1
Rem 2: Notation and Pauli Operators
▶
1.1
Definition of Pauli Operators
1.2
Multiplication
1.3
Products over Finite Sets
1.4
Supports
1.5
Support Characterizations
1.6
Identity Support
1.7
Single-Site Operator Supports
1.8
Product Support Characterizations
1.9
Support Union Characterization
1.10
Single-Site and Product Relationships
2
Rem 3: Notation for Stabilizer Codes
3
Rem 4: Cheeger Constant and Expander Graphs
4
Def 1: Boundary and Coboundary Maps
5
Rem 5: Exactness of Sequences
6
Def 2: Gauss’s Law and Flux Operators
7
Def 3: Deformed Operator
8
Rem 6: Noncommuting Operators Cannot Be Deformed
9
Def 4: Deformed Code
10
QEC1: Root Module
11
Lem 1: Deformed Code Checks
12
Rem 7: Codespace Dimension After Gauging
13
Rem 8: Freedom in Deformed Checks
▶
13.1
Path Difference Lies in Cycle Space
13.2
Pure-Z Edge Operator
13.3
Deformed Checks Differ by a Pure-Z Edge Operator
13.4
The Difference Operator Commutes with All Checks
13.5
Different Paths Yield the Same Stabilizer Group
13.6
Gauss’s Law and Flux Checks Are Fixed
13.7
Pure-Z Edge Operator Commutes with All Checks
13.8
Corollaries
14
Def 5: Gauging Measurement Algorithm
15
Thm 1: Gauging Measurement Correctness
16
Rem 9: Circuit Implementation
▶
16.1
CX Gate Conjugation on Pauli Operators
16.2
The Full Entangling Circuit Action
16.3
Symplectic Inner Product Preservation
16.4
Main Theorem: Entangling Circuit Transforms \(A_v\) to \(X_v\)
16.5
Effect on Edge \(Z\) Operators
16.6
Circuit Protocol Steps
16.7
Simultaneous Transformation and Consistency
17
Rem 10: Flexibility of Graph G
18
Rem 11: Desiderata for Graph \(G\)
▶
18.1
Desideratum 1: Short Paths for Deformation
18.2
Desideratum 2: Sufficient Expansion
18.3
Desideratum 3: Low-Weight Cycle Basis
18.4
Constant Degree and Edge Overhead
18.5
Gauss Check Weight Characterization
18.6
All Desiderata Satisfied
18.7
LDPC Preservation
18.8
Summary: Combined Consequences
19
Def 6: Cycle-Sparsified Graph
20
Lem 2: Decongestion Lemma Bound
▶
20.1
Edge Count Bounds for Constant-Degree Graphs
20.2
Cycle Space Dimension Bounds
20.3
Maximum Edge-Cycle Degree
20.4
Layer Assignment and Greedy Packing
20.5
Coarse Bound
20.6
BFS Ball Growth and Expander Diameter
20.7
Constructive Decongestion Lemma Bound
20.8
Universal Linear Bound
20.9
Main Theorem: Decongestion Lemma (Freedman–Hastings)
20.10
Consequences for the Sparsified Graph
20.11
Summary
21
Rem 12: Worst-Case Graph Construction
22
Lem 3: Space Distance
23
Rem 13: Optimal Cheeger Constant
▶
Lifting original operators to the extended system
Commutation of lifted operators with deformed code checks
Pure-\(X\) logicals lift to deformed code centralizer
Point 1: \(h(G) \geq 1\) is sufficient for \(d^* \geq d\)
Point 2: \(d^* \leq d\) via lifting
Point 2 combined with Point 1: \(d^* = d\) when \(h(G) \geq 1\)
Point 3: \(h(G) {\lt} 1\) causes distance loss
Optimality of \(h(G) = 1\)
Summary: Distance bound trichotomy
24
Rem 14: Parallel Gauging Measurement
▶
Part 1: Non-overlapping support
Part 2: Same-type overlapping support
Part 3: LDPC constraint for parallel gauging
Combined properties
25
Def 7: Space and Time Faults
26
Def 8: Detectors
27
Def 9: Syndrome
28
Def 10: Fault-Tolerant Gauging Procedure
29
Lem 4: Spacetime Code Detectors
