We formulate an adiabatic approximation for the imaginary-time Schroedinger equation. The obtained adiabatic condition consists of two inequalities, one of which coincides with the conventional adiabatic condition for the real-time Schroedinger equation, but the other does not. We apply this adiabatic approximation to the analysis of Markovian dynamics of the classical Ising model, which can be formulated as the imaginary-time Schrödinger equation, to obtain an asymptotic formula for the probability that the system reaches the ground state in the limit of a long annealing time in simulated annealing. Using this form, we amend the theory of Somma, Batista, and Ortiz for a convergence condition for simulated annealing.
This is a lecture note of Professor Shuichi Tasaki. In the last part of this manuscript, SA and FB demonstrate some recent results as well. As far as SA's knowledge, regular courses were given every two years for graduate students at Waseda university, and SA attended the course twice (probably in 2005 and 2007 or in 2006 and 2008). Some topics were added in the lectures of the later year, probably Professor Tasaki modified his lectures as he was discovering new results. A similar but intensive short course was given once as at Kyoto university. SA thinks Professor Tasaki had a unique philosophy of physics, which enabled him to give very attractive lectures that unfortunately it is not possible to attend any more. In Professor Tasaki's lectures, both beautiful mathematical views and clear physical intuition were always given together, which SA believe, is quite difficult. As many people know Professor Tasaki was very strong in mathematics, but he never forgot to think of physical interpretations. Professor Tasaki was writing a lecture note of this course based on the note taken by a student of Kyoto university. Unfortunately, Prof. Tasaki passed away before he finished writing the manuscript. This lecture note is based on the incomplete Japanese note[1] and a note taken by SA at Waseda university. It seems that main stories of the lectures are the same, but Professor Tasaki gave different materials in detail, and use different notations. This manuscript mainly follows the note of SA. SA and FB decided to write this manuscript (1) to complete a note of Professor Tasaki's lectures, and (2) to give an access for non-Japanese speaker to his lecture note. However, they are slow writer, and they could not finish it before the deadline of this volume. They could finish only parts before nonequilibrium states (only some of the results will be shown in this version), and some recent results about Landauer formula and sufficient condition on interaction for the existence of unique steady states. They hope to complete the lecture note part, and upload it on Arxiv in the future.
This is a set of lecture notes used in a graduate topic class in applied mathematics called ``Quantum Algorithms for Scientific Computation'' at the Department of Mathematics, UC Berkeley during the fall semester of 2021. These lecture notes focus only on quantum algorithms closely related to scientific computation, and in particular, matrix computation. The main purpose of the lecture notes is to introduce quantum phase estimation (QPE) and ``post-QPE'' methods such as block encoding, quantum signal processing, and quantum singular value transformation, and to demonstrate their applications in solving eigenvalue problems, linear systems of equations, and differential equations. The intended audience is the broad computational science and engineering (CSE) community interested in using fault-tolerant quantum computers to solve challenging scientific computing problems.