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International Journal of Modern Physics B

PATH INTEGRALS FOR QUBITS

Gian Fabrizio De Angelis

Published Feb 20, 2004• 0 citations
Publisher

Abstract

Feynman started the theory of quantum computation by observing that quantum mechanical amplitudes cannot be simulated efficiently on a computer, since path integrals are computationally hard. Path integrals are heuristic tools for Schrödinger's hamiltonians, at least for real time dynamics. On the contrary they have a rigorous mathematical meaning for quantum systems which are worth a finite but no matter how large number of qubits.

DOI
10.1142/s0217979204024239

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Feynman started the theory of quantum computation by observing that quantum mechanical amplitudes cannot be simulated efficiently on a computer, since path integrals are computationally hard. Path integrals are heuristic tools for Schrödinger's hamiltonians, at least for real time dynamics. On the contrary they have a rigorous mathematical meaning for quantum systems which are worth a finite but no matter how large number of qubits.
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The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to non-commuting operators. Generally, the quantum mechanics based on the Schrodinger equation is preferred for a system of particles. But path integral formulation of quantum mechanics finds applications in more complicated situations, particularly, in quantum field theory. The purpose of solving the Schrodinger equation lies in finding the time evolution operator which is able to generate the time translation of the given system. Richard Feynman noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. Dissipative phenomena with a quantum origin manifest themselves in many different situations and hence the study of dissipative systems and their quantization is of a great theoretical and practical value. Many dissipative systems can be modelled as a linearly damped harmonic oscillator.
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The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.
Introduction Preliminary remarks The path integral formalism is one of the most important tools in modern theoretical physics. Integration in functional spaces was considered for the first time by Norbert Wiener in his studies of diffusion processes [60, 61]. The corresponding integration measure is now known as the Wiener measure, i.e. the so-called Gaussian path integrals were studied. In quantum mechanics, functional integrals were introduced by R. Feynmann [62, 63] who was inspired by important observations due to P. A. M. Dirac concerning a relation between wave function and classical action [64, 65]. In modern physics the path integral formalism is applied not only to developing a quantum description for systems with infinitely many degrees of freedom as, for example, in quantum field theories or condensed matter physics, but also to ordinary quantum mechanical problems. Moreover there is a tendency to reformulate standard quantum mechanical problems (such as a particle in a “box” or potential well [66, 67], or a particle in the Coulomb field [68]) within the framework of the path integral method. The reason for doing this is rather obvious: physicists develop a new mathematical tool, a new language and, in order to comprehend all of its subtleties, it is necessary to investigate as many concrete examples as possible. Use of the path integral formalism is widespread in quantum field theory because it is the most adequate mathematical tool for a wide range of problems which allows, not only for elegant formulation of problems, but also for a great deal of simplification of their theoretical analysis.