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Optimal Quaternary Hermitian Linear Complementary Dual Codes for Entanglement-Assisted Quantum Error Correction

Maysara Al Jumaily

Published Nov 16, 2021• 0 citations

Abstract

The objective of this thesis is to find suboptimal and optimal parameters from classical codes and import them into entanglement-assisted quantum codes. The thesis begins by introducing classical error correction, followed by a detailed introduction to quantum computing. Topics that are discussed in the introduction include qubits, quantum phenomena, such as superposition and entanglement, and quantum gates/circuits. The thesis then reviews the basics of quantum error correction and provides Shor's code to reinforce the reader's understanding. Subsequently, the formalism of stabilizer codes is thoroughly examined. We then explain the generalized concept of stabilizer codes which is entanglement-assisted quantum codes. They do not require generators to satisfy the commutativity property. Rather, they utilize the usage of ebits to resolve the anti-commutativity constraint. Next, the thesis explains quaternary field and then the Java program implemented to find the optimal parameters. Lastly, the thesis concludes with presenting the parameters of the new codes that were obtained throughout the research. We have found the suboptimal largest distance for quaternary hermitian linear complementary dual codes that can be imported as entanglement-assisted quantum error correction for parameters [22, 9, 9 or 10]₄, [22, 12, 7 or 8]₄, [23, 8, 11 or 12]₄, [23, 10, 9 or 10]₄, [23, 13, 7 or 8]₄, [24, 10, 10 or 11]₄, [24, 11, 9 or 10]₄, [24, 14, 7 or 8]₄, [25, 12, 9 or 10]₄, [25, 13, 8 or 9]₄, as well as the optimal largest distance for [17, 11, 5]₄ and [17, 13, 3]₄.

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The objective of this thesis is to find suboptimal and optimal parameters from classical codes and import them into entanglement-assisted quantum codes. The thesis begins by introducing classical error correction, followed by a detailed introduction to quantum computing. Topics that are discussed in the introduction include qubits, quantum phenomena, such as superposition and entanglement, and quantum gates/circuits. The thesis then reviews the basics of quantum error correction and provides Shor's code to reinforce the reader's understanding. Subsequently, the formalism of stabilizer codes is thoroughly examined. We then explain the generalized concept of stabilizer codes which is entanglement-assisted quantum codes. They do not require generators to satisfy the commutativity property. Rather, they utilize the usage of ebits to resolve the anti-commutativity constraint. Next, the thesis explains quaternary field and then the Java program implemented to find the optimal parameters. Lastly, the thesis concludes with presenting the parameters of the new codes that were obtained throughout the research. We have found the suboptimal largest distance for quaternary hermitian linear complementary dual codes that can be imported as entanglement-assisted quantum error correction for parameters [22, 9, 9 or 10]₄, [22, 12, 7 or 8]₄, [23, 8, 11 or 12]₄, [23, 10, 9 or 10]₄, [23, 13, 7 or 8]₄, [24, 10, 10 or 11]₄, [24, 11, 9 or 10]₄, [24, 14, 7 or 8]₄, [25, 12, 9 or 10]₄, [25, 13, 8 or 9]₄, as well as the optimal largest distance for [17, 11, 5]₄ and [17, 13, 3]₄.

We introduce some methods for constructing quaternary Hermitian self-orthogonal (HSO) codes, and construct quaternary [n, 5] HSO for 342≤n≤492. Furthermore, we present methods of constructing Hermitian linear complementary dual (HLCD) codes from known HSO codes, and obtain many HLCD codes with good parameters. As an application, 31 classes of entanglement-assisted quantum error correction codes (EAQECCs) with maximal entanglement can be obtained from these HLCD codes. These new EAQECCs have better parameters than those in the literature.

The largest minimum weights among quaternary Hermitian linear complementary dual codes are known for dimension 2. In this paper, we give some conditions on the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum weights. As a consequence, we completely determine the largest minimum weights for dimension 3, by using a classification of some quaternary codes. In addition, for a positive integer s, an entanglement-assisted quantum error-correcting [[21s + 5, 3, 16s + 3; 21s + 2]] code with maximal entanglement is constructed for the first time from a quaternary Hermitian linear complementary dual [26, 3, 19] code.

The entanglement-assisted (EA) formalism generalizes the standard stabilizer formalism. All quaternary linear codes can be transformed into entanglement-assisted quantum error correcting codes (EAQECCs) under this formalism. In this work, we discuss construction of EAQECCs from Hermitian non-dual containing primitive Bose–Chaudhuri–Hocquenghem (BCH) codes over the Galois field GF(4). By a careful analysis of the cyclotomic cosets contained in the defining set of a given BCH code, we can determine the optimal number of ebits that needed for constructing EAQECC from this BCH code, rather than calculate the optimal number of ebits from its parity check matrix, and derive a formula for the dimension of this BCH code. These results make it possible to specify parameters of the obtained EAQECCs in terms of the design parameters of BCH codes.

Known quantum error correction schemes are typically able to take advantage of only a limited class of classical error-correcting codes. Entanglement-assisted quantum error correction is a partial solution which made it possible to exploit any classical linear codes over the binary or quaternary finite field. However, the known entanglement-assisted scheme requires noiseless qubits that help correct quantum errors on noisy qubits, which can be too severe an assumption. We prove that a more relaxed and realistic assumption is sufficient by presenting encoding and decoding operations assisted by qubits on which quantum errors of one particular kind may occur. As in entanglement assistance, our scheme can import any binary or quaternary linear codes. If the auxiliary qubits are noiseless, our codes become entanglement-assisted codes, and saturate the quantum Singleton bound when the underlying classical codes are maximum distance separable.

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.