in the interior of the triangle, then at least one stabilizer
plaquette supported on that corner tetron would yield non-zero syndrome. Figure B.5(a)
illustrates this with an example.
2. Suppose that a convex corner of the hull lies on a triangle edge, and an obtuse angle is
formed outside the hull by the triangle edge and one hull edge which meet at the convex
corner. Then at least one stabilizer plaquette supported on the corner tetron would yield
non-zero syndrome. Figure B.5(b) illustrates this with an example.
The connected error configuration may be classified into seven classes of geometry, as shown in
Fig. B.6, based on whether the affected tetrons lie on the vertices, edges or interior of the triangle:
1. The errors exist in the triangle interior.
2. The errors exist on one edge. The errors optionally exist in the triangle interior.
98

Figure B.6: The seven geometry classes of error configurations.
3. The error exists on one vertex. The errors optionally exist in the two adjacent edges and/or
the triangle interior.
4. The errors exist on two edges. The errors may optionally exist in the triangle interior.
5. The errors exist on two vertices. The errors also exist on the intermediate edge and/or the
triangle interior. The errors optionally exist on other edges.
6. The errors exist on one vertex, the triangle interior and the edge opposite to the vertex. The
errors may optionally exist on other edges.
7. The errors exist on all three edges. The errors may optionally exist in the triangle interior
and the three vertices.
If a syndromeless error has one of the first four geometry classes, then the same error configuration will produce zero syndrome in larger codes as well, as shown in Fig. B.7. Thus, these errors
have O(1) weight. In contrast, each of the last three geometries has weight proportional to the
edge length.
But we can show that each of the first four geometry classes must yield non-zero syndrome:
1. For the class 1 errors, the convex hull must have all its convex corners in the interior of the
triangle. So, some stabilizer plaquettes would anticommute with the error configuration at
these convex corners. Hence, it must yield non-zero syndrome.
99

Figure B.7: If an error configuration with one of the first four geometries produce zero syndrome,
then the same configuration would continue to yield zero syndrome in larger codes, without
requiring the error configuration to be scaled up with code size.
2. The convex hull of class 2 errors would take the form of either a one-dimensional line segment
at the edge, or a two-dimensional polygon. If it is shaped as a line segment, then the two
endpoints of the line segment would anticommute with at least two stabilizers. If it is shaped
as a polygon, then some convex corners would lie in the triangle interior, and hence the
syndrome would be non-zero.
3. The convex hull of class 3 errors would take the form of either a one-dimensional line segment,
or a two-dimensional polygon. If it is shaped as a line segment with the first endpoint at the
triangle vertex and the second endpoint at the edge or the triangle interior, then at least one
stabilizer would anticommute with the error configuration at the second endpoint. If the hull
is shaped as a polygon, then consider the lower hull perimeter extending from one edge to
another. If this perimeter is a straight line, then at least one of the angles formed between
the hull edge and the triangle edge must be obtuse. Else this perimeter may contain one or
more convex corners in the triangle interior. In both cases, the syndrome is non-zero.
4. The convex hull of class 4 errors would take the form of either a one-dimensional line segment
spanning from one edge to another, or a two-dimensional polygon. If it is shaped as a line
segment, then at least one of the angles formed between the hull edge and the triangle edge
must be obtuse, and hence the syndrome should be non-zero. If the hull is shaped as a
polygon, then consider the lower hull perimeter extending from one edge to another. If this
perimeter is a straight line, then at least one of the angles formed between the hull edge and
100

a triangle edge must be obtuse. Else this perimeter may contain one or more convex corners
in the triangle interior. In both cases, we would obtain non-zero syndrome.
Thus, for a subsystem fermion code derived from an Jn, 1, dbK triangular CSS code, a syndromeless error configuration cannot have one of the first four geometries, and hence they cannot
have O(1) weight. However, they can possibly have the geometry classes 5, 6, or 7. Thus, the
fermionic error correction capacity of the partitions is proportional to the least weight of these
three geometry classes, which is Ω(db). Hence, we have proved that the fermionic distance of the
subsystem code must scale with the code size.
Since the bosonic code distance db scales with the code size, and the fermionic error correction
capacity of a partition scales with code size, so both steps of the fermionic error correction scheme
can be scaled up for larger codes.
We should note that although the minimum fermionic error correction capacity scales with the
code size, the actual fermionic code distance is often higher than this theoretical minimum, and
this actual value might not have a smooth variation with the code size. For example, let us take a
look at the three codes given below:
In the subsystem fermion code derived from the J7, 1, 3bK code, the least weight syndromeless
error configuration has weight 3. It has geometry class 7, and has the same support as the
first gauge operator of gauge qubit 1, shown in Fig. B.1(b).
In the subsystem fermion code derived from the J19, 1, 5bK code, the least weight syndromeless
error configuration has weight 9. It has geometry class 7, and has the same support as the
first gauge operator shown in Fig. B.3(b).
In the subsystem fermion code derived from the J37, 1, 7bK code, the least weight syndromeless
error configuration has weight 9. It has geometry class 7, and has the same support as the
first gauge operator of gauge qubit 2, shown in Fig. B.4(b).
101

Although the fermionic code distance has sharp variations, its minimum limit is Ω(db), and it
scales with the code size.
B.9 Code capacity
We consider a biased noise model, where fermionic errors are η times more likely to occur than
bosonic errors. There are 3 possible modes for bosonic errors - X, Y, Z errors, each of which occurs
with equal probability pX = pY = pZ =
p
3(η+1) =
pb
3
. There are 4 possible modes for fermionic errors
- γa, γb, γc, γd, each of which occurs with equal probability pγa = pγb = pγc = pγd =
pη
4(η+1) =
pf
4
.
We use the BPOSD decoder to obtain the logical error rates for various bias values. We obtain the
pseudothreshold by the intersection of the logical error rate and the physical error rate. Note that
the physical error rate is given by pb +
3pf
4
since a tetron qubit is affected by all bosonic errors but
only 3 out of 4 fermionic errors. The variation of pseudothreshold with noise bias is provided in
Fig. B.8, and the code capacity logical error plots for bias values η = 0.1, 1, 10 are provided in
Fig. B.9.
14, 1, 2, df =3 code
38, 1, 1, df = 9 code
74, 1, 3, df = 9 code
Pseudothreshold
0
0.1
0.2
0.3
0.4
0.5
Bias
103 0.01 0.1 1 10 100 1000
Figure B.8: The figure shows the variation of pseudothreshold with noise bias for J14, 1, 2, df = 3K,
J38, 1, 1, df = 9K and J74, 1, 3, df = 9K subsystem codes.
102

