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Generalized toric codes on twisted tori for quantum error correction

Zijian Liang, Ke Liu, Hao Song, Yu-An Chen

arXiv:2503.03827v3quant-phcond-mat.str-elmath-phmath.MPmath.QA

TL;DR

Efficient analysis of quantum error-correcting codes has become important as bivariate bicycle codes show promise on small tori. This paper uses ring-theoretic and anyon methods to analyze topological CSS codes and construct generalized toric codes on twisted tori, reporting optimal codes through n ≤400 and a more localized [[360, 12, ≤24]] realization. The framework connects logical dimension to anyon structure and supports code analysis under twisted periodic boundaries.

  • Problem

    Growing interest in efficient analysis and characterization of quantum error-correcting codes accompanies promising bivariate bicycle-code results on small tori.

    Source: §I · p. 1

  • Method

    The paper applies a Laurent-polynomial ring-theoretic framework and Gröbner-basis anyon analysis to two-dimensional topological CSS codes.

    Source: §I · p. 2

  • Results

    The paper presents optimal generalized toric codes on twisted tori for n ≤400 and reduces the [[360, 12, ≤24]] code’s stabilizer range from 9 in Ref. to 3.

    Source: §I · p. 2

  • Takeaways & Limitations

    Counting independent anyon types, equivalently the dimension of R/I for I = ⟨f(x, y), g(x, y)⟩, determines logical dimension within a framework that incorporates twisted periodic boundaries.

    Source: §IV · p. 13

    For n of a few hundred and d >20, the probabilistic distance algorithm may be unreliable and yield only an upper bound.

    Source: §IV · p. 13

Abstract

from arXiv · show

The Kitaev toric code is widely considered one of the leading candidates for error correction in fault-tolerant quantum computation. However, direct methods to increase its logical dimensions, such as lattice surgery or introducing punctures, often incur prohibitive overheads. In this work, we introduce a ring-theoretic approach for efficiently analyzing topological CSS codes in two dimensions, enabling the exploration of generalized toric codes with larger logical dimensions on twisted tori. Using Gr\"obner bases, we simplify stabilizer syndromes to efficiently identify anyon excitations and their geometric periodicities, even under twisted periodic boundary conditions. Since the properties of the codes are determined by the anyons, this approach allows us to directly compute the logical dimensions without constructing large parity-check matrices. Our approach provides a unified method for finding new quantum error-correcting codes and exhibiting their underlying topological orders via the Laurent polynomial ring. This framework naturally applies to bivariate bicycle codes. For example, we construct optimal weight-6 generalized toric codes on twisted tori with parameters $[[ n, k, d ]]$ for $n \leq 400$, yielding novel codes such as $[[120,8,12]]$, $[[186,10,14]]$, $[[210,10,16]]$, $[[248, 10, 18]]$, $[[254, 14, 16]]$, $[[294, 10, 20]]$, $[[310, 10, \leq 22]]$, and $[[340, 16, 18]]$. Moreover, we present a new realization of the $[[360, 12, \leq 24]]$ quantum code using the $(3,3)$-bivariate bicycle code on a twisted torus defined by the basis vectors $(0,30)$ and $(6,6)$, improving stabilizer locality relative to the previous construction. These results highlight the power of the topological order perspective in advancing the design and theoretical understanding of quantum low-density parity-check (LDPC) codes.

I. INTRODUCTION

The introduction motivates efficient analysis of topological quantum codes and presents a Laurent-polynomial, ring-theoretic approach for constructing generalized toric codes on twisted tori. It reports optimal codes through n ≤400 and a [[360, 12, ≤24]] realization with more localized stabilizers than a previous construction.

I. INTRODUCTION

  • Growing interest in efficient code analysis follows progress on bivariate bicycle codes, which have shown promising performance on small tori, in some cases up to an order of magnitude better than the Kitaev toric code.The introduction also notes that high-distance quantum LDPC codes can reduce the physical-to-logical-qubit ratio while maintaining comparable error suppression.Source: §I · p. 1
  • The paper develops a ring-theoretic approach in the Laurent polynomial formalism to analyze two-dimensional topological CSS codes and construct generalized toric codes on twisted tori with n ≤400.The tabulated polynomials and lattice vectors are selected for having the most localized stabilizers among parameter-equivalent solutions.Source: §I · p. 2
  • For the [[360, 12, ≤24]] code, the twisted-torus construction reduces stabilizer range from 9 in Ref. to 3.The authors describe the reduced range as more practical for experimental implementation.Source: §I · p. 2

II. ALGEBRAIC METHODS FOR ERROR-CORRECTING CODES

The algebraic methods represent Pauli operators and translation-invariant CSS stabilizers with Laurent polynomials, then use anyons and Gröbner-basis methods to analyze code properties. The approach relates logical operators to torus-wrapping anyon strings and checks topological order through local operators and stabilizer products.

