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arXiv:1701.06963 · IEEE International Symposium on Information Theory (ISIT)

Codes for simultaneous transmission of quantum and classical information

Markus Grassl, Sirui Lu, Bei Zeng

Published Jun 1, 2017• 18 citations
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Abstract

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.

DOI
10.1109/isit.2017.8006823

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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.
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.
A hybrid code can simultaneously encode classical and quantum information into quantum digits such that the information is protected against errors when transmitted through a quantum channel. It is shown that a hybrid code has the remarkable feature that it can detect more errors than a comparable quantum code that is able to encode the classical and quantum information. Weight enumerators are introduced for hybrid codes that allow to characterize the minimum distance of hybrid codes. Surprisingly, the weight enumerators for hybrid codes do not obey the usual MacWilliams identity.
In this paper, we prove how to extend a subset of quantum stabilizer codes into a qudit hybrid code storing log⁡2p\log_2 plog2​p classical bits over a qudit space with dimension ppp, with ppp prime. Our proof also gives an explicit procedure for finding the entire collection of stabilizer algebras for all of the subcodes of the hybrid code. This allows extra classical information to be transmitted without having to arduously search for additional codes and their associated codewords, and also provides a first lower bound to the amount of classical information able to be transmitted in a qudit hybrid code, but unfortunately only allows for log⁡2p\log_2 plog2​p classical bits to be decoded by a receiver.
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.