Chapter 14 Differential Equations for Quantum Observables

Valerio Magnasco

Henrik Bruus, Karsten Flensberg

Abstract In the previous chapters, we saw how various physical observables can be expressed in terms of retarded Green’s functions. In many cases we need to calculate the time dependence of these functions. There are several ways of attacking this problem, one of which is the equation of motion technique. The basic idea of this method is to generate a series of coupled differential equations by differentiating the function at hand a number of times. If these equations close, the problem is in principle solvable; and if not, one needs to invoke physical arguments to truncate the set of equations in a reasonable fashion. For example, one can neglect certain correlations. We shall study examples of both situations in this chapter.

Alexander M. Dalzell, Sam McArdle, Mario Berta +10 more

This chapter covers quantum algorithms for numerically solving differential equations and the areas of application where such capabilities might be useful, such as computational fluid dynamics, semiconductor chip design, and many engineering workflows. We focus mainly on algorithms for linear differential equations (covering both partial and ordinary linear differential equations), but we also mention the additional nuances that arise for nonlinear differential equations. We discuss important caveats related to both the data input and output aspects of an end-to-end differential equation solver, and we place these quantum methods in the context of existing classical methods currently in use for these problems.

Vasily Evgenevich Tarasov

Efstratios Manousakis

Abstract This chapter begins by studying the one-dimensional time-dependent Schrödinger equation. In order to do this the chapter discretizes the spatial coordinates, and thus the differential equation becomes a set of coupled difference equations first order in time. Dealing with a finite number or a numerable set of states makes it easier to understand the concepts of Hilbert space, its measure, and the overlap between states. Furthermore, the chapter only has to deal with the Kronecker symbol as opposed to the Dirac delta function. Later, in Chapter 3, it will consider the continuum limit where we will naturally recover the Schrödinger differential equation, the state normalization condition as an integration condition, and the delta function.

Vasily Evgenevich Tarasov