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IEEE Transactions on Information Theory

Quantum codes [[6, 2, 3]]/sub p/ and [[7, 3, 3]]/sub p/ (p ≥ 3) exist

Keqin Feng

Published Aug 1, 2002• 51 citations
Publisher

Abstract

We prove that nonbinary quantum stabilizer codes with parameters [[n, k, d]]/sub p/ = [[6, 2, 3]]/sub p/ and [[7, 3, 3]]/sub p/ exist for all odd primes p by using graph machinery, given by Schlingemann and Werner(see Phys. Rev. A, vol.65, no.012308, 2001), with a little number theory and combinatorics.
DOI
10.1109/tit.2002.800469

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In this paper, we present a systematic way based on the nonbinary graph state of constructing good nonbinary quantum codes, both additive and nonadditive, for systems with integer dimensions. With a computer search, which results in many interesting codes including some nonadditive codes meeting the Singleton bounds, we are able to construct explicitly four families of optimal codes, namely, [[6,2,3]]p{[[6,2,3]]}_{p}[[6,2,3]]p​, [[7,3,3]]p{[[7,3,3]]}_{p}[[7,3,3]]p​, [[8,2,4]]p{[[8,2,4]]}_{p}[[8,2,4]]p​, and [[8,4,3]]p{[[8,4,3]]}_{p}[[8,4,3]]p​ for any odd dimension p\mathit{p}p and a family of nonadditive codes ((5,p,3))p{((5,p,3))}_{p}((5,p,3))p​ for arbitrary p>3p>3p>3. In the case of composite numbers as dimensions, we also construct a family of stabilizer codes ((6,2p2,3))2p{((6,2{p}^{2},3))}_{2p}((6,2p2,3))2p​ for odd p\mathit{p}p, whose coding subspace is not of a dimension that is a power of the dimension of the physical subsystem.
We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes are verified in graph-theoretic terms as originally advocated by Schlingemann and Werner. Finally, possible generalizations of our scheme for the graphical construction of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum codes are briefly addressed.