14, 1, 2, df = 3, bias = 0.1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5 0.6
(a)
14, 1, 2, df = 3, bias = 1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5 0.6
(b)
14, 1, 2, df = 3, bias = 10
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5 0.6
(c)
38, 1, 1, df = 9, bias = 0.1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5 0.6
(d)
38, 1, 1, df = 9, bias = 1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5 0.6
(e)
38, 1, 1, df = 9, bias = 10
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5 0.6
(f)
74, 1, 3, df = 9, bias = 0.1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5 0.6
(g)
74, 1, 3, df = 9, bias = 1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5 0.6
(h)
74, 1, 3, df = 9, bias = 0.1
Physical error rate
Logical error rate
0
0.2
0.4
0.6
0.8
1
Bosonic error rate + fermionic error rate (p)
0 0.1 0.2 0.3 0.4 0.5 0.6
(i)
Figure B.9: Code capacity logical error plots for fermionic subsystem codes derived from color
codes. The top row shows the plot for the 7 tetron code, the middle row shows the plot for the 19
tetron code, and the bottom row shows the plot for the 37 tetron code. In these plots, the 95%
confidence interval bars are smaller than the marker size.
103

B.10 Fault tolerance
Fault-tolerance can be achieved by careful ordering of stabilizer measurements, and might
require some redundant stabilizer measurements.
For example, Fig. B.10 shows a fault-tolerant sequence for the J14, 1, 2, df = 3K fermionic
subsystem code, which utilizes four additional redundant stabilizer measurements. This sequence
can tolerate 1 bosonic or 1 fermionic error, either at the input or at any intermediate stage.
We analyze this sequence against noise bias = η, bosonic error rate pb =
p
η+1 , fermionic error
rate pf =
pη
η+1 , and measurement error rate p. The fault-tolerance threshold for this sequence is
illustrated in Fig. B.11 for bias values of 0.1, 1 and 10.
104

(a) X1 X2 X3 X4 (b) X2 X4 X6 X7 (c) X3 X4 X5 X6 (d) Z1 Z2 Z3 Z4
(e) Z2 Z4 Z6 Z7 (f) Z3 Z4 Z5 Z6 (g) X1 X2
′ X3
′ X4 (h) X3
′ X4 X5 X6
′
(i) X2
′ X4 X6
′ X7 (j) Z1
′ Z2 Z3 Z4
′ (k) Z3 Z4
′ Z5
′ Z6 (l) Z2 Z4
′ Z6 Z7
′
(m) Y1 Y2 Y3 Y4 (n) Y3 Y4
′
Y5
′
Y6 (o) Y2 Y4
′
Y6 Y7
′

Figure B.10: Figures (a) to (o) demonstrate a sequence of fault-tolerant measurements for the
J14, 1, 2, df = 3K fermionic subsystem code. This sequence can tolerate one bosonic error or one
fermionic error. The stabilizer being measured is highlighted in color, and it is supported on four
operators at four vertices. The tetron operators follow the same notations as Fig. 2.1(d).
× 10-4
14, 1, 2, df = 3, bias = 0.1
Physical error rate
Logical error rate
0
2×104
4×104
6×104
8×104
103
Bosonic error rate + fermionic error rate (p)
012345
(a)
× 10-4
14, 1, 2, df = 3, bias = 1
Physical error rate
Logical error rate
0
2×104
4×104
6×104
8×104
103
Bosonic error rate + fermionic error rate (p)
012345
(b)
× 10-4
14, 1, 2, df = 3, bias = 10
Physical error rate
Logical error rate
0
2×104
4×104
6×104
8×104
103
Bosonic error rate + fermionic error rate (p)
012345
(c)
Figure B.11: The logical error plots for the fault-tolerant implementation of the J14, 1, 2, df = 3K
code are given in figures (a), (b) and (c) for bias values η = 0.1, 1, 10 respectively. The graphs also
show the 95% confidence intervals.
105

Appendix C
Arbitrary measurements on one octon
We have seen that two classical bits a1 and a2 can be stored in two parts of the octon. If both
parts have parity c, then state of the three octon qubits are given by:
Hc
|a1⟩ ⊗ Hc
|a2⟩ ⊗ |c⟩
We can perform both strong and weak 2-MZM measurements. So we can measure the operators
Z1, Z1Z3, X1, X1Z3, Z2, Z2Z3, X2, X2Z3 with arbitrary strength. Other 2-MZM operators can also
be measured, but they are not helpful in recovering the classical bits, and hence not considered.
C.1 Basics of weak measurement
Given an initial state |ψi⟩, we can weakly measure an operator T with ϵ strength to obtain
outcomes η = ±1. Note that η = +1 eigenvalue corresponds to even parity measurement, and
η = −1 eigenvalue corresponds to odd parity measurement.
We can obtain the probability p
T
η of outcome η, as well as the final resultant state |ψ
′
i
⟩ in the
following manner:
E
T
η =

A
T
η
2
=

1 + ϵ
2

π
T
η +

1 − ϵ
2

π
T
−η
10

A
T
η =
r
1 + ϵ
2
π
T
η +
r
1 − ϵ
2
π
T
−η
p
T
η = ⟨ψi
|E
T
η
|ψi⟩
|ψ
′
i
⟩ = A
T
η
|ψi⟩ ÷ q
p
T
η
The integrity of the final state is given by |⟨ψi
|ψ
′
i
⟩|2
. The integrity is equal to the probability
of obtaining the correct result when the final state is measured in the correct basis.
C.2 Sequence of measurement rounds
Each measurement round measures a pair of operators. We can measure any of these four
pairs: (a) Z1 and Z1Z3, (b) X1 and X1Z3, (c) Z2 and Z2Z3, or (d) X2 and X2Z3.
We can perform zero or more rounds of weak measurements to query |c⟩ which stores the basis
information. If we perform strong measurements in the correct basis, then we can obtain a1 and/or
a2. We would like to know the best sequence of strong and weak measurements in order to know
maximum information about a1 and a2.
A general sequence of measurement rounds would be: n1 rounds of weak measurements →
first round of strong measurement → n2 rounds of weak measurements → second round of strong
measurement → n3 rounds of weak measurements.
Now we can see that the n3 rounds of weak measurements would have no utility, because the
two rounds of strong measurements have already collapsed the two qubits, and any information
yielded by the n3 rounds of weak measurements would only pertain to the collapsed states which
we already know.
Furthermore, we can also see that the n2 rounds of weak measurements serve no purpose. Each
round of weak measurement has slightly revealed the parity and slightly damaged the state. But
the parity has already been revealed by the first round of strong measurement, and hence any
subsequent weak measurements are redundant.
107

