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arXiv:quant-ph/0008027 · Physical Review A

Error correction for mutually interacting qubits

Julio R. Gea-Banacloche

Published Nov 14, 2000• 5 citations
arXivHTMLPDF

Abstract

For a simple model of mutually interacting qubits it is shown how the errors induced by mutual interactions can be eliminated using concatenated coding. The model is solved exactly for arbitrary interaction strength, for two well-known codes, one and two levels deep: this allows one to see under which circumstances error amplitudes add coherently or incoherently. For deeper concatenation, approximate results are derived that make it possible to calculate an approximate ``threshold'' value for the product of interaction strength and free evolution time, below which the failure probability for an encoded qubit decreases exponentially with the depth of the encoding. The results suggest that concatenated coding could fully handle the errors arising from mutual interactions, at no extra cost, in terms of resources needed, than what would be required to deal with random environmental errors.

DOI
10.1103/physreva.62.062313

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For a simple model of mutually interacting qubits it is shown how the errors induced by mutual interactions can be eliminated using concatenated coding. The model is solved exactly for arbitrary interaction strength, for two well-known codes, one and two levels deep: this allows one to see under which circumstances error amplitudes add coherently or incoherently. For deeper concatenation, approximate results are derived that make it possible to calculate an approximate ``threshold'' value for the product of interaction strength and free evolution time, below which the failure probability for an encoded qubit decreases exponentially with the depth of the encoding. The results suggest that concatenated coding could fully handle the errors arising from mutual interactions, at no extra cost, in terms of resources needed, than what would be required to deal with random environmental errors.
Estimates of the quantum accuracy threshold often tacitly assume that it is possible to interact arbitrary pairs of qubits in a quantum computer with a failure rate that is independent of the distance between them. None of the many physical systems that are candidates for quantum computing possess this property. Here we study the performance of a concatenated error-detection code in a system that permits only nearest-neighbor interactions in one dimension. We make use of a message-passing scheme that maximizes the number of errors that can be reliably corrected by the code. Our numerical results indicate that arbitrarily accurate universal quantum computation is possible if the probability of failure of each elementary physical operation is below approximately ${10}^{\ensuremath{-}5}$. This threshold is three orders of magnitude lower than the highest known.
It is known that quantum error correction via concatenated codes can be done with exponentially small failure rate if the error rate for physical qubits is below a certain accuracy threshold. Other, unconcatenated codes with their own attractive features such as improved accuracy threshold, local operations, have also been studied. By iteratively distilling a certain two-qubit entangled state it is shown how to perform an encoded Toffoli gate, important for universal computation, on codes whose C-NOT operation is bitwise. For certain codes, over a very large range of block sizes, this technique requires at most one concatenation.
Concatenating quantum error correction codes scales error correction capability by driving logical error rates down double-exponentially across levels. However, the noise structure shifts under concatenation, making it hard to choose an optimal code sequence. We automate this choice by estimating the effective noise channel after each level and selecting the next code accordingly. In particular, we use learning-based methods to tailor small, non-additive encoders when the noise exhibits sufficient structure, then switch to standard codes once the noise is nearly uniform. In simulations, this level-wise adaptation achieves a target logical error rate with far fewer qubits than concatenating stabilizer codes alone--reducing qubit counts by up to two orders of magnitude for strongly structured noise. Therefore, this hybrid, learning-based strategy offers a promising tool for early fault-tolerant quantum computing.
It has been known that error-correction via concatenated codes can be done with exponentially small failure rate if the error rate for physical qubits is below a certain accuracy threshold (probably ∼ 10 −3 –10 −6). Other, un-concatenated codes with their own attractive features—e.g., an accuracy threshold ∼ 10 −2 —have also been studied. A method to obtain universal computation is presented here which does not rely on any concatenated structure within the code itself, but instead emulates this structure with logical qubits in order to construct an encoded Toffoli gate. This realizes ∼ 10 −2 as a threshold for fault-tolerant quantum computation. 1 QEC codes and universal computation In the “space ” of all possible quantum error-correcting codes, much recent work has focused on a relatively small class, namely concatenated codes [1]–[7]. The basic idea behind these is to improve the results of a given few-qubit code by
When a logical qubit is protected using a quantum error-correcting code, the net effect of coding, decoherence (a physical channel acting on qubits in the codeword) and recovery can be represented exactly by an effective channel acting directly on the logical qubit. In this paper we describe a procedure for deriving the map between physical and effective channels that results from a given coding and recovery procedure. We show that the map for a concatenation of codes is given by the composition of the maps for the constituent codes. This perspective leads us to an efficient means for calculating the exact performance of quantum codes with arbitrary levels of concatenation. We present explicit results for single-bit Pauli channels. For certain codes under the symmetric depolarizing channel, we use the coding maps to compute exact threshold error probabilities for achievability of perfect fidelity in the infinite concatenation limit.