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arXiv:2012.05896 · International Journal of Quantum Information

Encoding classical information in gauge subsystems of quantum codes

Andrew Nemec, Andreas Klappenecker

Published Jan 12, 2022• 2 citations
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Abstract

In this paper, we show how to construct hybrid quantum-classical codes from subsystem codes by encoding the classical information into the gauge qudits using gauge fixing. Unlike previous work on hybrid codes, we allow for two separate minimum distances, one for the quantum information and one for the classical information. We give an explicit construction of hybrid codes from two classical linear codes using Bacon–Casaccino subsystem codes, as well as several new examples of good hybrid code.

DOI
10.1142/s0219749921500416

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In this paper, we show how to construct hybrid quantum-classical codes from subsystem codes by encoding the classical information into the gauge qudits using gauge fixing. Unlike previous work on hybrid codes, we allow for two separate minimum distances, one for the quantum information and one for the classical information. We give an explicit construction of hybrid codes from two classical linear codes using Bacon–Casaccino subsystem codes, as well as several new examples of good hybrid code.
Quantum synchronizable codes are quantum error correcting codes that can correct not only Pauli errors but also errors in block synchronization. The code can be constructed from two classical cyclic codesC\mathcal {C}C,D\mathcal {D}DsatisfyingC⊥⊂C⊂D\mathcal {C}^{\perp } \subset \mathcal {C} \subset \mathcal {D}C⊥⊂C⊂Dthrough the Calderbank-Shor-Steane (CSS) code construction. In this work, we establish connections between quantum synchronizable codes, subsystem codes, and hybrid quantum-classical codes constructed from the same pair of classical cyclic codes. We also propose a method to construct a synchronizable hybrid subsystem code which can correct both Pauli and synchronization errors, is resilient to gauge errors by virtue of the subsystem structure, and can transmit both classical and quantum information, all at the same time. The trade-offs between the number of synchronization errors that the code can correct, the number of gauge qubits, and the number of logical classical bits of the code are also established. In addition, we propose general methods to construct hybrid and hybrid subsystem codes of CSS type from classical codes, which cover relevant codes from our main construction.
Hybrid codes simultaneously encode both quantum and classical information into physical qubits. We give several general results about hybrid codes, most notably that the quantum codes comprising a genuine hybrid code must be impure and that hybrid codes can always detect more errors than comparable quantum codes. We also introduce the weight enumerators for general hybrid codes, which we then use to derive linear programming bounds. Finally, inspired by the construction of some families of nonadditive codes, we construct several infinite families of genuine hybrid codes with minimum distance two and three.
We consider the characterization as well as the construction of quantum codes that allow to transmit both quantum and classical information, which we refer to as `hybrid codes'. We construct hybrid codes [![n,k:m,d]!]q[\\![n,k{: }m,d]\\!]_q[![n,k:m,d]!]q​ with length nnn and distance ddd, that simultaneously transmit kkk qudits and mmm symbols from a classical alphabet of size qqq. Many good codes such as [![7,1:1,3]!]2[\\![7,1{: }1,3]\\!]_2[![7,1:1,3]!]2​, [![9,2:2,3]!]2[\\![9,2{: }2,3]\\!]_2[![9,2:2,3]!]2​, [![10,3:2,3]!]2[\\![10,3{: }2,3]\\!]_2[![10,3:2,3]!]2​, [![11,4:2,3]!]2[\\![11,4{: }2,3]\\!]_2[![11,4:2,3]!]2​, [![11,1:2,4]!]2[\\![11,1{: }2,4]\\!]_2[![11,1:2,4]!]2​, [![13,1:4,4]!]2[\\![13,1{: }4,4]\\!]_2[![13,1:4,4]!]2​, [![13,1:1,5]!]2[\\![13,1{: }1,5]\\!]_2[![13,1:1,5]!]2​, [![14,1:2,5]!]2[\\![14,1{: }2,5]\\!]_2[![14,1:2,5]!]2​, [![15,1:3,5]!]2[\\![15,1{: }3,5]\\!]_2[![15,1:3,5]!]2​, [![19,9:1,4]!]2[\\![19,9{: }1,4]\\!]_2[![19,9:1,4]!]2​, [![20,9:2,4]!]2[\\![20,9{: }2,4]\\!]_2[![20,9:2,4]!]2​, [![21,9:3,4]!]2[\\![21,9{: }3,4]\\!]_2[![21,9:3,4]!]2​, [![22,9:4,4]!]2[\\![22,9{: }4,4]\\!]_2[![22,9:4,4]!]2​ have been found. All these codes have better parameters than hybrid codes obtained from the best known stabilizer quantum codes.
We consider the characterization as well as the construction of quantum codes that allow to transmit both quantum and classical information, which we refer to as `hybrid codes'. We construct hybrid codes [n, k:m, d]qwith length n and distance d, that simultaneously transmit k qudits and m symbols from a classical alphabet of size q. Many good codes such as [7,1:1, 3]2, [9,2:2,3]2, [10, 3:2, 3]2, [11,4:2, 3]2, [11,1:2,4]2, [13,1:4,4]2, [13,1:1, 5]2, [14,1:2, 5]2, [15,1:3, 5]2, [19, 9:1,4]2, [20, 9:2,4]2, [21, 9:3, 4]2, [22, 9:4,4]2have been found. All these codes have better parameters than hybrid codes obtained from the best known stabilizer quantum codes.
Clifford codes are a natural generalization of quantum stabilizer codes based primarily on representation theory. This class of codes has previously been extended to the setting of quantum subsystem codes. We formulate a two-fold generalization of Clifford codes, for both the hybrid classical and quantum information and projective representation theory settings. This leads to new classes of hybrid subspace and subsystem Clifford codes. We extend the fundamental representation theoretic quantum error correction theorem to include these codes, based on the operator algebra quantum error correction framework. We also discuss several examples throughout the presentation, of both stabilizer and non-stabilizer type.
The essential insight of quantum error correction was that quantum information can be protected by suitably encoding this quantum information across multiple independently erred quantum systems. Recently it was realized that, since the most general method for encoding quantum information is to encode it into a subsystem, there exists a novel form of quantum error correction beyond the traditional quantum error correcting subspace codes. These new quantum error correcting subsystem codes differ from subspace codes in that their quantum correcting routines can be considerably simpler than related subspace codes. Here we present a class of quantum error correcting subsystem codes constructed from two classical linear codes. These codes are the subsystem versions of the quantum error correcting subspace codes which are generalizations of Shor's original quantum error correcting subspace codes. For every Shor-type code, the codes we present give a considerable savings in the number of stabilizer measurements needed in their error recovery routines.