On Constructing and Decoding Quantum Triorthogonal Codes

• URL: http://arxiv.org/abs/2605.24519v1
• Date: Sat, 23 May 2026 11:13:06 GMT
• Title: On Constructing and Decoding Quantum Triorthogonal Codes
• Abstract summary: A triorthogonal code is a binary quantum computation Calderbank-Shor-Steane code.<n>We study the construction and decoding of triorthogonal codes with prescribed dual-distance properties.
• Score: 8.986356590623483
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A triorthogonal code is a binary quantum Calderbank-Shor-Steane (CSS) code defined by a triorthogonal matrix. Triorthogonal codes are a key ingredient in magic-state distillation, since they allow for transversal $\mathsf{T}$ gates, a non-Clifford logical operation useful for achieving universal fault-tolerant quantum computation. Their construction is challenging because it must satisfy simultaneous pairwise and triple-wise overlap constraints, as well as row-weight requirements. In this work, we study the construction and decoding of triorthogonal codes with prescribed 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 new nontrivial triorthogonal codes that are not necessarily generated by classical triply-even codes. The decoding performance of high-distance triorthogonal codes obtained via the doubling construction is then evaluated over the dephasing channel. We compare bounded-distance decoding, belief propagation plus ordered-statistics post-processing, and a GRAND-based decoder adapted to the quantum setting, which turns out to be a promising option.

Related papers

• Coupled-Layer Codes: Beyond Quantum Product Constructions [0.664167116861767]
  Product codes are an important class of quantum error-correcting codes constructed from multiple input codes.<n>We show how coupled-layer codes reproduce the X-cube and Chamon models.<n>In particular, we show how balancing by a ZX-duality can reproduce non-CSS codes such as the fermionic toric code and the 3-fermion Walker-Wang model in 3D.
  arXiv  Detail & Related papers  (2026-08-10T18:00:01Z)
• Quantum codes from classical annealing [0.15293427903448023]
  Search for quantum error-correcting codes with high encoding rates and large distances.<n>For block lengths of up to $50$ physical qubits, our search finds state-of-the-art CSS and SWEL codes.<n>Codes found in this work are promising candidates for high-rate code demonstrations on near-term quantum computing hardware.
  arXiv  Detail & Related papers  (2026-07-21T21:01:37Z)
• Explicit Block Encodings of Discrete Laplacians with Mixed Boundary Conditions [45.88028371034407]
  Block encoding provides the standard method for accessing matrix data within quantum circuits.<n>We present a unified framework for efficiently block encoding finite-difference discretizations of the Laplacian.
  arXiv  Detail & Related papers  (2026-03-12T19:35:16Z)
• Low Overhead Universal Quantum Computation with Triorthogonal Codes [8.516351290711953]
  We study the use of triorthogonal codes for universal fault-tolerant quantum computation.<n>We propose two methods to circumvent the Eastin-Knill theorem, which prohibits any single quantum error-correcting code from supporting both universality and a gate set.
  arXiv  Detail & Related papers  (2025-10-07T09:17:10Z)
• Quantum error correction beyond $SU(2)$: spin, bosonic, and permutation-invariant codes from convex geometry [48.254879700836376]
  We develop a framework for constructing quantum error-correcting codes and logical gates for three types of spaces.<n>We prove that many codes and their gates in $SU(q)$ can be inter-converted between the three state spaces.<n>We present explicit constructions of codes with shorter length or lower total spin/excitation than known codes with similar parameters.
  arXiv  Detail & Related papers  (2025-09-24T20:21:30Z)
• Magic tricycles: Efficient magic state generation with finite block-length quantum LDPC codes [1.0792624191049491]
  We introduce a class of finite block-length quantum LDPC codes which we name tricycle codes.<n>These codes can support constant-depth physical circuits that implement logical $CCZ$ gates between three code blocks.<n>We show that tricycle codes enable single-shot state-preparation and error correction, leading to a highly efficient magic-state generation protocol.
  arXiv  Detail & Related papers  (2025-08-14T14:56:50Z)
• Universal fault tolerant quantum computation in 2D without getting tied in knots [1.796950496605906]
  We show how to perform fault-tolerant non-Clifford gates in two dimensions.<n>We formulate a path framework which provides both a macroscopic picture for different logical gates as well as a way to derive the associated microscopic circuits.
  arXiv  Detail & Related papers  (2025-03-19T23:59:46Z)
• Geometric structure and transversal logic of quantum Reed-Muller codes [51.11215560140181]
  In this paper, we aim to characterize the gates of quantum Reed-Muller (RM) codes by exploiting the well-studied properties of their classical counterparts. A set of stabilizer generators for a RM code can be described via $X$ and $Z$ operators acting on subcubes of particular dimensions.
  arXiv  Detail & Related papers  (2024-10-10T04:07:24Z)
• Triorthogonal Codes and Self-dual Codes [16.728666221394857]
  We give an algorithm to construct binary triorthogonal matrices from binary self-dual codes. We generalize to this setting the classical coding techniques of shortening and extending.
  arXiv  Detail & Related papers  (2024-08-19T03:47:09Z)
• Homological Quantum Rotor Codes: Logical Qubits from Torsion [47.52324012811181]
  homological quantum rotor codes allow one to encode both logical rotors and logical qudits in the same block of code.<n>We show that the $0$-$pi$-qubit as well as Kitaev's current-mirror qubit are indeed small examples of such codes.
  arXiv  Detail & Related papers  (2023-03-24T00:29:15Z)
• Finding the disjointness of stabilizer codes is NP-complete [77.34726150561087]
  We show that the problem of calculating the $c-disjointness, or even approximating it to within a constant multiplicative factor, is NP-complete. We provide bounds on the disjointness for various code families, including the CSS codes,$d codes and hypergraph codes. Our results indicate that finding fault-tolerant logical gates for generic quantum error-correcting codes is a computationally challenging task.
  arXiv  Detail & Related papers  (2021-08-10T15:00:20Z)
• Realization of arbitrary doubly-controlled quantum phase gates [62.997667081978825]
  We introduce a high-fidelity gate set inspired by a proposal for near-term quantum advantage in optimization problems. By orchestrating coherent, multi-level control over three transmon qutrits, we synthesize a family of deterministic, continuous-angle quantum phase gates acting in the natural three-qubit computational basis.
  arXiv  Detail & Related papers  (2021-08-03T17:49:09Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.