Lorenzo Leone, Salvatore F.E. Oliviero, Alioscia Hamma, Jens Eisert, Lennart Bittel (May 16 2025).
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=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 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=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 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.
