Dense outputs from quantum simulations

TL;DR

This paper introduces algorithms for efficiently computing time-dependent observables in quantum simulations, leveraging linearization techniques and complexity analysis to optimize performance on quantum computers.

Contribution

It presents novel quantum algorithms for dense output problems, including a linearization approach with near-optimal complexity and a finite-dimensional closure related to Koopman theory.

Findings

Linearization approach nearly achieves optimal complexity $\\mathcal{O}(T/\epsilon)$ for low-rank dense outputs.

Provides a finite-dimensional closure that exactly represents the original states.

Connects the dense output problem to Koopman Invariant Subspace theory.

Abstract

The quantum dense output problem is the process of evaluating time-accumulated observables from time-dependent quantum dynamics using quantum computers. This problem arises frequently in applications such as quantum control and spectroscopic computation. We present a range of algorithms designed to operate on both early and fully fault-tolerant quantum platforms. These methodologies draw upon techniques like amplitude estimation, Hamiltonian simulation, quantum linear Ordinary Differential Equation (ODE) solvers, and quantum Carleman linearization. We provide a comprehensive complexity analysis with respect to the evolution time $T$ and error tolerance $\epsilon$. Our results demonstrate that the linearization approach can nearly achieve optimal complexity $\mathcal{O}(T/\epsilon)$ for a certain type of low-rank dense outputs. Moreover, we provide a linearization of the dense output problem that yields an exact and finite-dimensional closure which encompasses the original states. This formulation is related to the Koopman Invariant Subspace theory and may be of independent interest in nonlinear control and scientific machine learning.