Local random quantum circuits are approximate polynomial-designs
Fernando G. S. L. Brandao, Aram W. Harrow, Michal Horodecki
TL;DR
This work shows that local random quantum circuits of polynomial size on n qubits form approximate unitary poly(n)-designs, bridging random circuit models with the Haar measure through a chain of techniques: tensor-product expanders, spectral-gap analysis of local Hamiltonians, and Markov-chain coupling. The authors derive explicit bounds for both Haar-uniform and universal gate sets, and extend the results to parallel random circuits, providing scalable construction of pseudo-random unitaries with applications to cryptography, equilibration, and topological order. The approach unifies representation-theoretic and many-body methods, delivering both rigorous design guarantees and insights into the limits of randomness extraction from quantum circuits. These results imply that typical random circuits are hard to distinguish from Haar randomness by shallow or structured observers, enabling robust pseudo-randomness with practical circuit sizes.
Abstract
We prove that local random quantum circuits acting on n qubits composed of O(t^{10} n^2) many nearest neighbor two-qubit gates form an approximate unitary t-design. Previously it was unknown whether random quantum circuits were a t-design for any t > 3. The proof is based on an interplay of techniques from quantum many-body theory, representation theory, and the theory of Markov chains. In particular we employ a result of Nachtergaele for lower bounding the spectral gap of frustration-free quantum local Hamiltonians; a quasi-orthogonality property of permutation matrices; a result of Oliveira which extends to the unitary group the path-coupling method for bounding the mixing time of random walks; and a result of Bourgain and Gamburd showing that dense subgroups of the special unitary group, composed of elements with algebraic entries, are infty-copy tensor-product expanders. We also consider pseudo-randomness properties of local random quantum circuits of small depth and prove that circuits of depth O(t^{10}n) constitute a quantum t-copy tensor-product expander. The proof also rests on techniques from quantum many-body theory, in particular on the detectability lemma of Aharonov, Arad, Landau, and Vazirani. We give applications of the results to cryptography, equilibration of closed quantum dynamics, and the generation of topological order. In particular we show the following pseudo-randomness property of generic quantum circuits: Almost every circuit U of size O(n^k) on n qubits cannot be distinguished from a Haar uniform unitary by circuits of size O(n^{(k-9)/11}) that are given oracle access to U.