The non-Clifford cost of random unitaries

• URL: http://arxiv.org/abs/2505.10110v2
• Date: Mon, 26 May 2025 09:52:57 GMT
• Title: The non-Clifford cost of random unitaries
• Authors: Lorenzo Leone, Salvatore F. E. Oliviero, Alioscia Hamma, Jens Eisert, Lennart Bittel,
• Abstract summary: We explore the ensemble of $t$-doped Clifford circuits on $n$ qubits.<n>We establish rigorous convergence bounds towards unitary $k$-designs.<n>We derive analytic expressions for the operator twirling over the ensemble of random doped Clifford circuits.
• Score: 0.2796197251957244
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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