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bibkey: khesinli2024equivalence
authors: Andrey Boris Khesin, Alexander M. Li
year: 2024
title: Equivalence Classes of Quantum Error-Correcting Codes
doi: 10.48550/arXiv.2406.12083
url: https://arxiv.org/abs/2406.12083v1
claim: Conjecture 7.1 states that connected equivalence classes of normalized [n,k] encoders have sizes divisible by 3^k for positive integers n > k; Conjecture 8.3 states that bipartite forms in one equivalence class are related by output permutations, input permutations and row operations alone.
strata_touched:
  - D5/S3/Quantum/Codes/KhesinLiEquivalenceClasses
  - D5/S3/Quantum/Codes/KhesinLiBipartiteEncoderEquivalence
license: citation-only
triage: anchor

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DOI: 10.48550/arXiv.2406.12083

Source: https://arxiv.org/abs/2406.12083v1

Definition 2.7, p. 6:

By performing a local complementation about vertex v, all edges connecting two vertices in N(v) are toggled.

Conjecture 6.3, p. 17, lists:

1. Local complementing about any vertex of the graph.
2. Permuting the output vertices.
3. Permuting the input vertices.
4. Performing row operations on the adjacency matrix of input to output edges.
5. Removing an input-input edge.
6. Applying local operations on the output edges.

Section 6, p. 18:

Operation 4 consists of adding rows of the adjacency matrix between input and output nodes in modulo 2.

Claim 6.4, p. 19:

To find the distinct encoders in an equivalence class of the set of Clifford codes, it is sufficient to find the distinct encoders in the equivalence class satisfying the following constraints:

• The input-output adjacency matrix is in RREF.
• The pivot nodes are the lowest-numbered output nodes, and they are ordered to match the order of the input nodes.
• Local Clifford gates on the free output edges are all removed, leaving only output Z nodes with phase 0.
• Diagrams with pivot-pivot edges are not included.

Section 7, p. 19:

The code takes an integer representation, say n, of an encoder graph, performs one operation from Conjecture 6.3 on the encoder graph, then merges the disjoint sets of n and the integer representing the resulting encoder graph.

Section 7, p. 19, describes the normalization:

To fix this, we first employ operation 5 from Conjecture 6.3 to set all input-input edges to 0.

Operation 2 is used to put the pivots back into their fixed positions, so they once again correspond to their input vertices.

Conjecture 7.1, p. 20:

For positive integers n > k, the equivalence classes of [n,k] codes with a single connected component have sizes divisible by 3^k.

The encoder vertices are Fin k ⊕ Fin n: inputs are Sum.inl i, outputs are Sum.inr j. Pivot outputs have indices below k and match inputs in order. Graphs have no input-input or pivot-pivot edges; input-pivot edges form the identity matrix. Local Clifford decorations are removed, so operation 6 contributes no graph operation. A normalized step is one of operations 1–5 followed by a finite sequence of operations 2–5, with normalized endpoints. The equivalence closure counts distinct labelled normalized graphs, rather than isomorphism classes. A connected class means that every normalized member is connected, following the restriction used in the source’s figures.

The repository proof rotates each input-pivot pair with order three. These rotations commute, and connectedness excludes a zero free-row pair. The resulting free action of (ZMod 3)^k partitions every such class into orbits of size 3^k. The proof does not establish completeness of the equivalence operations in Conjecture 6.3 or formulas for the number of classes. Completeness of the search for later literature settlements is ASSUMED-UNVERIFIED.

Conjecture 8.3

Section 8 introduces bipartite forms, encoder graphs “with no edges among output nodes”, as canonical forms, and conjectures:

In an equivalence class with bipartite forms, all bipartite forms in the class can be transformed from one to another using only output permutations, input permutations, and row operations on the input-output adjacency matrix.

The repository reads the equivalence classes as those generated by operations 1–5 of Conjecture 6.3 on decoration-free encoders (the classes of Section 7) and a bipartite form as an encoder graph with only input–output edges. It proves the conjecture through the output cut-rank profile, which these operations preserve up to the output permutation and which determines the kernel, hence the row space, of a bipartite form’s input–output adjacency matrix. The quantum code equivalence of Definition 6.2 and Conjecture 6.3 itself are outside this conclusion.