# Interpolation approach to Hamiltonian-varying quantum systems and the adiabatic theorem

- Yu Pan
^{1}Email author, - Zibo Miao
^{2}, - Nina H Amini
^{3}, - Valery Ugrinovskii
^{4}and - Matthew R James
^{1}

**Received: **30 June 2015

**Accepted: **7 November 2015

**Published: **25 November 2015

## Abstract

Quantum control could be implemented by varying the system Hamiltonian. According to adiabatic theorem, a slowly changing Hamiltonian can approximately keep the system at the ground state during the evolution if the initial state is a ground state. In this paper we consider this process as an interpolation between the initial and final Hamiltonians. We use the mean value of a single operator to measure the distance between the final state and the ideal ground state. This measure resembles the excitation energy or excess work performed in thermodynamics, which can be taken as the error of adiabatic approximation. We prove that under certain conditions, this error can be estimated for an arbitrarily given interpolating function. This error estimation could be used as guideline to induce adiabatic evolution. According to our calculation, the adiabatic approximation error is not linearly proportional to the average speed of the variation of the system Hamiltonian and the inverse of the energy gaps in many cases. In particular, we apply this analysis to an example in which the applicability of the adiabatic theorem is questionable.

## Keywords

## 1 Introduction

Adiabatic process is aimed at stabilizing a parameter-varying quantum system at its eigenstate. This process has many applications in the engineering of quantum systems [1–5], and in particular plays the fundamental role in adiabatic quantum computation (AQC) [6–8]. The adiabatic theorem [9, 10] states that a system will undergo adiabatic evolution given that the system parameter varies slowly.

Quantifying the applicability of adiabatic approximations is an interesting topic of current research efforts. On the one hand, this kind of research has been spurred by so-called shortcuts to adiabaticity [11], and on the other hand recent insights from thermodynamics haven put adiabatic processes back into focus [12, 13]. In particular, the validity of the adiabatic theorem has been under intensive studies both theoretically and experimentally since it was proposed, and much of these efforts were devoted to the rigorous description of the sufficient quantitative conditions of adiabatic theorem, and the estimation of the error accumulated over a long time [10, 14–16]. Once the exact knowledge on the adiabatic process is available, it is straightforward to apply the results to the optimal design of adiabatic control on specific systems [17, 18]. The most interesting progress is that the validity of the adiabatic theorem itself has been challenged in the recent decade [19–27], both by strict analysis and counter-examples. According to these findings, the errors induced by the adiabatic approximation could accumulate over time despite certain quantitative condition is satisfied [19–21, 24, 25], e.g., when there exists an additional perturbation or driving that is resonant with the system. Particularly as indicated in [24], it is not new that resonant driving can cause population transfer between eigenstates. Also, a proof can be found in [25] stating that only a resonant perturbation whose amplitude gradually decays to zero can result in a violation of a well-known sufficient condition.

*λ*is a dimensionless quantity. Δ

*H*is a fixed operator and so the direction of the variation is fixed. We assume \(H_{1}\), \(H_{2}\), and Δ

*H*are bounded operators throughout this paper.

*T*is the evolution time. The transition of the system from \(H_{1}\) to \(H_{2}\) can be described using an interpolating function \(f(t)\) so that

*λ*should be smaller than a threshold value. It is worth mentioning that the classical adiabatic theorem was proved also using a perturbative analysis, which cannot be applied directly to a large variation of Hamiltonian. Therefore, our analysis in this paper is not concerned with the adiabatic evolution for a large variation of Hamiltonian. However, our analysis provides a rigorous estimation of the error accumulated during this small-variation evolution for an arbitrarily given interpolation.

Our work is different from the previous works in two ways. First, instead of studying the evolution of the eigenstates and their corresponding probability amplitudes, the mean value of a Hermitian operator is defined as a measure of the error. For example, in the context of adiabatic quantum computation where one wants to prepare the ground state of a target Hamiltonian \(\hat{H}_{2}\geq0\) whose ground-state energy is 0, \(\epsilon =\langle\hat{H}_{2}\rangle_{\rho_{t}}\) serves as a good measure of the distance between the real-time state \(\rho_{t}\) and the ground state. This measure resembles the excitation energy or excess work performed during the process, as studied in thermodynamics [12]. In this paper we only consider the error accumulated over the entire process, which means we are only interested in \(\epsilon=\langle \hat{H}_{2}\rangle_{\rho_{T}}\). The second difference is that the error, or the excitation energy or excess work performed during the process, can be estimated with a sufficient precision for arbitrarily given interpolating functions. As a result, the parameters which are related to the suppression of the error can be easily identified. For example, we have \(\epsilon=O(\frac{\lambda^{2}}{T^{2}\lambda_{2}^{3}})\) as \(\lambda\rightarrow0\) in the case of linear interpolation. Here \(\lambda_{2}\) is the energy gap between the ground and first-excited states of the initial Hamiltonian. However for the interpolation in the counterexample [19, 25], the scaling of *ϵ* is not so simple.

