Time Evolution of Complexity: A Critique of Three Methods

TL;DR

The paper proposes a testing framework using Loschmidt echo and fidelity to differentiate three notions of quantum circuit complexity in a simple lattice QFT: Fubini-Study, covariance-matrix, and Nielsen-style circuit complexity. It shows that only the Nielsen circuit approach captures time evolution differences between LE and Fidelity, while the other two methods yield identical results for these overlaps. A universal feature is found where complexity scales with the number of distinct Hamiltonian evolutions applied to a reference state, and more evolutions imply greater complexity, hinting at resource hierarchies. The study also reveals that non-local (higher α) theories slow the early-time growth of complexity, linking non-locality to extended information-processing timescales and suggesting connections to holographic and entanglement structures.

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.

Figures (10)

• Figure 1: An illustration for different distances between reference ($\psi_R$) and target state ($\psi_T$) in the Hilbert-space. Complexity (from Fubini-Study approach) will correspond to the length of the geodesic shown in blue.
• Figure 2: An illustration for Loschmidt echo and Fidelity
• Figure 3: LE vs F Test for Fubini-Study
• Figure 4: LE vs F Test for Circuit Complexity
• Figure 5: Difference plot for circuit complexity method. Dashed lines corresponds to the value around which $|\mathcal{C}_{\text{LE}}(\tilde{U})-\mathcal{C}_{\text{F}}(\tilde{U})|$ fluctuates at late times which are approximately, 3.2 ( for N=1000, corresponding to the red dashed line) and 2.2 (for N=500, corresponding to the black dashed line) respectively.