Bridging Krylov Complexity and Universal Analog Quantum Simulator

TL;DR

This paper introduces generalized Krylov complexity as a quantitative measure to evaluate the complexity of implementing quantum operations in analog quantum simulators, aiding in designing efficient control protocols.

Contribution

It develops a framework linking Krylov complexity to synthesis time in analog quantum simulation, providing a new predictive tool for quantum control.

Findings

Krylov complexity predicts the minimum realization time of quantum operations.

The framework applies to systems like Rydberg atom arrays.

Krylov complexity organizes operator space for native interactions.

Abstract

Quantum simulation of complex many-body systems beyond classical computational capabilities provides a promising route toward understanding novel quantum phases and their transitions. In particular, analog quantum simulators with global control fields have attracted considerable attention due to their potential to simulate arbitrary Hamiltonians and perform quantum computing tasks. However, a clear, quantitative measure for the complexity of implementing specific quantum operations in such systems is still lacking. In this Letter, we address this challenge by introducing generalized Krylov complexity, a concept originating from operator growth dynamics, as a direct diagnosis for this synthesis complexity. We construct the block Krylov basis generated by a set of Hamiltonians, which naturally organizes the operator space achievable through the simulator's native interactions and their nested commutators. By analyzing representative systems including Rydberg atom arrays, we demonstrate that the generalized Krylov complexity of a target operation serves as a strong predictor of the minimum time required for its realization. Our results establish Krylov complexity as an intuitive and predictive tool for designing efficient control protocols in analog quantum simulators.