Heuristic Search for Minimum-Distance Upper-Bound Witnesses in Quantum APM-LDPC Codes

Abstract: This paper investigates certified upper bounds on the minimum distance of an explicit family of Calderbank-Shor-Steane quantum LDPC codes constructed from affine permutation matrices. All codes considered here have active Tanner graphs of girth eight. Rather than attempting to prove a general lower bound for the full code distance, we focus on constructing low-weight non-stabilizer logical representatives, which yield valid upper bounds once they are verified to lie in the opposite parity-check kernel and outside the stabilizer row space. We develop a unified framework for such witnesses arising from latent row relations, restricted-lift subspaces including block-compressed, selected-fiber, and CRT-stripe constructions, cycle- 8 elementary trapping-set structures, and decoder-failure residuals. In every case, search is used only to generate candidates; the reported bounds begin only after explicit kernel and row-space exclusion tests have been passed. For the latent part, we also identify a block-compression criterion under which the certification becomes exact. Applying these methods to representative APM-LDPC codes sharpens previously reported upper bounds and provides concrete certified values across the explored parameter range.

• The paper presents a heuristic search framework that constructs explicit logical operator witnesses to certify minimum-distance upper bounds.
• It details how active Tanner graphs, algebraic kernel tests, and restricted-lift protocols contribute to verifying code performance.
• The study demonstrates method separation through latent block-compression and cycle-8 trapping set techniques for precise bound estimation.

Heuristic Witness Search for Minimum-Distance Upper Bounds in Quantum APM-LDPC Codes

Overview and Motivation

The manuscript presents a comprehensive framework for certified upper bounds on the minimum distance of Calderbank-Shor-Steane (CSS) quantum LDPC codes constructed via affine permutation matrices (APM-LDPC). Every code studied adheres to an active Tanner graph of girth eight, imposing constraints on local code structure and influencing both decoding performance and minimum-weight logical operator profiles.

Unlike previous approaches directed at proving general lower bounds or asymptotic scaling for minimum distance, the work focuses on explicit upper bounds generated from a suite of heuristic search methods. The reported bounds are always based on explicit logical operator witnesses, each satisfying carefully staged kernel (syndrome) and stabilizer-exclusion tests to guarantee their status as nontrivial logical representatives. Multiple operational subspaces—latent row relations, restricted-lift patterns (block-compressed, selected-fiber, CRT-stripe), and cycle-8 trapping sets—are explored, along with failure artifacts from decoders. Each method produces candidates, certified after exact algebraic checks; no bound is claimed without a complete verification step.

Code Structure and Search Framework

CSS Construction and Affine Permutation Matrices

Quantum CSS codes require two parity-check matrices, $H_X$ and $H_Z$, satisfying a mutual orthogonality constraint, tailored here for the APM-LDPC family. The matrices are structured using blockwise affine permutation rules—$x \mapsto ax + b$—with modular parameters designed so that the active part sustains high girth by design, while the latent rows permit controlled, algebraically nontrivial commutation. The parameter selection, particularly the non-prime-power modulus and CRT-split block factors, is critical, as prime powers are proven incompatible with the construction's commutation patterns.

Decomposition: Parent, Active, Latent, and Witness Spaces

The code's parity-check structure decomposes into active and latent row spaces, facilitating classification of logical operators into latent (contained within the latent row space) and non-latent (outside latent row space) representatives. Minimum distance is thus split into latent and non-latent contributions, with explicit characterization and kernel criteria for each.

Candidates are generated from latent row combinations (with block-compression exactness under specified algebraic criteria), as well as from broader restricted-lift subspaces. The latter include block-compressed lifts, selected-fiber structures, and CRT-stripe-based restrictions, each formalized using appropriate linear maps and quotient mechanics.

Upper Bound Witness Methods

Latent Upper Bounds and Block Compression

Latent representatives are identified by solving kernel equations involving the active–latent mixed product matrices. When block-compressed structure is exact (verified via rank tests and kernel analyses), latent witnesses provide not only upper bounds but exact certifications for the latent contribution, tied to the block factor.

