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Non-Clifford Cost of Random Unitaries

Lorenzo Leone1,2,*, Salvatore F.E. Oliviero3,2,†, Alioscia Hamma4,5,‡, Jens Eisert2,§, and Lennart Bittel2,‖

  • *Contact author: loleone@unisa.it
  • †Contact author: s.oliviero@fu-berlin.de
  • ‡Contact author: alioscia.hamma@unina.it
  • §Contact author: jense@fu-berlin.de
  • ‖Contact author: l.bittel@fu-berlin.de
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PRX Quantum 7, 020321 – Published 4 May, 2026

DOI: https://doi.org/10.1103/25v1-my1x

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 toward unitary k-designs, revealing the intrinsic cost in terms of non-Clifford resources in various flavors. First, we analyze the kth 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=Θ~(k2), 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 toward state k-designs. Second, we derive tight bounds on the convergence of t-doped Clifford circuits toward relative-error k-designs, showing that t=Θ~(nk) is both necessary and sufficient for the ensemble to form a relative ε-approximate k-design. Similarly, t=Θ~(n) is required to generate pseudorandom 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 analyze high-order 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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Popular Summary

Randomness is a foundational feature of quantum mechanics and an essential resource across quantum information science, underpinning applications such as cryptography, simulation, benchmarking, and quantum algorithms. However, generating truly random unitaries drawn from the uniform (Haar) distribution is infeasible for large quantum systems due to the exponential number of gates required. To overcome this, quantum protocols often rely on unitary k-designs, structured ensembles of unitaries that replicate the first k statistical moments of the Haar distribution. A related but more relaxed notion is computational pseudorandomness, where unitaries are indistinguishable from Haar-random ones under efficient computational tests.

This work investigates the power and limitations of generating such randomness using quantum circuits composed primarily of Clifford gates, which are efficient and fault-tolerant, together with a small number t of non-Clifford gates—referred to as non-Clifford doping. It introduces a general framework for computing ensemble averages over doped Clifford circuits, enabling precise quantification of how well these ensembles approximate ideal randomness. The results demonstrate that, to approximate the frame potential rigorous measure of closeness to a k-designan amount of non-Clifford doping that scales quadratically with k is both necessary and sufficient. In contrast, stronger forms of approximation such as relative-error k-designs require doping that scales linearly with both the number of qubits and the design order.

These findings establish fundamental bounds on the resource cost of generating quantum randomness under fault-tolerant constraints. They show that while additive-error designs can be achieved with modest non-Clifford resources, attaining high-quality randomness as relative error designsneeded for computational pseudorandomness and cryptographic applications requires substantially more. This delineates the practical limits of randomness generation in current and near-term quantum architectures and motivates the development of more resource-efficient circuit constructions that go beyond the regime of classical simulability.

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PRX Quantum 7, 020321– Published 4 May, 2026
  • Received 12 June 2025
  • Revised 15 October 2025
  • Accepted 3 February 2026
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