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arXiv:1806.06472 · Physical Review A

Calderbank-Shor-Steane holographic quantum error-correcting codes

Robert J. Harris, Nathan A. McMahon, Gavin K. Brennen, Thomas M. Stace

Published Nov 1, 2018• 51 citations
arXivHTMLPDF

Abstract

We expand the class of holographic quantum error-correcting codes by developing the notion of block perfect tensors, a wider class that includes previously defined perfect tensors. The relaxation of this constraint opens up a range of other holographic codes. We demonstrate this by introducing the self-dual Calderbank-Shor-Steane (CSS) heptagon holographic code, based on the 7-qubit Steane code. Finally, we show promising thresholds for the erasure channel by applying a straightforward, optimal erasure decoder to the heptagon code and benchmark it against existing holographic codes.

DOI
10.1103/physreva.98.052301

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We expand the class of holographic quantum error-correcting codes by developing the notion of block perfect tensors, a wider class that includes previously defined perfect tensors. The relaxation of this constraint opens up a range of other holographic codes. We demonstrate this by introducing the self-dual Calderbank-Shor-Steane (CSS) heptagon holographic code, based on the 7-qubit Steane code. Finally, we show promising thresholds for the erasure channel by applying a straightforward, optimal erasure decoder to the heptagon code and benchmark it against existing holographic codes.
Calderbank-Shor-Steane (CSS) quantum error-correcting codes are based on pairs of classical codes which are mutually dual containing. Explicit constructions of such codes for large blocklengths and with good error correcting properties are not easy to find. In this paper we propose a construction of CSS codes which combines a classical code with a two-universal hash function. We show, using the results of Renner and Koenig, that the communication rates of such codes approach the hashing bound on tensor powers of Pauli channels in the limit of large block-length. While the bit-flip errors can be decoded as efficiently as the classical code used, the problem of efficiently decoding the phase-flip errors remains open.
A valid Calderbank-Shor-Steane (CSS) error correction code requires two classical linear codes for the preparation of the initial state (codewords). This code allow to correct for certain errors caused by an unwanted interaction which produces a degraded quantum state. However, this initial seven qubits encoding can be obtained from a maximally entangled Bell state (0000000>+|1111111)/√ through an operation Hint whose explicit expression is derived in the present work. The price the CSS syndrome has to pay due to its classical grounds is that the operator Hint is not unitary. In other words, Hint is not a valid quantum gate i. e. this does not represent a logical operation. Consequently, the final state is not completely robust for the standard cryptography of Quantum Computation. Besides to be a non unitary operator, Hint, is not reversible introducing with this dissipative effects that destroy the coherence in the quantum computer. Additionally, this operator is not invariant under rotations of the protector qubits inducing then preferred directions of the propagation of the logical information. These are indeed the reasons that prompt us for extending the semi classical CSS quantum error correction codes formalism to a pure quantum Hamming codes.
This chapter deals with the subject of quantum error correction and the related codes (QECC), which can be applied to noisy quantum channels and quantum memories with the purpose of preserving or protecting the information integrity. I first describe the basics of quantum repetition codes , as applicable to bit-flip and phase-flip quantum channels. Then I consider the 9-qubit Shor code , which has the capability of diagnosing and correcting any combination of bit-flip and phase-flip errors, up to one error of each type. Furthermore, it is shown that the Shor code is, in fact, capable of fully restoring qubit integrity under a continuum of bit or phase errors, a property that has no counterpart in the classical world of error-correction codes. But the exploration of QECC does not stop here! We shall discover the elegant Calderbank–Shor–Steane (CSS) codes, which have the capability of correcting any number of errors t , both bit-flip and phase-flip. As an application of the CSS code, I then describe the 7-qubit Hadamard–Steane code , which can correct up to one error on single qubits. A corresponding quantum circuit, based on an original generator-matrix example, is presented. Quantum repetition code In Chapter 11, we saw that the simplest form of error-correction code (ECC) is the repetition code , based on the principle of majority logic . The background assumption is that in a given message sequence, or bit string, the probability of a bit error is sufficiently small for the majority of bits to be correctly transmitted through the channel.
A method for concatenating quantum error-correcting codes is presented. The method is applicable to a wide class of quantum error-correcting codes known as Calderbank-Shor-Steane (CSS) codes. As a result, codes that achieve a high rate in the Shannon theoretic sense and that are decodable in polynomial time are presented. The rate is the highest among those known to be achievable by CSS codes. Moreover, the best known lower bound on the greatest minimum distance of codes constructible in polynomial time is improved for a wide range.
Stabilizer codes lie at the heart of modern quantum-error-correcting codes (QECCs). Of particular importance is a class called Calderbank-Shor-Steane (CSS) codes, which includes many important examples such as toric codes, color codes, and fractons. Recent studies have revealed that the decoding transition for these QECCs could be intrinsically captured by calculating information-theoretic quantities from the mixed state. Here we perform a simple analytic calculation of the coherent information for general CSS codes under local incoherent Pauli errors via diagonalization of the density matrices and mapping to classical statistical mechanical (SM) models. Our result establishes a rigorous connection between the decoding transition of the quantum code and the phase transition in the random classical SM model. It is also directly confirmed for CSS codes that exact error correction is possible if and only if the maximum-likelihood decoder always succeeds in the thermodynamic limit. Thus, the fundamental threshold is saturated by the optimal decoder.
A method for concatenating quantum error-correcting codes is presented. The method is applicable to a wide class of quantum error-correcting codes known as Calderbank-Shor-Steane (CSS) codes. As a result, codes that achieve a high rate in the Shannon-theoretic sense and that are decodable in polynomial time are presented. The rate is the highest among those known to be achievable by CSS codes. Moreover, the best known lower bound on the greatest minimum distance of codes constructible in polynomial time is improved for a wide range.