On Scalable Pseudorandom Unitaries and the Unitary Synthesis Problem

2026-05-11

AI Summary

This work investigates the existence of scalable pseudorandom unitaries (PRUs) and their connection to the unitary synthesis problem. In the random oracle model (ROM), it establishes for the first time a theoretical link between statistically secure ROM-PRUs and unitary synthesis, proving that any unitary synthesis algorithm must query a classical oracle on inputs of length at least $(2 - o(1))\log d$. This result rules out the security of all known candidates for scalable PRUs. By integrating tools from approximate unitary designs, $\varepsilon$-nets over the unitary group, and quantum complexity theory, the paper positions ROM-PRUs as an idealized model for analyzing pseudorandom unitary operators and demonstrates that existing constructions fail to meet the required security guarantees.

Abstract

We consider the task of constructing pseudorandom unitaries (PRUs) with scalable security, i.e. families in which the security parameter may vary independently of the dimension (or input bit-length). It is not known whether scalable PRUs can be constructed. In this work we show that if scalable PRUs can be constructed via the prevailing paradigm for analyzing PRUs, then there would be a positive solution to the Aaronson-Kuperberg unitary synthesis problem, a longstanding question in quantum complexity theory about whether implementing arbitrary unitaries can be efficiently reduced to computing a Boolean function. Specifically, we formalize the notion of ROM-PRUs, which are statistically secure PRUs in the random oracle model (ROM). All prior known constructions of cryptographically secure PRUs are based on a ROM-PRU construction. We prove novel connections between ROM-PRUs, approximate unitary designs, epsilon-nets over the unitary group, and the unitary synthesis problem. In particular, we prove that any unitary synthesis algorithm (and thus any ROM-PRU) must use a classical oracle with input length (2 - o(1)) log d bits, where d is the dimension of the unitary to be implemented. This bound rules out all existing candidates for scalable PRUs in the literature. Together, these connections indicate that ROM-PRUs provide a fruitful idealized model for studying pseudorandom unitaries.

Problem

• pseudorandom unitaries
• scalable security
• unitary synthesis problem
• random oracle model
• quantum complexity theory

Innovation

• pseudorandom unitaries
• unitary synthesis problem
• random oracle model
• approximate unitary designs
• scalable security