So, the general sequence should be modified to:
n1 rounds of weak measurements → first round of strong measurement → second round of strong
measurement.
C.3 Information about classical bits
Assume that several rounds of weak measurements have been performed on the two octon
parts, and we have gained d knowledge about the correct parity (stored in qubit 3). This means if
we make an informed decision based on past outcomes, then we can guess the correct parity in d
fraction of cases, and we will obtain the wrong parity in (1 − d) fraction of cases.
However, the past weak measurements have partially damaged the states of qubit 1 and qubit
2, and their state integrity have reduced to t1 and t2. Here we discuss the amount of information
about the two classical bits that can be obtained at this stage. We have two options:
1. We can now perform one round of strong measurement on the top part, followed by strong
measurement on the bottom part. For strong measurement on the top part, we can guess
the correct parity with d probability, and then obtain bit a1 with t1 probability. We can
also guess the wrong parity for top part with (1 − d) probability, and then guess bit a1 with
1
2
probability of success. So, the expected information about a1 = dt1 + (1 − d)
1
2
. We will
also obtain exact information about the parity c. Then we perform strong measurement on
the bottom part in the correct basis. The expected information about a2 is t2. Thus, the
expected amount of total information = dt1 + (1 − d)
1
2 + t2.
2. We can now perform one round of strong measurement on the top part, followed by strong
measurement on the bottom part. Then the expected amount of total information =
dt2 + (1 − d)
1
2 + t1.
In the subsequent sections, we will see how d, t1 and t2 can vary based on the number of prior
measurement rounds.
108

C.4 Parity knowledge after 1 round of measurement
In this section, we choose either qubit 1 or qubit 2. Suppose qubit i is in an arbitrary state
|ψi⟩ = α|0⟩ + β|1⟩. We perform weak measurements of ϵ strength on the operator pair (Zi and
ZiZ3) or (Xi and XiZ3). Then we check how much information about parity c has been revealed
in this round of measurement.
We use the below positive operator valued measures:
E
Zi
+1 =

A
Zi
+12
=

1 + ϵ
2

|0i⟩⟨0i
| +

1 − ϵ
2

|1i⟩⟨1i
|
E
Zi
−1 =

A
Zi
−1
2
=

1 + ϵ
2

|1i⟩⟨1i
| +

1 − ϵ
2

|0i⟩⟨0i
|
E
Xi
+1 =

A
Xi
+12
=

1 + ϵ
2

|+i⟩⟨+i
| +

1 − ϵ
2

|−i⟩⟨−i
|
E
Xi
−1 =

A
Xi
−1
2
=

1 + ϵ
2

|−i⟩⟨−i
| +

1 − ϵ
2

|+i⟩⟨+i
|
When c = 0,
E
ZiZ3
+1 =

A
ZiZ3
+1 2
=

1 + ϵ
2

|0i03⟩⟨0i03| +

1 − ϵ
2

|1i03⟩⟨1i03| = E
Zi
+1
Similarly, EZiZ3
−1 = E
Zi
−1
, EXiZ3
+1 = E
Xi
+1 , EXiZ3
−1 = E
Xi
−1
.
When c = 1,
E
ZiZ3
+1 =

A
ZiZ3
+1 2
=

1 + ϵ
2

|1i13⟩⟨1i13| +

1 − ϵ
2

|0i13⟩⟨0i13| = E
Zi
−1
Similarly, EZiZ3
−1 = E
Zi
+1, EXiZ3
+1 = E
Xi
−1
, EXiZ3
−1 = E
Xi
+1 .
Note that
E
Zi
+12
+

E
Zi
−1
2
=

E
Xi
+12
+

E
Xi
−1
2
=
1 + ϵ
2
2
× identity matrix.
This result shall be useful later.
109

Case 1A: Suppose |ψi⟩ = α|0⟩ + β|1⟩ and |c⟩ = |0⟩. We weakly measure Zi and ZiZ3 to obtain
their parity. The probability to obtain correct parity is evaluated below.
We get correct parity if Zi = +1, ZiZ3 = +1 or if we get Zi = −1, ZiZ3 = −1
Say we obtain Zi = +1 with probability p
Zi
+1 .
The post-measurement state is: |ψ
′
i⟩ =
A
Zi
+1|ψi⟩
r
p
Zi
+1
.
The probability of obtaining ZiZ3 = +1 is:
p
ZiZ3
+1 =
D
ψ
′
i

E
ZiZ3
+1

ψ
′
i
E
=
D
ψ
′
i

E
Zi
+1

ψ
′
i
E
⇒ p
ZiZ3
+1 =
1
p
Zi
+1
ψi

E
Zi
+12

ψi

⇒ p
Zi
+1 × p
ZiZ3
+1 =

ψi

E
Zi
+12

ψi

Say we obtain Zi = −1 with probability p
Zi
−1
.
The post-measurement state is: |ψ
′
i⟩ =
A
Zi
−1
|ψi⟩
r
p
Zi
−1
.
The probability of obtaining ZiZ3 = −1 is:
p
ZiZ3
−1 =
D
ψ
′
i

E
ZiZ3
−1

ψ
′
i
E
=
D
ψ
′
i

E
Zi
−1

ψ
′
i
E
⇒ p
ZiZ3
−1 =
1
p
Zi
−1

ψi

E
Zi
−1
2

ψi

⇒ p
Zi
−1 × p
ZiZ3
−1 =

ψi

E
Zi
−1
2

ψi

Thus, for case 1A, the proability to get the correct parity is
p
Zi
+1 × p
ZiZ3
+1 + p
Zi
−1 × p
ZiZ3
−1 =

