Quantum error-correction code and circuit construction is an active field of research, motivating automated methods for identifying equivalent designs. We develop and evaluate automated strategies for deciding whether two descriptions of quantum error-correction codes are equivalent under specified transformations. We formalize permutation equivalence for stabilizer and CSS codes and local-Clifford equivalence for stabilizer codes within a unified framework that separates physical from representation freedom. We develop three decision strategies (group-action search, isomorphism-based methods, and constraint solving), together with invariants and signatures for preprocessing. The empirical evaluation shows that no paradigm dominates across equivalence notions, code families, and parameter ranges. We therefore combine their complementary strengths in input-dependent hybrid checkers following a staged Refute-Refine-Decide architecture. Mean hybrid runtimes remain below 10 s for most evaluated structured-code cases with $n \leq 37$; for inequivalent code instances with $n=144$, they are about 310 s for permutation and 16 s for local-Clifford equivalence.
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22 pages, 16 main text + 2 references + 4 appendix, 19 figures
Quantum error-correction code and circuit construction is an active field of research, motivating automated methods for identifying equivalent designs. We develop and evaluate automated strategies for deciding whether two descriptions of quantum error-correction codes are equivalent under specified transformations. We formalize permutation equivalence for stabilizer and CSS codes and local-Clifford equivalence for stabilizer codes within a unified framework that separates physical from representation freedom. We develop three decision strategies (group-action search, isomorphism-based methods, and constraint solving), together with invariants and signatures for preprocessing. The empirical evaluation shows that no paradigm dominates across equivalence notions, code families, and parameter ranges. We therefore combine their complementary strengths in input-dependent hybrid checkers following a staged Refute-Refine-Decide architecture. Mean hybrid runtimes remain below 10 s for most evaluated structured-code cases with n≤37; for inequivalent code instances with n=144, they are about 310 s for permutation and 16 s for local-Clifford equivalence.
Quantum error-correcting codes (QECC's) are needed to combat the inherent noise affecting quantum processes. Using ZX calculus, we represent QECC's in a form called a ZX diagram, consisting of a tensor network. In this paper, we present canonical forms for CSS codes and CSS states (which are CSS codes with 0 inputs), and we show the resulting canonical forms for the toric code and certain surface codes. Next, we introduce the notion of prime code diagrams, ZX diagrams of codes that have a single connected component with the property that no sequence of rewrite rules can split such a diagram into two connected components. We also show the Fundamental Theorem of Clifford Codes, proving the existence and uniqueness of the prime decomposition of Clifford codes. Next, we tabulate equivalence classes of ZX diagrams under a different definition of equivalence that allows output permutations and any local operations on the outputs. Possible representatives of these equivalence classes are analyzed. This work expands on previous works in exploring the canonical forms of QECC's in their ZX diagram representations.
Verifying whether two quantum circuits are equivalent is a central challenge in the compilation and optimization of quantum programs. We introduce \textsc{Manjushri}, a new automated framework for scalable quantum-circuit equivalence checking. \textsc{Manjushri} uses local projections as discriminative circuit fingerprints, implemented with weighted binary decision diagrams (WBDDs), yielding a compact and efficient symbolic representation of quantum behavior. We present an extensive experimental evaluation that, for random 1D Clifford+T circuits, explores the trade-off between \textsc{Manjushri} and \textsc{ECMC}, a tool for equivalence checking based on a much different approach. \textsc{Manjushri} is much faster up to depth 30 (with the crossover point varying from 39--49, depending on the number of qubits and whether the input circuits are equivalent or inequivalent): when inputs are equivalent, \textsc{Manjushri} is about 10× faster (or more); when inputs are inequivalent, \textsc{Manjushri} is about 8× faster (or more). For both kinds of equivalence-checking outcomes, \textsc{ECMC}'s success rate out to depth 50 is impressive on 32- and 64-qubit circuits: on such circuits, \textsc{ECMC} is almost uniformly successful. However, \textsc{ECMC} struggled on 128-qubit circuits for some depths. \textsc{Manjushri} is almost uniformly successful out to about depth 38, before tailing off to about 75\% at depth 50 (falling to 0\% at depth 48 for 128-qubit circuits that are equivalent). These results establish that \textsc{Manjushri} is a practical and scalable solution for large-scale quantum-circuit verification, and would be the preferred choice unless clients need to check equivalence of circuits of depth $>$38.
Calderbank--Shor--Steane (CSS) codes support decoding and fault-tolerant logical operations through separate pure X and pure Z stabilizer checks. Generator choices and single-qubit Clifford transformations can obscure this structure. We give a deterministic polynomial-time algorithm that decides whether an arbitrary stabilizer code is locally Clifford-equivalent to a CSS code and constructs the required transformation and CSS checks whenever such a form exists. For m possibly redundant generator rows on n qubits, the algorithm uses O(mn2+n4) binary field operations. We reduce the problem to affine linear systems by characterizing local projectors that preserve the span of the binary stabilizer labels. The conversion preserves code parameters and check sparsity for quantum low-density parity-check (qLDPC) codes. We demonstrate CSS recovery and decoder reuse and resolve previously open LC-to-CSS classification questions for published code constructions. For graph states, LC-to-CSS equivalence corresponds to the existence of a bipartite graph in the local-complementation orbit. We derive a necessary-and-sufficient adjacency-matrix criterion for this property and construct such a graph whenever one exists. Our algorithm makes CSS decoding and fault-tolerant constructions accessible throughout the local Clifford equivalence classes of CSS codes.