II. ALGEBRAIC METHODS FOR ERROR-CORRECTING CODES

  • Gröbner-basis methods classify anyons, whose count determines the torus ground-state degeneracy in the stated topological-order framework.The paper presents this as a way to streamline analysis of code properties.Source: §II · p. 2
  • Logical operators are represented as Wilson-line anyon strings wrapping the torus’s non-contractible cycles.The paper describes this representation when the logical dimension reaches its maximum value.Source: §II · p. 2
  • The two-edge unit cell represents translated Pauli operators by Laurent-polynomial vectors, with multiplication by x^n y^m implementing lattice shifts.Products of Pauli operators correspond to sums of their four-dimensional vectors.Source: §II · p. 2, Figure 1 · p. 3
  • The Kitaev and generalized toric codes are expressed through polynomial stabilizer generators, with generalized-code parameters specifying the added monomial terms.The generalized toric code is identified as a specific type of bivariate bicycle code.Source: §II · p. 2
  • Single-edge Pauli errors are mapped to syndrome polynomials involving f(x, y) or g(x, y).The displayed excitation maps assign the two X errors to [0, g(x, y)] and [0, f(x, y)], and the two Z errors to [f(x, y), 0] and [g(x, y), 0].Source: §II · p. 2
  • The topological-order check requires each local operator commuting with the stabilizers to be expressible as a finite product of stabilizers.The supplied text also states that the condition entails f(x, y) and g(x, y) are coprime.Source: §II · pp. 3–4

A. Classification of anyons on an infinite plane

The paper classifies anyon types through quotient rings and uses Gröbner bases to make the classification systematic. Counting independent monomials in the resulting ring gives the maximal logical dimension.

A. Classification of anyons on an infinite plane

B. Effects of finite geometries on tori

This section examines how finite torus geometries and boundary conditions shape anyon periodicities, logical dimensions, and code realizations for bivariate bicycle codes. It also describes a local approach to finding logical operators for frustrated values of k.

B. Effects of finite geometries on tori

  • An anyon disappears on a finite torus if its period does not divide the torus length, so maximal logical dimension requires compatible lengths.Gröbner bases yield univariate polynomials whose divisibility conditions determine the minimal lengths; irreducible factors can be checked separately for their shortest periodicities.Source: B. Effects of finite geometries on tori · p. 7
  • A 12 × 12 torus gives the (3, 3)-BB code its maximal logical dimension, k = kmax = 16.The minimal period is 12 in each direction, with x-y symmetry of the stabilizers.Source: B. Effects of finite geometries on tori · p. 8
  • The (3, −3)-BB code reaches k = 26 on a twisted torus with vectors (0, 762) and (6, 360), versus an untwisted 762 × 762 torus.The twisted torus achieves the same logical dimension with a significantly smaller system size.Source: B. Effects of finite geometries on tori · p. 8
  • For the (4, −4)-BB code, maximal dimension k = kmax = 40 on an untwisted torus requires size 69,905 × 1,048,575.The minimal lengths are Ly = 1,048,575 and Lx = 69,905.Source: B. Effects of finite geometries on tori · p. 8
  • Theorem 3 computes logical dimension on twisted tori from the quotient by the stabilizer and boundary-condition polynomials, avoiding costly parity-check rank calculations.At maximal logical dimension, logical X and Z operators are anyon strings wrapping the torus’s two non-contractible loops.Source: B. Effects of finite geometries on tori · pp. 9–10
  • On a 12 × 6 torus, the (3, −3)-BB code gives [[144, 12, 12]], while a 12 × 12 torus gives [[288, 12, 18]] with more-local stabilizers than Ref..The [[288, 12, 18]] code differs from the Ref. construction in its defining polynomials.Source: B. Effects of finite geometries on tori · p. 10
  • On a twisted 9 × 15 torus with basis vectors (0, 15) and (9, 6), the (3, −3)-BB example has k = 8 and d = 20, yielding [[270, 8, 20]].The same stabilizers on a twisted torus produce a different code; on a 12 × 12 torus they yield [[288, 12, 18]].Source: B. Effects of finite geometries on tori · p. 10
  • The cited passage reports these code parameter sets, but does not identify their construction, assign them to a BB-code family, or state that they are optimal.[[90, 8, 10]], [[108, 8, 10]], [[144, 12, 12]], [[162, 8, 14]], [[168, 8, 14]], [[180, 8, 16]], [[234, 8, 18]], [[270, 8, 20]], [[288, 12, 18]], [[306, 8, ≤22]], [[324, 8, ≤22]], [[342, 8, ≤22]], and [[396, 8, ≤26]].Source: B. Effects of finite geometries on tori · p. 10
  • The (3, 3)-BB code likewise generates a large family of codes on finite tori, including [[72, 8, 8]] and [[360, 12, ≤24]].Other reported examples are [[108, 8, 10]], [[144, 12, 12]], [[162, 8, 14]], [[180, 8, 16]], [[192, 8, 16]], [[234, 8, 18]], [[270, 8, 20]], and [[282, 4, ≤24]].Source: B. Effects of finite geometries on tori · p. 11