ψi

E
Zi
+12
+

E
Zi
−1
2

ψi

=
1 + ϵ
2
2
.
Case 1B: Suppose |ψi⟩ = α|0⟩ + β|1⟩ and |c⟩ = |0⟩. We weakly measure Xi and XiZ3 to obtain
their parity. The probability to obtain correct parity is evaluated below.
We get correct parity if Xi = +1, XiZ3 = +1 or if we get Xi = −1, XiZ3 = −1
Say we obtain Xi = +1 with probability p
Xi
+1 .
The post-measurement state is: |ψ
′
i⟩ =
A
Xi
+1|ψi⟩
r
p
Xi
+1
.
The probability of obtaining XiZ3 = +1 is:
p
XiZ3
+1 =
D
ψ
′
i

E
XiZ3
+1

ψ
′
i
E
=
D
ψ
′
i

E
Xi
+1

ψ
′
i
E
⇒ p
XiZ3
+1 =
1
p
Xi
+1
ψi

E
Xi
+12

ψi

⇒ p
Xi
+1 × p
XiZ3
+1 =

ψi

E
Xi
+12

ψi

Say we obtain Xi = −1 with probability p
Xi
−1
.
The post-measurement state is: |ψ
′
i⟩ =
A
Xi
−1
|ψi⟩
r
p
Xi
−1
.
The probability of obtaining XiZ3 = −1 is:
p
XiZ3
−1 =
D
ψ
′
i

E
XiZ3
−1

ψ
′
i
E
=
D
ψ
′
i

E
Xi
−1

ψ
′
i
E
⇒ p
XiZ3
−1 =
1
p
Xi
−1

ψi

E
Xi
−1
2

ψi

⇒ p
Xi
−1 × p
XiZ3
−1 =

ψi

E
Xi
−1
2

ψi

Thus, for case 1B, the proability to get the correct parity is
p
Xi
+1 × p
XiZ3
+1 + p
Xi
−1 × p
XiZ3
−1 =

ψi

E
Xi
+12
+

E
Xi
−1
2

ψi

=
1 + ϵ
2
2
.
110

Case 2A: Suppose |ψi⟩ = α|0⟩ + β|1⟩ and |c⟩ = |1⟩. We weakly measure Zi and ZiZ3 to obtain
their parity. The probability to obtain correct parity is evaluated below.
We get correct parity if Zi = +1, ZiZ3 = −1 or if we get Zi = −1, ZiZ3 = +1
Say we obtain Zi = +1 with probability p
Zi
+1 .
The post-measurement state is: |ψ
′
i⟩ =
A
Zi
+1|ψi⟩
r
p
Zi
+1
.
The probability of obtaining ZiZ3 = −1 is:
p
ZiZ3
−1 =
D
ψ
′
i

E
ZiZ3
−1

ψ
′
i
E
=
D
ψ
′
i

E
Zi
+1

ψ
′
i
E
⇒ p
ZiZ3
−1 =
1
p
Zi
+1
ψi

E
Zi
+12

ψi

⇒ p
Zi
+1 × p
ZiZ3
−1 =

ψi

E
Zi
+12

ψi

Say we obtain Zi = −1 with probability p
Zi
−1
.
The post-measurement state is: |ψ
′
i⟩ =
A
Zi
−1
|ψi⟩
r
p
Zi
−1
.
The probability of obtaining ZiZ3 = +1 is:
p
ZiZ3
+1 =
D
ψ
′
i

E
ZiZ3
+1

ψ
′
i
E
=
D
ψ
′
i

E
Zi
−1

ψ
′
i
E
⇒ p
ZiZ3
+1 =
1
p
Zi
−1

ψi

E
Zi
−1
2

ψi

⇒ p
Zi
−1 × p
ZiZ3
+1 =

ψi

E
Zi
−1
2

ψi

Thus, for case 2A, the proability to get the correct parity is
p
Zi
+1 × p
ZiZ3
+1 + p
Zi
−1 × p
ZiZ3
−1 =

ψi

E
Zi
+12
+

E
Zi
−1
2

ψi

=
1 + ϵ
2
2
.
Case 2B: Suppose |ψi⟩ = α|0⟩ + β|1⟩ and |c⟩ = |1⟩. We weakly measure Xi and XiZ3 to obtain
their parity. The probability to obtain correct parity is evaluated below.
We get correct parity if Xi = +1, XiZ3 = −1 or if we get Xi = −1, XiZ3 = +1
Say we obtain Xi = +1 with probability p
Xi
+1 .
The post-measurement state is: |ψ
′
i⟩ =
A
Xi
+1|ψi⟩
r
p
Xi
+1
.
The probability of obtaining XiZ3 = −1 is:
p
XiZ3
−1 =
D
ψ
′
i

E
XiZ3
−1

ψ
′
i
E
=
D
ψ
′
i

E
Xi
+1

ψ
′
i
E
⇒ p
XiZ3
−1 =
1
p
Xi
+1
ψi

E
Xi
+12

ψi

⇒ p
Xi
+1 × p
XiZ3
−1 =

ψi

E
Xi
+12

ψi

Say we obtain Xi = −1 with probability p
Xi
−1
.
The post-measurement state is: |ψ
′
i⟩ =
A
Xi
−1
|ψi⟩
r
p
Xi
−1
.
The probability of obtaining XiZ3 = +1 is:
p
XiZ3
+1 =
D
ψ
′
i

E
XiZ3
+1

ψ
′
i
E
=
D
ψ
′
i

E
Xi
−1

ψ
′
i
E
⇒ p
XiZ3
+1 =
1
p
Xi
−1

ψi

E
Xi
−1
2

ψi

⇒ p
Xi
−1 × p
XiZ3
+1 =

ψi

E
Xi
−1
2

ψi

Thus, for case 2B, the proability to get the correct parity is
p
Xi
+1 × p
XiZ3
−1 + p
Xi
−1 × p
XiZ3
+1 =