30
Def 11: Spacetime Logical Fault
31
Def 12: Spacetime Fault-Distance
32
Lem 5: Spacetime Stabilizers
▶
Part I: Generator Predicates
Part II: Model-Theoretic Foundation
Part III: Algebraic Classification
▶
\(Z_e\) Commutation Properties
\(X_e\) Commutation Properties
Vertex Pauli Commutation Properties
Edge Endpoints
\(Z_e\) Non-Gauss Commutation
Part IV: Generator Stabilizer Proofs
Part V: Listed Generator Classification
Part VI: Main Theorem
Part VII: Algebraic Justifications
Part VIII: Completeness
Part IX: Corollaries
Part X: Space-Fault Cleaning and Centralizer Properties
33
Lem 6: Time Fault-Distance
34
Lem 7: Space-Time Decoupling
35
Thm 2: Fault-Tolerant Gauging Distance
▶
35.1
Space-Fault Witness Construction
35.2
Upper Bound: \(d_{\text{spacetime}} \leq d\)
35.3
Lower Bound: \(d_{\text{spacetime}} \geq d\)
35.4
Main Theorem
35.5
Corollaries
36
Rem 15: Flux Check Measurement Frequency
37
Rem 16: Boundary Rounds Overkill
▶
Point 1: The All-or-None Property is Purely Phase 2
Point 2: Boundary Detectors Connect Phases
Point 3: Only One Boundary Round Needed for Boundary Coverage
Point 4: \(d\) Rounds Used in Theorem 2’s Clean Proof
Point 5: Space-Faults at \(t_i\) Caught by Phase 2
Summary
38
Cor 1: Worst-Case Qubit Overhead
▶
Edge Overhead from Sparsified Graph Construction
Desiderata Satisfied with \(O(W \log ^2 W)\) Overhead
Distance Preservation
Main Corollary
Concrete Overhead Characterization
39
Rem 17: Hypergraph Generalization
40
Rem 18: Relation to Lattice Surgery
41
Rem 19: Bivariate Bicycle Code Notation
42
Rem 20: The Gross Code Definition
43
Rem 21: Gross Code Gauging Measurement Construction
44
Rem 22: The Double Gross Code Definition
▶
44.1
Code Parameters
44.2
Check Commutation
44.3
The Double Gross Stabilizer Code
44.4
Logical Operator Polynomial
44.5
Logical X Operators
44.6
Support and Weight
44.7
Commutation with Checks
44.8
The Gauging Graph
44.9
Edge Counts
44.10
Expansion Edge Validity
44.11
Graph Connectivity and Cycle Rank
44.12
Overhead Calculation
44.13
Tanner Expansion Property
44.14
Check Weight
44.15
Summary
45
Rem 23: Generalizations Beyond Pauli
▶
45.1
Qudit Generalization: Boundary Maps over \(\mathbb {Z}_p\)
45.2
Transpose Properties
45.3
Chain Complex Property
45.4
Specialization to \(\mathbb {Z}_2\)
45.5
Abelian Group Charge Determination
45.6
Nonabelian Groups: Product Order Dependence
45.7
Qudit Gauss’s Law Operators (Generalized)
45.8
Nonabelian Local vs. Global Charge
45.9
Summary
46
Rem 24: Shor-Style Measurement as Gauging
▶
46.1
Shor-Style Graph Construction
46.2
Extended Logical Operator
46.3
Measurement Sign with Dummies
46.4
Gauging Structure on the Shor Graph
46.5
Circuit Correspondence
46.6
Edge Structure
46.7
Structural Correspondence
46.8
Degree Analysis
46.9
Optimization Flexibility
47
Rem 25: Steane-Style Measurement as Gauging
▶
47.1
CSS Stabilizer Codes
47.2
Step 1: State Preparation via Hypergraph Gauging
47.3
Step 2: Entangling via Perfect Matching Graph
47.4
Step 2: Entangling Gauss Operators
47.5
Step 3: Readout via Z Measurements
47.6
Composition: Steane-Style Measurement as Gauging
47.7
Ancilla Code Properties
47.8
Degree Analysis
47.9
Summary
48
Rem 26: Cohen et al. Scheme Recovery
