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arXiv:2404.18302 · IEEE International Conference on Quantum Computing and Engineering (QCE)

GNarsil: Splitting Stabilizers into Gauges

Oskar Novak, Narayanan Rengaswamy

Published Sep 15, 2024• 0 citations
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Abstract

Quantum subsystem codes have been shown to improve error-correction performance, ease the implementation of logical operations on codes, and make stabilizer measurements easier by decomposing stabilizers into smaller-weight gauge operators. In this paper, we present two algorithms that produce new subsystem codes from a “seed” CSS code. They replace some stabilizers of a given CSS code with smaller-weight gauge operators that split the remaining stabilizers, while being compatible with the logical Pauli operators of the code. The algorithms recover the well-known Bacon-Shor code computationally as well as produce a new [9, 1, 2, 2] rotated surface subsystem code with weight-3 gauges and weight-4 stabilizers. We illustrate using a [100, 25, 3] subsystem hypergraph product (SHP) code that the algorithms can produce more efficient gauge operators than the closed-form expressions of the SHP construction. However, we observe that the stabilizers of the lifted product quantum LDPC codes are more challenging to split into small-weight gauge operators. Hence, we introduce the subsystem lifted product (SLP) code construction and develop a new [775, 124, 20] code from Tanner's classical quasi-cyclic LDPC code. The code has high-weight stabilizers but all gauge operators that split stabilizers have weight 5, except one. In contrast, the LP stabilizer code from Tanner's code has parameters [1054, 124, 20]. This serves as a novel example of new subsystem codes that outperform stabilizer versions of them. Finally, based on our experiments, we share some general insights about non-locality's effects on the performance of splitting stabilizers into small-weight gauges.

