发表机构
The Institute for Fundamental Study (IF), Naresuan University; Chula Intelligent and Complex Systems Center of Excellence, Department of Physics, Faculty of Science, Chulalongkorn University; School of Physical and Mathematical Sciences, Nanyang Technological University（纳瑞苏安大学基础研究所; 朱拉隆功大学理学院物理系查拉智能与复杂系统卓越中心; 南洋理工大学物理与数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出一种自适应协议，以最优的 $\epsilon^{-1}$ 样本复杂度认证任意费米子高斯态，并扩展到相位修饰态，其效率由马尔可夫链谱隙控制。
AI 中文摘要
费米子高斯态是基于量子比特的费米子系统量子模拟的基准参考态，然而现有的认证协议要么通过保真度估计进行，导致在目标精度 $\epsilon$ 下的样本复杂度次优，要么仅适用于Haar典型态，要么需要对大部分量子比特进行自适应测量。我们给出一个自适应协议，该协议使用 $O(d^2\epsilon^{-1})$ 个副本、单量子比特测量（每个副本仅对一个量子比特进行自适应测量）以及每个副本 $O(d^3)$ 的经典处理时间，来认证任意 $d$ 模纯费米子高斯态。$\epsilon^{-1}$ 的依赖对于认证问题是最优的，即使在采用纠缠测量的策略中也是如此。样本复杂度由 $d\leftrightarrow d-2$ 降升游走的谱隙控制——这是高维扩展器理论中研究的一个马尔可夫链，等价于统计物理中的双位点格劳伯动力学——将认证效率与平衡弛豫时间联系起来。该协议的最坏情况界是紧的，并由物理相关态达到，包括完全二聚化的Su-Schrieffer-Heeger（SSH）链的基态和BCS配对态。相比之下，直到 $d=14$ 的数值模拟表明，$O(d \epsilon^{-1})$ 个副本足以认证Haar典型高斯态，这比最坏情况界改善了 $d$ 倍。最后，由于该界仅依赖于目标的计算基分布，该协议可扩展到有效相位修饰态——即通过向高斯态注入有效可计算的相位对角相位而获得的态——并具有相同的样本复杂度。该类别包括用于matchgate计算的连续四模非高斯魔法态族。
英文摘要
Fermionic Gaussian states are the workhorse reference states for qubit-based quantum simulation of fermionic systems, yet existing certification protocols either proceed via fidelity estimation, leading to suboptimal sample complexity in the target precision $ε$, only apply to Haar-typical states, or require adaptive measurements on a large fraction of qubits. We give an adaptive protocol that certifies any $d$-mode pure fermionic Gaussian state using $O(d^2ε^{-1})$ copies, single-qubit measurements with only one qubit measured adaptively per copy, and $O(d^3)$ classical processing time per copy. The $ε^{-1}$ dependence is optimal for the certification problem, even among strategies using entangled measurements. The sample complexity is controlled by the spectral gap of a $d\leftrightarrow d-2$ down-up walk---a Markov chain studied in the theory of high-dimensional expanders, equivalent to a two-site Glauber dynamics in statistical physics---relating certification efficiency to relaxation time to equilibrium. The worst-case bound is tight for this protocol and is attained by physically relevant states, including ground states of the fully dimerized Su-Schrieffer-Heeger (SSH) chain and BCS pair states. In contrast, numerics up to $d=14$ suggest that $O(d ε^{-1})$ copies suffice to certify Haar-typical Gaussian states, a factor of $d$ improvement over the worst-case bound. Finally, because the bound depends only on the target's computational-basis distribution, the protocol extends to efficiently phase-dressed states---states obtained by injecting an efficiently computable diagonal phase to Gaussian states---with the same sample complexity. The class includes a continuous family of four-mode non-Gaussian magic states for matchgate computation.
Comments29 pages, 2 figures, comments welcome
