Dense outputs from quantum simulations
Jin-Peng Liu, Lin Lin
TL;DR
The paper tackles the dense-output problem in quantum dynamics: estimating time-accumulated observables $J = \int_0^T \langle \psi(t) | O(t) | \psi(t) \rangle \mathrm{d} t$ on quantum devices. It develops a suite of algorithms spanning an early fault-tolerant Hadamard-test approach, fault-tolerant amplitude estimation (biased and unbiased), a quantum linear ODE solver for non-unitary dynamics, and a quantum Carleman linearization for nonlinear dynamics with low-rank observables, including a padding technique. Complexity results show a spectrum of scaling: $\mathcal{O}(\|H\| T^3 \log(1/\delta)/\epsilon^2)$ for the early method; $\mathcal{O}(\|H\| T^3 \log(1/\delta)/\epsilon)$ (biased) and $\mathcal{O}(\|H\| T^{2.5} \log(1/\delta)/\epsilon)$ (unbiased); $\mathcal{O}(\|H\| T^2 \log(1/\delta)/\epsilon)$ via global amplitude estimation with a linear ODE solver; and a near-optimal $\mathcal{O}(\|H\| T \Gamma \log(1/\delta)/\epsilon)$ using Carleman linearization with $\Gamma = (|J(T)|^2+1)/|J(T)|$. A key theoretical contribution is an exact finite-dimensional closure under Koopman Invariant Subspace theory for certain nonlinear dynamics, with potential impact on KOOC and DMD in scientific ML. The framework applies to quantum control and spectroscopy, offering end-to-end quantum routines and connections to variational and learning-based control methods. Overall, the work advances practical quantum algorithms for time-dependent observables and clarifies their trade-offs across fault-tolerant paradigms and nonlinear dynamics.
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 $ε$. Our results demonstrate that the linearization approach can nearly achieve optimal complexity $\mathcal{O}(T/ε)$ 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.