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.
Specifically, we study 1D noisy random circuit sampling as a simple model for exploring the adverse effects of the realistic gate errors. The key takeaway from our work is that in noisy settings, there exists a characteristic system size, determined solely by the gate error rate, above which adding more qubits does not bring about an exponential growth of the computing power of a noisy quantum system. That is, quality limits the utility of quantity. In particular, we demonstrate that matrix product operators are able to describe the dynamics of a 1D noisy system in a reliable and compressed way.
Our work provides a framework for assessing the utility of near-term quantum computing technologies based on noisy intermediate-scale quantum (NISQ) devices. Going beyond the chain architecture and investigating more general settings (e.g., planar architecture) would be a fruitful future research direction.
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