Basic Sciences Journal of Textile Universities

The research on quantum adiabatic theorem

Cao Huaixi

Published Jan 1, 2015• 0 citations

Abstract

Quantum adiabatic theorem(QAT)is one of the important principles in quantum theory.With the discovery of the inconsistency of the QAT in application,it has received much more attention from physicists and mathematicians.Based on a brief review about the recent researches on QAT,some discussions on this topic are introduced,including the concepts of uniformly slowly evolving,CPT-frame,PT-symmetric operator,quantum adiabatic evolution theorem,quantum adiabatic approximation theorem,and error estimation of adiabatic approximation.

Related papers

INTRODUCING ADIABATIC QUANTUM COMPUTATION

Huazhong Li

Recent applications of quantum adiabatic approximation to quantum computation are reviewed.We discuss the adiabatic quantum algorithm and the recent discovery of inconsistency of quantum adiabatic theorem giving impact to this application.

Progress in physicsJan 1, 20080 citations

Validity of the quantum adiabatic theorem

Zhaoyan Wu, Hui Yang

arXiv:quant-ph/0410118Physical Review AJul 22, 200555 citations

Reexamination of the quantum adiabatic theorem

Yan Zhao

Physical Review AMar 21, 200823 citations

Invalidity of the quantitative adiabatic condition and general conditions for adiabatic approximations

Dafa Li

Laser Physics LettersApr 12, 20162 citations

On the consistency in the adiabatic theorem and quantum geometric phase

Huazhong Li

The recent discovery of inconsistency (MS inconsistency) in the adiabatic approximation is discussed. In particular, the so-called, inconsistency in Berry phase is analyzed. On the contrary to some authors, we found that the MS inconsistency persisted and the so-called inconsistency in Berry phase does not exist.

arXiv:quant-ph/0702145Feb 14, 20070 citations

Chapter 9: Optimization with the adiabatic theorem

Giacomo Nannicini

Society for Industrial and Applied Mathematics eBooksDec 17, 20250 citations

Basic Sciences Journal of Textile Universities

The research on quantum adiabatic theorem

Cao Huaixi

Published Jan 1, 2015• 0 citations

OpenAlex

Abstract

Related papers

INTRODUCING ADIABATIC QUANTUM COMPUTATION

Huazhong Li

Progress in physicsJan 1, 20080 citations

Validity of the quantum adiabatic theorem

Zhaoyan Wu, Hui Yang

The consistency of the quantum adiabatic theorem has been doubted recently. It is shown in the present paper that the difference between the adiabatic solution and the exact solution to the Schr\"odinger equation with a slowly changing driving Hamiltonian is small; while the difference between their time derivatives is not small. This explains why substituting the adiabatic solution back into the Schr\"odinger equation leads to ``inconsistency'' of the adiabatic theorem. Physics is determined completely by the state vector, and not by its time derivative. Therefore the quantum adiabatic theorem is physically correct.The consistency of the quantum adiabatic theorem has been doubted recently. It is shown in the present paper that the difference between the adiabatic solution and the exact solution to the Schr\"odinger equation with a slowly changing driving Hamiltonian is small; while the difference between their time derivatives is not small. This explains why substituting the adiabatic solution back into the Schr\"odinger equation leads to ``inconsistency'' of the adiabatic theorem. Physics is determined completely by the state vector, and not by its time derivative. Therefore the quantum adiabatic theorem is physically correct.

arXiv:quant-ph/0410118Physical Review AJul 22, 200555 citations

Reexamination of the quantum adiabatic theorem

Yan Zhao

The quantum adiabatic theorem is a widely applied result in quantum mechanics. However, recently, counterexamples were produced, indicating that the transition probabilities in some systems do not converge as predicted by the adiabatic theorem [K. P. Marzlin and B. C. Sanders, Phys. Rev. Lett. 93, 160408 (2004)]. In this paper, we examine a proof of the adiabatic theorem, paying specific attention to the validity of the assumptions made in the proof. We show that the convergence of the transition probabilities is path dependent when the change in the Hamiltonian is large. Finally, we consider the application of this reformulated form of the adiabatic theorem to the adiabatic evolution paradigm in quantum computation.The quantum adiabatic theorem is a widely applied result in quantum mechanics. However, recently, counterexamples were produced, indicating that the transition probabilities in some systems do not converge as predicted by the adiabatic theorem [K. P. Marzlin and B. C. Sanders, Phys. Rev. Lett. 93, 160408 (2004)]. In this paper, we examine a proof of the adiabatic theorem, paying specific attention to the validity of the assumptions made in the proof. We show that the convergence of the transition probabilities is path dependent when the change in the Hamiltonian is large. Finally, we consider the application of this reformulated form of the adiabatic theorem to the adiabatic evolution paradigm in quantum computation.

