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arXiv:quant-ph/0008049 · Physical Review A

Correlated errors in quantum-error corrections

Won Young Hwang, Doyeol Ahn, Sung Woo Hwang

Published Jan 11, 2001• 24 citations
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Abstract

We show that errors are not generated correlatedly provided that quantum bits do not directly interact with (or couple to) each other. Generally, this no-qubit-interaction condition is assumed except for the case where two-qubit gate operation is being performed. In particular, the no-qubit-interaction condition is satisfied in the collective decoherence models. Thus, errors are not correlated in the collective decoherence. Consequently, we can say that current quantum error correcting codes that correct single-qubit errors will work in most cases including the collective decoherence.

DOI
10.1103/physreva.63.022303

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We show that errors are not generated correlatedly provided that quantum bits do not directly interact with (or couple to) each other. Generally, this no-qubit-interaction condition is assumed except for the case where two-qubit gate operation is being performed. In particular, the no-qubit-interaction condition is satisfied in the collective decoherence models. Thus, errors are not correlated in the collective decoherence. Consequently, we can say that current quantum error correcting codes that correct single-qubit errors will work in most cases including the collective decoherence.
We show a way to correct collective error through a three-qubit quantum code. The encoding and decoding (i.e., error recovery) operations for such a code, which protects one qubit of quantum information perfectly (with infinite ``distance'') against collective decoherence, are presented. The code is actually a manifestation of a noiseless subsystem first reported by Knill, Laflamme, and Viola [Phys. Rev. Lett. 84, 2525 (2000)], and provides a good illustration of the relationship between noiseless subsystems and error-correcting codes already noted by these authors. We also show in detail how this code can be used to preserve an arbitrary joint state (including entangled states) of two qubits, by pairing each of them with two other qubits, even in the case in which the two resulting clusters interact in nonequivalent ways with a common environment.
In order to stabilize quantum algorithms against decoherence one has to fulfill two requirements. Firstly, one has to develop an appropriate quantum error correcting code. Secondly, one has to find a set of suitable unitary quantum transformations acting on the physical qubits which preserve the properties of this error correcting quantum code and which allow the implementation of a universal set of quantum gates. This is a challenging task in particular if we restrict ourselves to a limited class of two-particle interactions by which the physical qubits can be controlled. For the special cases of four and six physical qubits we discuss a set of such quantum gates which satisfy these two conditions for the recently developed detected-jump correcting quantum codes [1]. These quantum codes are capable of stabilizing distinguishable qubits against decoherence arising from spontaneous decay processes.
We study the effect of continuous quantum error correction in the case where each qubit in a codeword is subject to a general Hamiltonian interaction with an independent bath. We first consider the scheme in the case of a trivial single-qubit code, which provides useful insights into the workings of continuous error correction and the difference between Markovian and non-Markovian decoherence. We then study the model of a bit-flip code with each qubit coupled to an independent bath qubit and subject to continuous correction, and find its solution. We show that for sufficiently large error-correction rates, the encoded state approximately follows an evolution of the type of a single decohering qubit, but with an effectively decreased coupling constant. The factor by which the coupling constant is decreased scales quadratically with the error-correction rate. This is compared to the case of Markovian noise, where the decoherence rate is effectively decreased by a factor which scales only linearly with the rate of error correction. The quadratic enhancement depends on the existence of a Zeno regime in the Hamiltonian evolution which is absent in purely Markovian dynamics. We analyze the range of validity of this result and identify two relevant time scales. Finally, we extend the result to more general codes and argue that the performance of continuous error correction will exhibit the same qualitative characteristics.
Abstract Quantum computing is an emerging technology that, with its exponential computational speedup, will allow us to solve a new level of hard problems. From rapid vaccine development, efficient discovery of exotic materials, and complex AI models, to quickly cracking RSA encryption, quantum computers have the potential to revolutionize science. However, for quantum computing to solve these problems, quantum error correction methods are needed to eliminate qubit errors. An underlying assumption of current quantum error correction is that qubit errors are not correlated. If correlated qubit errors are prevalent, quantum error correction will be difficult, if not impossible. In this work, I developed a method to identify correlated errors between pairs of qubits and used this method to determine if correlations exist on current small-scale quantum computers. My method uses independence of error probabilities P(qa) ⨉ P(qb) = P(qaqb) to determine correlation. After running 40 million experiments to collect error probabilities by varying the quantum gates, number of gates, and pairs of qubits on multiple quantum computers, I found that these computers have a high occurrence of correlated qubit errors when qubits are topological neighbors. For example, I found that Hadamard gate operations can cause correlated errors, supported by a p-value of 0.0160. My work shows that the assumption that qubit errors are not correlated is incorrect for current quantum computers. In the future, scientists can use these methods to test for correlated errors as they develop new quantum computers, enabling quantum computing to achieve its full potential.
The goal of this work is to analyse the performance of quantum error correction codes in regard to fixing independent errors in several qubits. We say that a quantum code does not fix a quantum computing error if its application does not reduce the variance of the error. We restrict our study to the classical 5−qubit quantum error correcting code (proposed by Laflamme and some collaborators), which is able to correct arbitrary errors in a single qubit and is also fault-tolerant. We show that this code does not fix qubit independent errors, even assuming that the correction circuit does not introduce new errors. We also prove, for qubit independent errors, that if the correction circuit of the 5−qubit quantum code detects an error, then the corrected state has central symmetry and, as a consequence, its variance is maximal. We have been able to obtain these results thanks to the high symmetry of the 5−qubit quantum code. Although the calculations needed to extend our proofs to less symmetric codes seem to be extremely complicated, we nevertheless think that the results obtained for the 5−qubit quantum code reveal a general behavior pattern of quantum error correcting codes against qubit independent errors.