Non-Clifford Cost of Random Unitaries

TL;DR

This paper analyzes the non-Clifford resource cost in random quantum circuits, establishing bounds on how many non-Clifford gates are needed for various levels of unitary design approximation, highlighting the high cost of generating pseudo-random unitaries.

Contribution

It provides rigorous bounds on the non-Clifford resources required for random circuit ensembles to approximate unitary designs, refining understanding of their complexity and classical intractability.

Findings

Quadratic doping level $t = ilde{\Theta}(k^2)$ suffices for unitary $k$-designs.

Linear doping $t = ilde{\Theta}(nk)$ is necessary for relative-error $k$-designs.

Generating pseudo-random unitaries requires doping $t = \tilde{\Theta}(n)$.

Abstract

Recent years have enjoyed a strong interest in exploring properties and applications of random quantum circuits. In this work, we explore the ensemble of $t$-doped Clifford circuits on $n$ qubits, consisting of Clifford circuits interspersed with $t$ single-qubit non-Clifford gates. We establish rigorous convergence bounds towards unitary $k$-designs, revealing the intrinsic cost in terms of non-Clifford resources in various flavors. First, we analyze the $k$-th order frame potential, which quantifies how well the ensemble of doped Clifford circuits is spread within the unitary group. We prove that a quadratic doping level, $t = \tilde{\Theta}(k^2)$, is both necessary and sufficient to approximate the frame potential of the full unitary group. As a consequence, we refine existing upper bounds on the convergence of the ensemble towards state $k$-designs. Second, we derive tight bounds on the convergence of $t$-doped Clifford circuits towards relative-error $k$-designs, showing that $t = \tilde{\Theta}(nk)$ is both necessary and sufficient for the ensemble to form a relative $\varepsilon$-approximate $k$-design. Similarly, $t = \tilde{\Theta}(n)$ is required to generate pseudo-random unitaries. All these results highlight that generating random unitaries is extremely costly in terms of non-Clifford resources, and that such ensembles fundamentally lie beyond the classical simulability barrier. Additionally, we introduce doped-Clifford Weingarten functions to derive analytic expressions for the twirling operator over the ensemble of random doped Clifford circuits, and we establish their asymptotic behavior in relevant regimes.