ψi

E
Xi
+12
+

E
Xi
−1
2

ψi

=
1 + ϵ
2
2
.
111

Thus, in all cases, one round of weak measurement reveals the parity c with probability d =
1 + ϵ
2
2
.
C.5 Parity knowledge after n rounds of measurement
Suppose we perform n rounds of weak measurements, distributed over both parts of the octon.
In each round we measure one of these four pairs: (a) Z1 and Z1Z3, (b) X1 and X1Z3, (c) Z2 and
Z2Z3, or (d) X2 and X2Z3.
In each round, we have d probability of obtaining the correct parity. The probability of
obtaining the correct parity for k times, as well as obtaining the wrong parity for (n − k) times is
nCk × d
k × (1 − d)
n−k
. Furthermore, our knowledge about the correct parity is k/n. Thus, the
expected knowledge about the correct parity is Xn
k=0
nCk × d
k × (1 − d)
n−k ×
k
n
= d. So, we obtain
the same knowledge about |c⟩ from a single round as from multiple rounds.
C.6 Variation in state integrity due to one round of weak
measurement
We choose either qubit 1 or qubit 2, and suppose that it is an arbitrary state |ψi⟩ = α|0⟩ + β|1⟩.
We weakly measure the operators (Zi and ZiZ3) or (Xi and XiZ3) with ϵ strength. The postmeasurement state is given below:
1. If the measurements yield the wrong parity, then the state |ψi⟩ remains unchanged.
2. If the measurements yield the correct parity, then the state |ψi⟩ = α|0⟩ + β|1⟩ is damaged:
If the correct parity is revealed by weak measurements Zi = s and ZiZ3 = t, then the
new state would be given by α
′
|0⟩ + β
′
|1⟩, where α
′ = α(1 + sϵ) and β
′ = β(1 − sϵ),
with additional normalization factor.
112

If the correct parity is revealed by weak measurements Xi = s and XiZ3 = t, then the
new state would be given by α
′
|0⟩ + β
′
|1⟩, where α
′ = α + sϵβ and β
′ = β + sϵα, along
with normalization factor.
For example, Table C.1 shows all possible ways in which the pure states |0⟩, |1⟩, |+⟩, |−⟩ are
damaged by one round of weak measurements.
Initial state Weak measurements of ϵ strength Final state

1
0

−−−−−−−−−→ Xi = +1
−−−−−−−−−−−→ XiZ3 = +1
1
ϵ

1
0

−−−−−−−−−→ Xi = −1
−−−−−−−−−−−→ XiZ3 = −1

1
−ϵ

0
1

−−−−−−−−−→ Xi = +1
−−−−−−−−−−−→ XiZ3 = +1
ϵ
1

0
1

−−−−−−−−−→ Xi = −1
−−−−−−−−−−−→ XiZ3 = −1

−ϵ
1

1
1

−−−−−−−−−→ Zi = +1
−−−−−−−−−−−→ ZiZ3 = −1

1 + ϵ
1 − ϵ

1
1

−−−−−−−−−→ Zi = −1
−−−−−−−−−−−→ ZiZ3 = +1
1 − ϵ
1 + ϵ

1
−1

−−−−−−−−−→ Zi = +1
−−−−−−−−−−−→ ZiZ3 = −1

1 + ϵ
−1 + ϵ

1
−1

−−−−−−−−−→ Zi = −1
−−−−−−−−−−−→ ZiZ3 = +1
1 − ϵ
−1 − ϵ

Table C.1: There are eight ways in which pure states can be damaged by a pair of weak measurements. Normalization constants are not shown in the final damaged states.
In all eight cases, the state integrity reduces from 1 to 1/

1 + ϵ
2

. In other cases, for example
when the measurements are performed in the correct basis, or when the obtained parity is wrong,
then the final state remains undamaged and does not suffer any integrity loss.
11

C.7 Variation in state integrity due to multiple rounds of
weak measurements
Now let us choose either qubit 1 or qubit 2, and suppose that it begins in a pure state |ψi⟩. We
perform multiple rounds of weak measurement on the operators (Zi and ZiZ3) and/or (Xi and
XiZ3). Depending on the parity output of the preceding rounds, we can probabilistically choose
the basis for the subsequent rounds of weak measurement. Such a strategy progressively damages
the state integrity, and is shown in the Monte Carlo plot of Fig. C.1.
ϵ = 0.1
ϵ = 0.2
ϵ = 0.3
ϵ = 0.4
ϵ = 0.5
State integrity
0.5
0.6
0.7
0.8
0.9
1
Number of weak measurement rounds
0 20 40 60 80 100
Figure C.1: A Monte Carlo simulation of variation in state integrity over multiple rounds of weak
measurements.
114

C.8 Optimized sequence of measurement rounds
Appendix C.2 showed that the general measurement sequence should be:
n1 rounds of weak measurements → first round of strong measurement → second round of strong
measurement.
In Appendix C.3 we showed that the expected total information about a1 and a2 is of the form
dti + (1 − d)
1
2 + t3−i
. In Appendix C.4 and Appendix C.5 , we showed that one round of ϵ-weak
measurement reveals the same parity knowledge as multiple rounds of ϵ-weak measurements over
the top part and/or the bottom part of octon. Furthermore, Appendix C.7 showed that multiple
rounds of weak measurements reduce the state integrity. Hence the optimal strategy to gain
information about a1 and a2 should not have multiple rounds of weak measurements one after the
other. Thus, n1 should be either 0 or 1.
So the possible measurement strategies are:
1. Weak measurement on part i → Strong measurement on part i → Strong measurement on
part (3 − i)
2. Weak measurement on part i → Strong measurement on part (3 − i) → Strong measurement
on part i
3. Strong measurement on part i → Strong measurement on part (3 − i)
115

