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arXiv:quant-ph/0508228 · Physical Review Letters

Decoherence by Correlated Noise and Quantum Error Correction

E. Novais, Harold U. Baranger

Published Jul 24, 2006• 41 citations
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Abstract

We study the decoherence of a quantum computer in an environment which is inherently correlated in time and space. We first derive the nonunitary time evolution of the computer and environment in the presence of a stabilizer error correction code, providing a general way to quantify decoherence for a quantum computer. The general theory is then applied to the spin-boson model. Our results demonstrate that effects of long-range correlations can be systematically reduced by small changes in the error correction codes.

DOI
10.1103/physrevlett.97.040501

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We study the decoherence of a quantum computer in an environment which is inherently correlated in time and space. We first derive the nonunitary time evolution of the computer and environment in the presence of a stabilizer error correction code, providing a general way to quantify decoherence for a quantum computer. The general theory is then applied to the spin-boson model. Our results demonstrate that effects of long-range correlations can be systematically reduced by small changes in the error correction codes.
We analyze the long-time behavior of a quantum computer running a quantum error correction (QEC) code in the presence of a correlated environment. Starting from a Hamiltonian formulation of realistic noise models, and assuming that QEC is indeed possible, we find formal expressions for the probability of a given syndrome history and the associated residual decoherence encoded in the reduced density matrix. Systems with nonzero gate times (``long gates'') are included in our analysis by using an upper bound on the noise. In order to introduce the local error probability for a qubit, we assume that propagation of signals through the environment is slower than the QEC period (hypercube assumption). This allows an explicit calculation in the case of a generalized spin-boson model and a quantum frustration model. The key result is a dimensional criterion: If the correlations decay sufficiently fast, the system evolves toward a stochastic error model for which the threshold theorem of fault-tolerant quantum computation has been proven. On the other hand, if the correlations decay slowly, the traditional proof of this threshold theorem does not hold. This dimensional criterion bears many similarities to criteria that occur in the theory of quantum phase transitions.
In the scope of this work the coherence of quantum information, which is encoded into a qubit register, is analysed. The qubit register is modelled by a spin chain with finite inter-spin distance. In most physically relevant realisations this spin chain irreversibly interacts with a surrounding environment, such that a spin-boson model is used to describe the setting. Due to the interaction decoherence occurs among the qubits register and quantum information gets lost. Mechanisms to slow down this decoherence process are investigated. For that purpose, the techniques of encoding qubits into decoherence-reduced subspaces and quantum error correction are used. In both cases only a linear subspace of the complete available Hilbert space of the spin chain is used as quantum code. The stability of such a code against decoherence has to be evaluated. This evaluation is performed on average over all states within the code by a code fidelity.
We analyze the problem of a quantum computer in a correlated environment protected from decoherence by quantum error correction using a perturbative renormalization group approach. The scaling equation obtained reflects the competition between the dimension of the computer and the scaling dimension of the correlations. For an irrelevant flow, the error probability is reduced to a stochastic form for a long time and/or a large number of qubits; thus, the traditional derivation of the threshold theorem holds for these error models. In this way, the "threshold theorem" of quantum computing is rephrased as a dimensional criterion.
Quantum error correction will be an indispensable ingredient of large-scale quantum computations. Conventional quantum error correction codes (QECC) have been devised with an independent-error model in mind, but one may expect that the noise affecting a system of qubits will, in general, exhibit nonzero correlations in time, or space, or both. This talk will present a brief introduction to the principles of quantum error correction, followed by a discussion of the performance of conventional QECCs in the presence of correlated noise.
Quantum error correction is a solution to preserve the fidelity of quantum information encoded in physical systems subject to noise. However, unfavorable correlated errors could be induced even for non-interacting qubits through the environment (bath), when they are "packed" together. The question is, to what extent can we treat the noise induced by the bath as independent? In the context of the spin-boson model, we show that, under some reasonable constraints, the independent noise approximation could be valid. On the other hand, in the strongly correlated limit, we show how the method of decoherence free subspace can be made applicable. Combining these two methods makes fault-tolerant quantum computation promising in fighting against correlated errors.