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Quantum Information and Computation

Performance of topological quantum error correction in the presence of correlated noise

Muhammad Ahsan, Syed Abbas Zilqurnain Naqvi

Published Aug 1, 2018• 3 citations
Publisher

Abstract

We investigate the efficacy of topological quantum error-correction in correlated noise model which permits collective coupling of all the codeword qubits to the same non-Markovian environment. In this noise model, the probability distribution over set of phase-flipped qubits, decays sub-exponentially in the size of the set and carries non-trivial likelihood of the occurring large numbers of qubits errors. We find that in the presence of noise correlation, one cannot guarantee arbitrary high computational accuracy simply by incrementing the codeword size while retaining constant noise level per qubit operation. However, if instead, per-operation qubit error probability in an n-qubits long codeword is reduced O(\sqrt{n}) times below the accuracy threshold, arbitrarily accurate quantum computation becomes feasible with acceptable scaling of the codeword size. Our results suggest that progressively reducing noise level in qubits and gates is as important as continuously integrating more qubits to realize scalable and reliable quantum computer.

DOI
10.26421/qic18.9-10-2

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We investigate the efficacy of topological quantum error-correction in correlated noise model which permits collective coupling of all the codeword qubits to the same non-Markovian environment. In this noise model, the probability distribution over set of phase-flipped qubits, decays sub-exponentially in the size of the set and carries non-trivial likelihood of the occurring large numbers of qubits errors. We find that in the presence of noise correlation, one cannot guarantee arbitrary high computational accuracy simply by incrementing the codeword size while retaining constant noise level per qubit operation. However, if instead, per-operation qubit error probability in an n-qubits long codeword is reduced O(\sqrt{n}) times below the accuracy threshold, arbitrarily accurate quantum computation becomes feasible with acceptable scaling of the codeword size. Our results suggest that progressively reducing noise level in qubits and gates is as important as continuously integrating more qubits to realize scalable and reliable quantum computer.
We simulate the performance of toplogical error-correcting codes when all the encoding qubits share same non-Markovian bath. We find that scalable and reliable quantum computing may still be possible if error-rates decline as n\sqrt{n}n​ where nnn is the codeword size.
We simulate the performance of toplogical error-correcting codes when all the encoding qubits share same non-Markovian bath. We find that scalable and reliable quantum computing may still be possible if error-rates decline as n\sqrt{n}n​ where nnn is the codeword size.
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.
We prove a new version of the quantum accuracy threshold theorem that applies to non-Markovian noise with algebraically decaying spatial correlations. We consider noise in a quantum computer arising from a perturbation that acts collectively on pairs of qubits and on the environment, and we show that an arbitrarily long quantum computation can be executed with high reliability in D spatial dimensions, if the perturbation is sufficiently weak and decays with the distance r between the qubits faster than 1/r(D).
Topological quantum error correction codes are known to be able to tolerate arbitrary local errors given sufficient qubits. This includes correlated errors involving many local qubits. In this work, we quantify this level of tolerance, numerically studying the effects of many-qubit errors on the performance of the surface code. We find that if increasingly large-area errors are at least moderately exponentially suppressed, arbitrarily reliable quantum computation can still be achieved with practical overhead. We furthermore quantify the effect of nonlocal two-qubit correlated errors, which would be expected in arrays of qubits coupled by a polynomially decaying interaction, and when using many-qubit coupling devices. We surprisingly find that the surface code is very robust to this class of errors, despite a provable lack of a threshold error rate when such errors are present.
We analyze noise in the circuit model of quantum computers when the qubits are coupled to a common bosonic bath and discuss the possible failure of scalability of quantum computation. Specifically, we investigate correlated (super-radiant) decay between the qubit energy levels from a two- or three-dimensional array of qubits without imposing any restrictions on the size of the sample. We first show that regardless of how the spacing between the qubits compares with the emission wavelength, correlated decay produces errors outside the applicability of the threshold theorem. This is because the sum of the norms of the two-body interaction Hamiltonians (which can be viewed as the upper bound on the single-qubit error) that decoheres each qubit scales with the total number of qubits and is unbounded. We then discuss two related results: (1) We show that the actual error (instead of the upper bound) on each qubit scales with the number of qubits. As a result, in the limit of large number of qubits in the computer, $N\ensuremath{\rightarrow}\ensuremath{\infty}$, correlated decay causes each qubit in the computer to decohere in ever shorter time scales. (2) We find the complete eigenvalue spectrum of the exchange Hamiltonian that causes correlated decay in the same limit. We show that the spread of the eigenvalue distribution grows faster with NNN compared to the spectrum of the unperturbed system Hamiltonian. As a result, as $N\ensuremath{\rightarrow}\ensuremath{\infty}$, quantum evolution becomes completely dominated by the noise due to correlated decay. These results argue that scalable quantum computing may not be possible in the circuit model in a two- or three- dimensional geometry when the qubits are coupled to a common bosonic bath.