We develop a decoder-agnostic logical-class diagnostic layer for abelian Pauli stabilizer codes, motivated by fault-tolerant quantum computing. An ancilla fermionic register defines a nilpotent supercharge whose clean degree-zero harmonic sector is the code space. Projected-return blocks derived from graded commutators organize the recovery test of whether a decoder-selected correction preserves every encoded input for every error in a declared same-syndrome family; the passing syndromes form the certified faithful window. Demoting the supercharge to an exact binary diagnostic formulation classifies residuals modulo stabilizers, separating a check violation, a harmless stabilizer residual, and a syndrome-silent nontrivial logical action, with a two-sector form for CSS codes. Because the unknown fault cancels, the relative logical class of two record-compatible candidates is computable online. We verify the diagnostic formalisms on the Steane code and specify a static--circuit bridge audit protocol for the Gross bivariate-bicycle quantum low-density parity-check (qLDPC) code. The method supplements existing decoders without additional physical qubits for software use.
We develop a decoder-agnostic logical-class diagnostic layer for abelian Pauli stabilizer codes, motivated by fault-tolerant quantum computing. An ancilla fermionic register defines a nilpotent supercharge whose clean degree-zero harmonic sector is the code space. Projected-return blocks derived from graded commutators organize the recovery test of whether a decoder-selected correction preserves every encoded input for every error in a declared same-syndrome family; the passing syndromes form the certified faithful window. Demoting the supercharge to an exact binary diagnostic formulation classifies residuals modulo stabilizers, separating a check violation, a harmless stabilizer residual, and a syndrome-silent nontrivial logical action, with a two-sector form for CSS codes. Because the unknown fault cancels, the relative logical class of two record-compatible candidates is computable online. We verify the diagnostic formalisms on the Steane code and specify a static--circuit bridge audit protocol for the Gross bivariate-bicycle quantum low-density parity-check (qLDPC) code. The method supplements existing decoders without additional physical qubits for software use.
Small high-rate non-CSS stabilizer codes provide compact platforms for encoded quantum computation, but mixed-Pauli checks and limited native transversal logical gates complicate fault-tolerant dynamics. Block-level constructions offer an alternative by mapping an entire logical block to a physical circuit rather than compiling separately protected logical gates. We investigate this approach using the high-rate [[8,3,3]] non-CSS code and logical Trotter circuits as a testbed. We construct flagged syndrome-extraction circuits and establish a circuit-level memory pseudo-threshold near 1.5×10−3. We then apply our symplectic-transvection construction, which maps a logical Trotter circuit to a physical circuit with the same block pattern for any stabilizer code. Although this mapping preserves the intended unitary algebraically, encoded Trotter circuits exhibit asymmetry between logical-X and logical-Z failure channels. Single-fault analysis identifies the mechanism: a fault on the shared parity ancilla can propagate through the uncomputation network into an undetectable logical operator, reducing the effective circuit distance in the affected sector. We evaluate flag-conditioned recovery, biased-noise decoding, CliNR resource verification, flag postselection, and asymmetric gate-noise models. These methods suppress propagated faults but do not simultaneously suppress both logical sectors in the realistic configurations studied. A diagnostic protected limit removing the identified malignant first-order locations restores pseudo-threshold behavior in both sectors, approaching memory performance. These results demonstrate the potential of block-level logical constructions for non-CSS codes without rich native transversal gate sets and the joint protection of the parity network, analog rotation, and recovery required to preserve fault-tolerant distance.
Quantum low-density parity-check (QLDPC) codes promise scalable quantum memories, but their decoders receive only sparse check outcomes that may themselves be wrong. We prove that one such record supports logical list decoding near the Singleton radius in a fixed folded-block metric. For any fixed rate and positive gap below that radius, we construct bounded-locality Calderbank--Shor--Steane (CSS) QLDPC families with a randomized near-linear-time decoder. The decoder returns a constant number of Pauli candidates, independent of blocklength. An exact record gives zero-residual coverage of every compatible logical class within the target radius. If an adversarial fraction below a fixed threshold of outcomes is faulty, every compatible error within that radius has stabilizer-reduced relative distance at most a constant times that fraction from some returned candidate. The main advance is to reduce the high-weight logical ambiguity before residual correction begins. Each local syndrome outcome cuts the error space to an affine slice. We prove the quotient-compatibility conditions that let the published Alon--Edmonds--Luby (AEL) average-radius bound control distinct logical classes, leaving only a constant number of alternatives. We then turn the observed slices into the agreement graphs used by fast regularity enumeration. An existing single-shot quantum-Tanner decoder handles the small remainder. The resulting front end preserves the sparse measurement record, and its residual grows linearly with syndrome corruption below the stated threshold.
