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arXiv:2208.05353 · Quantum Information Processing

New quantum codes from CSS codes

Markus Grassl

Published Jan 23, 2023• 14 citations
arXivHTMLPDF

Abstract

Abstract We present a new propagation rule for CSS codes. Starting with a CSS code $$[\![n,k,d]\!]_q$$ [ [ n , k , d ] ] q , we construct a CSS code with parameters $$[\![n-2,k,d-1]\!]_q$$ [ [ n - 2 , k , d - 1 ] ] q . In general, one would only obtain a code with parameters $$[\![n-2,k,d-2]\!]_q$$ [ [ n - 2 , k , d - 2 ] ] q . The construction applies to asymmetric quantum codes from the CSS construction as well.

DOI
10.1007/s11128-023-03835-3

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Abstract We present a new propagation rule for CSS codes. Starting with a CSS code [ ⁣[n,k,d] ⁣]q[\![n,k,d]\!]_q[[n,k,d]]q​ [ [ n , k , d ] ] q , we construct a CSS code with parameters [ ⁣[n−2,k,d−1] ⁣]q[\![n-2,k,d-1]\!]_q[[n−2,k,d−1]]q​ [ [ n - 2 , k , d - 1 ] ] q . In general, one would only obtain a code with parameters [ ⁣[n−2,k,d−2] ⁣]q[\![n-2,k,d-2]\!]_q[[n−2,k,d−2]]q​ [ [ n - 2 , k , d - 2 ] ] q . The construction applies to asymmetric quantum codes from the CSS construction as well.
In this work, we derive achievability (upper) bounds for the logical error probability (LEP) of finite-length CSS codes for two asymmetric channels. In other words, we prove the existence of [[n, k]] CSS codes whose LEP does not exceed certain bounds that depend on the code and channel parameters. The bounds are derived with the help of finding the average LEPs of CSS codes from different code ensembles.The obtained bounds allow us to optimize code parameters for specific quantum channels. We show that such optimization leads to drastic performance improvements in asymmetric channels. We further show that expurgating bad codes from the ensemble of all CSS codes also leads to large performance improvements in certain channels.
We propose a new systematic construction of CSS-T codes from any given CSS code using a map ϕ. When ϕ is the identity map I, we retrieve the construction of hu2021mitigating and use it to prove the existence of asymptotically good binary CSS-T codes, resolving a previously open problem in the literature, and of asymptotically good quantum LDPC CSS-T codes. We analyze the structure of the logical operators corresponding to certain non-Clifford gates supported by the quantum codes obtained from this construction (ϕ=I), concluding that they always result in the logical identity. An immediate application of these codes in dealing with coherent noise is discussed. We then develop a new doubling transformation for obtaining triorthogonal codes, which generalizes the doubling construction presented in jain2024. Our approach permits using self-orthogonal codes, instead of only doubly-even codes, as building blocks for triorthogonal codes. This broadens the range of codes available for magic state distillation.
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.
We generalize the unified framework in [Phys. Rev. A 113, 022438 (2026)] to accommodate arbitrary stabilizer codes (referred as non-CSS for emphasis). The generalization has immediate consequences in areas including logical measurement, quantum weight reduction and Euclidean embedding. For example, currently CSS (pure X,ZX,ZX,Z-type) logical measurements are well-understood based on the (height-1) cone, while non-CSS (e.g., YYY-type) logical measurements are ad hoc (e.g., rely on local Clifford transform). Similarly, quantum weight reduction and optimal Euclidean embeddings only exist for CSS input codes. Here, we show that our generalized framework addresses this issue, and thus many constructions, including qLDPC surgery, Layer Codes and quantum weight reduction generalize to non-CSS codes in a relatively straightforward fashion. The previously mentioned examples are derived in detail for clarity.
Using the Calderbank-Shor-Steane (CSS) construction, pure qqq-ary asymmetric quantum error-correcting codes attaining the quantum Singleton bound are constructed. Such codes are called pure CSS asymmetric quantum maximum distance separable (AQMDS) codes. Assuming the validity of the classical MDS Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for.