哈密顿量学习与认证：基于本征相位工程

中文摘要 AI 辅助

我们研究大步长下$k$-局域哈密顿量的学习问题，仅使用固定时间间隔整数倍的前向时间演化。针对该任务，我们提出的新技术——称为本征相位工程——在步长达到对数因子意义下最优的同时，实现了海森堡极限的学习精度。对于未知的$s$-稀疏、局域性为$k$且满足$\\|H\\|_{\mathrm{op}}\leq\Lambda$的哈密顿量，我们的算法以总演化时间$\widetilde{O}(s/\varepsilon)$将其泡利系数恢复到$\ell_2$误差$\varepsilon$，与已知最优标度匹配，同时允许近最大步长$\widetilde{\Theta}(1/\Lambda)$。作为独立贡献，我们将演化时间下界加强为$\Omega_k(s^{1-1/(2k)}/\varepsilon)$，使得我们的算法在因子$\widetilde{O}(s^{1/(2k)})$内接近最优。本征相位工程不同于基于动力学近似或迭代抵消未知哈密顿量的方法。它将能量期望差异编码在实际受控演化的本征相位的微扰响应中，然后通过相位估计和高阶外推提取这些信息。这避免了随着目标精度提高而需要多项式更短演化步长的问题。仅使用单量子比特操作的版本在$n$量子比特的$\ell_2$学习时间上保持相同步长，但带有$\sqrt n$的额外开销。该方法还支持容错的哈密顿量认证以及指定泡利系数的估计，后者无需局域性或稀疏性假设。这些结果确立了本征相位工程作为一种以低控制频率提取哈密顿量信息的多功能方法。

英文摘要

We study Hamiltonian learning for $k$-local Hamiltonians at large step size, using only forward time evolution at integer multiples of a fixed time interval. For this task, our new technique, which we call eigenphase engineering, achieves Heisenberg-limited learning with a step size that is optimal up to logarithmic factors. For an unknown $s$-sparse Hamiltonian with fixed locality $k$ and $\|H\|_{\mathrm{op}}\leqΛ$, our algorithm recovers its Pauli coefficients to $\ell_2$ error $\varepsilon$ in total evolution time $\widetilde{O}(s/\varepsilon)$, matching the best known scaling while allowing a near-maximal step size $\widetildeΘ(1/Λ)$. As an independent contribution, we strengthen the evolution-time lower bound to $Ω_k(s^{1-1/(2k)}/\varepsilon)$, placing our algorithm within a factor $\widetilde{O}(s^{1/(2k)})$ of being optimal. Eigenphase engineering departs from approaches based on dynamical approximation or iteratively canceling the unknown Hamiltonian. It encodes energy expectation differences in the perturbative response of eigenphases of the actual controlled evolution, then extracts them through phase estimation and high-order extrapolation. This avoids the need for polynomially shorter evolution steps as the target precision improves. A version using only single-qubit operations retains the same step size with a $\sqrt n$ overhead in $\ell_2$ learning time for $n$ qubits. The method also enables tolerant Hamiltonian certification and estimation of a specified Pauli coefficient, the latter without locality or sparsity assumptions. These results establish eigenphase engineering as a versatile approach to extracting Hamiltonian information with infrequent control.

发表机构

• School of Computing, Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院计算学院）
• Institute for Quantum Information and Matter, California Institute of Technology（加州理工学院量子信息与物质研究所）
• Department of Electrical and Computer Engineering, Duke University（杜克大学电气与计算机工程系）
• Department of Mathematics, Duke University（杜克大学数学系）
• Duke Quantum Center, Duke University（杜克大学量子中心）

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