arXiv subjects

Olai Å. Mostad

Publications and source records attributed to Olai Å. Mostad.

5 recordsLinked to original sources

arXiv2026

On Constructing and Decoding Quantum Triorthogonal Codes

Triorthogonal codes are a basic ingredient in magic-state distillation, as they enable transversal implementation of logical $\mat{T}$ gates. As a non-Clifford operation, the $\mat{T}$ gate enables universal fault-tolerant quantum computation when combined with Clifford operations. In this work, we study the construction and decoding of triorthogonal codes with given dual distance properties. We derive an existence criterion for even-weight triorthogonal generator matrices with a target dual minimum distance. The criterion combines triorthogonality constraints with MacWilliams identities via Krawtchouk polynomial conditions on the dual weight distribution, yielding an integer linear programming formulation for the construction problem. We find novel nontrivial triorthogonal codes (degenerate and non-degenerate), that are not necessarily constructed from triply-even binary linear codes. We also evaluate the performance of the obtained high-distance triorthogonal codes over the dephasing channel using different decoding methods, showing that automorphism ensemble decoding and guessing random additive-noise decoding (GRAND) are promising options.

quant-ph↗

arXiv2026

Improved Decoding of Quantum Tanner Codes Using Generalized Check Nodes

We study the decoding problem for quantum Tanner codes and propose to exploit the underlying local code structure by grouping check nodes into more powerful generalized check nodes for enhanced iterative belief propagation (BP) decoding by decoding the generalized checks using a maximum a posteriori (MAP) decoder as part of the check node processing of each decoding iteration. We mainly study the finite-length setting and show that the proposed enhanced generalized BP decoder for quantum Tanner codes significantly outperforms the standard quaternary BP decoder with memory effects, as well as the recently proposed Relay-BP decoder, even outperforming generalized bicycle (GB) codes with comparable parameters in some cases. For other classes of quantum low-density parity-check (qLDPC) codes, we propose a greedy algorithm to combine checks for generalized BP decoding. However, for GB codes, bivariate bicycle codes, hypergraph product codes, and lifted-product codes, there seems to be limited gain by combining simple checks into more powerful ones. To back up our findings, we also provide a theoretical cycle analysis for the considered qLDPC codes. This leads us to a construction criterion for quantum Tanner codes that produces codes particularly well-suited for BP decoding.

quant-ph↗

arXiv2026

Local Automorphism-Aware Syndrome Compilation for General Quantum LDPC Codes

Low-depth syndrome extraction for Calderbank-Shor-Steane (CSS) quantum low-density parity-check codes can be formulated as a proper ordered edge-coloring problem subject to quantum parity constraints. A proper edge-coloring of the CSS Tanner graph ensures that each data or ancilla qubit participates in at most one two-qubit gate per layer, but does not guarantee a valid interleaving of the X- and Z-check measurements as for every overlapping X/Z check pair, the number of shared data qubits on which the X interaction precedes the Z interaction must be even. The minimum number of colors in a proper ordered edge-coloring satisfying the quantum parity constraints equals the minimum two-qubit depth when each stabilizer check is measured with a single ancilla. We introduce local automorphism-aware syndrome compilation (LocalASC), which reduces the constraint system to edge-orbit variables under a subgroup of the Tanner graph automorphisms and lifts each feasible orbit assignment to the full graph. Although the $6$-layer degree lower bound is unattainable for the published weight-$6$ IBM bivariate bicycle codes, we show that this is not universal among two-block CSS codes. Among code instances for which the maximum check weight equals the maximum Tanner graph degree, LocalASC finds depth-optimal syndrome-extraction schedules for several two-block CSS codes with odd component weights, including instances with unequal odd weights. We also obtain lower-bound-saturating syndrome-extraction schedules for several quantum Tanner codes satisfying the same degree condition. To obtain the subgroups used by LocalASC without computing the full automorphism group of the Tanner graph, we construct translation subgroups for two-block group-algebra CSS codes over abelian groups. For quantum Tanner codes, we give conditions under which square-complex symmetries extend to Tanner graph automorphisms.

quant-ph↗

arXiv2025

Advancing Finite-Length Quantum Error Correction Using Generalized Bicycle Codes

Generalized bicycle (GB) codes have emerged as a promising class of quantum error-correcting codes with practical decoding capabilities. While numerous asymptotically good quantum codes and quantum low-density parity-check code constructions have been proposed, their finite block-length performance often remains unquantified. In this work, we demonstrate that GB codes exhibit comparable or superior error correction performance in finite-length settings, particularly when designed with higher or unrestricted row weights. Leveraging their flexible construction, GB codes can be tailored to achieve high rates while maintaining efficient decoding. We evaluate GB codes against other leading quantum code families, such as quantum Tanner codes and single-parity-check product codes, highlighting their versatility in practical finite-length applications.

quant-ph↗

arXiv2024

Generalizing Quantum Tanner Codes

In this work, we present a generalization of the recently proposed quantum Tanner codes by Leverrier and Zémor, which contains a construction of asymptotically good quantum LDPC codes. Quantum Tanner codes have so far been constructed equivalently from groups, Cayley graphs, or square complexes constructed from groups. We show how to enlarge this to group actions on finite sets, Schreier graphs, and a family of square complexes which is the largest possible in a certain sense. Furthermore, we discuss how the proposed generalization opens up the possibility of finding other families of asymptotically good quantum codes.

cs.IT↗