Formal verification of quantum circuits becomes difficult as circuit size grows because exhaustive reasoning over the Hilbert space is computationally expensive. In previous work [1], we introduced a bit-vector abstraction that separates superposition tracking from basis-state behavior, enabling SMT-based verification of fixed-size error-detection and entanglement circuits built from H, X, and C-NOT gates. This paper extends the framework in five directions. First, we generalise correctness properties to parametrised repetition and concatenated phase-flip codes with up to 101 logical elements. Second, we verify syndrome extraction for CSS codes specified by arbitrary parity-check matrices, including the Steane, [[15,7,3]] Hamming, [[15,1,3]] Reed-Muller, and rotated surface codes with distances up to 101. We consider arbitrary error patterns, detection up to a specified weight, and recovery using a classical decoder, with counterexamples beyond each code's correction or detection capability. Third, we characterize the {H,X,C-NOT} fragment for which the abstraction faithfully represents Hilbert-space semantics. We prove soundness, demonstrate necessity with counterexamples, validate the characterisation against state-vector simulation, and prove the six lemmas from the original work. Fourth, a mutation study using 13 operators identifies equivalent mutants and detects all 622 non-equivalent mutants within the evaluated fault model. Fifth, a direct Z3 implementation verifies GHZ circuits with up to 262,144 qubits across three topologies on a 16GB laptop. Random H+C-NOT networks show verification cost rises sharply with depth when outputs depend on many inputs. The approach does not target general stabilizer circuits, Y-type stabilizers, fault-tolerant protocols, or physical noise; we identify extensions needed to address these cases.
Abstract Quantum stabilizer codes are essential for protecting quantum information, but systematic searches for high-distance codes become computationally prohibitive as code length increases. Existing code constructions exhibit structural regularities, yet no general framework quantifies how such constraints reduce the stabilizer-code search space or incorporates them into code searches. This project asks whether structural properties of known quantum codes can be formulated as mathematical constraints that quantitatively reduce search spaces while remaining useful for discovering high-distance codes. We develop a unified framework based on q-Galois matrices and define the search-space retention factor as the asymptotic fraction of logarithmic search space retained after imposing a constraint. We derive retention factors for weight-spectrum, qudit-participation, overlap, support, Pauli-distribution, local non-commutation, graph-theoretic, and symmetry constraints, including combinations. We introduce finite-size correction factors C and C′ and find empirically that they decay approximately as logn/n and 1/n, respectively. The framework is evaluated using 58,902 quantum-code parameter sets, including 34,775 codes with usable matrix data across q=2,3,4,5,7,8; 1,909 semi-perfect quantum codes are analyzed. We develop a general method to empirically estimate the weight-spectrum and average-weight for unresolved semi-perfect quantum codes, achieving runtime speedups ranging from 108.3608 to 10193.9298 across 4 distinct unresolved codes. Runtime analysis validates that these reductions translate into substantial computational savings. Applying the framework to a (14,3) qubit search produced a code detecting 91,765 of 91,770 errors of weight at most 4 and achieving 99.9999933% and 99.9733% of the theoretically optimal performance of [[14,3,5]]2 code at p=0.01. Thus, the framework converts structural properties into explicit candidate-space reductions that can be imposed during code enumeration, providing a quantitative basis for reducing the computational cost of high-distance stabilizer-code searches.
Fault-tolerant quantum computation requires error-correcting codes that not only protect quantum information but also support useful logical operations with practical physical realizations. Here we develop a general framework in which the stabilizer syndromes of commuting Pauli rotations directly organize logical-gate design. We derive an exact codespace-preservation condition valid at arbitrary rotation angles, providing a common stabilizer-level description for CSS and non-CSS codes and for both Clifford and non-Clifford operations. A particularly useful equal-syndrome specialization reduces Clifford gate design to stabilizer factorization, yields a general generator weight bound of ⌈d/2⌉ for distance-d codes, and systematically extends to logical rotations higher in the Clifford hierarchy. Applied to bivariate-bicycle (BB) and non-CSS Mirror qLDPC codes, the framework gives explicit low-weight Clifford and controlled-phase gates. Our implementations yield new low-weight gates, including an addressed T gate on the [[16,4,4]] BB code, T-type rotations on the [[64,8,8]] BB code, and CCZ-, CS-, and T-type operations on the [[36,6,6]] Mirror code. We also develop analytic constructions using Clifford algebra bases, which turn the syndrome constraints into explicit algebraic relations and provide a systematic route to logical addressability. We use this approach to analytically construct addressed CNOT, SWAP, controlled, and CCZ-type gates on recursive hypercube codes. An independent construction on middle quantum Reed--Muller codes yields CrZ gates that saturate the equal-syndrome weight bound. These results establish stabilizer syndromes as a constructive link between the stabilizer structure of a quantum code, the logical computations it can directly support, and the fault-tolerance properties of their physical realizations.