论文标题
带有粒子数对称的费米的经典阴影
Classical shadows of fermions with particle number symmetry
论文作者
论文摘要
我们考虑使用$ n $模式的$η$粒子的费米式波函数的经典阴影。我们证明，所有最多使用的所有$ k $ dreduced密度矩阵（RDMS）可以同时估计为$ε^{2} $的平均差异最多使用$ \binomη{k} \ big（1- \ frac {η-k} {n} \ big） $ \ MATHCAL {O}（k^2η）$经典复杂性。我们的样本复杂性是对$ \ Mathcal {o}（\ binom {n} {k} \ frac {\ sqrt {k}} {k}} {ε^{2}}} $缩放先前的方法，因为$ n $比$ n $η$更大，这是自然问题。我们的方法在最严重的半填充中仍然提供了$ 4^{k} $在样品复杂性中的优势，还估计所有$η$还原密度矩阵，适用于与所有单个Slater确定因素估算重叠，最多可与最多$ \ Mathcal {O}（O}（O frac} $} $} $} $} $} {2 $η$。
We consider classical shadows of fermion wavefunctions with $η$ particles occupying $n$ modes. We prove that all $k$-Reduced Density Matrices (RDMs) may be simultaneously estimated to an average variance of $ε^{2}$ using at most $\binomη{k}\big(1-\frac{η-k}{n}\big)^{k}\frac{1+n}{1+n-k}/ε^{2}$ measurements in random single-particle bases that conserve particle number, and provide an estimator for any $k$-RDM with $\mathcal{O}(k^2η)$ classical complexity. Our sample complexity is a super-exponential improvement over the $\mathcal{O}(\binom{n}{k}\frac{\sqrt{k}}{ε^{2}})$ scaling of prior approaches as $n$ can be arbitrarily larger than $η$, which is common in natural problems. Our method, in the worst-case of half-filling, still provides a factor of $4^{k}$ advantage in sample complexity, and also estimates all $η$-reduced density matrices, applicable to estimating overlaps with all single Slater determinants, with at most $\mathcal{O}(\frac{1}{ε^{2}})$ samples, which is additionally independent of $η$.