This paper is organized as follows. In Section 2, we introduce the model of this paper. In Section 3, we give the estimation of the error for linear interpolation. In Section 4, we present the general algorithm to estimate the error for an arbitrarily given interpolating function. We discuss three examples in Section 5. Conclusion is given in Section 6.

## 2 Definitions and preliminaries

The system is defined on an *N*-dimensional Hilbert space. We set Dirac constant \(\hbar=1\). \(\Vert \cdot \Vert \) denotes the matrix norm. Two real functions \(f_{1}(x)\) and \(f_{2}(x)\) can be denoted as \(f_{1}(x)=O(f_{2}(x))\), \(x\rightarrow\infty\), if and only if there exists a positive real number *M* and a real number \(x_{0}\) such that \(\vert f_{1}(x)\vert \leq M\vert f_{2}(x)\vert \), \(x\geq x_{0}\), where \(\vert \cdot \vert \) denotes the absolute value.

Let \(\{\omega_{i}:i=1,2,\ldots, N\}\) be the monotonically increasing sequence of eigenvalues of \(H_{1}\), so that \(\omega_{i}\geq\omega_{j}\) when \(i>j\), and \(\{|i\rangle\}\) be the corresponding eigenstates. We denote the energy gap between the *i*th eigenstate and the ground state as \(\lambda_{i}=\omega_{i}-\omega_{1}\). Similarly, we define the increasing sequence of eigenvalues of \(H_{2}\), \(\{\omega _{i}':i=1,2,\ldots, N\}\) and \(\{\lambda_{i}'\}\), \(\{|i'\rangle\}\) correspondingly.

For convenience, we also introduce two offset Hamiltonians, \(\hat{H}_{1}\) and \(\hat{H}_{2}\). The Hamiltonian \(\hat{H}_{1}\) is defined as \(\hat{H}_{1}=H_{1}-\omega_{1}\), i.e., by offsetting the Hamiltonian of the system at \(t=0\) by a constant operator \(\omega_{1}\) so that \(\hat{H}_{1}\geq0\). By \(\hat{H}_{1}\geq0\) we mean \(\hat{H}_{1}\) is positive semidefinite and its the smallest eigenvalue of \(\hat{H}_{1}\) is zero. Similarly, we define \(\hat{H}_{2}=H_{2}-\omega'_{1}\geq0\) by offsetting the system Hamiltonian by a constant operator \(\omega'_{1}\). Let \(\rho _{t}\) denote the system state at time *t* and let \(\rho_{g}\) be the initial state of the system at \(t=0\). We always assume that \(\rho_{g}\) is the ground state of \(\hat{H}_{1}\), and so we have \(\langle\hat{H}_{1}\rangle_{\rho_{g}}=0\).

The measure of adiabaticity is proposed as follows

### Definition 1

Obviously, if the evolution is adiabatic, i.e., \(\rho_{T}\) is the ground state of \(H_{2}\), then we have \(\epsilon=0\). In particular, *ϵ* is closely related to the fidelity of the final state and ground state in the Schrödinger picture (see Appendix C). A small error *ϵ* implies a large fidelity.

In this paper we also call *ϵ* the adiabatic approximation error, as *ϵ* reflects how well we can approximate the evolution as a perfect adiabatic process.

*λ*such that \(\rho_{t}\), \(t\in[0,T]\) can be expanded using Magnus series in the interaction picture. For more details about the expansion in the interaction picture, please refer to Appendix A. If the series expansion is valid in the interaction picture, we can transform back to the Schrödinger picture and write the evolution of the state as (see Appendix A)

Our aim is to estimate an asymptotic behaviour of *ϵ* provided \(\lambda\to0\). Furthermore, we will use the obtained estimate to analyze several cases of the adiabatic theorem including those where some difficulties with adiabatic approximation have been encountered.

