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.

翻译：量子密集输出问题是指利用量子计算机从含时量子动力学中评估时间累积可观测量的过程。该问题在量子控制与光谱计算等应用中频繁出现。我们提出了一系列适用于早期及完全容错量子平台的算法。这些方法借鉴了振幅估计、哈密顿量模拟、量子线性常微分方程求解器以及量子卡莱曼线性化等技术。我们针对演化时间$T$和误差容限$\epsilon$提供了全面的复杂度分析。结果表明，对于特定类型的低秩密集输出，线性化方法几乎能达到最优复杂度$\mathcal{O}(T/\epsilon)$。此外，我们给出了密集输出问题的一种线性化形式，该形式能够产生包含原始状态的精确有限维闭包。该形式与库普曼不变子空间理论相关，在非线性控制与科学机器学习领域可能具有独立的研究价值。