Efficiency of Producing Random Unitary Matrices with Quantum Circuits

TL;DR

This paper investigates how quickly a random unitary circuit ensemble (UCE) approximates CUE statistics as a function of the number of qubits and gates. It analyzes the full distribution of matrix elements, their moments up to order 16, and multi-element correlators within a column, comparing against CUE. The main finding is that these quantities converge to CUE with a gate count scaling at most as $n_q \log(n_q/\epsilon)$, much faster than general upper bounds would suggest. The results imply that properties of CUE that require exponentially many gates are likely of a more intricate nature, while many practical statistics can be efficiently generated by UCE. This supports the use of pseudo-random quantum circuits in simulating Haar-like statistics for quantum information tasks.

Abstract

We study the scaling of the convergence of several statistical properties of a recently introduced random unitary circuit ensemble towards their limits given by the circular unitary ensemble (CUE). Our study includes the full distribution of the absolute square of a matrix element, moments of that distribution up to order eight, as well as correlators containing up to 16 matrix elements in a given column of the unitary matrices. Our numerical scaling analysis shows that all of these quantities can be reproduced efficiently, with a number of random gates which scales at most as $n_q\log (n_q/ε)$ with the number of qubits $n_q$ for a given fixed precision $ε$. This suggests that quantities which require an exponentially large number of gates are of more complex nature.

Figures (10)

• Figure 1: A random UCE circuit. The two different angles $\hbox{\boldmath{$\theta$}}$ and $\hbox{\boldmath{$\theta$}}'$ mean two different random U(2) gates.
• Figure 2: Convergence of $\tilde{P}(l)$ to $P(l)$ (dashed line) for 4 qubits with $n_g=$5, 10, 20 and 50 for an ensemble of $10^4$ matrices
• Figure 3: The distance $D_P(n_{g})$ between the distributions $P(l)$ and $\tilde{P}(l)$ as function of the number of gates $n_g$ for $n_q=2,3,\ldots,28$ qubits (from left to right).
• Figure 4: (Color online) The number of gates $n^*$ needed to achieve $D_p\le \epsilon$ for $\ln(\epsilon)=$0,-1,-2,-3,-4 and -5 ($\ast$, $+$, $\triangle$, $\diamond$, $\Box$, $\circ$ respectively) and $n_q=2\ldots 28$. Straight lines are fits to the functions $f_1$, $f_2$ and $f_3$ (1st, 2nd and 3rd plot respectively). The last plot shows $\chi^2$ for these fits ($f_1$ ($\circ$), $f_2$ ($\Box$)), and $f_3$ ($\diamond$).
• Figure 5: (Color online) The coefficients $a_1$ (green squares) and $a_2$ (red circles) as a function of $\ln(1/\epsilon)$.