Abstract
The class of commuting quantum circuits known as IQP (instantaneous quantum polynomial-time) has been shown to be hard to simulate classically, assuming certain complexity-theoretic conjectures. Here we study the power of IQP circuits in the presence of physically motivated constraints. First, we show that there is a family of sparse IQP circuits that can be implemented on a square lattice of n qubits in depth O(sqrt(n) log n), and which is likely hard to simulate classically. Next, we show that, if an arbitrarily small constant amount of noise is applied to each qubit at the end of any IQP circuit whose output probability distribution is sufficiently anticoncentrated, there is a polynomial-time classical algorithm that simulates sampling from the resulting distribution, up to constant accuracy in total variation distance. However, we show that purely classical error-correction techniques can be used to design IQP circuits which remain hard to simulate classically, even in the presence of arbitrary amounts of noise of this form. These results demonstrate the challenges faced by experiments designed to demonstrate quantum supremacy over classical computation, and how these challenges can be overcome.
Highlights
Over the last few years there has been significant attention devoted to devising experimental demonstrations of quantum supremacy [33]: namely a quantum computer solving a computational task that goes beyond what a classical machine could achieve
This is, in part, driven by the hope that a clear demonstration of quantum supremacy can be performed with a device that is intermediate between the small quantum circuits that can currently be built and a full-scale quantum computer
We show that for any IQP circuit C on n qubits, we can produce a new IQP circuit C on O(n) qubits in polynomial time such that, if depolarising noise is applied to every qubit of the output of C, we can sample from a distribution which is close to p up to arbitrarily small 1 distance
Summary
Over the last few years there has been significant attention devoted to devising experimental demonstrations of quantum supremacy [33]: namely a quantum computer solving a computational task that goes beyond what a classical machine could achieve. There are several intermediate quantum computing models which could be used to demonstrate quantum supremacy, including simple linear-optical circuits (the boson sampling problem [1]); the one clean qubit model [31]; and commuting quantum circuits, a model known as “IQP” [41, 10] In each of these cases, it has been shown that efficient classical simulation of the simple quantum computations involved is not possible, assuming that the polynomial hierarchy does not collapse. If p is the distribution that the noise-free quantum circuit would produce, it is hard for the classical machine to sample from any distribution p such that p − p 1 ≤ , for some small , where the size of depends on the conjectures one is willing to assume These results imply that a fault-tolerant implementation of IQP sampling or boson sampling can be made resilient to noise while (potentially) maintaining a quantum advantage. Such long-range interactions incur significant physical resource overheads for most computational architectures
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