Time Evolution of Complexity: A Critique of Three Methods

TL;DR

This paper introduces a new testing procedure using information-theoretic measures to evaluate different methods of computing quantum complexity, revealing that only wave function-based circuit complexity accurately reflects time evolution.

Contribution

It proposes a novel, basis-independent testing method for quantum complexity approaches and demonstrates the unique sensitivity of wave function-based circuit complexity to time evolution.

Findings

Wave function-based circuit complexity is sensitive to time evolution.

Complexity is proportional to the number of distinct Hamiltonian evolutions.

Non-local theories exhibit prolonged complexity growth.

Abstract

In this work, we propose a testing procedure to distinguish between the different approaches for computing complexity. Our test does not require a direct comparison between the approaches and thus avoids the issue of choice of gates, basis, etc. The proposed testing procedure employs the information-theoretic measures Loschmidt echo and Fidelity; the idea is to investigate the sensitivity of the complexity (derived from the different approaches) to the evolution of states. We discover that only circuit complexity obtained directly from the wave function is sensitive to time evolution, leaving us to claim that it surpasses the other approaches. We also demonstrate that circuit complexity displays a universal behaviour---the complexity is proportional to the number of distinct Hamiltonian evolutions that act on a reference state. Due to this fact, for a given number of Hamiltonians, we can always find the combination of states that provides the maximum complexity; consequently, other combinations involving a smaller number of evolutions will have less than maximum complexity and, hence, will have resources. Finally, we explore the evolution of complexity in non-local theories; we demonstrate the growth of complexity is sustained over a longer period of time as compared to a local theory.