Investigation of Perturbation Theory with Variational Quantum Algorithm

We know that efficiently simulating quantum systems with high degrees of freedom is hard for classical computers due to the exponential growth of variables for characterizing these systems [1]. Quantum computers were proposed to solve such an exponential explosion problem, ranging from optimization to materials design, and the algorithms used in quantum computers have made great strides in the calculation and efficiency of various issues [2, 3, 4, 5]. Among the different approaches to quantum computing, the near-term quantum devices are mostly center around quantum simulations, which consists of a relatively low-depth quantum circuit by variational quantum algorithms. The variational algorithms were recently attracting a lot of attention, designed to utilize both quantum and classical resources to solve specific optimization tasks not accessible to traditional classical computers [6, 7, 8, 9, 10]. The main idea of this method is to divide the problem into two parts that each of performing a single task and can be implemented efficiently on a classical and a quantum computer. The significant benefit of this method is that it gives rise to a setup that can have much less strict hardware requirements and promising for NISQ [11] and devices typically have on the order of fewer qubits(contain from $10$ to $10^{3}$ of qubits) with high gate fidelity and not fault-tolerant error correction. From a practical point of view, most of the current efforts concentrate on analog quantum computing methods such as quantum annealing [12, 13], and quantum adiabatic simulation[14, 15]. Recently, several variational quantum algorithms for specific tasks have been developed, and analog approaches can be approximately solved using gate model NISQ devices. These algorithms and their applications are progressing in various fields such as variations quantum eigensolver (VQE) which is a hybrid algorithm to approximate the ground state eigenvalues for quantum simulations [16, 4], quantum approximate optimization algorithm (QAOA) for finding an approximate solution of an optimization problem[5, 17], dissipative-system Variational Quantum Eigensolver(dVQE) to simulate Non-equilibrium steady states an open system [18], molecular simulations on a quantum computer [19],variational quantum state diagonalization (VQSD) [20] , also to simulate dynamics of open and closed systems[21, 22]. A complete overview of these algorithms is provided in Ref.[23, 24]. In quantum mechanics, few problems have exact solution, whether or not Hamiltonian is time-dependent. Hence, approximate methods have been proposed[25]. The approximate methods play a significant role in the study of different quantum systems. In general, we have three approximate methods; they are Perturbation Theory (PT)[26], Variational theory[27] and WKB approximation[28]. Here we focuse on PT, suppose there is a problem the Hamiltonian of which is represented by $H$, for which the exact solution is not known. The Hamiltonian of the system can be described in terms of two terms; the Hamiltonian is written $H=H_{0}+\lambda V$, in which $H_{0}$ is called as the unperturbed Hamiltonian, and $V$ is called as the perturbation Hamiltonian, and $\lambda$ is a continuous real parameter. It is important to realize that the system actually can be split in terms of these two Hamiltonians, and that should be known. In addition, there are two more conditions; one is that solution of the unperturbed Hamiltonian is completely known. The following condition, which is a stringen condition, that $\lambda$ has to be much smaller than 1 $(\lambda<<1)$, which means the perturbation terms are much smaller compared to the original Hamiltonian $H_{0}$. In general, the PT can be divided into two categories, Time-Independent Perturbation Theory(TIPT) and Tim-Dependent Perturbation Theory(TDPT). In TIPT perturbation Hamiltonian, $V$ is independent of time, and in TDPT $V$ is dependent of time. Here, we consider TIPT, and TDPT, and investigate them with the variational quantum algorithm. In fact, in TDPT, we are dealing with a time-dependent Hamiltonian and try to find its spectrum (In the language of mathematics described in detail in Section 2). In this paper, we prospect the possibility of using a variational quantum algorithm to simulate quantum dynamics in TDPT and, TIPT. For this purpose, we employ time-dependent variational quantum algorithm introduced in Ref[21, 22, 29, 30, 31]. In this algorithm, the time-dependent quantum state is approximated by a parametrized quantum state and using McLachlan’s principle, the equation of motion for the variational parameters is obtained(described in detail in Section 3). The paper is organized as follows: In Section 2, we introduce Time-Independent and Time-Dependent Perturbation Theory. The variational quantum algorithm for quantum dynamics is described in Section 3. In Section 4, we present the numerical results. Finally, Section 5, gives the conclusions.

Perturbation theory is widely used and plays an important role in describing real quantum systems, because it is impossible to find exact solutions to the Schrodinger equation for Hamiltonians even with moderate complexity. In general, the PT can be divided into two categories, TIPT and TDPT. In this section, we describe these two categories in detail.

