Quantum error correction beyond $SU(2)$: spin, bosonic, and permutation-invariant codes from convex geometry

• URL: http://arxiv.org/abs/2509.20545v2
• Date: Sat, 27 Sep 2025 18:48:49 GMT
• Title: Quantum error correction beyond $SU(2)$: spin, bosonic, and permutation-invariant codes from convex geometry
• Authors: Arda Aydin, Victor V. Albert, Alexander Barg,
• Abstract summary: 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.
• Score: 48.254879700836376
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We develop a framework for constructing quantum error-correcting codes and logical gates for three types of spaces -- composite permutation-invariant spaces of many qubits or qudits, composite constant-excitation Fock-state spaces of many bosonic modes, and monolithic nuclear state spaces of atoms, ions, and molecules. By identifying all three spaces with discrete simplices and representations of the Lie group $SU(q)$, we prove that many codes and their gates in $SU(q)$ can be inter-converted between the three state spaces. We construct new code instances for all three spaces using classical $\ell_1$ codes and Tverberg's theorem, a classic result from convex geometry. We obtain new families of quantum codes with distance that scales almost linearly with the code length $N$ by constructing $\ell_1$ codes based on combinatorial patterns called Sidon sets and utilizing their Tverberg partitions. This improves upon the existing designs for all the state spaces. We present explicit constructions of codes with shorter length or lower total spin/excitation than known codes with similar parameters, new bosonic codes with exotic Gaussian gates, as well as examples of short codes with distance larger than the known constructions.

Related papers

• Calderbank-Shor-Steane codes on group-valued qudits [1.744249132777104]
  Calderbank-Shor-Steane (CSS) codes are a versatile quantum error-correcting family built out of commuting $X$- and $Z$-type checks.<n>We introduce CSS-like codes on $G$-valued qudits for any finite group $G$ that reduce to qubit CSS codes for $G = mathbbZ$ yet generalize the Kitaev quantum double model for general groups.
  arXiv  Detail & Related papers  (2026-02-23T07:08:00Z)
• Poincaré Duality and Multiplicative Structures on Quantum Codes [11.11194917284133]
  We build circuits composed of $mathrmCmathrmCZ$ gates as well as for higher order controlled-$Z$ gates.<n>We conjecture that they generate nontrivial logical actions, pointing towards fault-tolerant non-Clifford gates on nearly optimal qLDPC sheaf codes.
  arXiv  Detail & Related papers  (2025-12-26T08:38:08Z)
• Near-Asymptotically-Good Quantum Codes with Transversal CCZ Gates and Sublinear-Weight Parity-Checks [18.20811830109862]
  We construct the first known quantum codes with linear dimension and distance supporting non-Clifford gates.<n>We design an efficient decoding algorithm for these codes.<n>Our results can be viewed as a new generalization of Prony's method for reconstructing a function from partial access to its transform.
  arXiv  Detail & Related papers  (2025-10-08T09:27:41Z)
• 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)
• Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates [23.22566380210149]
  We construct quantum codes that support $CCZ$ gates over qudits of arbitrary prime power dimension $q$. The only previously known construction with such linear dimension and distance required a growing alphabet size $q$.
  arXiv  Detail & Related papers  (2024-08-17T16:54:51Z)
• Expansion of higher-dimensional cubical complexes with application to quantum locally testable codes [4.034916061859703]
  We introduce a high-dimensional cubical complex, for any dimension t>0, and apply it to quantum locally testable codes.<n>For t=4 our construction gives a new family of "almost-good" quantum LTCs -- with constant relative rate, inverse-polylogarithmic relative distance and soundness, and constant-size parity checks.
  arXiv  Detail & Related papers  (2024-02-12T08:32:13Z)
• Quotient Space Quantum Codes [0.0]
  In this paper, I establish the quotient space codes to construct quantum codes. These new codes unify additive codes and codeword stabilized codes and can transmit classical codewords. I also present new bounds for quantum codes and provide a simple proof of the quantum Singleton bound.
  arXiv  Detail & Related papers  (2023-11-13T12:03:59Z)
• A family of permutationally invariant quantum codes [54.835469342984354]
  We show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes. For small $t$, these conditions can be used to construct new examples of codes by computer.
  arXiv  Detail & Related papers  (2023-10-09T02:37:23Z)
• Geometrically Local Quantum and Classical Codes from Subdivision [10.357542321841887]
  A geometrically local quantum code is an error correcting code situated within $mathbbRD$, where the checks only act on qubits within a fixed spatial distance. Recently, Portnoy made a significant breakthrough with codes achieving optimal dimension and distance up to polylogs. This paper bypasses this step and streamlines the construction by noticing that a family of good quantum low-density parity-check codes, balanced product codes, naturally carries a two-dimensional structure.
  arXiv  Detail & Related papers  (2023-09-28T02:12:38Z)
• 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)
• Quantum spherical codes [55.33545082776197]
  We introduce a framework for constructing quantum codes defined on spheres by recasting such codes as quantum analogues of the classical spherical codes. We apply this framework to bosonic coding, obtaining multimode extensions of the cat codes that can outperform previous constructions.
  arXiv  Detail & Related papers  (2023-02-22T19:00:11Z)
• Morphing quantum codes [77.34726150561087]
  We morph the 15-qubit Reed-Muller code to obtain the smallest known stabilizer code with a fault-tolerant logical $T$ gate. We construct a family of hybrid color-toric codes by morphing the color code.
  arXiv  Detail & Related papers  (2021-12-02T17:43:00Z)
• Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians [7.092674229752723]
  We introduce a framework that takes it any classical code and explicitly constructs the corresponding QEC code. A concrete advantage is that the desirable properties of a classical code are automatically incorporated in the design of the resulting quantum code.
  arXiv  Detail & Related papers  (2020-12-02T19:00:19Z)

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.