Basic Theory of Quantum Annealing

模拟退火是一种广泛应用于近似求解各种组合优化问题的方法。在模拟退火过程中，需要适当地控制温度以达到最优状态。在本研究计划中，我们建立了用量子涨落代替热涨落进行态间跃迁的方法，并将这种技术应用于旅行商问题。我们还提出了一种利用量子涨落进行图像恢复的新观点，在基态横场中的伊辛模型的性质在许多方面与有限温度下没有横场的伊辛模型的性质相似.因此，可以预期的是，横向场不仅在平衡态方面，而且在动力学性质方面也起着与温度类似的作用。事实上，我们已经证实了这一猜想，通过数值计算的伊辛模型的基态与时间相关的横向场，我们已经表明，量子退火给出了一个更大的概率系统达到真正的基态比热搅动。我们将这一思想应用于一个典型的优化问题，旅行商问题，和随机图像恢复，并取得了非常满意的结果。特别是，在后者的应用中，我们发现，适当的热和量子波动的组合产生恢复协议对超参数估计的不确定性鲁棒。

Simulated annealing is a widely used method to approximately solve various combinatorial optimization problems. It is necessary to control the temperature appropriately in simulated annealing to achieve the optimal state. In the present research project, we have established the method to use quantum fluctuations instead of thermal fluctuations for transitions between states, and applied this technique the traveling salesman problem. We have also introduced a new point of view to image restoration by using quantum fluctuations.Properties of the Ising model in a transverse field at the ground state is in many respects similar to those at finite temperatures without transverse field. It is thus expected that the transverse field plays similar roles as the temperature not just in equilibrium aspects but in dynamical properties as well. Indeed we have confirmed this conjecture by numerically calculating the ground state of the Ising model with time-dependent transverse field; we have shown that the quantum annealing gives a larger probability for the system to reach the true ground state than the thermal agitations. We have applied this idea to a typical optimization problem, the traveling salesman problem, and stochastic image restoration and have obtained very satisfactory results. In particular, in the latter application, we have found that appropriate combinations of thermal and quantum fluctuations yield restoration protocol robust against uncertainties in hyperparameter estimation.