We analyze the first sequence in Fig. C.2. The first sequence has probability of obtaining
ai =
3
4
and the probability of obtaining a3−i = 1.
= ݌
1
2
߰௜ Integrity = 1
Choose ܿ
ܪ = Basis
௖ܼ
Choose ҧܿ = 1 െ ܿ
ܪ = Basis
௖ҧܼ
Measure
ܪ
ܪ and௖ܼ௜
௖ܼ௜Zଷ
with strength = ߳
Measure
ܪ
௖ҧܼ௜ and ܪ
௖ҧܼ௜Zଷ
with strength = ߳
Result
Measured basis = ҧܿ
Integrity = 1
Result
Measured basis = ܿ
Integrity = 1
Result
Measured basis = ҧܿ
Integrity = 1
Result
Measured basis = ܿ
Integrity =
ଵ
ଵାఢమ
Measure
ܪ
ܪ and௖ܼ௜
௖ܼ௜Zଷ
with strength = 1
Measure
ܪ
௖ҧܼ௜ and ܪ
௖ҧܼ௜Zଷ
with strength = 1
Result
Got ܽ௜ with ݌ = 1
Got c with ݌ = 1
Result
Got ܽ௜ with ݌1/2 =
Got ܿ with ݌ = 1
Measure
ܪ
௖ܼଷି௜
with strength = 1
Result
Got ܽଷି௜ with ݌ = 1
= ݌
1 + ߳
ଶ
= ݌ 2
1 െ ߳ଶ
= ݌ 2
1 + ߳
ଶ
2
= ݌
1 െ ߳ଶ
2
= ݌
1
2
Measure
ܪ
௖ܼଷି௜
with strength = 1
Result
Got ܽଷି௜ with ݌ = 1
Measure
ܪ
ܪ and௖ܼ௜
௖ܼ௜Zଷ
with strength = 1
Measure
ܪ
௖ҧܼ௜ and ܪ
௖ҧܼ௜Zଷ
with strength = 1
Result
Got ܽ௜ with ݌=
ଵ
ଵାఢమ
Got c with ݌ = 1
Result
Got ܽ௜ with ݌1/2 =
Got ܿ with ݌ = 1
Measure
ܪ
௖ܼଷି௜
with strength = 1
Result
Got ܽଷି௜ with ݌ = 1
Measure
ܪ
௖ܼଷି௜
with strength = 1
Result
Got ܽଷି௜ with ݌ = 1
Figure C.2: According to the first sequence, we perform weak measurement on part i followed by
strong measurement on part i and finally strong measurement on part (3 − i). Here p denotes
probability.
116

We analyze the second sequence in Fig. C.3. The second sequence has probability of obtaining
ai =
4−ϵ
2
4
and the probability of obtaining a3−i =
3+ϵ
2
4
.
= ݌
1
2
߰௜ Integrity = 1
Choose ܿ
ܪ = Basis
௖ܼ
Choose ҧܿ = 1 െ ܿ
ܪ = Basis
௖ҧܼ
Measure
ܪ
ܪ and௖ܼ௜
௖ܼ௜Zଷ
with strength = ߳
Measure
ܪ
௖ҧܼ௜ and ܪ
௖ҧܼ௜Zଷ
with strength = ߳
Result
Measured basis = ҧܿ
Integrity = 1
Result
Measured basis = ܿ
Integrity = 1
Result
Measured basis = ҧܿ
Integrity = 1
Result
Measured basis = ܿ
Integrity =
ଵ
ଵାఢమ
Measure
ܪ
௖ܼଷି௜ and ܪ
௖ܼଷି௜Zଷ
with strength = 1
Measure
ܪ
௖ҧܼଷି௜ and ܪ
௖ҧܼଷି௜Zଷ
with strength = 1
Result
Got ܽଷି௜ with ݌ = 1
Got c with ݌ = 1
Result
Got ܽଷି௜ with ݌1/2 =
Got ܿ with ݌ = 1
Measure
ܪ
௖ܼ௜
with strength = 1
Result
Got ܽ௜ with ݌ = 1
= ݌
1 + ߳
ଶ
= ݌ 2
1 െ ߳ଶ
2
= ݌
1
2
Measure
ܪ
௖ܼ௜
with strength = 1
Result
Got ܽ௜ with ݌ = 1
Measure
ܪ
௖ܼଷି௜ and ܪ
௖ܼଷି௜Zଷ
with strength = 1
Measure
ܪ
௖ҧܼଷି௜ and ܪ
௖ҧܼଷି௜Zଷ
with strength = 1
Result
Got ܽଷି௜ with ݌ = 1
Got c with ݌ = 1
Result
Got ܽଷି௜ with ݌1/2 =
Got ܿ with ݌ = 1
Measure
ܪ
௖ܼ௜
with strength = 1
Result
Got ܽ௜ with ݌=
ଵ
ଵାఢమ
Measure
ܪ
௖ܼ௜
with strength = 1
Result
Got ܽ௜ with ݌ = 1
= ݌
1 െ ߳ଶ
= ݌ 2
1 + ߳
ଶ
2
Figure C.3: According to second sequence, we perform weak measurement on part i followed by
strong measurement on part (3 −i) and finally strong measurement on part i. Here p is probability.
The third sequence was previously discussed in Section 4.2, where we showed that the output
probability of ai
is 3
4
and the output probability of a3−i
is 1.
Thus, we have shown that (probability of obtaining bit a1) + (probability of obtaining bit
a2) ≤ 1.75 for any arbitrary sequence of weak and strong measurements.
117

Appendix D
Arbitrary measurements on cluster
D.1 Individual bit availability
Consider a cluster of k octons, each isolated from the other. The i
th octon stores the classical
bits a1i and a2i
in the top part and bottom part respectively. Assume that some arbitrary weak
and/or strong measurements have already occurred on this octon, and our current knowledge
about parity is di ∈

1
2
, 1

. It means that we can choose the correct basis with di probability, and
the wrong basis with (1 − di) probability. But the past measurements have reduced the state
integrity of qubits 1 and 2 to (t1i
, t2i) ∈

1
2
, 1
2
.
Now if we wish to obtain bit a1i by strong measurements on the top part:
We can choose the correct basis (with probability di), and obtain bit a1i with t1i probability.
We can choose the wrong basis (with probability 1 − di), and guess the bit ai with 1
2
probability.
Thus, the availability of bit a1i
is αi = (di × t1i × 1) +
(1 − ki) ×
1
2

.
We can reorder the probabilities and substitute g1i = 2t1i−1, where g1i ∈ [0, 1] is the probability
of finding the top bit in a state of good integrity (such that measurement in correct basis always
reveals correct result), and (1 − g1i) is the probability of finding the top bit in a state of bad
integrity (such that measurement in correct basis yields correct result and wrong result with equal
118

probability). Similarly g2i
is the probability of finding the bottom bit in a state of good integrity.
Then the availability for the top bit and bottom bit are given by αi and βi below:
αi =
1 + dig1i
2
, βi =
1 + dig2i
2
In a cluster of k octons, we can correctly obtain the top bit a1 as follows:
If we can guess the correct basis for all k octons, and if all the top bits are in a state of good
integrity, then we can obtain all the top bits with probability 1.
If we choose the wrong basis for at least one octon and/or if at least one of the top bits is in
a state of bad integrity, then we can only guess the top bit a1 with probability 1
2
.
Thus we can obtain top bit a1 with probability α given below:
α =


