Non-Haar random circuits form unitary designs as fast as Haar random circuits

The unitary design formation in random circuits has attracted considerable attention due to its wide range of practical applications and relevance to fundamental physics. While the formation rates in Haar random circuits have been extensively studied in previous works, it remains an open question how these rates are affected by the choice of local randomizers. In this work, we prove that the circuit depths required for general non-Haar random circuits to form unitary designs are upper bounded by those for the corresponding Haar random circuits, up to a constant factor independent of the system size. This result is derived in a broad range of circuit structures, including one- and higher-dimensional lattices, geometrically non-local configurations, and even extremely shallow circuits with patchwork architectures. We provide specific applications of these results in randomized benchmarking and random circuit sampling, and also discuss their implications for quantum many-body physics. Our work lays the foundation for flexible and robust randomness generation in real-world experiments, and offers new insights into chaotic dynamics in complex quantum systems.

💡 Research Summary

The paper addresses a fundamental question in quantum information: how does the choice of local random gates affect the rate at which random quantum circuits converge to unitary t‑designs? While the convergence of Haar‑random circuits has been thoroughly characterized, real‑world devices typically employ discrete, non‑Haar gate sets. The authors prove that for a broad class of circuit architectures—including single‑layer connected, multilayer‑connected (e.g., brickwork), and patchwork circuits—the depth required to achieve an ε‑approximate unitary t‑design with non‑Haar local gates is bounded above by the depth required for the corresponding Haar‑random circuit, up to a constant factor that depends only on the spectral gap of the two‑qubit gate ensemble and is independent of the system size N.

The technical core rests on two observations. First, the moment operator M^{(t)} of any ensemble can be decomposed into the Haar projector plus a residual term whose norm is directly linked to the spectral gap Δ^{(t)}. Lemma 1 shows that after L convolutions the deviation from Haar decays as exp(−L ∑Δ^{(t)}i). For a single‑layer connected circuit the authors lower‑bound Δ^{(t)} by the product of the minimal two‑qubit gap Δ^{(t)}{loc,ν} and the known Haar gap Δ^{(t)}H, yielding a depth L = 2·(Δ^{(t)}{loc,ν})^{−1}·L_H.

Second, for fixed‑architecture multilayer circuits the classic detectability lemma, previously applicable only to Haar gates, is extended to non‑Haar ensembles. This yields a lower bound Δ^{(t)}A ≥ (Δ^{(t)}{loc,ν_A})^{l}·L_B(t)_A for an l‑layer block, where L_B(t)A is the Haar‑circuit block gap from prior work. Consequently the overall depth becomes L_A = (Δ^{(t)}{loc,ν_A})^{−l}·L_H^A.

Finally, the patchwork construction, which stitches together small random subcircuits of size O(log N), is shown to inherit the same depth scaling: each patch needs only a constant‑factor larger depth than its Haar counterpart (the factor being (Δ^{(t)}_{loc,ν_bw})^{−2}). Thus O(log N) depth suffices even with non‑Haar gates.

The results imply that as long as the local gate ensemble possesses a non‑vanishing spectral gap—true for any universal gate set augmented with the identity—the formation of unitary designs is essentially insensitive to whether the gates are drawn from the Haar measure. This has immediate practical impact: randomized benchmarking, random circuit sampling, and studies of quantum chaos (scrambling, operator growth) can be performed with experimentally realistic gate sets without sacrificing the asymptotic efficiency of design generation. The work therefore bridges a gap between idealized theoretical models and the constraints of near‑term quantum hardware, providing a robust foundation for flexible randomness generation in many‑body quantum systems.