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IEEE Transactions on Information Theory

Upper bounds on the size of quantum codes

Alexei Ashikhmin, S. Litsyu

Published May 1, 1999• 76 citations
Publisher

Abstract

This paper is concerned with bounds for quantum error-correcting codes. Using the quantum MacWilliams (1972, 1977) identities, we generalize the linear programming approach from classical coding theory to the quantum case. Using this approach, we obtain Singleton-type, Hamming-type, and the first linear-programming-type bounds for quantum codes. Using the special structure of linear quantum codes, we derive an upper bound that is better than both Hamming and the first linear programming bounds on some subinterval of rates.

DOI
10.1109/18.761270

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This paper is concerned with bounds for quantum error-correcting codes. Using the quantum MacWilliams (1972, 1977) identities, we generalize the linear programming approach from classical coding theory to the quantum case. Using this approach, we obtain Singleton-type, Hamming-type, and the first linear-programming-type bounds for quantum codes. Using the special structure of linear quantum codes, we derive an upper bound that is better than both Hamming and the first linear programming bounds on some subinterval of rates.
Linear programming approaches have been applied to derive upper bounds on the size of classical and quantum codes. In this paper, we derive similar results for general quantum codes with entanglement assistance by considering a type of split weight enumerator. After deriving the MacWilliams identities for these enumerators, we are able to prove algebraic linear programming bounds, such as the Singleton bound, the Hamming bound, and the first linear programming bound. Our Singleton bound and Hamming bound are more general than the previous bounds for entanglement-assisted quantum stabilizer codes. In addition, we show that the first linear programming bound improves the Hamming bound when the relative distance is sufficiently large. On the other hand, we obtain additional constraints on the size of Pauli subgroups for quantum codes, which allow us to improve the linear programming bounds on the minimum distance of quantum codes of small length. In particular, we show that there is no [[27, 15, 5]] or [[28, 14, 6]] stabilizer code. We also discuss the existence of some entanglement-assisted quantum stabilizer codes with maximal entanglement. As a result, the upper and lower bounds on the minimum distance of maximal-entanglement quantum stabilizer codes with length up to 20 are significantly improved.
In this paper, we define two split weight enumerators for general quantum codes with entanglement assistance, including nonadditive codes. We show that they obey a MacWilliams identity, which allows us to prove algebraic linear programming bounds, such as the Singleton bound, the Hamming bound, and the first linear programming bound. On the other hand, we derive additional constraints on the size of Pauli subgroups for quantum codes, which helps to improve the linear programming bounds on the minimum distance of quantum codes of small length.
In this final chapter we present a number of bounds on the minimum dis­ tance of a quantum error correcting code (QECC). Sections 7.1-7.3 derive the quantum Hamming, Gilbert-Varshamov, and Singleton bounds, respectively. The corresponding classical bounds were derived in Section 1.2.5. Section 7.4 introduces quantum generalizations of the classical MacWilliams identities, and of weight and shadow enumerators [1,2]. With these tools in hand, the problem of finding an optimal upper bound on the code distance is reduced to solving a problem in linear programming [3-5]. Finally, we close in Section 7.5 by showing how QECCs can be connected to the process of entanglement pu­ rification [6,7]. This final section allows our study of QECCs to segue into the developing field of quantum information theory [8,9], and provides a good place to bring the discussion in this book to a close.
The quantum Hamming bound arises from a direct dimension estimate for exact quantum error-correcting codes. For nondegenerate codes, distinct correctable errors map the code space into orthogonal subspaces, so dimension counting directly gives the bound; degeneracy allows these subspaces to overlap and invalidates the argument. Although linear-programming, asymptotic, and structural approaches have advanced the problem from several directions, the general local-dimension case has remained open. Here we prove the quantum Hamming bound for every local dimension q≥3q\ge3q≥3. Building on the exact two-center Fourier reduction, we develop new overlap estimates for nonbinary Hamming spaces. For q≥4q\ge4q≥4, Jacobi estimates and monotonicity give a uniform gap; qutrits form the boundary case where the gap vanishes and are handled by a filtered correction. This completes the finite-length nonbinary case; together with the existing qubit result, it establishes the quantum Hamming bound for arbitrary local dimension.
In 1997, Shor and Laflamme defined the weight enumerators for quantum error-correcting codes and derived a MacWilliams identity. We extend their work by introducing our double weight enumerators and complete weight enumerators for qubit codes and then investigate the MacWilliams identities for these enumerators. Based on the generalized MacWilliams identities, we solve an open problem, namely, the Singleton-type bound for asymmetric quantum codes (AQCs). Besides, the Hamming-type and the first linear-programming-type bounds for the AQCs are deduced similarly.