Quantum circuits consisting of single and two-qubit gates selected at random from a universal gate set are examined. Specifically, the asymptotic rate for large numbers of qubits n and large circuit depth k at which t-order statistical moments of the matrix elements of the resulting random unitary transformation converge to their values with respect to the invariant Haar measure on U(2 ⁿ) are determined. The asymptotic convergence rate is obtained from the spectral gap of a superoperator describing the action of the circuit on t-copies of the system Hilbert space. For a class of random quantum circuits that are reversible and invariant under permutation of the qubit labels, the gap and hence the asymptotic convergence rate is shown to scale as ∼ 1/n for sufficiently large n, with a coefficient that may in general depend on t. Bounds are derived between the convergence rates for a broader class of reversible random quantum circuits and the convergence rates of second order moments of irreversible random quantum circuits are examined through a mapping to a Markov chain.

Weak constraints are constructed for finite moments of matrix elements of local observables with respect to the eigenvectors of general many-body Hamiltonians in the thermodynamic limit. This is accomplished by means of an expansion in terms of polynomials which are orthogonal with respect to the density of states. The way in which such constraints are satisfied is explored in connection to non-integrability and is argued to provide a general framework for analyzing many-body quantum chaos. Hamiltonians consisting of the XX-interaction between spin-1/2 particles (qubits) which are nearest neighbors on a 3-regular random graph (non-integrable), and an open chain (integrable), are diagonalized numerically to illustrate how the weak constraints can be satisfied. The entanglement content of the eigenvectors of chaotic many-body Hamiltonians is discussed as well as the connection between quantum chaos and thermalization in closed quantum systems.