Pseudorandom unitaries with non-adaptive security
Tony Metger, Alexander Poremba, Makrand Sinha, Henry Yuen
TL;DR
The paper addresses constructing pseudorandom unitaries (PRUs) that cannot be distinguished from Haar-random unitaries by quantum polynomial-time attackers with parallel queries. It proposes a simple, efficiently implementable PRU: $U_k = P_{k_1} F_{k_2} C_{k_3}$, formed from a quantum-secure pseudorandom permutation, a quantum-secure pseudorandom binary phase, and a random Clifford unitary; under quantum-secure one-way functions, this construction is secure against non-adaptive distinguishers. The key technical approach combines Schur-Weyl duality with $t$-wise twirls, showing that random Clifford projection pushes inputs into a distinct subspace and that the permutation-phase twirl closely mimics the Haar $t$-twirl on this subspace, yielding a trace-distance bound of $O(t/\sqrt{2^n})$ for polynomial $t$. The authors discuss extending to adaptive security and exploring applications in quantum cryptography and modeling chaotic quantum dynamics, highlighting PRUs as a foundational primitive bridging Haar randomness and efficient quantum operations. Overall, the work provides a concrete, security-guaranteed PRU construction under standard cryptographic assumptions and advances the theoretical understanding of quantum pseudorandomness.
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.