哈密顿模拟线性组合的最优核函数
Optimal kernel functions for linear combination of Hamiltonian simulation
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中文总结 AI 辅助
针对哈密顿模拟线性组合，在更广核函数类中构造唯一最优核，使查询代价最小，其首项系数比Low和Somma经验值提高约2.1倍，并证明离散化后改进仍保持。
中文摘要 AI 辅助
哈密顿模拟的线性组合（LCHS）为线性非酉动力学的量子模拟提供了一个渐近最优的框架。LCHS方法通过由核函数确定的积分表示来近似实现Peano--Baker传播子。我们针对一类更广泛的容许核函数建立了一个LCHS逼近定理，该类函数由自然的复解析条件定义。在标准的时间无关块编码访问模型中，我们针对每个目标误差$\varepsilon$显式构造了该类中使查询代价泛函最小化的唯一核函数，并证明该泛函的最小值具有渐近展开式$\tfrac{2\mathrm{e}}{\pi}\big(\log\frac{1}{\varepsilon}-\log\log\frac{1}{\varepsilon}+o(1)\big)$（当${\varepsilon\downarrow0}$时）。可证明最优的首项系数$2\mathrm{e}/\pi$比Low和Somma通过对其最优缩放核函数族进行数值最小化而经验获得的系数提高了约2.1倍。我们构造了一个谱收敛的求积公式，并给出了算子误差和酉线性组合（LCU）次归一化的显式界，证明这种渐近改进在离散化为适合实现的有限酉线性组合后仍然保持。我们的结果基于Hardy空间理论和凸分析。
英文摘要
Linear combination of Hamiltonian simulation (LCHS) provides an asymptotically optimal framework for the quantum simulation of linear non-unitary dynamics. The LCHS approach approximately implements the Peano--Baker propagator through an integral representation determined by a kernel function. We establish an LCHS approximation theorem for a broadened class of admissible kernel functions, defined by natural complex-analytic conditions. In the standard time-independent block-encoding access model, we explicitly construct, for each target error $\varepsilon$, the unique kernel function in this class minimizing the query-cost functional and show that the minimum of this functional has the asymptotic expansion $\tfrac{2\mathrm{e}}π\big(\log\frac{1}{\varepsilon}-\log\log\frac{1}{\varepsilon}+o(1)\big)$ as ${\varepsilon\downarrow0}$. The provably optimal leading coefficient $2\mathrm{e}/π$ improves by a factor of approximately 2.1 on the coefficient obtained empirically by Low and Somma through numerical minimization over their family of optimally scaling kernel functions. We construct a spectrally convergent quadrature with explicit bounds on operator error and linear combination of unitaries (LCU) subnormalization, proving that this asymptotic improvement persists after discretization into a finite linear combination of unitaries suitable for implementation. Our results are based on Hardy space theory and convex analysis.
发表机构
- PsiQuantum
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