Quantum supremacy and hardness of estimating output probabilities of quantum circuits

TL;DR

This paper proves that approximating output probabilities of random quantum circuits is computationally hard for classical computers, supporting the claim of quantum supremacy through complexity theory under standard assumptions.

Contribution

It establishes the hardness of estimating output probabilities of random quantum circuits within specific error bounds, extending to various circuit depths and architectures, using novel proof techniques.

Findings

Approximating output probabilities is (P) hard under standard complexity assumptions.

Hardness extends to circuits of various depths including constant, (n), and (/2n) depths.

Results apply to arbitrary circuits, including trivial ones, and do not rely on standard proof techniques.

Abstract

Motivated by the recent experimental demonstrations of quantum supremacy, proving the hardness of the output of random quantum circuits is an imperative near term goal. We prove under the complexity theoretical assumption of the non-collapse of the polynomial hierarchy that approximating the output probabilities of random quantum circuits to within $\exp(-\Omega(m\log m))$ additive error is hard for any classical computer, where $m$ is the number of gates in the quantum computation. More precisely, we show that the above problem is $\#\mathsf{P}$-hard under $\mathsf{BPP}^{\mathsf{NP}}$ reduction. In the recent experiments, the quantum circuit has $n$-qubits and the architecture is a two-dimensional grid of size $\sqrt{n}\times\sqrt{n}$. Indeed for constant depth circuits approximating the output probabilities to within $2^{-\Omega(n\log{n})}$ is hard. For circuits of depth $\log{n}$ or $\sqrt{n}$ for which the anti-concentration property holds, approximating the output probabilities to within $2^{-\Omega(n\log^2{n})}$ and $2^{-\Omega(n^{3/2}\log n)}$ is hard respectively. We then show that the hardness results extend to any open neighborhood of an arbitrary (fixed) circuit including the trivial circuit with identity gates. We made an effort to find the best proofs and proved these results from first principles, which do not use the standard techniques such as the Berlekamp--Welch algorithm, the usual Paturi's lemma, and Rakhmanov's result.