Convergence rates for arbitrary statistical moments of random quantum circuits

TL;DR

This work considers 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 determines how quickly averages of arbitrary finite-degree polynomials in the matrix elements of the resulting unitary converge to Haar measure averages.

Abstract

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 ε-approximate unitary t-designs.

Figures (2)

• Figure 1: The moment space $\mathcal{H}_{M_t}$ may be visualized as an array of $2nt$ qubits, in such a way 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 $\mathcal{H}_{l_t}$, whereas the ovals correspond to a unitary $U$ acting non-trivially on the first two qubits.
• Figure 2: (Color online) 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.