AF: Small: Alternative Models of Quantum Computing for Optimization and Simulation of Quantum Systems

量子计算机可以比传统计算机更快地解决许多问题。然而，所有已知的加速量子算法都使用了少数几种成功的技术中的一种。这个项目追求新的量子方法来解决问题，特别是量子系统的优化和模拟。优化的方法将包括信号分析和滤波器设计中的线性变换，以推广量子计算中的关键傅里叶变换。将特别关注分数傅立叶变换和小波变换，以及它们在计算函数特定性质的量子黑盒算法中的应用。将开发实现这些变换的高效量子电路，并分析它们在量子算法设计中的影响。量子系统的模拟方法将使用绝热状态变换，这是通过缓慢演化获得物理学中的能态的关键工具。设计了计算有限温度性质的绝热量子算法，并将其计算成本与传统蒙特卡罗方法进行了比较。还将得到复杂性下界。此外，还将探讨所设计的量子方法在量子信息的其他领域，特别是在量子计量学和参数估计方面的影响。

Quantum computers could solve a number of problems substantially faster than their conventional counterparts. However, all known quantum algorithms exhibiting speedups use one of a small number of techniques that are responsible for their success. This project pursues novel quantum methods for problem solving, particularly for optimization and simulation of quantum systems. The methods for optimization will include linear transformations in signal analysis and filter design that generalize the Fourier transform, which is key in quantum computing. Special interest will be given to the fractional Fourier transform and wavelet transforms, and to their applications in quantum black-box algorithms that compute specific properties of a function. Efficient quantum circuits that implement these transformations will be developed and their impact in quantum-algorithmic design will be analyzed. The methods for the simulation of quantum systems will use adiabatic state transformation, which is a key tool for accessing energy states in physics via slow evolution. Adiabatic quantum algorithms to compute finite-temperature properties will be designed, and their computational cost will be compared to that of conventional Monte Carlo methods. Complexity lower bounds will also be obtained. In addition, the impact of the devised quantum methods in other areas of quantum information, especially in quantum metrology and parameter estimation, will be explored.