发表机构
Pennsylvania State University（宾夕法尼亚州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种稀疏哈密顿量模拟算法，其查询复杂度对最大列欧几里得范数达到最优依赖，消除了先前算法的次多项式开销，并解决了黑盒酉算子实现的最优查询问题。
AI 中文摘要
我们给出一个量子算法，用于模拟一个 $d$-稀疏的厄米哈密顿量 $H$，假设已知其上界 $\Lambda$ 为其最大列欧几里得范数 $\\|H\\|_{1\to2}$。对于 $t\Lambda\ge1/2$，以算子范数误差 $\epsilon$ 进行模拟所需的稀疏预言查询次数为 $O\\!\left(t\Lambda\sqrt d+\sqrt d\log(2/\epsilon)\right)$。这消除了 Low 算法 [STOC 2019] 中的次多项式开销，取而代之的是一个加性的对数精度项。对于 $d>1$ 且 $t\Lambda\ge\log(2/\epsilon)$，该界限与最坏情况下的下界相匹配。已知的谱范数上界也可用于替代 $\Lambda$。1 量子比特和 2 量子比特门的数量与查询规模成线性关系，直至预言成本以及输入比特长度和对数精度参数的多项式开销。作为应用，我们在标准稀疏和态制备访问下，求解 $d$-稀疏量子线性系统（满足 $\\|A\\|\le1$ 且 $\\|A^{-1}\\|\le\kappa$）时，获得 $O(\kappa\sqrt d\\,\mathrm{polylog}(\kappa/\epsilon))$ 次查询。我们还给出了一个门高效的实现，用于模拟每行和每列至多有 $d$ 个非零项的黑盒酉算子，使用 $O(\sqrt d\log(2/\epsilon))$ 次查询，前提是对该酉算子及其共轭转置具有稀疏访问。在常数误差下，查询界限是最优的，并对任意 $N\times N$ 酉算子产生 $\Theta(\sqrt N)$ 次查询，解决了 Berry 和 Childs [QIC 2012] 提出的关于黑盒酉算子实现的开放问题。
英文摘要
We give a quantum algorithm for simulating a $d$-sparse Hermitian Hamiltonian $H$, assuming a known upper bound $Λ$ on its maximum column Euclidean norm $\|H\|_{1\to2}$. For $tΛ\ge1/2$, simulation with operator-norm error $ε$ uses \[ O\!\left(tΛ\sqrt d+\sqrt d\log(2/ε)\right) \] sparse-oracle queries. This removes the subpolynomial overhead in Low's algorithm [STOC 2019], replacing it with an additive logarithmic precision term. For $d>1$ and $tΛ\ge\log(2/ε)$, the bound matches the worst-case lower bound. A known spectral-norm upper bound may also be used in place of $Λ$. The number of 1- and 2-qubit gates is linear in the query scale, up to oracle costs and polynomial overhead in the input bit lengths and logarithmic precision parameters. As applications, we obtain $O(κ\sqrt d\,\mathrm{polylog}(κ/ε))$ queries for solving $d$-sparse quantum linear systems with $\|A\|\le1$ and $\|A^{-1}\|\leκ$, under standard sparse and state-preparation access. We also give a gate-efficient implementation of black-box unitaries with at most $d$ nonzero entries per row and column using $O(\sqrt d\log(2/ε))$ queries, given sparse access to the unitary and its adjoint. At constant error, the query bound is optimal and yields $Θ(\sqrt N)$ queries for arbitrary $N\times N$ unitaries, resolving the open question on black-box unitary implementation posed by Berry and Childs [QIC 2012].
Comments37 pages, no figures
