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arXiv:1602.00413 · IEEE Transactions on Information Theory

Linear Programming Bounds for Entanglement-Assisted Quantum Error-Correcting Codes by Split Weight Enumerators

Ching–Yi Lai, Alexei Ashikhmin

Published Jun 2, 2017• 103 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/tit.2017.2711601

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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.
Abstract—The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting from ex-changing the original code’s information qubits for its ebits and vice versa. As an introduction to this notion, we show how entanglement-assisted repetition codes and entanglement-assisted accumulator codes are dual to each other, much like their classical counterparts, and we give an explicit, general quantum shift-register circuit that encodes both classes of codes. We later show that our constructions are optimal, and this result completes our understanding of these dual classes of codes. We obtain linear programming bounds for EAQEC codes, by exploiting general forms of these dualities and corresponding MacWilliams identities. We establish the Gilbert-Varshamov bound and the Plotkin bound for EAQEC codes, and all of these bounds allow us to formulate a table of upper and lower bounds on the minimum distance of any maximal-entanglement EAQEC code with length up to 15 channel qubits. Finally, we provide an upper bound on the block error probability when transmitting maximal-entanglement EAQEC codes through the depolarizing channel. Index Terms—quantum dual code, entanglement-assisted quantum error correction, MacWilliams identity, linear programming bound, entanglement-assisted repetition codes, entanglement-assisted accumulator codes I.
Quantum computers need to be protected by quantum error-correcting codes against decoherence. One of the most interesting and useful classes of quantum codes is the class of quantum stabilizer codes. Entanglement-assisted (EA) quantum codes are a class of stabilizer codes that make use of preshared entanglement between the sender and the receiver. We provide several code constructions for entanglement-assisted quantum codes. The MacWilliams identity for quantum codes leads to linear programming bounds on the minimum distance. We find new constraints on the simplified stabilizer group and the logical group, which help improve the linear programming bounds on entanglement-assisted quantum codes. The results also can be applied to standard stabilizer codes. In the real world, quantum gates are faulty. To implement quantum computation fault-tolerantly, quantum codes with certain properties are needed. We first analyze Knill's postselection scheme in a two-dimensional architecture. The error performance of this scheme is better than other known concatenated codes. Then we propose several methods to protect syndrome extraction against measurement errors.
The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting from exchanging the original code’s information qubits for its ebits and vice versa. As an introduction to this notion, we show how entanglement-assisted repetition codes and entanglementassisted accumulator codes are dual to each other, much like their classical counterparts, and we give an explicit, general quantum shift-register circuit that encodes both classes of codes. We later show that our constructions are optimal, and this result completes our understanding of these dual classes of codes. We obtain linear programming bounds for non-degenerate EAQEC codes, by exploiting general forms of these dualities and corresponding MacWilliams identities. We establish the GilbertVarshamov bound and the Plotkin bound for EAQEC codes, and all of these bounds allow us to formulate a table of upper and lower bounds on the minimum distance of any maximalentanglement EAQEC code with length up to 15 channel qubits. Finally, we provide an upper bound on the block error probability when transmitting maximal-entanglement EAQEC codes through the depolarizing channel.
The entanglement-assisted (EA) formalism is a generalization of the standard stabilizer formalism, and it can transform arbitrary quaternary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using of shared entanglement between the sender and the receiver. Using the special structure of linear EAQECCs, we derive an EA-Plotkin bound for linear EAQECCs, which strengthens the previous known EA-Plotkin bound. This linear EA-Plotkin bound is tighter then the EA-Singleton bound, and matches the EA-Hamming bound and the EA-linear programming bound in some cases. We also construct three families of EAQECCs with good parameters. Some of these EAQECCs saturate this linear EA-Plotkin bound and the others are near optimal according to this bound; almost all of these linear EAQECCs are degenerate codes.