Quantum computations (course of lectures)

Abstract

This course of lectures has been taught for several years at the Lomonosov Moscow State University; its modified version in 2021 is read in the Zhejiang University (Hangzhou), in the framework of summer school on quantum computing. The course is devoted to a new type of computations based on quantum mechanics. Quantum computations are fundamentally different from classical ones in that they occur in the space of so-called quantum states, and not in ordinary binary strings. The physical implementation of quantum computing -a device called a quantum computer has already been partially created, and its technology continues to develop intensively. Quantum computing is a real process in which the mathematical description is inextricably linked with quantum physics. In particular, the quantum mechanics of complex systems, the model of which is a quantum computer, is currently only being created, so quantum computing is a fundamental direction to a greater extent than an applied one. Therefore, in the course -in general, mathematical, much attention is paid to the physical implementation of this new type of computations. Various forms of quantum computing are considered: the Feynman gate model, fermionic and adiabatic computations. A class of problems is described in which quantum computing is not only more efficient than classical ones, but also cannot be replaced by them. These are the most important tasks of describing complex processes at the predictive level. In particular, using the phenomenon of quantum nonlocality, discovered at the end of the 20th century. Estimates of the ultimate possibility of quantum computing are also given -lower estimates of quantum complexity. The course is designed for students of physics and mathematics and natural science specialties as well as all those interested in this subject. It requires familiarity with the basics of linear algebra and mathematical analysis in the first two courses of the university.