Efficient classical simulation of noisy random quantum circuits in one dimension

Abstract

Understanding the computational power of noisy intermediate-scale quantum (NISQ) devices is of both fundamental and practical importance to quantum information science. Here, we address the question of whether error-uncorrected noisy quantum computers can provide computational advantage over classical computers. Specifically, we study noisy random circuit sampling in one dimension (or 1D noisy RCS) as a simple model for exploring the effects of noise on the computational power of a noisy quantum device. In particular, we simulate the real-time dynamics of 1D noisy random quantum circuits via matrix product operators (MPOs) and characterize the computational power of the 1D noisy quantum system by using a metric we call MPO entanglement entropy. The latter metric is chosen because it determines the cost of classical MPO simulation. We numerically demonstrate that for the two-qubit gate error rates we considered, there exists a characteristic system size above which adding more qubits does not bring about an exponential growth of the cost of classical MPO simulation of 1D noisy systems. Specifically, we show that above the characteristic system size, there is an optimal circuit depth, independent of the system size, where the MPO entanglement entropy is maximized. Most importantly, the maximum achievable MPO entanglement entropy is bounded by a constant that depends only on the gate error rate, not on the system size. We also provide a heuristic analysis to get the scaling of the maximum achievable MPO entanglement entropy as a function of the gate error rate. The obtained scaling suggests that although the cost of MPO simulation does not increase exponentially in the system size above a certain characteristic system size, it does increase exponentially as the gate error rate decreases, possibly making classical simulation practically not feasible even with state-of-the-art supercomputers.

Classical simulation of lossy boson sampling using matrix product operators

“…Lastly, we briefly compare our analysis of boson sampling with a related previous work on 1D noisy RCS [22]. Both studies build on an observation that noise tends to reduce nontrivial correlation in quantum systems and use MPOs to more efficiently simulate such noisy systems than the brute force methods.…”

“…As a remark, in the case of nonuniform loss [52], we cannot simplify the problem by merging all loss channels as we did for uniform loss because nonuniform loss channels do not commute with beam splitters in general. Therefore, one needs to update an MPO by a completely positive trace-preserving map for a loss channel for each step, which requires more computational time [22]. In addition, we may not be able to take advantage from symmetry because loss channels do not preserve global U(1) symmetry.…”

“…Now, we present how to compute outcome probabilities and sample outcomes according to the probability distribution using MPS and MPO [22]. The probability of obtaining a given outcome n is written as…”

“…That is, many classical simulation methods become unavoidably ineffective for simulating large quantum systems (consisting of, e.g., 70 qubits) due to exponentially large Hilbert space, even if such systems are noisier than a smaller system which can be classically simulated. On the other hand, various efficient simulation methods based on matrix product state (MPS) and matrix product operator (MPO) [18] have recently been proposed for simulating large but noisy quantum systems [19][20][21][22]. These methods take advantage of the fact that noise in quantum circuits limits the growth of quantum entanglement in NISQ devices and thus use MPS or MPO to describe such NISQ systems with bounded entanglement in a compressed manner.…”

Low-rank density-matrix evolution for noisy quantum circuits

“…While our attention rests on the column space of density matrices, further speedups can be achieved by optimizing the representation with respect to the computational basis 14,74,75 .…”