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CRC Press eBooks

Bounds on Quantum Error Correcting Codes

Frank Gaitan

Published Feb 7, 2008• 0 citations
Publisher

Abstract

The previous two chapters laid out the theoretical framework for quantum error correcting codes, and especially for quantum stabilizer codes. Following Cleve and Gottesman [1], the present chapter will explain how to construct quantum circuits that efficiently encode and decode a quantum stabilizer code. Before entering into that discussion, it proves advantageous to introduce a standard form for quantum stabilizer codes. This is done in Section 4.1. Assuming that a quantum stabilizer code in standard form has been given, Sections 4.2 and 4.3 explain how to, respectively, encode and decode its code­ words. A brief summary of quantum circuits is given in Appendix C.

DOI
10.1201/b15868-8

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The previous two chapters laid out the theoretical framework for quantum error correcting codes, and especially for quantum stabilizer codes. Following Cleve and Gottesman [1], the present chapter will explain how to construct quantum circuits that efficiently encode and decode a quantum stabilizer code. Before entering into that discussion, it proves advantageous to introduce a standard form for quantum stabilizer codes. This is done in Section 4.1. Assuming that a quantum stabilizer code in standard form has been given, Sections 4.2 and 4.3 explain how to, respectively, encode and decode its code­ words. A brief summary of quantum circuits is given in Appendix C.
Quantum computers have the potential to provide exponential speedups over their classical counterparts. Quantum principles are being applied to fields such as communications, information processing, and artificial intelligence to achieve quantum advantage. However, quantum bits are extremely noisy and prone to decoherence. Thus, keeping the qubits error free is extremely important toward reliable quantum computing. Quantum error correcting codes have been studied for several decades and methods have been proposed to import classical error correcting codes to the quantum domain. Along with the exploration into novel and more efficient quantum error correction codes, it is also essential to design circuits for practical realization of these codes. This article serves as a tutorial on designing and simulating quantum encoder and decoder circuits for stabilizer codes. We first describe Shor’s 9-qubit code which was the first quantum error correcting code. We discuss the stabilizer formalism along with the design of encoding and decoding circuits for stabilizer codes such as the five-qubit code and Steane code. We also design nearest neighbor compliant circuits for the above codes. The circuits were simulated and verified using IBM Qiskit.
Quantum error-correcting codes are used to protect quantum information from decoherence. A raw state is mapped, by an encoding circuit, to a codeword so that the most likely quantum errors from a noisy quantum channel can be removed after a decoding process.A good encoding circuit should have some desired features, such as low depth, few gates, and so on. In this paper, we show how to practically implement an encoding circuit of gate complexity O(n(n - k + c)/ log n) for an [[n, k; c]] quantum stabilizer code with the help of c pairs of maximally-entangled states. For the special case of an [[n, k]] stabilizer code with c = 0, the encoding complexity is O(n(n-k)/ log n), which is previously known to be O(n2/ log n). For c > 0, this suggests that the benefits from shared entanglement come at an additional cost of encoding complexity.Finally we discuss decoding of entanglement-assisted quantum stabilizer codes and extend previously known computational hardness results on decoding quantum stabilizer codes.
Stabilizer codes form an important class of quantum error correcting codes which have an elegant theory, efficient error detection, and many known examples. Constructing stabilizer codes of length nnn is equivalent to constructing subspaces of Fpn×Fpn\mathbb{F}_p^n \times \mathbb{F}_p^nFpn​×Fpn​ which are "isotropic" under the symplectic bilinear form defined by ⟨(a,b),(c,d)⟩=aTd−bTc\left\langle (\mathbf{a},\mathbf{b}),(\mathbf{c},\mathbf{d}) \right\rangle = \mathbf{a}^{\mathrm{T}} \mathbf{d} - \mathbf{b}^{\mathrm{T}} \mathbf{c}⟨(a,b),(c,d)⟩=aTd−bTc. As a result, many, but not all, ideas from the theory of classical error correction can be translated to quantum error correction. One of the main theoretical contribution of this article is to study stabilizer codes starting with a different symplectic form. In this paper, we concentrate on cyclic codes. Modifying the symplectic form allows us to generalize the previous known construction for linear cyclic stabilizer codes, and in the process, circumvent some of the Galois theoretic no-go results proved there. More importantly, this tweak in the symplectic form allows us to make use of well known error correcting algorithms for cyclic codes to give efficient quantum error correcting algorithms. Cyclicity of error correcting codes is a "basis dependent" property. Our codes are no more "cyclic" when they are derived using the standard symplectic forms (if we ignore the error correcting properties like distance, all such symplectic forms can be converted to each other via a basis transformation). Hence this change of perspective is crucial from the point of view of designing efficient decoding algorithm for these family of codes. In this context, recall that for general codes, efficient decoding algorithms do not exist if some widely believed complexity theoretic assumptions are true.
Quantum error correcting code is an efficient tool to correct and prevent quantum error.It is the basic for quantum computation and quantum communications.Taking advantage of the cyclic difference set,we present a novel approach to construct quantum stabilizer codes in this paper.It's not difficult to construct the well-known quantum code by our approach.By analyzing the property of the quantum code constructed by this method,such as ,as well as ,we find all the constructed codes are the quantum Single-bound codes,and we can use this method to encode different length information.Besides,the code rate is high.