Interpolation Approach to Hamiltonian-varying Quantum Systems and the Adiabatic Theorem

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 could be taken as the error of adiabatic approximation. We prove under certain conditions, this error can be precisely 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 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 on which the applicability of the adiabatic theorem is questionable.

and the inverse of the energy gaps in many cases. In particular, we apply this analysis to an example on which the applicability of the adiabatic theorem is questionable.

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 [1,2,3,4,5], and in particular plays the fundamental role in adiabatic quantum computation (AQC) [6,7,8]. The adiabatic theorem [9,10] states that a system will undergo adiabatic evolution given that the system parameter varies slowly.
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 [10,11,12,13]. 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 [14,15]. The most interesting progress is that the validity of the adiabatic theorem itself has been challenged in the recent decade [16,17,18,19,20,21,22,23,24], 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 [16,17,18,21,22], e.g., when there exists an additional perturbation or driving that is resonant with the system. Particularly as indicated in [21], it is not new that resonant driving can cause population transfer between eigenstates. Also, a proof can be found in [22] stating that only a resonant perturbation whose amplitude gradually decays to zero can result in a violation of a well-known sufficient condition.
In this paper we consider the following process: The initial Hamiltonian is H 1 . The final Hamiltonian H 2 = H 1 + λ∆H is obtained by varying H 1 along a fixed direction ∆H. λ is the parameter which measures the maximum variation of H(t). We define an interpolating function T is the evolution time. We work under the condition that a valid perturbative analysis of the system evolution is available. This often means λ should be smaller than a threshold value. 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 H 2 ≥ 0, ǫ = H 2 ρt will serve as a good measure of the distance between the real-time state ρ t and the ground state. In this paper we only consider the error accumulated over the entire process, which means we are only interested in ǫ = H 2 ρ T . This operator approach provides another way to look at this time-dependent evolution and sometimes greatly simplifies the calculations. The second difference is that the error 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 ǫ = O( λ 2 as λ → 0 in the case of linear interpolation. However for the interpolation in the counterexample [16,22], the scaling of ǫ 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.

Definitions and Preliminaries
To begin with, we consider the process starting at t = 0. The system Hamiltonian at t = 0 is H 1 , and the system Hamiltonian at t = T is H 2 = H 1 + λ∆H, λ > 0. ∆H is a fixed operator and so the direction of the variation is fixed. T is the evolution time. We assume H 1 , H 2 , and ∆H are bounded operators throughout this paper.
Let {ω i : i = 1, 2, ...N} be the monotonically increasing sequence of eigenvalues of H 1 , so that ω i ≥ ω j when i > j, and {|i } be the corresponding eigenstates. We denote the energy gap between the ith eigenstate and the ground state as λ i = ω i − ω 1 . Similarly, we define the increasing sequence of eigenvalues of H 2 , {ω For convenience, we also introduce two offset Hamiltonians,Ĥ 1 andĤ 2 . The HamiltonianĤ 1 is defined asĤ 1 = H 1 − ω 1 , i.e., by offsetting the Hamiltonian of the system at t = 0 by a constant operator ω 1 so thatĤ 1 ≥ 0. ByĤ 1 ≥ 0 we meanĤ 1 is positive semidefinite and its the smallest eigenvalue ofĤ 1 is zero. Similarly, we definê H 2 = H 2 − ω ′ 1 ≥ 0 by offsetting the system Hamiltonian by a constant operator ω ′ 1 . Let ρ t denote the system state at time t and let ρ g be the initial state of the system at t = 0. We always assume that ρ g is the ground state ofĤ 1 , and so we have Ĥ 1 ρg = 0.
The transition of the system from H 1 to H 2 can be described using an interpolating function f (t) so that with f (0) = 0 and f (T ) = 1. In particular, the linear change from H 1 to H 2 along the direction ∆H, can be described using the increasing interpolating function f (t) = t T . The measure of adiabaticity is proposed as follows Interpolation Approach to Hamiltonian-varying Quantum Systems and the Adiabatic Theorem4 Definition 1 The distance between the final state and the ground state of H 2 is measured by Obviously, if the evolution is adiabatic, i.e., ρ T is the ground state of H 2 , then we have ǫ = 0. In particular, ǫ is closely related to the fidelity of the final state and ground state in the Schrödinger picture (See Appendix C). A small error ǫ implies a large fidelity.
In this paper we also call ǫ the adiabatic approximation error, as ǫ reflects how well we can approximate the evolution as a perfect adiabatic process.
In this paper we only consider λ such that ρ t , t ∈ [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) where we have ||R(t)|| = O(λ 2 ). A sufficient condition for the Magnus series to converge is given by (see Appendix A) Our aim is to estimate an asymptotic behaviour of ǫ provided λ → 0. 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.