Restricted-Lift Protocols

The unified restricted-lift framework covers:

• Block-compressed lifts (full-fiber): searching in quotient spaces induced by block factors $m$.
• Selected-fiber lifts: unions of coordinate fibers under specified patterns $S \subseteq Z/mZ$.
• CRT-stripe lifts: coordinates are parametrized as stripes from CRT decompositions, allowing further algebraic compression.

Each subspace restricts candidate generation for logical representatives, with kernel and stabilizer exclusion checks enforced. Method-specific bounds are maintained, notably keeping components (block, fiber, CRT) separated rather than optimizing over all restricted lifts.

Additional Witnesses: Direct CSS and Trapping Sets

Direct CSS witness search operates in the full syndrome kernel, scanning for low-weight representatives outside stabilizer row space without enforcing subspace structure.

For girth-eight codes, cycle-8 elementary trapping sets (ETS)—notably those with empty odd-check boundaries—provide graph-theoretic supports for logical operators. The work leverages ETS searches to construct explicit codewords, with corresponding kernel and stabilizer tests confirming their status as logical operators.

Decoder-failure residuals also supply upper bounds: whenever a residual produced by a failed error correction event lies outside the stabilizer row space and has zero syndrome, its weight is an upper bound on minimum distance.

Numerical Results and Method Separation

Upper bounds are reported for an explicit sequence of codes, with parameters ranging up to block size $P=768$. Method-specific bounds (latent, block, fiber, CRT, direct, ETS, decoder-failure) are presented for each code instance, always as the minimum weight among checked witnesses per method. Single-method optimizations are not compounded; each method's "best" witness is retained as a separate entry.

Concrete examples demonstrate the steps: kernel solving, block-compressed quotient analysis, fiber restriction, CRT-stripe parametrization, support enumeration, and stabilizer-exclusion verification. For select instances, latent block-compressor structure is rigorously certified via rank tests and compressed kernel analysis, excluding the existence of lower-weight latent witnesses.

Cycle-8 ETS witnesses and decoder-failure residuals are reported only upon verification; negative entries simply reflect the absence of a certified witness within the method's search scope.

Implications and Theoretical Context

The paper stops short of general lower bounds for minimum distance, focusing on upper bounds with explicit witness certification. Existing quantum LDPC lower-bound frameworks are inapplicable because of the peculiar active/latent decomposition and the stabilizer quotient structure of APM codes. Classical minimum-weight search algorithms are likewise limited for these code parameters.

The practical implication is that, for active girth-eight APM-LDPC codes with non-prime-power block structure, minimum-weight logical operators can be tightly upper-bounded with explicit witnesses, often exhibiting linear growth in weight relative to code dimension over observed parameter ranges (though this observation is contingent on computational sample coverage and cannot exclude the possibility of plateaus in true minimum distance).

From a theoretical perspective, the framework delineates a clear algebraic frontier for quantum code search methodologies—latent row spaces, quotient lifts, CRT compressions, trapping-set-based local supports, and decoder artefacts are structurally distinct and require different forms of algebraic certification.

The appendix details the rigorous latent certification under block-constant kernel hypotheses, providing exact lower bounds on latent distance using linear algebra and SAT/SMT techniques, contingent upon block-compressed kernel structure.

Conclusion

This work delivers an operational classification and certification architecture for minimum-distance upper bounds in quantum APM-LDPC codes, anchored in explicit logical operator witnesses, staged algebraic verification, and tailored subspace searches. The layered methodology circumvents asymptotic or ensemble-level arguments, providing tangible bounds for concrete codes. Future directions include further tightening of these bounds, increasing computational sample sizes, expanding methods to new block parameter regimes (notably CRT splits), and eventually integrating explicit lower-bound certification protocols into the non-latent contribution landscape.