DOI
10.1109/qce60285.2024.00022

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•Nov 28, 2025•0 citations
Quantum subsystem codes have been shown to improve error-correction performance, ease the implementation of logical operations on codes, and make stabilizer measurements easier by decomposing stabilizers into smaller-weight gauge operators. In this paper, we present two algorithms that produce new subsystem codes from a “seed” CSS code. They replace some stabilizers of a given CSS code with smaller-weight gauge operators that split the remaining stabilizers, while being compatible with the logical Pauli operators of the code. The algorithms recover the well-known Bacon-Shor code computationally as well as produce a new [9, 1, 2, 2] rotated surface subsystem code with weight-3 gauges and weight-4 stabilizers. We illustrate using a [100, 25, 3] subsystem hypergraph product (SHP) code that the algorithms can produce more efficient gauge operators than the closed-form expressions of the SHP construction. However, we observe that the stabilizers of the lifted product quantum LDPC codes are more challenging to split into small-weight gauge operators. Hence, we introduce the subsystem lifted product (SLP) code construction and develop a new [775, 124, 20] code from Tanner's classical quasi-cyclic LDPC code. The code has high-weight stabilizers but all gauge operators that split stabilizers have weight 5, except one. In contrast, the LP stabilizer code from Tanner's code has parameters [1054, 124, 20]. This serves as a novel example of new subsystem codes that outperform stabilizer versions of them. Finally, based on our experiments, we share some general insights about non-locality's effects on the performance of splitting stabilizers into small-weight gauges.
We provide a numerical investigation of two families of subsystem quantum codes that are related to hypergraph product codes by gauge-fixing. The first family consists of the Bravyi-Bacon-Shor (BBS) codes which have optimal code parameters for subsystem quantum codes local in 2-dimensions. The second family consists of the constant rate “generalized Shor” codes of Bacon and Cassicino [1], which we re-brand as subsystem hypergraph product (SHP) codes. We show that any hypergraph product code can be obtained by entangling the gauge qubits of two SHP codes. To evaluate the performance of these codes, we simulate both small and large examples. For circuit noise, a [21], [4], [3] BBS code and a [49], [16], [3] SHP code have pseudthresholds of 2 × 10-3and 8 × 10-4, respectively. Simulations for phenomenological noise show that large BBS and SHP codes start to outperform surface codes with similar encoding rate at physical error rates 1 × 10-6and 4 × 10-4, respectively.
High-rate bivariate bicycle (BB) codes are promising low-overhead quantum memories, but their stabilizer checks typically have weight 666 or higher, making syndrome extraction challenging. We introduce subsystem bivariate bicycle (SBB) codes, a translation-invariant CSS subsystem construction that realizes BB-code logical structure using local weight-444 gauge measurements. Their stabilizer syndromes are inferred by multiplying the corresponding gauge outcomes. We further show that nonlocal stabilizers in translation-invariant CSS subsystem codes can be detected using a determinantal-ideal criterion based on the gauge-operator commutation matrix. When this criterion excludes nonlocal stabilizers, a finite-depth Clifford circuit decouples gauge qubits and identifies the protected subsystem with a corresponding BB stabilizer code. An SBB code is topological, meaning that it has no nontrivial local logical operators, if and only if the corresponding BB code is topological. A finite search yields low-overhead examples including [[27,6,3]][[27,6,3]][[27,6,3]], [[75,10,5]][[75,10,5]][[75,10,5]], and [[108,12,6]][[108,12,6]][[108,12,6]]; the latter encodes six times more logical qubits than a subsystem surface code at the same block length and distance. These results show how gauge degrees of freedom can make high-rate BB logical structure compatible with local weight-444 syndrome extraction.
Quantum error correction codes are needed to protect quantum systems from interference. An important family of quantum error correction codes are stabilizer codes. Many stabilizer codes in the literature belong to a subclass of stabilizer codes, referred to as CSS codes since they are based on the so-called CSS formalism. CSS codes have received significant attention because the dual-containing condition for the CSS formalism is relatively straightforward to satisfy and also because it is easy to produce CSS codes from classical error correction codes. However, some stabilizer codes in the literature show superior error correction performance than existing CSS codes, and are based on binary low-density parity-check (LDPC) codes. Since it has been shown that CSS codes based on nonbinary LDPC codes have good performance over quantum qubit channels, in this paper we investigate stabilizer codes based on nonbinary LDPC codes. In particular, we propose constructions of stabilizer codes based on nonbinary quasi-cyclic low-density parity-check (QC-LDPC) codes for qubit channels. The simulation results show that our QC-LDPC stabilizer codes decoded by a nonbinary sum-product algorithm have better performance than their binary counterparts.
The CSS code construction is a powerful framework used to express features of a quantum code in terms of a pair of underlying classical codes. Its subsystem extension allows for similar expressions, but the general case has not been fully explored. Extending previous work of Aly, Klappenecker, and Sarvepalli \cite{AKS06}, we determine subsystem CSS code parameters, express codewords, and develop a Steane-type decoder using only data from the two underlying classical codes. Generalizing a result of Kovalev and Pryadko \cite{KP13}, we show that any subsystem stabilizer code can be "doubled" to yield a subsystem CSS code with twice the number of physical, logical, and gauge qudits and up to twice the code distance. This mapping preserves locality and is tighter than the Majorana-based mapping of Bravyi, Terhal, and Leemhuis \cite{BTL10}. Using Goursat's Lemma, we show that every subsystem stabilizer code can be constructed from two nested subsystem CSS codes satisfying certain constraints, and we characterize subsystem stabilizer codes based on the nested codes' properties.
A Bacon-Shor code is a subsystem quantum error-correcting code on an $L\ifmmode\times\else\texttimes\fi{}L$ lattice where the $2(L\ensuremath{-}1)$ weight-2L2L2L stabilizers are usually inferred from the measurements of $2L(L\ensuremath{-}1)$ weight-2 gauge operators. Here, we show that the stabilizers can be measured directly and fault tolerantly with bare ancillary qubits by constructing circuits that follow the pattern of gauge operators. We then examine the implications of this method for small quantum error-correcting codes by comparing distance-3 versions of the rotated surface code and the Bacon-Shor code with the standard depolarizing model and in the context of a trapped-ion quantum computer. We find that for a simple circuit of prepare, error correct, and measure, the Bacon-Shor code outperforms the surface code by requiring fewer qubits, taking less time, and having a lower error rate.
In classical coding theory, the lift of a code is a practical and natural method for generatingfamilies of LDPC codes from a single chosen input code. This strategy has proven to bevery successful in producing LDPC codes of various blocklengths, reaching good parametersand approaching the capacity of certain channels in the asymptotic regime. It is also nowapplied in the 5G New Radio [RK18].It is widely believed that a quantum computer will integrate a quantum error correctingscheme, and the last three decades have seen the emergence of a variety of code-basedquantum error correction known as CSS codes. It is natural to ask whether we can performlifts of quantum codes, with the hope of having the same benefits as for classical codes.However, the method to lift linear codes cannot be directly applied to quantum CSS codes,as the constituent linear codes usually lose their orthogonality. This dissertation presentsthree new results on lifts of quantum CSS codes.Firstly, we propose a general notion of lift for quantum CSS codes. It is based on theexistence of a canonical complex associated to any CSS code, that we call the Tanner cone-complex, and over which we generate covering spaces. We show that our notion generalizesclassical code lifting and generate important CSS code families. Our main application ofthe lift is to devise new types of quantum CSS codes. So far, most seriously consideredconstructions rely on topology or some kind of product operations. Here, we intend toexplore beyond these paradigmatic ideas and for that we construct new non-topologicaland non-product CSS codes. We then apply the prescription to generate their lifts andshow that this procedure can improve their relative parameters, reaching parameter regionsnot explored prior to this work.Secondly, we introduce new families of quantum Tanner codes, a class of quantum codeswhich first appeared in the work of Leverrier and Zémor [LZ22]. We build these codesby generalizing the lift of classical Tanner codes, based on graph coverings, to the case ofCSS codes, based on coverings of 2D geometrical complexes. We present several explicitfamilies, and identify instances of moderate length quantum codes which are degenerate,have low check weight, and for which the distance is greater than the square root of thecode length.Lastly, we present a topological approach to lifting a quantum CSS code. Our originaldefinition of CSS code lifting was inspired by the work of Freedman and Hastings [FH21],which associates CSS codes with handlebodies. This thesis is a good opportunity to showhow the handlebody realization of a code can also be used to perform code lifting. Weprovide a detailed discussion of why in many cases this is essentially equivalent to ourprevious approach.