## 3 Adiabatic approximation under linear interpolation of the Hamiltonian

*H*is the system Hamiltonian. Recall that \(\rho_{g}=|1\rangle \langle1|\). Since \(H_{1}|1\rangle=\omega_{1}|1\rangle\), \(\langle X(t)\rangle_{\rho_{g}}\) is a constant of motion under the action of \(H_{1}\):

*ϵ*. The time evolution of \(\langle\hat{H}_{2}\rangle_{\rho_{t}}\) is determined by its generator \(\frac{d}{dt}\langle\hat{H}_{2}\rangle _{\rho_{t}}=\langle-\mathrm{i}[\hat{H}_{2},H(t)]\rangle_{\rho_{t}}\). For linear interpolating function \(f(t)=\frac{t}{T}\), integration of \(\frac {d}{dt}\langle\hat{H}_{2}\rangle_{\rho_{t}}\) over \([0,T]\) results in the following expression (see details in Appendix B):

*ϵ*can be expressed as

*ϵ*to zero as

*λ*tends to zero in the case where \(f(t)\) defines a linear interpolation, as summarized in the following proposition:

### Proposition 1

*Assume*
\(\lambda_{2}>0\) (*the ground state of*
\(H_{1}\)
*is non*-*degenerate*) *and suppose*
\(f(t)=t/T\), *which corresponds to the linear interpolation of the Hamiltonian*. *The estimation of*
*ϵ*
*is given by*
\(\sum_{i\neq 1}\frac{4\lambda^{2}\sin^{2}(\lambda_{i}T/2)|\langle1|\Delta H|i\rangle |^{2}}{T^{2}\lambda_{i}^{3}}+O(\lambda^{3})\), *which is of the order*
\(O(\frac{\lambda^{2}}{T^{2}\lambda_{2}^{3}})\)
*as*
\(\lambda\rightarrow0\).

### Proof

*λ*and \(\omega_{1}\) [28]:

*ϵ*is estimated by

## 4 Error estimation for arbitrary interpolations

### Proposition 2

*For an arbitrarily given*\(f(t)\),

*the error estimation is given by*

*as*\(\lambda\rightarrow0\).

### Proof

*ϵ*is still calculated by (11), using \(\langle H_{2}\rangle_{\rho_{T}}-\langle H_{2}\rangle_{\rho_{g}}\) and \(\omega _{1}'-\langle H_{2}\rangle_{\rho_{g}}\). We have

It must be pointed out that \(A(T)\) is very easy to calculate with the aid of any softwares that can perform symbolic integration, and therefore it is straightforward to apply Proposition 2 to find the error estimation for a given interpolating function, as we are going to do in the next section.

## 5 Examples

### 5.1 Linear interpolation: \(f(t)=t/T\)

*H*are bounded, this error term is primarily determined by \(\frac{\lambda}{T}\) which is the average speed of the variation of the system Hamiltonian, and \(\frac{1}{\lambda_{i}}\) which is the inverse of the energy gap between the ground and

*i*th eigenstates of \(H_{1}\), as \(\lambda\rightarrow0\). In particular, we have

*ϵ*is estimated to be proportional to the square of the average speed of the variation of the Hamiltonian, which is \((\frac{\lambda}{T})^{2}\), as \(\lambda\rightarrow0\).

### 5.2 Quadratic interpolation: \(f(t)=t^{2}/T^{2}\)

*λ*, the error is estimated to be of order of \(\lambda^{2}\):

*ϵ*with respect of \((\frac{\lambda}{T})^{2}\) is primarily determined by the inverse of the energy gaps as \(\lambda\rightarrow0\). In the quadratic case, this scaling is primarily determined by a complex factor \([\frac{16\sin^{2}(\frac{T\lambda_{i}}{2})}{T^{2}\lambda_{i}^{5}}+\frac{4}{\lambda _{i}^{3}}-\frac{8\sin(T\lambda_{i})}{T\lambda_{i}^{4}}]\) which depends mainly on the inverse of the energy gaps \(\{\lambda_{i}\}\) and the inverse of the evolution time

*T*.

### 5.3 Interpolation with decaying resonant terms

*i*th term in (29) approaches

*T*being in its denominator. The error resulting from the

*i*th term is given by

*T*in an adiabatic evolution experiment, the adiabatic approximation error may not decrease as expected when one applies a slow evolution speed \(\frac{\lambda}{T}\).

*ϵ*to zero observed in this case and the quadratic case, as the speed of the adiabatic process (\(\lambda /T\)) reduces and the evolution horizon

*T*increases. The difference in the speed of convergence can be clearly seen using the ratio

*ϵ*goes to zero as \(\lambda\to0\) at a much slower rate than in the linear interpolation case or the quadratic interpolation case if

*T*is large. Furthermore, the larger

*T*is, the slower the convergence.

## 6 Conclusion

In this paper we provide a rigorous analysis of the time-dependent evolution of Hamiltonian-varying quantum systems. As we calculated, the adiabatic approximation error is not proportional to the average speed of the variation of the system Hamiltonian and the inverse of the energy gaps in many cases. The results in this paper may provide guidelines when applying complicated interpolation for adiabatic evolution.

## Declarations

### Acknowledgements

Yu Pan would like to thank Li Li and Charles Hill for their valuable suggestions. We gratefully acknowledge support by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (project number CE110001027), Australian Research Council Discovery Project (projects DP110102322 and DP140101779).

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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