1
Rem 1: Stabilizer Code Convention
2
Rem 2: Graph Convention
3
Rem 3: Binary Vector Notation
▶
3.1
Binary Vectors over \(\mathbb {Z}/2\mathbb {Z}\)
3.2
Characteristic Vector of a Set
3.3
Symmetric Difference and Vector Addition
3.4
Inverse Map: From Binary Vector to Finset
3.5
Algebraic Structure: Finsets as a Vector Space over \(\mathbb {Z}/2\mathbb {Z}\)
3.6
Properties for Graph Theory Applications
4
Rem 4: Z-Type Support Convention
5
Rem 5: Cheeger Constant Definition
6
Rem 6: Circuit Implementation of the Gauging Procedure
7
Def 1: Boundary and Coboundary Maps
8
Rem 7: Exactness of Boundary and Coboundary Maps
▶
8.1
Definitions and Basic Properties
8.2
All-Ones Vector and Nontrivial Kernel
8.3
Coboundary Characterization
8.4
Chain Complex Property
8.5
Cochain Complex Property
8.6
Zero Boundary Characterization
8.7
Connected Graph Kernel Classification
8.8
Exactness Properties
9
Rem 8: Desiderata for Graph \(G\) in Gauging Measurement
10
Rem 9: Worst-Case Graph Construction
11
Rem 10: Parallelization of Gauging Measurements
12
Def 2: Gauss’s Law Operators
13
Def 3: Flux Operators
14
Def 4: Deformed Operator
15
Def 5: Deformed Check
16
Def 6: Cycle-Sparsified Graph
17
Def 7: Space and Time Faults
18
Def 8: Detector
▶
18.1
Measurement Outcomes
18.2
Events: Initialization and Measurement
18.3
Detector Events
18.4
Detectors
18.5
Empty Detector
18.6
Single-Event Detectors
18.7
Combining Detectors
18.8
Fault-Free Execution
18.9
Effect of Faults on Detectors
18.10
Detector Violation Analysis
18.11
Detector Collection
19
Def 9: Syndrome
20
Def 10: Spacetime Logical Fault
▶
20.1
Syndrome
20.2
Identity Fault Has Empty Syndrome
20.3
Logical Effect Predicate
20.4
Spacetime Stabilizers
20.5
Spacetime Logical Faults
20.6
Partition of Empty-Syndrome Faults
20.7
Spacetime Stabilizers Form a Subgroup
20.8
Fault Equivalence
20.9
Equivalence Classes as Cosets
21
Def 11: Spacetime Fault-Distance
22
Def 12: Time Step Convention
23
Lem 1: Deformed Code Stabilizer Structure
▶
Part 1: Gauss’s Law Operators Become Stabilizers
Part 2: Flux Operators Are Stabilizers
Part 3: Deformed Checks Are Stabilizers
Part 4: Dimension Count
Complete Theorem: Deformed Code Structure
Corollaries
24
Lem 2: Space Distance Bound for Deformed Code
25
Lem 3: Spacetime Code Detectors
26
Lem 4: Spacetime Stabilizers
27
Lem 5: Time Fault Distance
28
Lem 6: Spacetime Decoupling
29
Lem 7: Spacetime Fault-Distance Lemma
30
Thm 1: Gauging Measurement
31
Thm 2: Fault Tolerance of Gauging Measurement
32
Cor 1: Overhead Bound
33
Cor 2: Cheeger Optimality
34
Rem 11: Initial and Final Boundary Conditions
35
Rem 12: Noncommuting Operators Cannot Be Deformed
36
Rem 13: Flux Check Measurement Frequency
37
Rem 14: Generalizations of the Gauging Measurement Procedure
38
Rem 15: Hypergraph Generalization of Gauging Measurement
39
Rem 16: Practical Measurement Rounds
▶
39.1
Round Requirements
39.2
Computation Context
39.3
Practical Round Configuration
39.4
Effective Code Parameters
39.5
Worst-Case vs Practical Guarantees
39.6
Practical Implementations
39.7
Connection to Boundary Conditions
39.8
Summary
40
Rem 17: Circuit Implementation Fault Tolerance
41
Rem 18: Lattice Surgery as Gauging
▶
41.1
Ladder Graph
41.2
Gauss’s Law Product Equals Logical Operator
41.3
Deformed Code Structure on the Merged Patch
41.4
Split Step
41.5
Non-Adjacent Patches via Dummy Grid
41.6
Generalized Surgery Graph
41.7
Relaxed Expansion
42
Rem 19: Shor-Style Measurement as Gauging
43
Rem 20: Cohen Scheme as Gauging
44
Rem 21: CSS Code Initialization as Gauging
▶
44.1
CSS Code Structure
44.2
Initialization Hypergraph
44.3
X-Checks in Kernel of \(H_Z\)
44.4
Dummy Vertex Structure
44.5
Gauging Procedure for Initialization
44.6
Steane-Style Measurement via Gauging
44.7
Steane Gauging Hypergraph
44.8
Three-Step Steane Procedure
44.9
Unification Under Gauging Framework
45
Def 13: Bivariate Bicycle Code
46
Def 14: Gross Code
47
Rem 22: Gross Code Gauging Example
Dependency graph
