从割多胞体几何实现快速哈密顿量工程
Fast Hamiltonian engineering from cut polytope geometry
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中文总结 AI 辅助
该研究提出一个统一框架，将哈密顿量工程中的时间最优问题转化为割多胞体问题，通过松弛和舍入生成脉冲，实现近最优量子运行时间，并证明其NP完全性，为模拟量子模拟器自动编程提供可证明保证。
中文摘要 AI 辅助
哈密顿量工程（HE）利用本征纠缠系统哈密顿量和受限控制来模拟目标哈密顿量下的量子动力学，其应用涵盖从量子门设计到模拟量子模拟。观察到对于许多系统，将HE表述为线性规划仅需控制脉冲与系统哈密顿量之间的特定对易关系，我们为广泛的$2$-局域量子比特、量子比特（qudit）和费米子系统提供了统一框架。我们将时间最优HE重新表述为复数$k$-割多胞体问题，其中$k$为这些对易关系中不同相位的数目，并证明对于每个有限$k$，该问题都是NP完全的。因此，我们将多胞体松弛为椭圆体（elliptope），并应用Krivine型舍入，产生由系统和目标哈密顿量信息驱动的脉冲，随后将其用于线性规划。我们推导出线性规划可靠可行所需的充分必要的信息脉冲数量。此外，混合均匀采样的脉冲可保证从$\mathrm{O}(m)$个脉冲（$m$为相互作用项数）得到接近松弛值的解。结合最优量子运行时间的上下界，这产生了$\mathrm{O}(\sqrt{m})$的近似比。在完全连接的Ising模型、用于qudit的手征时钟模型以及费米子Harper-Hofstadter模型的基准测试中，我们的方法在最优值可计算处达到接近最优的量子运行时间，在费米子情形下运行时间随晶格尺寸饱和，并以$\mathrm{O}(m)$个脉冲超越最先进方法。这为模拟量子模拟器的自动编程建立了具有可证明保证的统一方法。
英文摘要
Hamiltonian engineering (HE) simulates quantum dynamics under a target Hamiltonian using a native entangling system Hamiltonian and restricted control, with applications from quantum gate design to analog quantum simulation. Observing that, for many systems, formulating HE as a linear program only requires specific commutation relations between the control pulses and the system Hamiltonian, we provide a unified framework for broad classes of $2$-local qubit, qudit, and fermionic systems. We reformulate time-optimal HE as a complex $k$-cut polytope problem, with $k$ the number of distinct phases in these commutation relations, and prove it NP-complete for every finite $k$. We therefore relax the polytope to the elliptope and apply a Krivine-type rounding, yielding pulses informed by the system and target Hamiltonians, which we then use in a linear program. We derive how many informed pulses are necessary and sufficient for the linear program to be reliably feasible. Additionally mixing in uniformly sampled pulses guarantees a solution close to the relaxation value from $\mathrm{O}(m)$ pulses, with $m$ the number of interaction terms. Together with upper and lower bounds on the optimal quantum run time, this yields an $\mathrm{O}(\sqrt{m})$ approximation ratio. In benchmarks on a fully connected Ising model, a chiral clock model for qudits, and the fermionic Harper-Hofstadter model, our approach attains near-optimal quantum run times wherever the optimum is computable, reaches run times that saturate with the lattice size in the fermionic case, and outperforms state-of-the-art methods with $\mathrm{O}(m)$ pulses. This establishes a unified approach with provable guarantees to the automatic programming of analog quantum simulators.
发表机构
- Institute for Quantum Inspired and Quantum Optimization, Hamburg University of Technology（量子启发与量子优化研究所，汉堡工业大学）
- Center for Theoretical Physics -- a Leinweber Institute, Massachusetts Institute of Technology（理论物理中心——莱因韦伯研究所，麻省理工学院）
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