Quantum State Learning Implies Circuit Lower Bounds

TL;DR

This paper links quantum state tomography and synthesis to circuit lower bounds, showing that efficient algorithms for distinguishing or learning quantum states imply significant complexity-theoretic consequences.

Contribution

It establishes a novel connection between quantum state learning algorithms and circuit lower bounds, extending classical complexity results to the quantum setting.

Findings

Efficient quantum state distinguishing algorithms imply circuit lower bounds.

Quantum state tomography complexity relates to quantum state synthesis classes.

Distinguishing algorithms can lead to circuit lower bounds for decision problems.

Abstract

We establish connections between state tomography, pseudorandomness, quantum state synthesis, and circuit lower bounds. In particular, let $\mathfrak{C}$ be a family of non-uniform quantum circuits of polynomial size and suppose that there exists an algorithm that, given copies of $|\psi \rangle$, distinguishes whether $|\psi \rangle$ is produced by $\mathfrak{C}$ or is Haar random, promised one of these is the case. For arbitrary fixed constant $c$, we show that if the algorithm uses at most $O(2^{n^c})$ time and $2^{n^{0.99}}$ samples then $\mathsf{stateBQE} \not\subset \mathsf{state}\mathfrak{C}$. Here $\mathsf{stateBQE} := \mathsf{stateBQTIME}[2^{O(n)}]$ and $\mathsf{state}\mathfrak{C}$ are state synthesis complexity classes as introduced by Rosenthal and Yuen (ITCS 2022), which capture problems with classical inputs but quantum output. Note that efficient tomography implies a similarly efficient distinguishing algorithm against Haar random states, even for nearly exponential-time algorithms. Because every state produced by a polynomial-size circuit can be learned with $2^{O(n)}$ samples and time, or $O(n^{\omega(1)})$ samples and $2^{O(n^{\omega(1)})}$ time, we show that even slightly non-trivial quantum state tomography algorithms would lead to new statements about quantum state synthesis. Finally, a slight modification of our proof shows that distinguishing algorithms for quantum states can imply circuit lower bounds for decision problems as well. This help sheds light on why time-efficient tomography algorithms for non-uniform quantum circuit classes has only had limited and partial progress. Our work parallels results by Arunachalam et al. (FOCS 2021) that revealed a similar connection between quantum learning of Boolean functions and circuit lower bounds for classical circuit classes, but modified for the purposes of state tomography and state synthesis.