Physical Review AMar 21, 200823 citations

Invalidity of the quantitative adiabatic condition and general conditions for adiabatic approximations

Dafa Li

Abstract The adiabatic theorem was proposed about 90 years ago and has played an important role in quantum physics. The quantitative adiabatic condition constructed from eigenstates and eigenvalues of a Hamiltonian is a traditional tool to estimate adiabaticity and has proven to be the necessary and sufficient condition for adiabaticity. However, recently the condition has become a controversial subject. In this paper, we list some expressions to estimate the validity of the adiabatic approximation. We show that the quantitative adiabatic condition is invalid for the adiabatic approximation via the Euclidean distance between the adiabatic state and the evolution state. Furthermore, we deduce general necessary and sufficient conditions for the validity of the adiabatic approximation by different definitions.Abstract The adiabatic theorem was proposed about 90 years ago and has played an important role in quantum physics. The quantitative adiabatic condition constructed from eigenstates and eigenvalues of a Hamiltonian is a traditional tool to estimate adiabaticity and has proven to be the necessary and sufficient condition for adiabaticity. However, recently the condition has become a controversial subject. In this paper, we list some expressions to estimate the validity of the adiabatic approximation. We show that the quantitative adiabatic condition is invalid for the adiabatic approximation via the Euclidean distance between the adiabatic state and the evolution state. Furthermore, we deduce general necessary and sufficient conditions for the validity of the adiabatic approximation by different definitions.

Laser Physics LettersApr 12, 20162 citations

On the consistency in the adiabatic theorem and quantum geometric phase

Huazhong Li

arXiv:quant-ph/0702145Feb 14, 20070 citations

Chapter 9: Optimization with the adiabatic theorem

Giacomo Nannicini

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 $\mathcal H$ (Def. 6.1), and the initial state of the system is the eigenstate of $\mathcal H$ 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 $i\,\frac{\mathrm{d} | \psi(t) \rangle}{\mathrm{d}t} = \mathcal{H}(t) | \psi(t) \rangle$ , and $\mathcal{H}(t)$ is slowly changing. If, for any value of the time parameter $t$ , there is always a gap between the lowest eigenvalue of $\mathcal{H}(t)$ 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 $\mathcal{H}(t)$ varies from an initial (time-independent) Hamiltonian $\mathcal{H}_{\mathrm{I}}$ to a final Hamiltonian $\mathcal{H}_{\mathrm{F}}$ . If the change happens slowly, and the initial state of the system is an eigenstate of $\mathcal{H}_{\mathrm{I}}$ with minimum eigenvalue, the adiabatic theorem asserts that the evolution eventually leads to the eigenstate with minimum eigenvalue of the final Hamiltonian $\mathcal{H}_{\mathrm{F}}$ . 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 $\mathcal{H}_{\mathrm{I}}$ should be changed into $\mathcal{H}_{\mathrm{F}}$ . 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.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 $\mathcal H$ (Def. 6.1), and the initial state of the system is the eigenstate of $\mathcal H$ 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 $i\,\frac{\mathrm{d} | \psi(t) \rangle}{\mathrm{d}t} = \mathcal{H}(t) | \psi(t) \rangle$, and $\mathcal{H}(t)$ is slowly changing. If, for any value of the time parameter $t$, there is always a gap between the lowest eigenvalue of $\mathcal{H}(t)$ 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 $\mathcal{H}(t)$ varies from an initial (time-independent) Hamiltonian $\mathcal{H}_{\mathrm{I}}$ to a final Hamiltonian $\mathcal{H}_{\mathrm{F}}$. If the change happens slowly, and the initial state of the system is an eigenstate of $\mathcal{H}_{\mathrm{I}}$ with minimum eigenvalue, the adiabatic theorem asserts that the evolution eventually leads to the eigenstate with minimum eigenvalue of the final Hamiltonian $\mathcal{H}_{\mathrm{F}}$. 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 $\mathcal{H}_{\mathrm{I}}$ should be changed into $\mathcal{H}_{\mathrm{F}}$. 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.

Society for Industrial and Applied Mathematics eBooksDec 17, 20250 citations

\mathcal H

(Def. 6.1), and the initial state of the system is the eigenstate of

\mathcal 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

i\,\frac{\mathrm{d} | \psi(t) \rangle}{\mathrm{d}t} = \mathcal{H}(t) | \psi(t) \rangle

, and

\mathcal{H}(t)

is slowly changing. If, for any value of the time parameter

t

, there is always a gap between the lowest eigenvalue of

\mathcal{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

\mathcal{H}(t)

varies from an initial (time-independent) Hamiltonian

\mathcal{H}_{\mathrm{I}}

to a final Hamiltonian

\mathcal{H}_{\mathrm{F}}

. If the change happens slowly, and the initial state of the system is an eigenstate of

\mathcal{H}_{\mathrm{I}}

with minimum eigenvalue, the adiabatic theorem asserts that the evolution eventually leads to the eigenstate with minimum eigenvalue of the final Hamiltonian

\mathcal{H}_{\mathrm{F}}

. 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

\mathcal{H}_{\mathrm{I}}

should be changed into

\mathcal{H}_{\mathrm{F}}

. 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.