Dependency graph
MerLean-example
doxtor6
1
Rem 2: Notation and Pauli Operators
1.1
Definition of Pauli Operators
1.2
Multiplication
1.3
Products over Finite Sets
1.4
Supports
1.5
Support Characterizations
1.6
Identity Support
1.7
Single-Site Operator Supports
1.8
Product Support Characterizations
1.9
Support Union Characterization
1.10
Single-Site and Product Relationships
2
Rem 3: Notation for Stabilizer Codes
3
Rem 4: Cheeger Constant and Expander Graphs
4
Def 1: Boundary and Coboundary Maps
5
Rem 5: Exactness of Sequences
6
Def 2: Gauss’s Law and Flux Operators
7
Def 3: Deformed Operator
8
Rem 6: Noncommuting Operators Cannot Be Deformed
9
Def 4: Deformed Code
10
QEC1: Root Module
11
Lem 1: Deformed Code Checks
12
Rem 7: Codespace Dimension After Gauging
13
Rem 8: Freedom in Deformed Checks
13.1
Path Difference Lies in Cycle Space
13.2
Pure-Z Edge Operator
13.3
Deformed Checks Differ by a Pure-Z Edge Operator
13.4
The Difference Operator Commutes with All Checks
13.5
Different Paths Yield the Same Stabilizer Group
13.6
Gauss’s Law and Flux Checks Are Fixed
13.7
Pure-Z Edge Operator Commutes with All Checks
13.8
Corollaries
14
Def 5: Gauging Measurement Algorithm
15
Thm 1: Gauging Measurement Correctness
16
Rem 9: Circuit Implementation
16.1
CX Gate Conjugation on Pauli Operators
16.2
The Full Entangling Circuit Action
16.3
Symplectic Inner Product Preservation
16.4
Main Theorem: Entangling Circuit Transforms \(A_v\) to \(X_v\)
16.5
Effect on Edge \(Z\) Operators
16.6
Circuit Protocol Steps
16.7
Simultaneous Transformation and Consistency
17
Rem 10: Flexibility of Graph G
18
Rem 11: Desiderata for Graph \(G\)
18.1
Desideratum 1: Short Paths for Deformation
18.2
Desideratum 2: Sufficient Expansion
18.3
Desideratum 3: Low-Weight Cycle Basis
18.4
Constant Degree and Edge Overhead
18.5
Gauss Check Weight Characterization
18.6
All Desiderata Satisfied
18.7
LDPC Preservation
18.8
Summary: Combined Consequences
19
Def 6: Cycle-Sparsified Graph
20
Lem 2: Decongestion Lemma Bound
20.1
Edge Count Bounds for Constant-Degree Graphs
20.2
Cycle Space Dimension Bounds
20.3
Maximum Edge-Cycle Degree
20.4
Layer Assignment and Greedy Packing
20.5
Coarse Bound
20.6
BFS Ball Growth and Expander Diameter
20.7
Constructive Decongestion Lemma Bound
20.8
Universal Linear Bound
20.9
Main Theorem: Decongestion Lemma (Freedman–Hastings)
20.10
Consequences for the Sparsified Graph
20.11
Summary
21
Rem 12: Worst-Case Graph Construction
22
Lem 3: Space Distance
23
Rem 13: Optimal Cheeger Constant
Lifting original operators to the extended system
Commutation of lifted operators with deformed code checks
Pure-\(X\) logicals lift to deformed code centralizer
Point 1: \(h(G) \geq 1\) is sufficient for \(d^* \geq d\)
Point 2: \(d^* \leq d\) via lifting
Point 2 combined with Point 1: \(d^* = d\) when \(h(G) \geq 1\)
Point 3: \(h(G) {\lt} 1\) causes distance loss
Optimality of \(h(G) = 1\)
Summary: Distance bound trichotomy
24
Rem 14: Parallel Gauging Measurement
Part 1: Non-overlapping support
Part 2: Same-type overlapping support
Part 3: LDPC constraint for parallel gauging
Combined properties
25
Def 7: Space and Time Faults
26
Def 8: Detectors
27
Def 9: Syndrome
28
Def 10: Fault-Tolerant Gauging Procedure