MerLean-example
doxtor6
1
Rem 1: Stabilizer Code Convention
2
Rem 2: Graph Convention
3
Rem 3: Binary Vector Notation
3.1
Binary Vectors over \(\mathbb {Z}/2\mathbb {Z}\)
3.2
Characteristic Vector of a Set
3.3
Symmetric Difference and Vector Addition
3.4
Inverse Map: From Binary Vector to Finset
3.5
Algebraic Structure: Finsets as a Vector Space over \(\mathbb {Z}/2\mathbb {Z}\)
3.6
Properties for Graph Theory Applications
4
Rem 4: Z-Type Support Convention
5
Rem 5: Cheeger Constant Definition
6
Rem 6: Circuit Implementation of the Gauging Procedure
7
Def 1: Boundary and Coboundary Maps
8
Rem 7: Exactness of Boundary and Coboundary Maps
8.1
Definitions and Basic Properties
8.2
All-Ones Vector and Nontrivial Kernel
8.3
Coboundary Characterization
8.4
Chain Complex Property
8.5
Cochain Complex Property
8.6
Zero Boundary Characterization
8.7
Connected Graph Kernel Classification
8.8
Exactness Properties
9
Rem 8: Desiderata for Graph \(G\) in Gauging Measurement
10
Rem 9: Worst-Case Graph Construction
11
Rem 10: Parallelization of Gauging Measurements
12
Def 2: Gauss’s Law Operators
13
Def 3: Flux Operators
14
Def 4: Deformed Operator
15
Def 5: Deformed Check
16
Def 6: Cycle-Sparsified Graph
17
Def 7: Space and Time Faults
18
Def 8: Detector
18.1
Measurement Outcomes
18.2
Events: Initialization and Measurement
18.3
Detector Events
18.4
Detectors
18.5
Empty Detector
18.6
Single-Event Detectors
18.7
Combining Detectors
18.8
Fault-Free Execution
18.9
Effect of Faults on Detectors
18.10
Detector Violation Analysis
18.11
Detector Collection
19
Def 9: Syndrome
20
Def 10: Spacetime Logical Fault
20.1
Syndrome
20.2
Identity Fault Has Empty Syndrome
20.3
Logical Effect Predicate
20.4
Spacetime Stabilizers
20.5
Spacetime Logical Faults
20.6
Partition of Empty-Syndrome Faults
20.7
Spacetime Stabilizers Form a Subgroup
20.8
Fault Equivalence
20.9
Equivalence Classes as Cosets
21
Def 11: Spacetime Fault-Distance
22
Def 12: Time Step Convention
23
Lem 1: Deformed Code Stabilizer Structure
Part 1: Gauss’s Law Operators Become Stabilizers
Part 2: Flux Operators Are Stabilizers
Part 3: Deformed Checks Are Stabilizers
Part 4: Dimension Count
Complete Theorem: Deformed Code Structure
Corollaries
24
Lem 2: Space Distance Bound for Deformed Code
25
Lem 3: Spacetime Code Detectors
26
Lem 4: Spacetime Stabilizers
27
Lem 5: Time Fault Distance
28
Lem 6: Spacetime Decoupling
29
Lem 7: Spacetime Fault-Distance Lemma
30
Thm 1: Gauging Measurement
31
Thm 2: Fault Tolerance of Gauging Measurement
32
Cor 1: Overhead Bound
33
Cor 2: Cheeger Optimality
34
Rem 11: Initial and Final Boundary Conditions
35
Rem 12: Noncommuting Operators Cannot Be Deformed
36
Rem 13: Flux Check Measurement Frequency
37
Rem 14: Generalizations of the Gauging Measurement Procedure
38
Rem 15: Hypergraph Generalization of Gauging Measurement
39
Rem 16: Practical Measurement Rounds
39.1
Round Requirements
39.2
Computation Context
39.3
Practical Round Configuration
39.4
Effective Code Parameters
39.5
Worst-Case vs Practical Guarantees
39.6
Practical Implementations
39.7
Connection to Boundary Conditions
39.8
Summary
40
Rem 17: Circuit Implementation Fault Tolerance
41
Rem 18: Lattice Surgery as Gauging
41.1
Ladder Graph
41.2
Gauss’s Law Product Equals Logical Operator
41.3
Deformed Code Structure on the Merged Patch
41.4
Split Step
41.5
Non-Adjacent Patches via Dummy Grid
41.6
Generalized Surgery Graph
41.7
Relaxed Expansion
42
Rem 19: Shor-Style Measurement as Gauging
43
Rem 20: Cohen Scheme as Gauging
44
Rem 21: CSS Code Initialization as Gauging
44.1
CSS Code Structure
44.2
Initialization Hypergraph
44.3
X-Checks in Kernel of \(H_Z\)
44.4
Dummy Vertex Structure
44.5
Gauging Procedure for Initialization
44.6
Steane-Style Measurement via Gauging
44.7
Steane Gauging Hypergraph
44.8
Three-Step Steane Procedure
44.9
Unification Under Gauging Framework
45
Def 13: Bivariate Bicycle Code
46
Def 14: Gross Code
47
Rem 22: Gross Code Gauging Example