耗散非马尔可夫耦合经典振子的量子模拟
Quantum Simulation of Dissipative Non-Markovian Coupled Classical Oscillators
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中文总结 AI 辅助
提出量子算法模拟非马尔可夫耗散振子网络，通过Prony级数嵌入马尔可夫空间并利用哈密顿模拟线性组合估计能量，实现四倍加速，并证明经典困难性。
中文摘要 AI 辅助
我们提出了一种量子算法，用于模拟具有非马尔可夫耗散和时变材料特性的经典振子网络，将近期针对无阻尼谐波系统的加速扩展到粘声和粘弹性介质。我们通过用Prony级数近似记忆核，将历史依赖动力学嵌入到由非厄米算子控制的马尔可夫状态空间中。然后，我们使用哈密顿模拟的线性组合来估计在时间$t$时一组振子的瞬时动能和势能，误差为$\epsilon$，对振子系统的查询次数按$\widetilde{\mathcal{O}}(\alpha_{\rm tot} t/\epsilon)$缩放，其中$\alpha_{\rm tot}$是耗散强度、弹簧常数、逆质量和网络连接稀疏度的多项式。我们进一步表明，即使在强耗散存在的情况下，该能量估计任务在最坏情况下在经典上也是困难的（即相应的判定问题是$\mathsf{BQP}$完全的）。对于时变材料，我们证明材料特性的变化在能量表示中表现为有效耗散或增长。此外，我们通过一种适用于微分方程的新型Lieb-Robinson类界限，证明了在局部耦合拓扑中指数级量子优势的不可行性。这使得我们的量子算法能够为三维局部耦合阻尼振子系统提供四次加速，为实际阻尼振子网络和近似波动方程实现实际量子加速提供了可能性。
英文摘要
We present a quantum algorithm for simulating classical oscillator networks characterized by non-Markovian dissipation and time-varying material properties, extending recent speedups for undamped harmonic systems to viscoacoustic and viscoelastic media. We embed the history-dependent dynamics into a Markovian state space governed by a non-Hermitian operator by approximating memory kernels through a Prony series. We then use linear combination of Hamiltonian simulation to estimate the instantaneous kinetic and potential energy for a subset of oscillators at time $t$ within error $ε$ using a number of queries to the oscillator system that scales as $\widetilde{\mathcal{O}}(α_{\rm tot} t/ε)$, where $α_{\rm tot}$ is polynomial in the strength of the dissipation, spring constants, inverse masses, and sparsity of the connections in the network. We further show that this energy estimation task is in the worst-case classically hard (i.e., a corresponding decision problem is $\mathsf{BQP}$-complete), even in the presence of strong dissipation. For time-dependent materials, we show that changes in material properties appear as effective dissipation or growth in the energy representation. Additionally, we prove the infeasibility of exponential quantum advantages in locally coupled topologies through a novel form of Lieb-Robinson-like bounds that apply to differential equations. This allows our quantum algorithms to provide quartic speedups for locally coupled damped oscillator systems in three dimensions, raising the possibility of practical quantum speedups for realistically damped oscillator networks and approximated wave equations.
发表机构
- ETH Zürich（苏黎世联邦理工学院）
- University of Toronto（多伦多大学）
- Pacific Northwest National Laboratory（太平洋西北国家实验室）
- Princeton University（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。
