Simple constructions of linear-depth t-designs and pseudorandom unitaries
Tony Metger, Alexander Poremba, Makrand Sinha, Henry Yuen
TL;DR
This work tackles the challenge of implementing Haar-random unitaries by constructing efficient ensembles that mimic Haar statistics. The authors introduce the PFC ensemble—composing a random permutation, a random binary phase, and a random Clifford—to reproduce Haar moments to exponential order, enabling both information-theoretic and computational derandomisation. By replacing the random permutation and phase with $t$-wise independent or pseudorandom counterparts, they obtain (i) linear-depth diamond-approximate $t$-designs, (ii) non-adaptive PRUs, and (iii) pseudorandom isometries with adaptive security (PRIs), with a path toward adaptive PRUs under conjecture. The results unify t-designs and PRUs within a single simple framework and have implications for quantum cryptography, complexity, and the modeling of chaotic quantum dynamics, while leaving open questions about relative-error designs and full adaptivity. Key techniques include Schur–Weyl duality, $t$-wise twirls, Clifford twirls, and gate-teleportation-based reductions to distinct-subspaces. The constructions are explicit and scalable, relying on standard cryptographic primitives (quantum-secure PRFs/PRPs) and efficient $t$-wise independent families.
Abstract
Uniformly random unitaries, i.e. unitaries drawn from the Haar measure, have many useful properties, but cannot be implemented efficiently. This has motivated a long line of research into random unitaries that "look" sufficiently Haar random while also being efficient to implement. Two different notions of derandomisation have emerged: $t$-designs are random unitaries that information-theoretically reproduce the first $t$ moments of the Haar measure, and pseudorandom unitaries (PRUs) are random unitaries that are computationally indistinguishable from Haar random. In this work, we take a unified approach to constructing $t$-designs and PRUs. For this, we introduce and analyse the "$PFC$ ensemble", the product of a random computational basis permutation $P$, a random binary phase operator $F$, and a random Clifford unitary $C$. We show that this ensemble reproduces exponentially high moments of the Haar measure. We can then derandomise the $PFC$ ensemble to show the following: (1) Linear-depth $t$-designs. We give the first construction of a (diamond-error) approximate $t$-design with circuit depth linear in $t$. This follows from the $PFC$ ensemble by replacing the random phase and permutation operators with their $2t$-wise independent counterparts. (2) Non-adaptive PRUs. We give the first construction of PRUs with non-adaptive security, i.e. we construct unitaries that are indistinguishable from Haar random to polynomial-time distinguishers that query the unitary in parallel on an arbitary state. This follows from the $PFC$ ensemble by replacing the random phase and permutation operators with their pseudorandom counterparts. (3) Adaptive pseudorandom isometries. We show that if one considers isometries (rather than unitaries) from $n$ to $n + ω(\log n)$ qubits, a small modification of our PRU construction achieves general adaptive security.