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 mapping the superoperator that describes t order moments on n qubits to a multilevel SU(4^{t}) Lipkin-Meshkov-Glick Hamiltonian. We show that, for arbitrary fixed t, the ground-state manifold is exactly spanned by factorized eigenstates and, under the assumption that a mean-field ansatz accurately describes the low-lying excitations, the spectral gap scales as 1/n in the thermodynamic limit. Our results imply that random quantum circuits yield an efficient implementation of ϵ approximate unitary t designs.
Random many-body states and matrices are both a useful tool to model certain physical systems and an important asset for quantum computation. Realizing them, however, generally requires an exponential (in system size) amount of resources. Recent research has presented a way out by showing that one can generate random states and matrices, or more precisely a controlled approximation of them, by applying a quantum circuit built in terms of few-body unitary gates. Most of this research has been focussed on the case of quantum circuits composed by fully random unitary gates. Here we consider what happens for circuits that, instead, involve a minimal degree of randomness. Specifically, we concentrate on two different settings: (a) brickwork quantum circuits with a single one-qudit random matrix at a boundary; (b) brickwork quantum circuits with fixed interactions but random one-qudit gates everywhere. We show that, for any given initial state, (a) and (b) produce a distribution of states approaching the Haar distribution in the limit of large circuit depth. More precisely, we show that the moments of the distribution produced by our circuits can approximate the ones of the Haar distribution in a depth proportional to the system size. Interestingly we find that in both Cases (a) and (b) the relaxation to the Haar distribution occurs in two steps—this is in contrast with what happens in fully random circuits. Moreover, we show that choosing appropriately the fixed interactions, for example taking the local gate to be a dual-unitary gate with high enough entangling power, minimally random circuits produce a Haar random distribution more rapidly than fully random circuits. In particular, dual-unitary circuits with maximal entangling power—i.e. perfect tensors—appear to provide the optimal quantum state design preparation for any design number. Our results also suggest that the minimally random settings (a) and (b) give an optimal framework to prepare unitary designs, i.e., random unitary matrices.
Unitary
T-designs play an important role in quantum information, with diverse applications in quantum algorithms, benchmarking, tomography, and communication. Until now, the most efficient construction of unitary
T-designs for
n-qudit systems has been via random local quantum circuits, which have been shown to converge to approximate
T-designs in the diamond norm using
O(T5+o(1)n2) quantum gates. In this work, we provide a new construction of
T-designs via random matrix theory using
O~(T2n2) quantum gates. Our construction leverages two key ideas. First, in the spirit of central limit theorems, we approximate the Gaussian Unitary Ensemble (GUE) by an i.i.d. sum of random Hermitian matrices. Second, we show that the product of just two exponentiated GUE matrices is already approximately Haar random. Thus, multiplying two exponentiated sums over rather simple random matrices yields a unitary
T-design, via Hamiltonian simulation. A central feature of our proof is a new connection between the polynomial method in quantum query complexity and the large-dimension (
N) expansion in random matrix theory. In particular, we show that the polynomial method provides exponentially improved bounds on the high moments of certain random matrix ensembles, without requiring intricate Weingarten calculations. In doing so, we define and solve a new type of moment problem on the unit circle, asking whether a finite number of equally weighted points, corresponding to eigenvalues of unitary matrices, can reproduce a given set of moments.
We explore the implementation of pseudorandom single-qubit rotations and multiqubit pseudorandom circuits constructed only from Clifford gates and the
T gate, a phase rotation of $\ensuremath{\pi}/4$. Such a gate set would be appropriate for computations performed in a fault tolerant setting. For single-qubit rotations the distribution of parameters found for unitaries constructed from Clifford plus
T quickly approaches that of random rotations and requires significantly fewer gates than the construction of arbitrary single-qubit rotations. For Clifford-plus-
T pseudorandom circuits we find an exponential convergence to a random matrix element distribution and a Gaussian convergence to the higher-order moments of the matrix element distribution. In addition, the nearest-neighbor eigenangle statistics distribution almost immediately converges to that of random unitary matrices. All of these convergence rates are found to be insensitive to the number of qubits.
We study the amount of interference in random quantum algorithms using a recently derived quantitative measure of interference. To this end we introduce two random circuit ensembles composed of random sequences of quantum gates from a universal set, mimicking quantum algorithms in the quantum circuit representation. We show numerically that, concerning the interference distribution and the level spacing distribution, these ensembles converge to the well-known circular unitary ensemble (CUE) for general complex quantum algorithms, and to the Haar orthogonal ensemble (HOE) for real quantum algorithms. We provide exact analytical formulas for the average and typical interference in the circular ensembles, and show that for sufficiently large numbers of qubits a random quantum algorithm uses with probability close to one an amount of interference approximately equal to the dimension of the Hilbert space. As a by-product, we offer a new way of constructing approximate random unitary operators from the Haar measures of CUE or HOE in a high dimensional Hilbert space using universal sets of quantum gates.
We consider quantum circuit models where the gates are drawn from arbitrary gate ensembles given by probabilistic distributions over certain gate sets and circuit architectures, which we call stochastic quantum circuits. Of main interest in this work is the speed of convergence of stochastic circuits with different gate ensembles and circuit architectures to unitary t-designs. A key motivation for this theory is the varying preference for different gates and circuit architectures in different practical scenarios. In particular, it provides a versatile framework for devising efficient circuits for implementing
t-designs and relevant applications including random circuit and scrambling experiments, as well as benchmarking the performance of gates and circuit architectures. We examine various important settings in depth. A key aspect of our study is an "ironed gadget" model, which allows us to systematically evaluate and compare the convergence efficiency of entangling gates and circuit architectures. Particularly notable results include i) gadgets of two-qubit gates with KAK coefficients
(4π−81arccos(51),8π,81arccos(51)) (which we call
χ gates) directly form exact 2- and 3-designs; ii) the iSWAP gate family achieves the best efficiency for convergence to 2-designs under mild conjectures with numerical evidence, even outperforming the Haar-random gate, for generic many-body circuits; iii) iSWAP + complete graph achieve the best efficiency for convergence to 2-designs among all graph circuits. A variety of numerical results are provided to complement our analysis. We also derive robustness guarantees for our analysis against gate perturbations. Additionally, we provide cursory analysis on gates with higher locality and found that the Margolus gate outperforms various other well-known gates.