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

Abstract

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...

Hard decoding algorithm for optimizing thresholds under general Markovian noise

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