Quantum learning algorithms imply circuit lower bounds

TL;DR

This paper establishes a fundamental link between quantum learning algorithms and circuit lower bounds, showing that improvements in quantum learning could lead to breakthroughs in computational complexity theory.

Contribution

It introduces the first connection between quantum PAC learning algorithms and circuit lower bounds, extending classical techniques to the quantum setting with new constructions.

Findings

Quantum learning algorithms imply circuit lower bounds under certain conditions.

Constructed the first quantum-secure pseudorandom generator.

Extended local list-decoding algorithms to quantum circuits.

Abstract

We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, let $\mathfrak{C}$ be a class of polynomial-size concepts, and suppose that $\mathfrak{C}$ can be PAC-learned with membership queries under the uniform distribution with error $1/2 - \gamma$ by a time $T$ quantum algorithm. We prove that if $\gamma^2 \cdot T \ll 2^n/n$, then $\mathsf{BQE} \nsubseteq \mathfrak{C}$, where $\mathsf{BQE} = \mathsf{BQTIME}[2^{O(n)}]$ is an exponential-time analogue of $\mathsf{BQP}$. This result is optimal in both $\gamma$ and $T$, since it is not hard to learn any class $\mathfrak{C}$ of functions in (classical) time $T = 2^n$ (with no error), or in quantum time $T = \mathsf{poly}(n)$ with error at most $1/2 - \Omega(2^{-n/2})$ via Fourier sampling. In other words, even a marginal improvement on these generic learning algorithms would lead to major consequences in complexity theory. Our proof builds on several works in learning theory, pseudorandomness, and computational complexity, and crucially, on a connection between non-trivial classical learning algorithms and circuit lower bounds established by Oliveira and Santhanam (CCC 2017). Extending their approach to quantum learning algorithms turns out to create significant challenges. To achieve that, we show among other results how pseudorandom generators imply learning-to-lower-bound connections in a generic fashion, construct the first conditional pseudorandom generator secure against uniform quantum computations, and extend the local list-decoding algorithm of Impagliazzo, Jaiswal, Kabanets and Wigderson (SICOMP 2010) to quantum circuits via a delicate analysis. We believe that these contributions are of independent interest and might find other applications.