We consider a class of random quantum circuits where at each step a gate from a universal set is applied to a random pair of qubits, and determine how quickly averages of arbitrary finite-degree polynomials in the matrix elements of the resulting unitary converge to Haar measure averages. This is accomplished by establishing an exact mapping between the superoperator that describes t-order moments on n qubits and a multilevel SU (4 t ) Lipkin-Meshkov-Glick Hamiltonian. For arbitrary fixed t, we find that the spectral gap scales as 1/n in the thermodynamic limit. Our results imply that random quantum circuits yield an efficient implementation of ǫ-approximate unitary t-designs. [6] . That is, a unitary t-design faithfully simulates the Haar measure with respect to any test that uses at most t copies of a selected n-qubit unitary. Ramifications of the theory of t-designs [8] are being uncovered in problems as different as black hole evaporation and fast "scrambling" of information [9] , efficient quantum tomography and randomized gate benchmarking [10] , quantum channel capacity [11] , and the foundations of quantum statistical mechanics [12] . Prompted by the above advances, significant effort has been devoted recently to identifying efficient constructions of t-designs and characterizing their convergence properties [2, 6, [13] [14] [15] [16] . Harrow and Low established, in particular, the equivalence between approximate 2-designs and random quantum circuits as introduced in [4] , and conjectured that a random circuit consisting of k=poly(n, t) gates from a two-qubit universal gate set yields an approximate t-design [16] . While supporting numerical evidence was gathered in [17] for low-order moments, and efficient constructions of t-designs were reported in [16] for any t = O(n/ log n), the extent to which random quantum circuits could be used to implement an approximate t-design for arbitrary, fixed t remained open. In this Letter, we address this question by determining the rate at which, for sufficiently large circuit depth, statistical moments of arbitrary order converge to their limiting Haar values. Our strategy involves two steps: first, for given t, we show that the asymptotic convergence rate is determined by the spectral gap of a certain superoperator, which encapsulates moments up to order t; next, we compute this gap by mapping the t-moment superoperator to a multilevel version of the Lipkin-Meshkov-Glick (LMG) model, which is known to be exactly solvable, and whose low-energy spectrum is well understood in the thermodynamic limit n → ∞ [18]. Our approach ties together t-design theory with established mean-field techniques from many-body physics, extending earlier results by Znidaric [14] for t = 2. Furthermore, asymptotic convergence rates allow us to upper bound the convergence time (minimum circuit length, k c ) needed for a desired accuracy ǫ relative to the Haar measure to be reached. For any fixed t, we find that the scaling k c ∼ n log(1/ǫ) holds for sufficiently large n and small ǫ. Moment superoperator.− Let a random quantum circuit of length k be a sequence U k . . . U 1 of k unitary operators on an n-qubit Hilbert space H = ⊗ n j H qj , where each U i is selected from an ensemble {µ(U ), U }, for a probability distribution µ with support on a universal gate set. To analyze arbitrary t-order moments, we introduce a Hilbert space H Mt = H ⊗2t , which consists of 2t copies of H and we refer to as the moment space, with dim(H Mt ) ≡ D = N 2t , and a local moment space H lt , which results from grouping factors corresponding to the same qubit in H Mt . That is, t . Moments of order t may be described in terms of the following linear operator on H Mt : Physically, M t [µ] may be viewed as the superoperator induced by the action of t copies of the random circuit on nt-qubit density operators defined on H ⊗t (see Fig.  1 ). In line with standard practice in open-system theory [19], we shall introduce "operator kets" in H Mt , denoted |A ≡ A, and, correspondingly, A| = A † . Thus, a D 2 -dimensional operator ket transforms according to U ⊗t,t |A ≡ U AU † , under U ∈ U (N ). Once a basis for H Mt is chosen, the matrix representation of M t [µ] specifies a complete set of t-order moments.
Recently Brown and Viola REF proposed an interesting approach to the problem of random quantum circuits as unitary t-designs, based on mapping the convergence time of moments of the random circuit to the spectral gap of a mean-field quantum Hamiltonian.
Convergence rates for arbitrary statistical moments of random quantum circuits
Medicine Physics