Convergence Rates for Arbitrary Statistical Moments of Random Quantum Circuits

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.

Figure 1

The moment space ${\mathcal{H}}_{M_t}$ may be visualized as an array of $2nt$ qubits, such that $t$ copies support a ket in the state space on $nt$ qubits, and the remaining $t$ copies the corresponding bra. In this way, a unitary $U$ on $n$ qubits induces a transformation $U^{\otimes t,t}$ on density operators on $nt$ qubits. Dashed rectangles indicate the $2t$ qubits corresponding to a local moment space ${\mathcal{H}}_{l_t}$. Ovals correspond to a unitary $U$ acting only on the first two qubits.

Figure 2

Inverse spectral gap ${\Delta }_t^{-1}$ of $M_t[{\mu }_H]$ with $t=2$, 3 for a random circuit consisting of two-qubit gates selected according to the Haar measure on $U(4)$. The line with slope $5/6$ corresponds to the asymptotic result.