Quantum Statistical Simulations for Quantum circuits

George F. Viamontes, Igor L. Markov, John P. Hayes

Soichiro Nishio, Yuki Oba, Yuki Kurashige

In this work, we present a detailed analysis of statistical errors in reduced density matrices (RDMs) of active space wavefunctions sampled from quantum circuit simulation and the impact on results obtained by the multireference theories.

Andrew M. Projansky, Joshuah T. Heath, James Daniel Whitfield

The entanglement level statistics of a quantum state have recently been proposed to be a signature of universality in the underlying quantum circuit. This is a consequence of level repulsion in the entanglement spectra being tied to the integrability of entanglement generated. However, such studies of the level-spacing statistics in the entanglement spectrum have thus far been limited to the output states of Clifford and Haar random circuits on product state inputs. In this work, we provide the first example of a circuit which is composed of a simulable gate set, yet has a Wigner-Dyson distributed entanglement level spectrum without any perturbing universal element. We first show that, for matchgate circuits acting on random product states, Wigner-Dyson statistics emerge by virtue of a single SWAP gate, in direct analog to previous studies on Clifford circuits. We then examine the entanglement spectrum of matchgate circuits with varied input states, and find a sharp jump in the complexity of entanglement as we go from two- to three-qubit entangled inputs. Studying Clifford and matchgate hybrid circuits, we find examples of classically simulable circuits whose output states exhibit Wigner-Dyson entanglement level statistics in the absence of universal quantum gate elements. Our study thus provides strong evidence that entanglement spectrum is not strongly connected to notions of simulability in any given quantum circuit.

Masahiro Fujita, Yusuke Kimura

Michail A. Zak

Tobias Micklitz

We introduce a simulation-free method to estimate the fidelity of large quantum circuits based on the order statistics of measured output probabilities from highly entangled, chaotic states. The approach requires only the highest-probability output bitstrings -- the most frequently observed measurement outcomes -- and builds on exact analytical results for the order statistics of Haar-random quantum states derived here. Analyzing their modification under depolarizing noise, we propose a scalable fidelity estimator, validated on Google's 12-qubit Sycamore experiment and further supported by numerical simulations. We demonstrate its practicality for intermediate-scale quantum circuits, where cross-entropy benchmarking is costly and direct fidelity estimation is difficult.