一种用于估计费米子可观测量与哈密顿量的简单高效联合测量策略
A Simple and Efficient Joint Measurement Strategy for Estimating Fermionic Observables and Hamiltonians
AI总结：
本文提出一种基于随机化酉算符、费米子高斯酉采样、占据数测量和后处理的联合测量方案，能以与费米子经典阴影相当的测量轮数估计二次和四次 Majorana 单项式，并在矩形晶格等设置下以更低电路深度和双量子比特门数实现，为估计费米子可观测量和哈密顿量提供了新的竞争性替代方法。
摘要：
我们提出一种简单方案，用于估计量子化学和相关费米子系统中重要的费米子可观测量与哈密顿量。我们的方法基于实现一种测量，该测量联合测量一个 N 模式费米子系统中任意两个或四个 Majorana 算符乘积的含噪声版本。为实现该测量，我们使用：(i) 在一组可实现 Majorana 费米子算符乘积的酉算符上进行随机化；(ii) 从一组大小恒定的、适当选择的费米子高斯酉算符中随机采样一个酉算符；(iii) 测量费米子占据数；(iv) 适当的后处理。我们的方案可以分别使用 O(N log(N)/ε^2) 和 O(N^2 log(N)/ε^2) 轮测量，将所有二次和四次 Majorana 单项式的期望值估计到 ε 精度，与费米子经典阴影（fermionic classical shadows）所提供的性能相匹配。在某些设置下，例如通过 Jordan-Wigner 变换编码 N 模式费米子系统的矩形量子比特晶格，我们的方案可以在电路深度 O(N^{1/2}) 内、使用 O(N^{3/2}) 个双量子比特门实现，相比需要深度 O(N) 和 O(N^2) 个双量子比特门的费米子经典阴影和 matchgate 经典阴影有所改进。通过在示例性分子哈密顿量上对我们的方法进行基准测试，并观察到与费米子经典阴影相当的性能，我们展示了一种新颖的、具有竞争力的现有策略替代方案。
英文摘要：
We propose a simple scheme to estimate fermionic observables and Hamiltonians relevant in quantum chemistry and correlated fermionic systems. Our approach is based on implementing a measurement that jointly measures noisy versions of any product of two or four Majorana operators in an $N$ mode fermionic system. To realize our measurement we use: (i) a randomization over a set of unitaries that realize products of Majorana fermion operators; (ii) a unitary, sampled at random from a constant-size set of suitably chosen fermionic Gaussian unitaries; (iii) a measurement of fermionic occupation numbers; (iv) suitable post-processing. Our scheme can estimate expectation values of all quadratic and quartic Majorana monomials to $ε$ precision using $\mathcal{O}(N \log(N)/ε^2)$ and $\mathcal{O}(N^2 \log(N)/ε^2)$ measurement rounds respectively, matching the performance offered by fermionic classical shadows. In certain settings, such as a rectangular lattice of qubits which encode an $N$ mode fermionic system via the Jordan-Wigner transformation, our scheme can be implemented in circuit depth $\mathcal{O}(N^{1/2})$ with $\mathcal{O}(N^{3/2})$ two-qubit gates, offering an improvement over fermionic and matchgate classical shadows that require depth $\mathcal{O}(N)$ and $\mathcal{O}(N^2)$ two-qubit gates. By benchmarking our method on exemplary molecular Hamiltonians and observing performances comparable to fermionic classical shadows, we demonstrate a novel, competitive alternative to existing strategies.
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