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Scientific Reports

Logical error rate in the Pauli twirling approximation

Amara Katabarwa, Michael R. Geller

Published Sep 30, 2015• 22 citations
Publisher

Abstract

The performance of error correction protocols are necessary for understanding the operation of potential quantum computers, but this requires physical error models that can be simulated efficiently with classical computers. The Gottesmann-Knill theorem guarantees a class of such error models. Of these, one of the simplest is the Pauli twirling approximation (PTA), which is obtained by twirling an arbitrary completely positive error channel over the Pauli basis, resulting in a Pauli channel. In this work, we test the PTA's accuracy at predicting the logical error rate by simulating the 5-qubit code using a 9-qubit circuit with realistic decoherence and unitary gate errors. We find evidence for good agreement with exact simulation, with the PTA overestimating the logical error rate by a factor of 2 to 3. Our results suggest that the PTA is a reliable predictor of the logical error rate, at least for low-distance codes.

DOI
10.1038/srep14670

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The performance of error correction protocols are necessary for understanding the operation of potential quantum computers, but this requires physical error models that can be simulated efficiently with classical computers. The Gottesmann-Knill theorem guarantees a class of such error models. Of these, one of the simplest is the Pauli twirling approximation (PTA), which is obtained by twirling an arbitrary completely positive error channel over the Pauli basis, resulting in a Pauli channel. In this work, we test the PTA's accuracy at predicting the logical error rate by simulating the 5-qubit code using a 9-qubit circuit with realistic decoherence and unitary gate errors. We find evidence for good agreement with exact simulation, with the PTA overestimating the logical error rate by a factor of 2 to 3. Our results suggest that the PTA is a reliable predictor of the logical error rate, at least for low-distance codes.
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 $\ensuremath{\chi}$ matrix of process tomography, suggests that it can be used as a standard reference point for more refined error model constructions.
One of simplest and most widely used error model in working with quantum circuits is the Pauli Twirling Approximation (PTA). Restricting ourselves to analysis of free dynamics of qubits we show explicitly how application of PTA is equivalent to ignoring most of the quantum back action of the system and give a general argument as to why this approximation leads to low logical error rates in fault tolerant stabilizer circuits as compared to other quantum channels. We provide numerical evidence that PTA's performance in modeling noise gets worse as number of qubits increase.
Classical simulations of noisy stabilizer circuits are often used to estimate the threshold of a quantum error-correcting code. Physical noise sources are efficiently approximated by random insertions of Pauli operators. For a single qubit, more accurate approximations that still allow for efficient simulation can be obtained by including Clifford operators and Pauli operators conditional on measurement. We examine the feasibility of employing these expanded error approximations to obtain better threshold estimates. We calculate the level-1 pseudothreshold for the Steane [[7,1,3]] code for amplitude damping and dephasing along a non-Clifford axis. The expanded channels estimate the actual channel action more accurately than the Pauli channels before error correction. However, after error correction, the Pauli twirling approximation yields very accurate estimates of the performance of quantum error-correcting protocols in the presence of the actual noise channel.
We introduce a sparse classical representation, a truncation strategy and a shot-efficient sampling method to push the classical prediction of quantum error correction thresholds beyond Clifford operations and Pauli errors. As two illustrations of the potential of our method, we first show that coherent noise error thresholds, when computed at the circuit level (i.e taking into account full syndrome circuits) for distances up to d=9, are systematically overestimated (by a factor of about 4) by a Pauli-twirling approximation of the noise. We then apply our method to the recently introduced magic-state cultivation protocol. We show, through shot-efficient importance sampling, that, at distance d=5, the multiplicative factor between the T-gate and the S-gate injection error rate is not the one conjectured from low-d computations: it can be as large as 7.
We compare the effect of single qubit incoherent and coherent errors on the logical error rate of the Steane [[7,1,3]] quantum error correction code by performing an exact full-density-matrix simulation of an error correction step. We find that the effective 1-qubit process matrix at the logical level reveals the key differences between the error models and provides insight into why the Pauli twirling approximation is a good approximation for incoherent errors and a poor approximation for coherent ones. Approximate channels composed of Clifford operations and Pauli measurement operators that are pessimistic at the physical level result in pessimistic error rates at the logical level. In addition, we observe that the pseudothreshold can differ by a factor of 5 depending on whether the error is calculated using the fidelity or the distance.
Twirling is a technique widely used for converting arbitrary noise channels into Pauli channels in error threshold estimations of quantum error correction codes. It is vitally useful both in real experiments and in classical quantum simulations. Minimising the size of the twirling gate set increases the efficiency of simulations and in experiments it might reduce both the number of runs required and the circuit depth (and hence the error burden). Conventional twirling uses the full set of Pauli gates as the set of twirling gates. This article provides a theoretical background for Pauli twirling and a way to construct a twirling gate set with a number of members comparable to the size of the Pauli basis of the given error channel, which is usually much smaller than the full set of Pauli gates. We also show that twirling is equivalent to stabiliser measurements with discarded measurement results, which enables us to further reduce the size of the twirling gate set.