Abstract
Recently, Google announced the first demonstration of quantum computational supremacy with a programmable superconducting processor. Their demonstration is based on collecting samples from the output distribution of a noisy random quantum circuit, then applying a statistical test to those samples called Linear Cross-Entropy Benchmarking (Linear XEB). This raises a theoretical question: How hard is it for a classical computer to spoof the results of the Linear XEB test? In this short note, we adapt an analysis of Aaronson and Chen to prove a conditional hardness result for Linear XEB spoofing. Specifically, we show that the problem is classically hard, assuming that there is no efficient classical algorithm that, given a random $n$-qubit quantum circuit $C$, estimates the probability of $C$ outputting a specific output string, say $0^n$, with mean squared error even slightly better than that of the trivial estimator that always estimates $1/2^n$. Our result automatically encompasses the case of noisy circuits.
Highlights
Quantum computational supremacy refers to the solution of a well-defined computational task by a programmable quantum computer in significantly less time than is required by the best known algorithms running on existing classical computers, for reasons of asymptotic scaling
A research team based at Google has announced a demonstration of quantum computational supremacy, by sampling the output distributions of random quantum circuits [3]
While there is some support for the conjecture that no classical algorithm can efficiently sample from the output distribution of a random quantum circuit [5], less is known about the hardness of directly spoofing a test like Linear XEB
Summary
Quantum computational supremacy refers to the solution of a well-defined computational task by a programmable quantum computer in significantly less time than is required by the best known algorithms running on existing classical computers, for reasons of asymptotic scaling. A research team based at Google has announced a demonstration of quantum computational supremacy, by sampling the output distributions of random quantum circuits [3] To verify that their circuits were working correctly, they tested their samples using Linear Cross-Entropy Benchmarking (Linear XEB). While there is some support for the conjecture that no classical algorithm can efficiently sample from the output distribution of a random quantum circuit [5], less is known about the hardness of directly spoofing a test like Linear XEB. QUATH states that it is impossible for a polynomial-time classical algorithm to guess whether a specific output string like 0n has greater-than-median probability of being observed as the output of a given n-qubit quantum circuit, with success probability 1/2 + Ω(1/2n). Our result provides some explanation for why these improvements had to target the general problem of amplitude estimation, rather than doing anything specific to the problem of spoofing Linear XEB
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call