On the average-case complexity of learning output distributions of quantum circuits
Alexander Nietner, Marios Ioannou, Ryan Sweke, Richard Kueng, Jens Eisert, Marcel Hinsche, Jonas Haferkamp
TL;DR
This work establishes strong average-case hardness results for learning the output distributions of random quantum circuits in the SQ model, showing that as circuit depth grows beyond logarithmic scales the resource requirements for learning explode (super-polynomial to doubly-exponential in $n$). The authors connect Haar-random behavior to brickwork circuits through unitary-design approximations and Gaussian integration, deriving explicit bounds on the maximally distinguishable fraction and the far-from-uniform probability that drive lower bounds. The results have implications for quantum advantage proposals and the verifiability of sampling tasks, while also offering auxiliary insights that the circuit outputs are typically far from any fixed distribution in total variation distance with high probability. The work also outlines open questions about lower-depth thresholds, tighter constants, and extensions to learning-from-samples, underscoring the nuanced landscape between design-based and Haar-like randomness in quantum circuits.
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 $d=ω(\log(n))$, 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 $Ω(2^n)$ 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 $2^{2^{Ω(n)}}$ many queries to achieve a $2^{-2^{Ω(n)}}$ 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.