Pseudorandom unitaries with non-adaptive security

TL;DR

This paper introduces a new construction of pseudorandom unitaries that are indistinguishable from Haar random unitaries by quantum polynomial-time algorithms, assuming quantum-secure one-way functions, with proven security against non-adaptive distinguishers.

Contribution

The paper presents a simple, efficient PRU construction combining Clifford unitaries, pseudorandom phases, and permutations, with proven security against non-adaptive quantum distinguishers.

Findings

Secure against non-adaptive distinguishers assuming quantum-secure one-way functions

Construction is efficient and implementable

Conjectured security against adaptive distinguishers

Abstract

Pseudorandom unitaries (PRUs) are ensembles of efficiently implementable unitary operators that cannot be distinguished from Haar random unitaries by any quantum polynomial-time algorithm with query access to the unitary. We present a simple PRU construction that is a concatenation of a random Clifford unitary, a pseudorandom binary phase operator, and a pseudorandom permutation operator. We prove that this PRU construction is secure against non-adaptive distinguishers assuming the existence of quantum-secure one-way functions. This means that no efficient quantum query algorithm that is allowed a single application of $U^{\otimes \mathrm{poly}(n)}$ can distinguish whether an $n$-qubit unitary $U$ was drawn from the Haar measure or our PRU ensemble. We conjecture that our PRU construction remains secure against adaptive distinguishers, i.e. secure against distinguishers that can query the unitary polynomially many times in sequence, not just in parallel.