a minimum ; the value of the integral is the corresponding eigenvalue El of the Hamiltonian H. It is also well known that this property leads to a powerful method of approximating, at least qualitatively, 4,1 and Er, by minimizing (1) among a restricted class of functions. No similarly simple minimum property exists for the higher eigenvalues . One can obtain the nth eigenvalue by minimizing (1) with the subsidiary condition that >G be orthogonal to ¢l, 42, y n-r• However, this procedure becomes progressively more cumbersome as n increases and besides it is of no use unless 4,1, 02, • • • , 4'n-1 are very exactly known. For problems of thermal equilibrium one is often concerned with the free energy, rather than with the individual eigenvalues . It is the purpose of this note to draw attention to a simple minimum property of the free energy which may be considered as a generalization of the variation principle for the lowest eigenvalue. The free energy has the following property: If v1, W2 , • • • , pn, • • • are an arbitrary set of orthogonal and normalized functions, and
a minimum ; the value of the integral is the corresponding eigenvalue El of the Hamiltonian H. It is also well known that this property leads to a powerful method of approximating, at least qualitatively, 4,1 and Er, by minimizing (1) among a restricted class of functions. No similarly simple minimum property exists for the higher eigenvalues . One can obtain the nth eigenvalue by minimizing (1) with the subsidiary condition that >G be orthogonal to ¢l, 42, y n-r• However, this procedure becomes progressively more cumbersome as n increases and besides it is of no use unless 4,1, 02, • • • , 4'n-1 are very exactly known. For problems of thermal equilibrium one is often concerned with the free energy, rather than with the individual eigenvalues . It is the purpose of this note to draw attention to a simple minimum property of the free energy which may be considered as a generalization of the variation principle for the lowest eigenvalue. The free energy has the following property: If v1, W2 , • • • , pn, • • • are an arbitrary set of orthogonal and normalized functions, and
In most cases solving a quantum mechanical problem exactly is not always possible. In such a case, an analytic solution to the Schrödinger equation is appropriate. This results to numerical solutions that may not necessarily be the best approach. Interest sometimes may not necessarily be in the detailed eigenfunctions, but rather in the energy levels and the qualitative features of the eigenfunctions. So, the numerical solutions are usually less intuitively reasonable. It is possible to find the values of the energy levels to be slightly sensitive to the deviation of the wave function from its true form. So, the expectation value of the energy for an approximate wave function can be a very good estimate of the corresponding energy eigenvalue. Using an approximate wave function dependent on some set of variational parameters and minimizing its energy by these variational parameters yield such energy estimates. This technique is the so-called variational method due to this minimization procedure.
The adiabatic theorem is a powerful result concerning the evolution of quantum mechanical systems. Intuitively, it states the following: suppose we want to study the evolution of a quantum mechanical system governed by a given Hamiltonian
H (Def. 6.1), and the initial state of the system is the eigenstate of
H corresponding to its lowest eigenvalue. The laws of quantum physics tell us that the evolution of the system over time follows the Schrüdinger equation, Eq. (6.1). Suppose now that the Hamiltonian is not fixed, but rather, it changes very slowly over time, so that the evolution of the system is described by the differential equation
idtd∣ψ(t)⟩=H(t)∣ψ(t)⟩, and
H(t) is slowly changing. If, for any value of the time parameter
t, there is always a gap between the lowest eigenvalue of
H(t) and all other eigenvalues, then the adiabatic theorem asserts that the state of the system always remains in an eigenstate with the lowest eigenvalue throughout the evolution. One specific case of interest in this chapter occurs when the time-dependent Hamiltonian
H(t) varies from an initial (time-independent) Hamiltonian
HI to a final Hamiltonian
HF. If the change happens slowly, and the initial state of the system is an eigenstate of
HI with minimum eigenvalue, the adiabatic theorem asserts that the evolution eventually leads to the eigenstate with minimum eigenvalue of the final Hamiltonian
HF. This result can be used for optimization: imagine that the initial Hamiltonian encodes an easy problem for which we can construct the eigenstate with minimum eigenvalue, and the final Hamiltonian encodes a difficult optimization problem whose solution is given by the eigenstate with minimum eigenvalue. By the adiabatic theorem, evolving the eigenstate of the initial Hamiltonian while slowly changing the Hamiltonian into the final one leads to the solution of the difficult optimization problem. Of course we need to know how slowly
HI should be changed into
HF. The adiabatic theorem is at the heart of the quantum approximate optimization algorithm (QAOA), a framework that has been widely used to implement optimization algorithms on existing quantum hardware, because it has low requirements of quantum resources–although it does not guarantee improvements over classical algorithms in general. In this chapter we first discuss the adiabatic theorem, and then give an overview of QAOA.
For a general smooth one-dimensional potential well with N minima (containing N−1 barrier humps of arbitrary shape), a phase-integral quantization condition (including σ terms) is obtained, from which the energy eigenvalues can be calculated with an accuracy relative to the chosen order of the Fröman phase-integral approximation. The quantization equation is derived from a condition for total transmission through a general system of potential barriers of arbitrary shape, by means of a limiting procedure, in which the two outermost barriers are made to grow large in a particular way. Finally, the quantization condition is expressed by means of a concise recursion formula. As an illustration, the special case N=2 is treated in some detail. For N=2, the quantization condition turns out to be identical to a previously well-known phase-integral quantization condition for the double-minimum potential, the accuracy of which has been examined in several papers by other authors. Their results concerning a symmetric double-minimum model potential show that when the σ term is included, the third- and fifth-order energy eigenvalues are accurate enough for all practical purposes, even close to the barrier maximum, but if the σ term is neglected, the quantization condition breaks down near the barrier maximum, this breakdown becoming more dramatic with increasing order. A table displaying phase-integral eigenvalues for the model potential as compared with exact (numerical) quantum mechanical results is given here; the figures are quoted from the papers mentioned. Applying the results to the special case N=3, it will be given explicitly the quantization condition for an arbitrary triple-minimum potential.