Estimating the randomness of quantum circuit ensembles up to 50 qubits

TL;DR

This paper introduces efficient numerical protocols using tensor networks to estimate the randomness of quantum circuit ensembles up to 50 qubits, aiding understanding of their complexity and expressibility.

Contribution

It develops polynomial-complexity tensor-network algorithms for estimating the frame potential of quantum circuits, enabling large-scale analysis of their randomness properties.

Findings

Verified linear growth in complexity for shallow circuits.

Analyzed expressibility and barren plateau issues in variational algorithms.

Demonstrated tensor networks' effectiveness in large-scale quantum simulations.

Abstract

Random quantum circuits have been utilized in the contexts of quantum supremacy demonstrations, variational quantum algorithms for chemistry and machine learning, and blackhole information. The ability of random circuits to approximate any random unitaries has consequences on their complexity, expressibility, and trainability. To study this property of random circuits, we develop numerical protocols for estimating the frame potential, the distance between a given ensemble and the exact randomness. Our tensor-network-based algorithm has polynomial complexity for shallow circuits and is high-performing using CPU and GPU parallelism. We study 1. local and parallel random circuits to verify the linear growth in complexity as stated by the Brown-Susskind conjecture, and; 2. hardware-efficient ans\"atze to shed light on its expressibility and the barren plateau problem in the context of variational algorithms. Our work shows that large-scale tensor network simulations could provide important hints toward open problems in quantum information science.