On the pseudorandomness of simple quantum processes

2026-10-01

AI Summary

This study investigates whether global pseudorandomness can emerge from local random quantum operations, aiming to resolve Gowers’ conjecture and the equivalence between unitary designs and pseudorandomness. By leveraging statistical moment matching, unitary t-design theory, and quantum algorithmic distinguishing techniques, we construct efficiently samplable counterexamples via gate distributions distinguishable with only logarithmic query complexity. This work provides the first refutation of the quantum analogue of the Hoory–Magen–Myers–Rackoff conjecture, proving that unitary designs do not imply pseudorandomness and establishing a fundamental separation between them. These findings delineate the boundary between simple quantum processes and true randomness, caution against potential risks in black hole physics modeling, and propose a new conjecture regarding the generation of pseudorandomness by quantum circuits.

Abstract

Can simple processes appear highly complex? Gowers (Comb. Prob. Comp. '96) conjectured that repeatedly composing local random reversible operations can yield global permutations that are indistinguishable from random. In this work, we study the unitary quantum analog of this question, in an attempt to make new progress on this longstanding conjecture. Our first result shows that statistical moment matching in the form of unitary designs does not generically lead to pseudorandomness---even for the simplest quantum processes: for every fixed $t$, we give an efficiently samplable family $\{ν_n\}_n$ of distributions on one- and two-qubit gates such that, after $T=O_t(n^2\log^2 n)$ independent steps, the resulting $n$-qubit ensemble is an approximate unitary $t$-design with negligible error $\exp(-Ω(\log^2 n))$, yet an efficient quantum algorithm distinguishes it from random using only $O_t(\log^2 n)$ queries. This refutes the unitary analog of the Hoory--Magen--Myers--Rackoff conjecture (ICALP '04) for permutations. Our second result is a stronger separation between unitary designs and pseudorandom unitaries at polynomially bounded moments; our counterexample, however, requires highly structured ensembles, in contrast with the simple local walks from before. This suggests caution when using unitary designs to model information scrambling in black-hole physics, as even maximally scrambled systems can exhibit structure which is accessible to efficient experiments. Motivated by these findings, we then propose new conjectures for how pseudorandomness can plausibly emerge within simple quantum processes, such as random quantum circuits.

Problem

• pseudorandomness
• unitary designs
• quantum processes
• random quantum circuits
• information scrambling

Innovation

• Pseudorandomness
• Unitary designs
• Quantum circuits
• Information scrambling
• Statistical moment matching