Classically Simulating Quantum Circuits with Local Depolarizing Noise

Abstract

We study the effect of noise on the classical simulatability of quantum circuits defined by computationally tractable (CT) states and efficiently computable sparse (ECS) operations. Examples of such circuits, which we call CT-ECS circuits, are IQP, Clifford Magic, and conjugated Clifford circuits. This means that there exist various CT-ECS circuits such that their output probability distributions are anti-concentrated and not classically simulatable in the noise-free setting (under plausible assumptions). First, we consider a noise model where a depolarizing channel with an arbitrarily small constant rate is applied to each qubit at the end of computation. We show that, under this noise model, if an approximate value of the noise rate is known, any CT-ECS circuit with an anti-concentrated output probability distribution is classically simulatable. This indicates that the presence of small noise drastically affects the classical simulatability of CT-ECS circuits. Then, we consider an extension of the noise model where the noise rate can vary with each qubit, and provide a similar sufficient condition for classically simulating CT-ECS circuits with anti-concentrated output probability distributions.

A Bayesian Approach for Characterizing and Mitigating Gate and Measurement Errors

“…Here, U is a noisy-free unitary gate and X g /2 represents a bit-flip error with probability 1 2 g . We adopt the error model in [18] to describe the single bit-flip error. Of note, [18] does not provide a proof on the validity of this model, we complete the proof and extend it to multiple-error case in Section 2.3.2.…”

“…We adopt the error model in [18] to describe the single bit-flip error. Of note, [18] does not provide a proof on the validity of this model, we complete the proof and extend it to multiple-error case in Section 2.3.2. Using this method, we first consider the case when there is only one gate and one bit-flip error (on all qubits) in the circuit, as shown in Figure 1.…”

“…Here, the transition matrix can be constructed from conditional probabilities if only classical errors are considered, or from results of tomography if non-classical errors are also significant [6,10,14,4]. On the other hand, the study in [18] shows the possibility to simulate bit-flip gate error in some quantum circuit in a classical manner.…”

“…The paper is organized as following. In Section 2, we provide the measurement error model based on independent classical measurement error and expand the gate error model in [18] to multiple-error scenario. In Section 3, we introduce the use of Bayesian algorithm in [5] to infer the distributions of parameters of measurement error and gate error models.…”

Fault-tolerant quantum speedup from constant depth quantum circuits

“…At the same time, the role of noise in simplifying the simulation is ever more important, as systems grow, noise becomes more difficult to control, and it is a subtle question as to when it dominates; and even simple noise can very easily lead to breakdown of quantum speedup. Indeed, in [24][25][26][27][28][29][30] it was shown that noise generally renders the output probabilities of these devices (which in the noiseless case demonstrate quantum speedup) classically simulable efficiently. There is clearly a great need to understand better the effect of noise, and develop methods of mitigation.…”