Logical operators for generic k

  • For frustrated k, logical operators can pair anyons moving along two directions when the intrinsic anyon periodicity is incompatible with a cycle.Modified Gaussian elimination solves locally around the corresponding strips in O((α + β + γ)^3), versus O(n^3) = O((αβ)^3) via the parity-check matrix.Source: B. Effects of finite geometries on tori · p. 11

III. APPLICATIONS FOR QLDPC CODE CONSTRUCTIONS

The authors search generalized toric codes on twisted tori for n ≤ 400, using their parameter-computation method to identify optimal codes and favoring more localized stabilizers. The search is restricted to a specific code family after small-size tests found no better parameters from more general bivariate bicycle codes.

III. APPLICATIONS FOR QLDPC CODE CONSTRUCTIONS

  • The search covers generalized toric codes for n ≤ 400, enumerating even-n factorizations and twisted-torus lattice choices before computing code parameters.The basis vectors are (0, m) and (l, q), with 0 ≤ q < m.Source: §III · p. 11
  • Tests for n ≤ 108 found no better parameters from more general bivariate bicycle codes, so the search was restricted to generalized toric codes.The broader polynomials considered have two nontrivial monomials in g(x, y).Source: §III · p. 11
  • The computation of k reduces lattice relations modulo the stabilizer polynomials and uses Gaussian elimination on monomials with range bounded by O(k), rather than a parity-check matrix scaling with n.In over 90% of cases, the Gröbner basis immediately yields ⟨1⟩, implying k = 0; the computations reached n = 400 on a standard personal computer.Source: §III · p. 11
  • Among solutions with the same parameters, the authors select those with the most local stabilizers; distances are exact for d ≤ 20 and probabilistically upper-bounded for larger d.They report confidence that the tabulated parameters are optimal, based on dozens or hundreds of solutions usually found for each parameter set.Source: §III · p. 11

A. Novel [[n, k, d]] codes

The section highlights novel code parameters and comparisons with prior weight-6 codes. In particular, [[120, 8, 12]] is reported as the best known example below n = 144.

A. Novel [[n, k, d]] codes

  • [[120, 8, 12]] is the best known example below n = 144, where the gross code [[144, 12, 12]] had previously been proposed.The paper identifies kd^2/n as a measure of performance relative to the Kitaev toric code.Source: A. Novel [[n, k, d]] codes · pp. 11–12
  • The new weight-6 codes surpass the previously best-reported weight-6 code [[288, 12, 18]] for comparable system sizes, which achieves kd^2/n = 13.5.Source: A. Novel [[n, k, d]] codes · p. 12
  • Optimal codes for each n are frequently realized on twisted tori, including an implementation with lattice vectors (0, 31) and (5, 11).Source: A. Novel [[n, k, d]] codes · p. 12

B. Improved locality of stabilizers in comparison to previous constructions

Twisted-torus implementations can reduce stabilizer range compared with earlier constructions. The section contrasts a prior untwisted realization of the [[360, 12, 24]] code with a (3, 3)-BB realization on a twisted torus.

B. Improved locality of stabilizers in comparison to previous constructions

C. Relation to one-dimensional generalized bicycle codes

The section relates generalized toric codes on narrow twisted tori to one-dimensional generalized bicycle codes, including a polynomial reduction and comparisons with existing constructions.

C. Relation to one-dimensional generalized bicycle codes

  • [[254, 14, 16]] is presented on a twisted 1 × 127 torus and reduced to a non-local one-dimensional quantum code.The reduction cancels the x-dependence using xy^25 − 1.Source: C. Relation to one-dimensional… · p. 12
  • The one-dimensional logical-dimension calculation reduces to a univariate-polynomial gcd and matches Proposition 1 in Ref..The resulting generalized bicycle codes are summarized in Tables V–VIII.Source: C. Relation to one-dimensional… · p. 12
  • Table V lists one-dimensional generalized bicycle codes induced from generalized toric codes on twisted 1 × n^2 tori; red [[n, k, d]] entries match optimal two-dimensional parameters.The table caption specifies two qubits per unit cell.Source: Table V · p. 13
  • A length-127 generalized bicycle code in Ref. [126] uses weight-10 stabilizers for [[254, 28, 14 ≤ d ≤ 20]].This is a comparison with the weight-6 [[254, 14, 16]] example reported in the section.Source: C. Relation to one-dimensional… · pp. 12–13

IV. DISCUSSION AND FUTURE DIRECTIONS

The discussion presents a ring-theoretic, anyon-based account of logical dimensions and reports generalized toric-code searches up to 400 physical qubits. It also identifies distance estimation and circuit-level evaluation as boundaries for future work.

