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.
We expand upon the standard quantum adiabatic theorem, examining the time-dependence of quantum evolution in the near-adiabatic limit. We examine a Hamiltonian that evolves along some fixed trajectory from
H^0 to
H^1 in a total evolution-time
τ, and our goal is to determine how the final state of the system depends on
τ. If the system is initialized in a non-degenerate ground state, the adiabatic theorem says that in the limit of large
τ, the system will stay in the ground state. We examine the near-adiabatic limit where the system evolves slowly enough that most but not all of the final state is in the ground state, and we find that the probability of leaving the ground state oscillates in
τ with a frequency determined by the integral of the spectral gap along the trajectory of the Hamiltonian, so long as the gap is big. If the gap becomes exceedingly small, the final probability is the sum of oscillatory behavior determined by the integrals of the gap before and after the small gap. We confirm these analytic predictions with numerical evidence from barrier tunneling problems in the context of quantum adiabatic optimization.
Adiabatic passage employs a slowly varying time-dependent Hamiltonian to control the evolution of a quantum system along the Hamiltonian eigenstates. For processes of finite duration, the exact time-evolving state may deviate from the adiabatic eigenstate at intermediate times, but in numerous applications it is observed that this deviation reaches a maximum and then decreases significantly towards the end of the process. We provide a straightforward theoretical explanation for this welcome but often unappreciated fact. Our analysis emphasizes a separate adiabaticity criterion for high-fidelity state-to-state transfer and it points to new effective shortcut strategies for near-adiabatic dynamics.
The behavior of a quantum-mechanical system with a slowly modulated oscillatory Hamiltonian is characterized by an adiabatic theorem similar to that for a system with a slowly changing ``static'' Hamiltonian. Quasiperiodic states—solutions of the instantaneous Schrödinger equation with an oscillatory Hamiltonian—play the same role as eigenfunctions of the instantaneous Hamiltonian do in an adiabatic theorem for a nearly static Hamiltonian. As an example, the theorem is used to establish the correct wavefunction to be used in computing the refractive index of atomic hydrogen.
We prove the adiabatic theorem for quantum evolution without the traditional gap condition. We show that the theorem holds essentially in all cases where it can be formulated. In particular, our result implies that the adiabatic theorem holds also for eigenvalues embedded in the continuous spectrum. If there is information on the Hölder continuity of the spectral measure, then one can also estimate the rate at which the adiabatic limit is approached. The adiabatic theorem of Quantum Mechanics describes the long time behavior of the solutions of an initial value problem where the Hamiltonian generating the evolution depends slowly in time. Traditionally, the theorem is stated for Hamiltonians that have an eigenvalue which is separated by a gap from the rest of the spectrum. Folk wisdom is that a gap condition is a sine qua non for the adiabatic theorem to hold. In particular, according to this folk wisdom, one does not expect a general adiabatic theorem for Hamiltonians that have an eigenvalue embedded in, say, the continuous spectrum. Our purpose is to show that this folk wisdom is wrong, and there is a general adiabatic theorem without a gap condition. All one really needs for the adiabatic theorem is a spectral projection for the Hamiltonian that depends smoothly on time.