有限能量下的量子模拟算法
Algorithms for quantum simulation at finite energies
摘要：
我们介绍了两种量子算法来探索多体系统的微正则和正则性质。第一个是混合量子算法，给定一个有效的可压缩状态，计算其平均能量附近的有限能量区间内的期望值。该算法基于滤波算子，类似于量子相位估计，其投射出期望能量区间之外的能量。然而，代替对物理状态执行此操作，其通过执行干涉测量来恢复物理值，而无需准备经滤波状态。我们表明，计算时间尺度与量子位数，规定的方差的倒数，和逆误差的多项式。在实践中，该算法不需要长时间的演化，而是需要大量的测量以获得合理的结果。我们的第二个算法是一个量子辅助蒙特卡罗抽样方法来计算其他数量接近的期望值的微正则和正则系综。这种方法利用经典的蒙特卡罗技术和量子计算机作为资源，只要能够制备具有合适能量的态，就可以避免经典量子蒙特卡罗模拟中的符号问题。所有算法都可以与小型量子计算机和模拟量子模拟器一起使用，只要它们可以执行干涉测量。我们还表明，这最后一项任务可以大大简化的代价进行更多的测量。
英文摘要：
We introduce two kinds of quantum algorithms to explore microcanonical and canonical properties of many-body systems. The first one is a hybrid quantum algorithm that, given an efficiently preparable state, computes expectation values in a finite energy interval around its mean energy. This algorithm is based on a filtering operator, similar to quantum phase estimation, which projects out energies outside the desired energy interval. However, instead of performing this operation on a physical state, it recovers the physical values by performing interferometric measurements without the need to prepare the filtered state. We show that the computational time scales polynomially with the number of qubits, the inverse of the prescribed variance, and the inverse error. In practice, the algorithm does not require the evolution for long times, but instead a significant number of measurements in order to obtain sensible results. Our second algorithm is a quantum-assisted Monte Carlo sampling method to compute other quantities which approach the expectation values for the microcanonical and canonical ensembles. Using classical Monte Carlo techniques and the quantum computer as a resource, this method circumvents the sign problem that is plaguing classical Quantum Monte Carlo simulations, as long as one can prepare states with suitable energies. All algorithms can be used with small quantum computers and analog quantum simulators, as long as they can perform the interferometric measurements. We also show that this last task can be greatly simplified at the expense of performing more measurements.
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