慢解析含时哈密顿量的量子模拟
Quantum simulation of slow analytic time-dependent Hamiltonians
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中文总结 AI 辅助
本研究提出一种慢解析含时哈密顿量的量子模拟算法，通过周期Gevrey延拓、傅里叶界分析结合Floquet嵌入等技术，实现近乎加性的查询复杂度与低门开销，还可推广至Gevrey哈密顿量及半耗散线性微分方程模拟。
中文摘要 AI 辅助
我们提出了一种适用于慢解析哈密顿量$\tilde{H}(t)=H(t/T)$（满足$\norm{H(s)}\leq\boldsymbol{\u03b1}$）的量子算法，该算法可实现近乎加性的查询复杂度和较低的门开销。我们的核心技术贡献是对$H(s)$进行周期Gevrey延拓，并结合傅里叶分量衰减与截断界，以实现高效的有限维模拟。将该方法与Floquet嵌入及最优的不含时哈密顿量模拟技术相结合，在可相干访问$H'(s)$及端点导数的假设下，可得到$\tilde{\u210b}\u2061\u2009\u200b(\u03b1T+\u200b\boldsymbol{\u200b\u03b5}))$的查询复杂度，以及$\tilde{\u210b}\u2061\u2009\u200b((\u03b1T+\u200b\boldsymbol{\u200b\u03b5}))^2\boldsymbol{\u200b\u03b5}))$的额外门复杂度。对于慢解析控制哈密顿量，仅需不含时控制算符的块编码即可达到相同的查询复杂度，且门开销更低。我们的方法还可推广至Gevrey哈密顿量，并改善了慢解析半耗散线性微分方程模拟的精度依赖关系。
英文摘要
We develop a quantum algorithm for slow analytic Hamiltonians $\widetilde H(t)=H(t/T)$ with $\|H(s)\|\leqα$ that achieves nearly additive query complexity and low gate overhead. Our main technical contribution is a periodic Gevrey extension of $H(s)$, together with Fourier component decay and truncation bounds that enable an efficient finite-dimensional simulation. Combined with Floquet embedding and optimal time-independent Hamiltonian simulation technique, this gives query complexity $\widetilde{\mathcal O}\!\left(αT+\log(1/\varepsilon)\right)$ and additional gate complexity $\widetilde{\mathcal O}\!\left((αT+\log(1/\varepsilon))^2\log(1/\varepsilon)\right)$, assuming coherent access to $H'(s)$ and endpoint derivatives. For slow analytic control Hamiltonians, only block encodings of the time-independent control operators are required, with the same query complexity and lower gate overhead. Our method also extends to Gevrey Hamiltonians and improves the precision dependence for simulating slow analytic semi-dissipative linear differential equations.
