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The non-Clifford cost of random unitaries

Lorenzo Leone, Salvatore F. E. Oliviero, Alioscia Hamma, Jens Eisert, Lennart Bittel

TL;DR

We address the cost of generating quantum randomness using $t$-doped Clifford circuits on $n$ qubits, focusing on unitary $k$-designs and pseudo-random unitaries. By developing a doped-Clifford Weingarten framework and computing the twirling operator, we derive tight convergence bounds for both the frame potential and unitary designs, revealing a quadratic non-Clifford cost $t = \Theta(k^2)$ for Haar-like frame spread and a linear cost $t = \tilde{\Theta}(nk)$ for relative-error unitary $k$-designs (with a regime where $t = \tilde{\Theta}(n)$ suffices for pseudo-randomness). The analysis shows a fundamental trade-off between the strength of the approximation notion (additive vs relative) and the required non-Clifford resources, with relative-error designs becoming impractically costly in large regimes, implying limitations on classical simulability. The doped-Clifford Weingarten functions provide analytic tools for expressing the twirl over these ensembles and their asymptotics, offering a rigorous pathway to quantify randomness generation costs. These results have implications for fault-tolerant designs and the feasibility of implementing high-order random unitaries in quantum devices, as well as for constructing efficient approximations and net coverings in high-dimensional unitary spaces.

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, $t = \tildeΘ(k^2)$, 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 $t = \tildeΘ(nk)$ is both necessary and sufficient for the ensemble to form a relative $\varepsilon$-approximate $k$-design. Similarly, $t = \tildeΘ(n)$ 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.

The non-Clifford cost of random unitaries

TL;DR

We address the cost of generating quantum randomness using -doped Clifford circuits on qubits, focusing on unitary -designs and pseudo-random unitaries. By developing a doped-Clifford Weingarten framework and computing the twirling operator, we derive tight convergence bounds for both the frame potential and unitary designs, revealing a quadratic non-Clifford cost for Haar-like frame spread and a linear cost for relative-error unitary -designs (with a regime where suffices for pseudo-randomness). The analysis shows a fundamental trade-off between the strength of the approximation notion (additive vs relative) and the required non-Clifford resources, with relative-error designs becoming impractically costly in large regimes, implying limitations on classical simulability. The doped-Clifford Weingarten functions provide analytic tools for expressing the twirl over these ensembles and their asymptotics, offering a rigorous pathway to quantify randomness generation costs. These results have implications for fault-tolerant designs and the feasibility of implementing high-order random unitaries in quantum devices, as well as for constructing efficient approximations and net coverings in high-dimensional unitary spaces.

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 -doped Clifford circuits on qubits, consisting of Clifford circuits interspersed with single-qubit non-Clifford gates. We establish rigorous convergence bounds towards unitary -designs, revealing the intrinsic cost in terms of non-Clifford resources in various flavors. First, we analyze the -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, , 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 -designs. Second, we derive tight bounds on the convergence of -doped Clifford circuits towards relative-error -designs, showing that is both necessary and sufficient for the ensemble to form a relative -approximate -design. Similarly, 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.
Paper Structure (22 sections, 35 theorems, 139 equations, 1 figure)

This paper contains 22 sections, 35 theorems, 139 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of the results
  3. The spreading of doped Clifford circuits via frame potential
  4. Implications for approximate state $k$-designs
  5. The twirling operator over doped Clifford circuits
  6. Convergence of doped Clifford circuits to relative unitary designs
  7. Remarks on $t$-doped Clifford circuits generating pseudo-random unitaries
  8. Discussion, outlook, and open questions
  9. Structure of this work
  10. Preliminaries
  11. Pauli operators
  12. Unitary $k$-designs
  13. State $k$-designs and state frame potential
  14. Haar average over the Clifford group
  15. Twirling operator over the doped-Clifford group
  16. ...and 7 more sections

Key Result

Theorem 1

Let $n\ge \Omega(k^2+t\log k)$. There exists two constants $A,B=\Theta(1)$ such that the difference between the frame of $t$-doped Clifford circuits $\mathcal{F}_{t}^{(k)}$ and the Haar frame potential $\mathcal{F}_{\operatorname{Haar}}^{(k)}$ is upper and lower bounded as

Figures (1)

  • Figure 1: Pictorial representation of $t$-doped Clifford circuits. The orange gates are Clifford gates, while the teal gates are single qubit non-Clifford gates. Hence, non-Clifford gates are interleaved within a circuit that predominantly consists of feasible Clifford operations.

Theorems & Definitions (66)

  • Theorem : Frame potential convergence. Informal of \ref{['th:framepotentialconvergence']}
  • Theorem : Doped Clifford circuits producing state $k$-designs. See \ref{['lem:relativeframepotentialandframepotentialtdoped', 'cor:statekdesignconvergence']}
  • Theorem : Optimal convergence to relative unitary designs
  • Proposition 1: Pseudo-random $t$-doped Clifford circuits
  • Definition 1: Clifford group
  • Definition 2: Stabilizer states
  • Definition 3: $k$-fold channel
  • Lemma 1: Weingarten calculus
  • Lemma 2: Asymptotics of Weingarten functions weingarten_asymptotic_1978
  • Definition 4: Unitary $k$-design
  • ...and 56 more