发表机构
The University of Osaka; RIKEN Center for Quantum Computing (RQC); Kyoto University（大阪大学; 理化学研究所量子计算中心; 京都大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该工作推广无辅助量子比特双括号算法，用态依赖厄米算子嵌入非线性动力学，依据解的收缩、非扩张或扩张特性给出指数级查询上界，并在离散Gross–Pitaevskii方程上实现最优复杂度。
AI 中文摘要
利用量子计算机模拟非线性动力学日益受到关注。一般而言，此类模拟需要额外的量子资源，因为酉量子演化是线性的。一个基本问题是如何将非线性动力学嵌入到完全相干的、无辅助量子比特的酉电路中，以及动力学的复杂度如何决定所需的量子资源。在本工作中，我们将用于虚时演化的无辅助量子比特双括号量子算法推广，将其与态无关的哈密顿量替换为态依赖的厄米算子。我们的框架递归调用初始态制备预言机及其逆操作，并通过完全相干的、无辅助量子比特的酉嵌入，以任意指定精度制备目标解。我们将查询代价与非线性动力学的复杂度（特别是其对初始条件的敏感性）相关联。当解之间的距离至少以指数方式收缩（收缩型）、不增加（非扩张型）或至多以指数方式增长（扩张型）时，我们分别获得查询上界为$\exp(O(T))$、$\exp(O(T^2))$和$\exp(\exp(O(T)))$，其中$T$为目标演化时间。对于离散Gross–Pitaevskii方程，我们的无辅助量子比特双括号电路在指定的单量子比特初始态族上实现了最优最坏情况查询复杂度$\Theta(e^{gT/2})$，其中$g>0$是非线性强度。这些结果将非线性动力学的复杂度与相干量子模拟的查询代价联系起来，并为设计具有最优查询复杂度的无辅助量子比特酉嵌入提供了基础。
英文摘要
Simulating nonlinear dynamics with quantum computers has gained increasing attention. In general, such simulations require additional quantum resources because unitary quantum evolution is linear. A fundamental question is how nonlinear dynamics can be embedded into fully coherent, ancilla-free unitary circuits and how the complexity of the dynamics governs the required quantum resources. In this work, we generalize the ancilla-free double-bracket quantum algorithm for imaginary-time evolution by replacing its state-independent Hamiltonian with a state-dependent Hermitian operator. Our framework recursively calls an initial state preparation oracle and its inverse, and prepares the target solution to any prescribed accuracy using a fully coherent, ancilla-free unitary embedding. We relate the query cost to the complexity of the nonlinear dynamics, specifically their sensitivity to initial conditions. We obtain query upper bounds of $\exp(O(T))$, $\exp(O(T^2))$, and $\exp(\exp(O(T)))$ when the distance between solutions contracts at least exponentially (contractive), does not increase (nonexpansive), or grows at most exponentially (expansive), respectively, where $T$ is the target evolution time. For the discrete Gross--Pitaevskii equation, our ancilla-free double-bracket circuit achieves optimal worst-case query complexity $Θ(e^{gT/2})$ over a specified family of single-qubit initial states, where $g>0$ is the nonlinearity strength. These results connect the complexity of nonlinear dynamics to the query cost of coherent quantum simulation and provide a foundation for designing ancilla-free unitary embeddings with optimal query complexity.
Comments23 pages, 1 figure
