最优查询复杂度的含时哈密顿量模拟
Time-Dependent Hamiltonian Simulation with Optimal Query Complexity
AI总结：
该研究针对含时哈密顿量模拟问题，在HAM-T模型下提出查询最优算法，其查询复杂度与不含时情况匹配，且可作为量子比特化的替代方案。
AI中文摘要：
我们针对在区间[0,T]上的一般n量子比特含时哈密顿量H(t)，给出了一个查询最优模拟算法，前提是H满足利普希茨连续条件且||H(t)||≤α。在标准的HAM-T访问模型中，该算法使用$$O\left( \alpha T+\frac{\log(1/\varepsilon)} {\log(e+\log(1/\varepsilon)/(\alpha T))} \right)$$次HAM-T查询，将时间有序传播子U_H(T)近似到误差ε。这与不含时哈密顿量的已知查询下界相匹配，表明时间依赖性不会带来渐近查询开销。我们的方法首先构造一个单查询换能器，在给定辅助态的情况下，实现U_H(T)的近似并保持该态不变。对多次应用该换能器的电路进行加权组合，可使省略该态导致的误差呈阶乘衰减，从而得到所述的最优精度依赖关系。对于不含时哈密顿量，同一方法也能给出量子比特化的查询最优替代方案。
英文摘要：
We give a query-optimal algorithm for simulating a general $n$-qubit time-dependent Hamiltonian $H(t)$ on $[0,T]$, assuming that $H$ is Lipschitz continuous and $\|H(t)\|\leqα$. In the standard $\mathrm{HAM\mbox{-}T}$ access model, the algorithm approximates the time-ordered propagator $U_H(T)$ to error $\varepsilon$ using $$ O\left( αT+\frac{\log(1/\varepsilon)} {\log(e+\log(1/\varepsilon)/(αT))} \right) $$ $\mathrm{HAM\mbox{-}T}$ queries. This matches the known query lower bound for time-independent Hamiltonians, showing that time dependence incurs no asymptotic query overhead. Our method first constructs a one-query transducer that, given an auxiliary state, implements an approximation to $U_H(T)$ and returns the state unchanged. A weighted combination of circuits that apply the transducer different numbers of times makes the error caused by omitting this state decay factorially, yielding the stated optimal precision dependence. For time-independent Hamiltonians, the same method also gives a query-optimal alternative to qubitization.