Y
i∈[k]
dig1i

 · 1 +

1 −
Y
i∈[k]
dig1i

 ·
1
2
=
1 + Q
i∈[k]
dig1i
2
Similarly, the availability of bottom bit would be β =
1 + Q
i∈[k]
dig2i
2
.
D.2 Combined bit availability
α + β =
1 + Q
i∈[k]
dig1i
2
+
1 + Q
i∈[k]
dig2i
2
⇒ α + β = 1 +
Q
i∈[k]
xi +
Q
i∈[k]
yi
2
where xi
, yi
, x, y are defined below:
xi = dig1i ∈ [0, 1], GM(xi) = Q
i∈[k]
xi
1
k
∈ [0, 1], AM(xi) =
P
i∈[k] xi
k = x ∈ [0, 1]
yi = dig2i ∈ [0, 1], GM(yi) = Q
i∈[k]
yi
1
k
∈ [0, 1], AM(yi) =
P
i∈[k]
yi
k = y ∈ [0, 1]
119

We know that geometric mean ≤ arithmetic mean.
So,
Y
i∈[k]
xi ≤ x
k
and Y
i∈[k]
yi ≤ y
k
⇒ α + β ≤ 1 +
x
k + y
k
2
In Appendix C , we proved that αi + βi ≤ 1.75, so xi + yi ∈

0,
3
2

, and also x + y ∈

0,
3
2

.
For x ∈ [0, 1], y ∈ [0, 1], x + y ∈

0,
3
2

, the following inequality holds:
x
k + y
k ≤ 1 +
1
2
k
⇒ α + β ≤
3
2
+
1
2
k+1
Thus, we have shown that (probability of obtaining bit a1) + (probability of obtaining bit a2) ≲ 1.5
for any arbitrary sequence of weak and strong measurements.
120

Appendix E
Dishonest OTM recipient
An honest OTM recipient maximizes his probability of obtaining the chosen bit by taking
advantage of error correction. However, a malicious recipient can sacrifice error correction for the
chosen bit, in order to maximize the probability of obtaining the remaining bit.
E.1 Malicious strategy on 1
2 OTM using Steane code
Consider the 1
2 OTM encoded in the double layered J7, 1, 3K Steane code. It stores two classical
bits on 7 clusters, where each cluster comprises k octons.
On the top layer, the classical analogue of X stabilizers encode a logical X operator, whose
parity is equal to the first classical bit.
On the bottom layer, the classical analogue of Z stabilizers encode a logical Z operator,
whose parity is equal to the third classical bit.
Figure E.1 illustrates the stabilizer structure and the stabilizer matrix, which is common for
both X and Z stabilizers.
The malicious recipient can follow two courses of action on any one cluster:
Strategy 1: On a brown cluster, he correctly reads the top bit, by using the parity information
from the bottom part. This damages the bottom bit with some probability.
121

1
2
3 5
4 6 7
(a)
1 1 1 0 1 0 0
0 0 1 1 1 1 0
0 1 0 0 1 1 1
ª º
« »
¬ ¼
1 2 3 4 5 6 7
(b)
Figure E.1: Optimal malicious strategy used by a dishonest recipient of a 1/2 OTM encoded in
the Steane code. The recipient sequentially measures the clusters numbered 1 through 7. He reads
the top bits from the brown clusters, he reads the bottom bits from the blue clusters, and he can
choose to read any bit from the purple cluster.
(a) The clusters are shown on the stabilizer diagram of the Steane code. Both X and Z stabilizers
have the same structure. (b)The clusters are numbered on the stabilizer matrix of the Steane code.
Both X and Z stabilizers have the same matrix.
Strategy 2: On a blue cluster, he correctly reads the bottom bit, by using the parity
information from the top part. This damages the top bit with some probability.
The user sequentially measures the clusters 1 through 6, by using the strategy sequence 121212.
1) In the first step, he reads the top bit of cluster 1, and he can obtain both bits on cluster 1
with 1
2
k probability, and then he can obtain the first classical bit stored in the top layer on
clusters (1, 2, 7), and he can also obtain the second classical bit stored in the bottom layer
on clusters (1, 3, 4).
2) If the first step did not allow him to cheat, then in the second step, he reads the bottom
bit of cluster 2, and he can obtain both bits on cluster 2 with 1
2
k probability, and then he
obtains the first classical bit stored in the top layer on (1, 2, 7) clusters, and he also obtains
the second classical bit stored in the bottom layer on (2, 4, 5) clusters.
...) Thus, he continues to alternate between the first strategy and the second strategy for clusters
1 to 6.
122

7) Finally, on the seventh cluster, which is marked in purple, he can choose to read either the
top bit or the bottom bit. Based on his choice, he is guaranteed to obtain one of the two
classical bits encoded in the code.
This is an optimal cheating strategy for a malicious recipient, and offers him 7 chances to
cheat. Thus, the cheating probability is pc =
7
2
k + O

1
2
2k

. If errors do not occur, then he can
correctly obtain one chosen bit. The availability of the remaining bit is given by pc +
1
2
(1 − pc) =
1
2 +
1
2
·
7
2
k + O