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.
Quantum low-density parity-check (qLDPC) codes reduce the qubit overhead of fault-tolerant quantum computation by an order of magnitude, but their decoding is harder than its classical counterpart: because many physical errors are equivalent up to stabilizers, the degenerate maximum-likelihood (ML) decoder must compare the probabilities of entire equivalence classes of errors, that is, partition functions, rather than single errors. The workhorse decoder BP+OSD sidesteps degeneracy heuristically and offers no guarantees. We treat degenerate decoding as probabilistic inference in an undirected graphical model: the probability of each logical class is the partition function of an unconstrained, strictly positive Markov random field over the code's check variables, a construction that generalizes the random-bond Ising mapping of the surface code to arbitrary CSS codes and to spacetime decoding with measurement errors and circuit-level noise. On this model we build two decoders. The first estimates all class partition functions by annealed importance sampling with common random numbers and attaches to every decision a certificate of optimality: a paired bootstrap test, or, composed with constant-factor estimators such as WISH, an exact optimality proof. The second is region-based: the Bethe free energy, whose bias cancels between classes, reproduces exact ML decoding on every tested surface-code instance at millisecond cost, and enlarging the regions to elimination clusters makes exact degenerate ML decoding of the [[72,12,6]] bivariate bicycle code feasible. Across surface codes and the bivariate bicycle codes [[72,12,6]] and [[144,12,12]], under code-capacity, phenomenological, and circuit-level noise, the sampling decoder matches or exceeds BP+OSD while certifying the bulk of its decisions, and the certificate flags exactly the syndromes on which any fast decoder should be distrusted.
Quantum error correcting codes (QECC) are essential for constructing large-scale quantum computers that deliver faithful results. As strong competitors to the conventional surface code, quantum low-density parity-check (qLDPC) codes are emerging rapidly: they offer high encoding rates while maintaining reasonable physical-qubit connectivity requirements. Despite the existence of numerous code constructions, a notable gap persists between these designs -- some of which remain purely theoretical -- and their circuit-level deployment.
In this work, we propose Auto-Stabilizer-Check (ASC), a universal compilation framework that generates depth-optimal syndrome extraction circuits for arbitrary qLDPC codes. ASC leverages the sparsity of parity-check matrices and exploits the commutativity of X and Z stabilizer measurement subroutines to search for optimal compilation schemes. By iteratively invoking an SMT solver, ASC returns a depth-optimal solution if a satisfying assignment is found, and a near-optimal solution in cases of solver timeouts. Notably, ASC provides the first definitive answer to one of IBM's open problems: for all instances of bivariate bicycle (BB) code reported in their work, our compiler certifies that no depth-6 syndrome extraction circuit exists.
Furthermore, by integrating ASC with an end-to-end evaluation framework -- one that assesses different compilation settings under a circuit-level noise model -- ASC reduces circuit depth by approximately 50% and achieves an average 7x-8x suppression of the logical error rate for general qLDPC codes, compared with as-soon-as-possible (ASAP) and coloration-based scheduling. ASC thus substantially reduces manual design overhead and demonstrates its strong potential to serve as a key component in accelerating hardware deployment of qLDPC codes.
Quantum error-correcting code (QECC) implementations are hybrid software-control systems: measured syndromes drive decoder dispatch, recovery actions, and logical checks. Existing QECC verification techniques can prove formal obligations, and general quantum testing can expose final failures, but developers still lack a scalable testing workflow that generates concrete faults and explains where a QECC repair path first diverges. This paper presents ConQEC, to our knowledge the first concolic testing framework for QECC implementations. By combining concrete (Pauli) fault replay with symbolic repair-path constraints, ConQEC generates tests that expose not only final logical failures but also fault--syndrome--decoder--recovery (FSDR) diagnostics: the injected fault, observed syndrome, decoder decision, recovery action, oracle result, and earliest divergent stage. Its symbolic encoding of Pauli faults, syndrome predicates, decoder and recovery relations, and oracle checks enables this diagnostic workflow to scale beyond exponentially increasing amplitude-level quantum-state reasoning. We implemented ConQEC and evaluated it on generated source-level QECC families and public QECC artifacts with up to thousands of qubits. ConQEC detects the seeded bug in every experimental case and reports a replay-valid, fault-localizing FSDR witness for each.