Surface Code Threshold in the Presence of Correlated Errors

Novais, E.; Mucciolo, Eduardo R.

doi:10.1103/physrevlett.110.010502

Cited by 35 publications

(58 citation statements)

References 17 publications

(24 reference statements)

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“…The standard approach for mitigating these errros is to use a faulttolerant error-correction protocol, which enables arbitrarily large computations as long as the strength of the errors are below a threshold value [1-5] and are not overly correlated in space or time [6][7][8][9][10]. The fault-tolerant error threshold is a measure of the robustness of a quantum computing platform and an estimate of its value is one of the most important tasks for practical quantum computer design.…”

Section: Pauli Twirling Approximationmentioning

confidence: 99%

Efficient error models for fault-tolerant architectures and the Pauli twirling approximation

Geller

Zhou

2013

Phys. Rev. A

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The design and optimization of realistic architectures for fault-tolerant quantum computation requires error models that are both reliable and amenable to large-scale classical simulation. Perhaps the simplest and most practical general-purpose method for constructing such an error model is to twirl a given completely positive channel over the Pauli basis, a procedure we refer to as the Pauli twirling approximation (PTA). In this work we test the accuracy of the PTA for a small stabilizer measurement circuit relevant to fault-tolerant quantum computation, in the presence of both intrinsic gate errors and decoherence, and find excellent agreement over a wide range of physical error rates. The combined simplicity and accuracy of the PTA, along with its direct connection to the χ matrix of process tomography, suggests that it be used as a standard reference point for more refined error model constructions. I. PAULI TWIRLING APPROXIMATIONThe principal obstacle to large-scale quantum computation is the introduction errors caused by decoherence, noise, leakage to non-computational states, incorrect implementation of quantum gates, qubit loss, and inaccurate state initialization and measurement. The standard approach for mitigating these errros is to use a faulttolerant error-correction protocol, which enables arbitrarily large computations as long as the strength of the errors are below a threshold value [1-5] and are not overly correlated in space or time [6][7][8][9][10]. The fault-tolerant error threshold is a measure of the robustness of a quantum computing platform and an estimate of its value is one of the most important tasks for practical quantum computer design.A straightforward approach for calculating logical error rates and associated error thresholds would be to do a full Hilbert space simulation of quantum codes of increasing size, in the presence of decoherence and other errors, but this approach quickly becomes intractable. This difficulty can be circumvented by using special error models that are efficient to simulate classically. The existence of a broad class of these efficient error models, which includes the Pauli and Clifford channels, is provided by the Gottesman-Knill theorem: This theorem states that any quantum circuit consisting only of Clifford-group unitaries and measurement in the Pauli basis can be simulated classically in polynomial time [11,12]. The circuits used to implement stabilizer-based error detection and correction-in the absence of any errors-are important examples. However, after including decoherence and intrinsic gate errors, which are required to assess fault-tolerance and calculate error thresholds, the simulations are no longer efficient. By intrinsic we mean an error, such as a unitary qubit rotation by the wrong angle, which does not result from noise or decoherence. The resulting inefficiency of classical simulation is the main motivation for the widely used stochastic approach of modeling nonideal stabilizer circuits by a sequence of one-and two-qubit operations, ea...

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Section: Pauli Twirling Approximationmentioning

confidence: 99%

Efficient error models for fault-tolerant architectures and the Pauli twirling approximation

Geller

Zhou

2013

Phys. Rev. A

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“…In a recent paper [20], two of us showed that the time evolution of the surface code in the presence of a common bosonic bath can be mapped onto a statistical spin model. This mapping allows for the computation of the surface code fidelity much in the same way that one computes the partition function and expectation values in a spin model.…”

Section: Introductionmentioning

confidence: 99%

Fidelity of the surface code in the presence of a bosonic bath

2013

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We study the resilience of the surface code to decoherence caused by the presence of a bosonic bath. This approach allows us to go beyond the standard stochastic error model commonly used to quantify decoherence and error threshold probabilities in this system. The full quantum mechanical system-bath dynamics is computed exactly over one quantum error correction cycle. Since all physical qubits interact with the bath, space-time correlations between errors are taken into account. We compute the fidelity of the surface code as a function of the quantum error correction time. The calculation allows us to map the problem onto an Ising-like statistical spin model with two-body interactions and a fictitious temperature which is related to the inverse bath coupling constant. The model departs from the usual Ising model in the sense that interactions can be long ranged and can involve complex exchange couplings; in addition, the number of allowed configurations is restricted by the syndrome extraction. Using analytical estimates and numerical calculations, we argue that, in the limit of an infinite number of physical qubits, the spin model sustain a phase transition which can be associated to the existence of an error threshold in the surface code. An estimate of the transition point is given for the case of nearest-neighbor interactions.Comment: 15 pages, 5 figure

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“…Following Refs. [11,12], we consider in the following the perfect measurement case p m = 0 for definiteness and simplicity. Generalization to the more realistic case of p m > 0 is straightforward; it merely corresponds to replacing p c (orp c , see below) by a lower value.…”

Section: Problem and Overviewmentioning

confidence: 99%

“…We have focused our discussion on the specific combination (D = 2, r = 0) investigated in Ref. [11], which corresponds to an Ohmic bath. This combination is of particular interest since it is the only one for which the maximal QEC time vanishes (very slowly) in the thermodynamic limit.…”

Section: Code Sizementioning

confidence: 99%

“…Following Refs. [11,12], we have made several simplifying assumptions to make the actual problem analytically tractable. These are: a trivial Hamiltonian for the qubits; undamped and non-interacting bath modes; no residual bath correlations between different QEC periods; one type of errors only (bit-flips); immediate and flawless syndrome measurement and error correction (including no time cost for efficient classical computations).…”

Section: Code Sizementioning

confidence: 99%

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Breakdown of surface-code error correction due to coupling to a bosonic bath

Hutter

Loss

2014

Phys. Rev. A

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We consider a surface code suffering decoherence due to coupling to a bath of bosonic modes at finite temperature and study the time available before the unavoidable breakdown of error correction occurs as a function of coupling and bath parameters. We derive an exact expression for the error rate on each individual qubit of the code, taking spatial and temporal correlations between the errors into account. We investigate numerically how different kinds of spatial correlations between errors in the surface code affect its threshold error rate. This allows us to derive the maximal duration of each quantum error correction period by studying when the single-qubit error rate reaches the corresponding threshold. At the time when error correction breaks down, the error rate in the code can be dominated by the direct coupling of each qubit to the bath, by mediated subluminal interactions, or by mediated superluminal interactions. For a 2D Ohmic bath, the time available per quantum error correction period vanishes in the thermodynamic limit of a large code size L due to induced superluminal interactions, though it does so only like

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Surface Code Threshold in the Presence of Correlated Errors

Cited by 35 publications

References 17 publications

Efficient error models for fault-tolerant architectures and the Pauli twirling approximation

Efficient error models for fault-tolerant architectures and the Pauli twirling approximation

Fidelity of the surface code in the presence of a bosonic bath

Breakdown of surface-code error correction due to coupling to a bosonic bath

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