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 -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~O(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~O(nk) is both necessary and sufficient for the ensemble to form a relative \epsilon-approximate k-design. Similarly, t ~ O(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.

Personal information

Salvatore F. E. Oliviero is a postdoctoral researcher in Quantum Information Theory at Scuola Normale Superiore di Pisa. He earned his PhD under Alioscia Hamma at UMass Boston. His research covers quantum chaos and information scrambling; resource-theoretic approaches to magic and entanglement; quantum learning theory; and the role of computational complexity in quantum information. He investigates how quantum complexity underpins differences between classical vs quantum states in information processing.