Unified framework for efficiently computable quantum circuits

TL;DR

This paper presents a unified framework that clarifies why certain quantum circuits, like Clifford and matchgates, are classically efficiently simulatable by analyzing operator spread and complexity dynamics.

Contribution

It introduces a transparent, operator-based approach to understand and quantify the classical simulability of specific quantum circuits, unifying previously separate cases.

Findings

Complexity exhibits initial exponential growth, saturation, then decay with decoherence.

The framework can be implemented numerically with controllable errors.

Provides a unified understanding of efficient classical simulation for Clifford and matchgate circuits.

Abstract

Quantum circuits consisting of Clifford and matchgates are two classes of circuits that are known to be efficiently simulatable on a classical computer. We introduce a unified framework that shows in a transparent way the special structure that allows these circuits can be efficiently simulatable. The approach relies on analyzing the operator spread within a network of basis operators during the evolution of quantum circuit. Quantifying the complexity of a calculation by the number of operators with amplitude above a threshold value, we show that there is a generic form of the complexity curve involving an initial exponential growth, saturation, then exponential decay in the presence of decoherence. Our approach is naturally adaptable into a numerical procedure, where errors can be consistently controlled as a function of the complexity of the simulation.