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arXiv:2503.03827 · PRX Quantum

Generalized Toric Codes on Twisted Tori for Quantum Error Correction

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

Published May 13, 2025• 8 citations
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Abstract

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 Gröbner 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 ≤ 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 , ≤ 22 ] ] , and [ [ 340 , 16 , 18 ] ] . Moreover, we present a new realization of the [ [ 360 , 12 , ≤ 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.

DOI
10.1103/rmy6-9n89

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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 Gröbner 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 ≤ 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 , ≤ 22 ] ] , and [ [ 340 , 16 , 18 ] ] . Moreover, we present a new realization of the [ [ 360 , 12 , ≤ 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.
We study finite-length qudit quantum low-density parity-check (LDPC) codes from translation-invariant CSS constructions on two-dimensional tori with twisted boundary conditions. Recent qubit work [PRX Quantum 6, 020357 (2025)] showed that, within the bivariate-bicycle viewpoint, twisting generalized toric patterns can significantly improve the finite-size distance of qubit codes. Building on this insight, we extend the construction and the search to qudit codes over finite fields. Using algebraic methods based on Laurent-polynomial ideals and Gröbner-basis computations over Fq\mathbb{F}_qFq​, we compute the number of logical qudits on any twisted torus and identify compact codes with favorable rate--distance tradeoffs. Overall, for the finite sizes explored, twisted-torus qudit constructions typically achieve larger estimated distances than their untwisted counterparts and, at the same number of physical and logical qudits, larger estimated distances than previously reported twisted qubit instances. The best new codes are tabulated.
Generalized Zp\mathbb{Z}_pZp​ toric codes as qudit low-density parity-check codes
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Quantum low-density parity-check codes, such as the Kitaev toric code and bivariate bicycle codes, are often defined with periodic boundary conditions, which are difficult to realize in physical systems. In this paper, we present an algorithm for constructing all gapped boundaries and defects of two-dimensional Pauli stabilizer codes. Using the operator algebra formalism, we establish a one-to-one correspondence between the topological data, such as anyon fusion rules and topological spins, of two-dimensional bulk stabilizer codes and one-dimensional boundary anomalous subsystem codes. To make the operator algebra computationally accessible, we adapt Laurent polynomials and convert the tasks into matrix operations, e.g., the Hermite normal form for obtaining boundary anyons and the Smith normal form for determining fusion rules. This approach enables computers to automatically generate all possible gapped boundaries and defects for topological Pauli stabilizer codes through boundary anyon condensation and topological order completion. This streamlines the analysis of surface codes and associated logical operations for fault-tolerant quantum computation. Our algorithm applies to Zd\mathbb{Z}_dZd​ qudits for both prime and nonprime ddd, enabling exploration of topological phases beyond the Kitaev toric code. We have applied the algorithm and explicitly demonstrated the lattice constructions of 2 boundaries and 6 defects in the Z2\mathbb{Z}_2Z2​ toric code, 3 boundaries and 22 defects in the Z4\mathbb{Z}_4Z4​ toric code, 1 boundary and 2 defects in the double semion code, 1 boundary and 22 defects in the six-semion code, 6 boundaries and 270 defects in the color code, and 6 defects in the anomalous three-fermion code. Finally, we study the boundaries of bivariate bicycle codes, showing that they exhibit large logical dimensions and anyons with long translation periods.
Symmetry-enriched topological order and quasifractonic behavior in ZN\mathbb{Z}_NZN​ stabilizer codes
We study a broad class of qudit stabilizer codes, termed ZN\mathbb{Z}_NZN​ bivariate-bicycle (BB) codes, arising either as two-dimensional realizations of modulated gauge theories or as ZN\mathbb{Z}_NZN​ generalizations of binary BB codes. Our central finding, derived from the polynomial representation, is that the essential topological properties of these ZN\mathbb{Z}_NZN​ codes can be determined by the properties of their Zp\mathbb{Z}_pZp​ counterparts, where ppp are the prime factors of NNN, even when NNN contains prime powers (N=∏ipikiN = \prod_i p_i^{k_i}N=∏i​piki​​). This result yields a significant simplification by leveraging the well-studied framework of codes with prime qudit dimensions. In particular, this insight directly enables the generalization of the algebraic-geometric methods (e.g., the Bernstein-Khovanskii-Kushnirenko theorem) to determine anyon fusion rules in the general qudit situation. Moreover, we elucidate the symmetry-enriched topological (SET) order underlying the quasifractonic behavior in qudit BB codes (including the Delfino-Chamon-You model), resolving the associated anyon mobility puzzle. We also develop an efficient computational algebraic method, based on Gröbner bases over the ring of integers, to determine both the topological order and its SET properties.
Quantum low-density parity-check (QLDPC) codes offer a promising path to low-overhead fault-tolerant quantum computation but lack systematic strategies for exploration. In this Letter, we establish a topological framework for studying the bivariate-bicycle codes, a prominent class of QLDPC codes tailored for real-world quantum hardware. Our framework enables the investigation of these codes through universal properties of topological orders. In addition to efficient characterizations using Gröbner bases, we also introduce a novel algebraic-geometric approach based on the Bernstein-Khovanskii-Kushnirenko theorem. This approach allows us to analytically determine how the topological order varies with the generic choices of bivariate-bicycle codes under toric layouts. Novel phenomena are unveiled, including topological frustration, where ground-state degeneracy on a torus deviates from the total anyon number, and quasifractonic mobility, where anyon movement violates energy conservation. We demonstrate their intrinsic link to symmetry-enriched topological orders and derive an efficient method for generating finite-size codes. Furthermore, we extend the connection between anyons and logical operators using Koszul complex theory. Our Letter provides a rigorous theoretical basis for exploring the fault tolerance of QLDPC codes and deepens the interplay among topological order, quantum error correction, and advanced algebraic structures.
Quantum low-density parity-check (qLDPC) codes promise low-overhead fault-tolerant quantum memory, but many leading constructions are naturally defined on periodic lattices, whereas scalable quantum hardware favors planar architectures with open boundaries. In this work, we develop a general operator-algebraic framework and computational algorithms for constructing all gapped boundaries and defects of two-dimensional topological Pauli stabilizer codes. Starting from a truncated bulk code, our method identifies boundary gauge operators, boundary anyons, and their string operators, and then constructs microscopic boundary Hamiltonians through anyon condensation and topological-order completion, thereby eliminating local logical operators near the boundary. We establish a one-to-one bulk-boundary correspondence between bulk anyons and excitations of the anomalous boundary subsystem code, enabling fusion rules, topological spins, and braiding statistics to be extracted directly from the boundary operator algebra. For translation-invariant codes, Laurent-polynomial representations reduce these tasks to finite matrix computations based on Hermite and Smith normal forms, enabling automated constructions for Z d qudits of both prime and composite dimension. We demonstrate the framework on toric, color, double-semion, six-semion, and anomalous three-fermion codes, and apply it to bivariate bicycle codes, revealing large logical dimensions and unusually long translation periods for anyon strings. By systematically converting bulk topological data into explicit boundary and defect Hamiltonians, our framework can serve as a foundation for constructing low-overhead planar qLDPC codes with open boundaries and for engineering fault-tolerant logical operations.