用量子计算机计算多体系统的生成函数：技术挑战和在量子-经典混合方法中的应用
Calculation of generating function in many-body systems with quantum computers: technical challenges and use in hybrid quantum-classical methods
摘要：
哈密尔顿算子$H$的生成函数被定义为$F(t)=\langle e^{-itH}\rangle$，其中$t$是时间，并且其中期望值取在给定的初始量子态上。该函数允许访问不同阶数$K$的哈密顿量$\langle H^{K}\rangle$的不同矩。$F(t)$的实部和虚部可以分别在量子计算机上使用一个额外的辅助量子比特进行计算，其中每个时间值$t$具有一组测量。量子位的低成本使其在量子位数量有限的短期内非常有吸引力。假设生成函数可以使用量子设备精确计算，我们展示了如何在经典计算机上后验地使用该函数的信息内容来解决量子多体问题。经典的后处理的几种方法进行说明，其目的是预测近似的基态或激发态能量和/或近似的长时间演化。该后处理可以使用基于Krylov空间和/或基于与虚时间演化密切相关的$t$展开方法的方法来实现。在多体相互作用系统中，使用配对和费米-哈伯德模型的混合量子经典计算。
英文摘要：
The generating function of a Hamiltonian $H$ is defined as $F(t)=\langle e^{-itH}\rangle$, where $t$ is the time and where the expectation value is taken on a given initial quantum state. This function gives access to the different moments of the Hamiltonian $\langle H^{K}\rangle$ at various orders $K$. The real and imaginary parts of $F(t)$ can be respectively evaluated on quantum computers using one extra ancillary qubit with a set of measurement for each value of the time $t$. The low cost in terms of qubits renders it very attractive in the near term period where the number of qubits is limited. Assuming that the generating function can be precisely computed using quantum devices, we show how the information content of this function can be used a posteriori on classical computers to solve quantum many-body problems. Several methods of classical post-processing are illustrated with the aim to predict approximate ground or excited state energies and/or approximate long-time evolutions. This post-processing can be achieved using methods based on the Krylov space and/or on the $t$-expansion approach that is closely related to the imaginary time evolution. Hybrid quantum-classical calculations are illustrated in many-body interacting systems using the pairing and Fermi-Hubbard models.
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