Adiabatic approximation under linear interpolation of the Hamiltonian
The Heisenberg evolution of the expectation of an observable is written as where H is the system Hamiltonian. Recall that ρ g = |1 1|. Since H 1 |1 = ω 1 |1 , X(t) ρg is a constant of motion under the action of H 1 : for any Hermitian operator X(t).
We will need to study the dynamics of Ĥ 2 ρt = Ĥ 2 (t) ρg in order to solve for ǫ. The time evolution of Ĥ 2 ρt is determined by its generator d dt As we noted before, Ĥ 2 ρ T is exactly zero if ρ T is the ground state ofĤ 2 . If ρ T is not the ground state ofĤ 2 , we can determine the bound on ǫ = Ĥ 2 ρ T from the following equality The error ǫ can be expressed as With the aid of (8), we can exactly calculate ǫ in the case where f (t) defines a linear interpolation, as summarized in the following proposition: Proposition 1 Assume λ 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. ǫ is of the order O( λ 2 Proof 1 Referring to (11) and (9), we need to compute the difference between (8) and ω ′ 1 − 1|H 2 |1 . First we write (8) as Interpolation Approach to Hamiltonian-varying Quantum Systems and the Adiabatic Theorem6 DenoteH = max f (t)∈(0,1) ||H(t)||. Since is O(λ 3 ), we can further write (13) as Clearly, the term O( λ 2 ) dominates as λ → 0.
Next we will calculate ω The smallest eigenvalue ω ′ 1 of H 2 can be calculated using the first-order time-independent perturbation theory for non-degenerate system.
Assume H 1 is the unperturbed Hamiltonian and the perturbation is λ∆H, then the lowest eigenvalue of the perturbed Hamiltonian H 1 + λ∆H can be written as series in terms of λ and ω 1 [25]: Thus we conclude Comparing (16) and (19), the second order terms cancel and so the error ǫ is estimated by

Error Estimation for Arbitrary Interpolations
The approach derived in the previous section can be easily generalized for arbitrary given continuous interpolating functions. The generalization can simply be done by Interpolation Approach to Hamiltonian-varying Quantum Systems and the Adiabatic Theorem7 replacing the linear interpolation function with the given continuous function f (t) and then recalculating the double integration in (7). The error estimation can easily be obtained from the proof of Proposition 1:

Proposition 2
For an arbitrarily given f (t), the error estimation is given by as λ → 0.
Proof 2 ǫ is still calculated by (11), using H 2 ρ T − H 2 ρg and ω and 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.

Linear Interpolation: f (t) = t/T
By Proposition 1, the error estimation is ǫ = O( λ 2 T 2 λ 3 2 ) as λ → 0. This error term is determined by λ T which is the average speed of the variation of the system Hamiltonian, and 1 λ 2 which is the inverse of the energy gap between the ground and first-excited eigenstates of H 1 , as λ → 0. In particular, we have Therefore, since the inverse of the energy gap 1 λ 2 is a fixed value, the approximation error ǫ is estimated to be proportional to the square of the average speed of the variation of the Hamiltonian, which is ( λ T ) 2 , as λ → 0.
Replace f (t) with a nonlinear function f (t) = t 2 T 2 in (7) and we recalculate the integral to be By Proposition 2, for sufficiently small λ, the error is estimated to be of order of λ 2 : That is, in contrast to the linear interpolation case, we have This calculation shows that if the evolution speed is infinitely slow, then the system dynamics is adiabatic during t ∈ [0, T ]. However, the scaling of ǫ quad with respect of the square of the average evolution speed λ T is not as simple as in the linear case, where the scaling of ǫ with respect of ( λ T ) 2 is solely determined by the inverse of the energy gaps as λ → 0. In the quadratic case, this scaling is determined by a complex factor

Interpolation with Decaying Resonant Terms
Here we assume a linear interpolating function with an additional oscillating term that gradually decays to zero. That is, where λ c is the oscillating frequency of the perturbation. Ortigoso observed in [22] the inconsistency in the applicability of the adiabatic theorem when the Hamiltonian contains resonant terms whose amplitudes go asymptotically to zero. (7) and we recalculate the integral to be i =1 .
Q 1 is a function of four parameters. Obviously, (29) is singular when λ c = λ i for some i.
To be more precise, we can let λ c = λ i for an arbitrary i and find that the coefficient where Q(T ) is a complicated fraction with T being in its denominator. The error resulting from the ith term is given by as λ → 0. We have The scaling of ǫ i with respect of ( λ T ) 2 is additionally determined by T 2 and T 4 , as compared to the quadratic case. This is where adiabatic evolution may break down even if the average evolution speed is slow. In particular by (32), if one chooses a comparably large value for T in an adiabatic evolution experiment, the adiabatic approximation error may not decrease as expected when one applies a slow evolution speed λ T . In addition, we can compare this case with the quadratic case using the ratio Note that the inner limit is taken for T being constant. Therefore, the rate of convergence considered in this subsection is slower than that in the quadratic or linear case. i.e., ǫ goes to zero as λ → 0 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.

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.

Appendix C.
The state of the system will remain a pure state during the evolution. Therefore, we can express the final state as ρ T = |ψ ψ| with |ψ = N i=1 c i |i ′ . Using this expression, the error measure ǫ defined by (2) can be written as (C.1) The fidelity of the final state and the ground state |1 ′ is calculated by (C.2)