IV. DISCUSSION AND FUTURE DIRECTIONS

  • The logical dimension k is obtained by counting independent anyon types, equivalent to the dimension of the quotient ring R/I for I = ⟨f(x, y), g(x, y)⟩.Gröbner-basis techniques support analysis of generalized toric codes up to n ≤ 400 physical qubits.Source: §IV · p. 13
  • For system sizes of a few hundred with d > 20, the probabilistic distance algorithm may be unreliable and yield only an upper bound.The authors identify distance computation as the primary bottleneck to searching larger systems.Source: §IV · p. 13
  • Treating π := xy as one variable expands the exponent interval to 0…n and substantially increases computational-resource requirements.The authors suggest comparing the resulting code parameters with those presented in this work.Source: §IV · pp. 13–14
  • Circuit-based error suppression, syndrome-measurement depth, physical realization, and logical-gate implementations remain to be investigated for the listed codes.The paper notes that circuit-level distance is typically smaller than nominal code distance.Source: §IV · p. 14

Appendix A: One-dimensional generalized bicycle codes

The appendix connects generalized toric codes on twisted 1 × n^2 tori with one-dimensional generalized bicycle codes and discusses their geometric representation.

Appendix A: One-dimensional generalized bicycle codes

  • Generalized toric codes on twisted 1 × n^2 tori reduce to one-dimensional quantum codes, specifically generalized bicycle codes.The induced codes are tabulated separately from the two-dimensional constructions.Source: Appendix A · p. 14
  • For these one-dimensional codes, the logical dimension can be obtained from Eq. (101), a special case of Theorem 3.The supplied passage identifies the theorem relation but does not state a numerical dimension.Source: Appendix A · p. 14
  • Twisted 1 × n^2 tori can have two-dimensional realizations with unit-cell aspect ratios close to one, despite appearing as long, narrow strips.The passage gives equivalent lattice-vector descriptions, including (0, 29), (1, 17) and (2, 5), (7, 3).Source: Appendix A · pp. 14–15
  • The illustrated twisted torus is equivalent to a rotated toric code, making lattice rotation a special case of the twisted-torus construction.The figure also depicts boundary identification and sequential qubit numbering.Source: Figure 4 · p. 15

Appendix B: Implementation of Pauli operators on twisted tori

The appendix defines a binary-vector convention for representing twisted-torus Pauli operators and constructing their X- and Z-type parity-check matrices.

Appendix B: Implementation of Pauli operators on twisted tori

  • The convention converts stabilizers f(x, y) and g(x, y) into n × n parity-check matrices HX and HZ for X- and Z-type checks.The matrices have a generalized cyclic structure.Source: Appendix B · p. 15
  • Qubits are numbered row by row, listing horizontal then vertical edges, with indices from 0 to n − 1 for n = 2αβ.The alternative torus representation makes this labeling systematic.Source: Appendix B · p. 15
  • A y-shift adds 2β per unit with wraparound, while x-shifts increment the index except at a boundary, where the lattice twist returns the edge inside.The x-shift rule applies the lattice vector (−β, −γ) at the boundary.Source: Appendix B · pp. 15–16
  • For the 36-qubit Kitaev toric-code example, each of HX and HZ has rank 17 because one stabilizer in each set is redundant.The construction translates origin stabilizers across the lattice to form the parity-check matrices.Source: Appendix B · pp. 16–17

Appendix C: Algorithm

Appendix C gives an algorithm for computing the logical dimension k from stabilizer polynomials using a Gröbner basis. The procedure handles codes on a twisted torus and constructs a matrix through a translation duplication map when the basis is not ⟨1⟩.

Appendix C: Algorithm

  • The algorithm computes logical dimension k from stabilizer polynomials using a Gröbner basis.It is presented as the pseudocode for the method stated in Theorem 3.Source: Appendix C · p. 18
  • For the twisted torus, it combines f(x, y), g(x, y), and the lattice-periodicity polynomials y^α − 1 and x^βy^γ − 1 before computing the Gröbner basis.The lattice vectors are a1 = (0, α) and a2 = (β, γ).Source: Appendix C · p. 18
  • If the resulting Gröbner basis is ⟨1⟩, the algorithm returns k = 0; otherwise, it counts independent monomials using degree bounds and a matrix built via the translation duplication map.The matrix construction uses the map TD_mx,my defined in Ref..Source: Appendix C · p. 18

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