29
Lem 4: Spacetime Code Detectors
30
Def 11: Spacetime Logical Fault
31
Def 12: Spacetime Fault-Distance
32
Lem 5: Spacetime Stabilizers
Part I: Generator Predicates
Part II: Model-Theoretic Foundation
Part III: Algebraic Classification
\(Z_e\) Commutation Properties
\(X_e\) Commutation Properties
Vertex Pauli Commutation Properties
Edge Endpoints
\(Z_e\) Non-Gauss Commutation
Part IV: Generator Stabilizer Proofs
Part V: Listed Generator Classification
Part VI: Main Theorem
Part VII: Algebraic Justifications
Part VIII: Completeness
Part IX: Corollaries
Part X: Space-Fault Cleaning and Centralizer Properties
33
Lem 6: Time Fault-Distance
34
Lem 7: Space-Time Decoupling
35
Thm 2: Fault-Tolerant Gauging Distance
35.1
Space-Fault Witness Construction
35.2
Upper Bound: \(d_{\text{spacetime}} \leq d\)
35.3
Lower Bound: \(d_{\text{spacetime}} \geq d\)
35.4
Main Theorem
35.5
Corollaries
36
Rem 15: Flux Check Measurement Frequency
37
Rem 16: Boundary Rounds Overkill
Point 1: The All-or-None Property is Purely Phase 2
Point 2: Boundary Detectors Connect Phases
Point 3: Only One Boundary Round Needed for Boundary Coverage
Point 4: \(d\) Rounds Used in Theorem 2’s Clean Proof
Point 5: Space-Faults at \(t_i\) Caught by Phase 2
Summary
38
Cor 1: Worst-Case Qubit Overhead
Edge Overhead from Sparsified Graph Construction
Desiderata Satisfied with \(O(W \log ^2 W)\) Overhead
Distance Preservation
Main Corollary
Concrete Overhead Characterization
39
Rem 17: Hypergraph Generalization
40
Rem 18: Relation to Lattice Surgery
41
Rem 19: Bivariate Bicycle Code Notation
42
Rem 20: The Gross Code Definition
43
Rem 21: Gross Code Gauging Measurement Construction
44
Rem 22: The Double Gross Code Definition
44.1
Code Parameters
44.2
Check Commutation
44.3
The Double Gross Stabilizer Code
44.4
Logical Operator Polynomial
44.5
Logical X Operators
44.6
Support and Weight
44.7
Commutation with Checks
44.8
The Gauging Graph
44.9
Edge Counts
44.10
Expansion Edge Validity
44.11
Graph Connectivity and Cycle Rank
44.12
Overhead Calculation
44.13
Tanner Expansion Property
44.14
Check Weight
44.15
Summary
45
Rem 23: Generalizations Beyond Pauli
45.1
Qudit Generalization: Boundary Maps over \(\mathbb {Z}_p\)
45.2
Transpose Properties
45.3
Chain Complex Property
45.4
Specialization to \(\mathbb {Z}_2\)
45.5
Abelian Group Charge Determination
45.6
Nonabelian Groups: Product Order Dependence
45.7
Qudit Gauss’s Law Operators (Generalized)
45.8
Nonabelian Local vs. Global Charge
45.9
Summary
46
Rem 24: Shor-Style Measurement as Gauging
46.1
Shor-Style Graph Construction
46.2
Extended Logical Operator
46.3
Measurement Sign with Dummies
46.4
Gauging Structure on the Shor Graph
46.5
Circuit Correspondence
46.6
Edge Structure
46.7
Structural Correspondence
46.8
Degree Analysis
46.9
Optimization Flexibility
47
Rem 25: Steane-Style Measurement as Gauging
47.1
CSS Stabilizer Codes
47.2
Step 1: State Preparation via Hypergraph Gauging
47.3
Step 2: Entangling via Perfect Matching Graph
47.4
Step 2: Entangling Gauss Operators
47.5
Step 3: Readout via Z Measurements
47.6
Composition: Steane-Style Measurement as Gauging
47.7
Ancilla Code Properties
47.8
Degree Analysis
47.9
Summary
48
Rem 26: Cohen et al. Scheme Recovery