On the average-case complexity of learning output distributions of quantum circuits

TL;DR

This paper demonstrates that learning the output distributions of brickwork random quantum circuits is computationally hard in the average case, requiring exponentially many queries even at moderate depths, with implications for quantum complexity theory.

Contribution

It establishes average-case hardness results for learning quantum circuit output distributions in the statistical query model, confirming a conjecture by Aaronson and Chen.

Findings

Learning becomes super-polynomially hard at super-logarithmic depths.

Exponential query complexity at linear depth for high success probability.

Double-exponential query complexity at infinite depth.

Abstract

In this work, we show that learning the output distributions of brickwork random quantum circuits is average-case hard in the statistical query model. This learning model is widely used as an abstract computational model for most generic learning algorithms. In particular, for brickwork random quantum circuits on $n$ qubits of depth $d$, we show three main results: - At super logarithmic circuit depth $d=\omega(\log(n))$, any learning algorithm requires super polynomially many queries to achieve a constant probability of success over the randomly drawn instance. - There exists a $d=O(n)$, such that any learning algorithm requires $\Omega(2^n)$ queries to achieve a $O(2^{-n})$ probability of success over the randomly drawn instance. - At infinite circuit depth $d\to\infty$, any learning algorithm requires $2^{2^{\Omega(n)}}$ many queries to achieve a $2^{-2^{\Omega(n)}}$ probability of success over the randomly drawn instance. As an auxiliary result of independent interest, we show that the output distribution of a brickwork random quantum circuit is constantly far from any fixed distribution in total variation distance with probability $1-O(2^{-n})$, which confirms a variant of a conjecture by Aaronson and Chen.