1
2
2k

.
E.2 Malicious strategy on 1
2 OTM using Golay code
1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 0 1 0
0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 0 1
0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0
0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1
0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 1 1
0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1
0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 0
0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 0 1 0 1
ª º
« »
« »
« »
« »
« »
« »
« »
« »
« »
« »
« »
« »
« »
« »
« »
¬ ¼
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 23 22
Figure E.2: Optimal malicious strategy used by a dishonest recipient of a 1/2 OTM encoded in
the Golay code. The recipient sequentially measures the clusters numbered 1 through 23. He reads
the top bits from the brown clusters, and he reads the bottom bits from the blue clusters. He
can choose either strategy for the purple cluster. The clusters are numbered and colored on the
stabilizer matrix of the Golay code. Both X and Z stabilizers have the same matrix.
Similarly, we can consider the 1
2 OTM encoded in the J23, 1, 7K Golay code. Figure E.2 shows
the stabilizer matrix which is common for both X and Z stabilizers of the Golay code. Each column
of the stabilizer matrix corresponds to a cluster, which is numbered at the top of the matrix. A
dishonest OTM recipient can forgo error correction, and choose a malicious strategy to maximize
availability of the remaining bit. Figure E.2 illustrates an optimal malicious strategy, denoted by
1

the numbers and colors on top of the stabilizer matrix. The recipient would sequentially measure
the clusters 1 through 23, and use strategy 1 on the brown clusters, and strategy 2 on the blue
clusters. On the final purple cluster, he can choose either strategy 1 or strategy 2. This provides
him a cheating chance of 23
2
k + O

1
2
2k

. The chosen bit can be correctly obtained in the absence of
errors. The remaining bit can be obtained with probability 1
2 +
1
2
·
23
2
k + O

1
2
2k

.
E.3 Malicious strategy on 1
3 OTM using Steane code
Now we shall give an example of an optimal cheating strategy by a malicious recipient of a
1
3 OTM. Consider the three layered J7, 1, 3K Steane code that encodes a 1
3 OTM. The malicious
recipient wishes to obtain all three logical bits, and he forgoes error correction on the chosen bit
to increase the probability for obtaining the other bits. He would have a priority order for the
three logical bits, for example, he may prioritize the first bit over the second, and the second bit
over the third. Then, the dishonest recipient would employ a similar malicious strategy as the 1
2
OTM on the Steane code. He would sequentially measure the clusters 1 through 7 by reading the
top bit from the first cluster, the middle bit from the cluster 2, the top bit from cluster 3, the
middle bit from cluster 4, the top bit form cluster 5, the middle bit from cluster 6, and finally the
top bit from cluster 7. This is illustrated in Fig. E.3. It allows him to correctly obtain the top bit
if there is no error, and he gets seven chances to cheat and get the second logical bit. Thus, the
probability of the second logical bit being undamaged by cheating is pc2 =
7
2
k + O

1
2
2k

.
Observe that the first four measurements offer him a chance to cheat and get all three logical
bits.
1. In step 1, if he measures cluster 1 and obtains all three bits (with 1
2
2k probability), then he
can obtain the first logical bit from clusters (1, 3, 4), he can obtain the second logical bit
from clusters (1, 5, 6), and he can obtain the third logical bit from clusters (1, 2, 7).

1
2
3 5
4 6 7
(a)
1 1 1 0 1 0 0
0 0 1 1 1 1 0
0 1 0 0 1 1 1
ª º
« »
¬ ¼
1 2 3 4 5 6 7
(b)
Figure E.3: Optimal malicious strategy used by a dishonest recipient of a 1/3 OTM encoded in
the Steane code. The recipient sequentially measures the clusters numbered 1 through 7. He reads
the top bits from the brown clusters, and the bottom bits from the blue clusters. (a) The clusters
are shown on the stabilizer diagram of the Steane code. Both X and Z stabilizers have the same
structure. (b) The clusters are numbered on the stabilizer matrix of the Steane code. Both X and
Z stabilizers have the same matrix.
2. If the first step failed but the second cluster measurement yields all three bits (with 1
2
2k
probability), then he can obtain the first logical bit from clusters (1, 2, 7), he can obtain the
second logical bit from clusters (2, 3, 6), and he can obtain the third logical bit from clusters
(2, 4, 5).
3. If the previous steps failed, but the third cluster measurement yields all three bits (with 1
2
2k
probability), then he obtains the first logical bit from clusters (1, 3, 4), he obtains the second
logical bit from clusters (2, 3, 6), and he obtains the third logical bit from clusters (3, 5, 7).
4. If the previous steps failed, but the fourth cluster measurement yields all three bits (with 1
2
2k
probability), then he obtains the first logical bit from clusters (1, 3, 4), he obtains the second
logical bit from clusters (2, 4, 5), and he obtains the third logical bit from clusters (4, 6, 7).
Thus, he has four chances to cheat and get the third logical bit. The probability of the third
logical bit being undamaged by his cheating strategy is pc3 =
4
2
2k + O

1
2
3k

.
So, the availability of the first logical bit is 1, as long as errors do not occur.
The availability of the second logical bit = pc2 +
1
2
(1 − pc2) = 1
2 +
1
2
·
7
2
k + O

1
2
2k

.
The availability of the third logical bit = pc3 +
1
2
(1 − pc3) = 1
2 +
1
2
·
4
2
2k + O

1
2
3k

.

Abstract (if available)

Abstract Majorana zero modes (MZMs) are promising candidates for topological quantum computation. Karzig et al. have proposed a tetron architecture for measurement-based quantum computation using Majorana zero modes. In this architecture, each tetron island hosts four MZMs, and possible measurements are constrained to span zero or two MZMs per island. Such measurements are known to be sufficient for correcting “bosonic errors”, which infect an even number of MZMs per tetron. We demonstrate that such measurements are also sufficient for correcting “fermionic errors”, which infect an odd number of MZMs in a tetron. We propose several strategies for constructing “fermionic codes” that can correct fermionic errors in addition to bosonic errors.

Then we extrapolate the nanowire architecture and consider octon islands, where each island contains eight MZMs. We show that octons can be used to construct a cryptographic primitive called one-time memory (OTM). One-time memories have been shown to allow for secure classical and quantum computations. An OTM is a memory device that stores two classical bits, such that either bit can be retrieved, but not both. We prove that a malicious recipient performing an arbitrary sequence of strong and weak measurements can not obtain more information than an honest recipient performing only strong measurements. We show that errors in the two stored bits can be corrected by a pair of classical codes derived from a quantum CSS code. Finally, we show that the one bit out of two bit OTM construction generalizes to 1-out-of-n and (n − 1)-out-of-n OTMs.

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Creator Kundu, Sourav (author)

Core Title Error correction and cryptography using Majorana zero modes

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Degree Conferral Date 2024-05

Publication Date 04/17/2024

Defense Date 03/20/2024

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Tag Majorana zero mode,OAI-PMH Harvest,one-time memory,quantum cryptography,quantum error correction,quantum fault-tolerance

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