For simulate the time-inpendent Hamiltonian dynamics, we employ variational quantum algorithm introduced in Ref[21, 22, 29, 30, 31]. In this algorithm, state $|\psi(t)\rangle$ is approximated by a parametrized state $|\phi(\vec{\lambda})\rangle$, i.e.

$$|\psi(t)\rangle=exp(-iHt)|\psi(0)\rangle\approx|\phi(\vec{\lambda})\rangle,$$

where $\vec{\lambda}=\{\lambda_{1}(t),\lambda_{2}(t)....\}$ are variational parameters. By using time-dependent variational principle corresponding to the Schrödinger equation we have

$$\mathcal{L}=\langle\psi(t)|\frac{i\partial}{\partial t}-H|\psi(t)\rangle$$

(8)

where, $\mathcal{L}$ is Lagrangian. By considering $|\psi(t)\rangle\approx|\phi(\vec{\lambda})\rangle$ and using the Euler-Lagrange equation we have

$$\sum_{q}M_{pq}\dot{\lambda_{q}}=V_{p},$$

(9)

where

$$M_{p,q}=i\frac{\partial\langle\phi(\vec{\lambda})|}{\partial\lambda_{k}}\frac{\partial|\phi(\vec{\lambda})\rangle}{\partial\lambda_{q}}+H.C$$

(10)

$$V_{p}=\frac{\partial\langle\phi|}{\partial\lambda_{p}}H|\phi(\vec{\lambda})\rangle+H.C$$

(11)

For simulate the time-dependent Hamiltonian dynamics, we employ variational quantum algorithm introduced in Ref[21, 22, 29, 30, 31]. In this algorithm, state $|\psi(t)\rangle$ is approximated by a parametrized state $|\phi(\vec{\lambda})\rangle$, i.e.

$$|\psi(t)\rangle=Texp(-i\int^{t}_{0}H(t^{\prime})dt^{\prime})|\psi(0)\rangle\approx|\phi(\vec{\lambda})\rangle,$$

where $\vec{\lambda}=\{\lambda_{1}(t),\lambda_{2}(t)....\}$ are variational parameters and $T$ is the time-ordering operator. By using McLachlan’s principle,( or Dirac and Frenkel variational principle) the equation of motion for the variational parameters is obtained by minimizing the quantity $||(i\frac{\partial}{\partial t}-H)|\phi(\vec{\lambda})\rangle||$, so we have

$$\lambda(t_{n+1})=\lambda(t_{n})+\dot{\lambda}\delta t$$

$$\sum_{i}M_{ki}\dot{\lambda_{i}}=V_{k}.$$

(12)

where

$$\begin{split}M_{ki}=Re\langle\frac{\partial\phi(\vec{\lambda})}{\partial\lambda_{k}}|\frac{\partial\phi(\vec{\lambda})}{\partial\lambda_{i}}\rangle\\ V_{k}=Im\langle\phi(\vec{\lambda})|H|\frac{\partial\phi(\vec{\lambda})}{\partial\lambda_{k}}\rangle.\end{split}$$

(13)

The coefficients of the differential Eq.( 13) are determined using a quantum computer, while each propagation step is carried out by classically solving the differential equation. To obtain high accuracy of algorithm, we must be sensitive in choosing the wavefunction ansatz. Here we consider an ansatz of the form

$$|\phi(\vec{\lambda})\rangle=U(\vec{\lambda})|\phi_{0}\rangle=U(\lambda_{1})U(\lambda_{2})...U(\lambda_{i})...U(\lambda_{N})|\phi_{0}\rangle$$

The evolution operator is unitary, so it is equivalent to a certain rotation in the Hilbert space of states .i.e.

$$U(\lambda_{i})=exp(-i\lambda_{i}\Lambda_{i})=exp(-i\sum_{j}\lambda_{i}s_{i,j}\hat{\sigma}_{i,j}),$$

(14)

where $\Lambda_{i}=\sum_{j}s_{i,j}\hat{\sigma}_{i,j}$ where $\hat{\sigma}_{i,j}$ are Pauli operators. So, we have,

$$\begin{split}\frac{\partial U(\vec{\lambda})}{\partial\lambda_{i}}=U(\lambda_{1})U(\lambda_{2})...-i\sum_{j}s_{i,j}\hat{\sigma}_{i,j}U(\lambda_{i})....U(\lambda_{N})\\ \frac{\partial U^{\dagger}(\vec{\lambda})}{\partial\lambda_{i}}=U^{\dagger}(\lambda_{N})....i\sum_{j}s^{*}_{i,j}\hat{\sigma}_{i,j}U^{\dagger}(\lambda_{i})...U^{\dagger}(\lambda_{2})U^{\dagger}(\lambda_{1}),\end{split}$$

(15)

therefore the matrix elements of $M$ and $V$ are

$$\begin{split}M_{ki}=Re(\langle\phi_{0}|U(\lambda_{1})^{\dagger}...U(\lambda_{N})^{\dagger}\Lambda_{k}^{\dagger}...U(\lambda_{l})^{\dagger}\Lambda_{i}U(\lambda_{i})....U(\lambda_{1})|\phi_{0}\rangle)\\ V_{k}=Im(i\sum_{j}c_{j}\langle\phi_{0}|U(\lambda_{1})^{\dagger}...U(\lambda_{N})^{\dagger}h_{j}U(\lambda_{N})...\Lambda_{k}U(\lambda_{k})...U(\lambda_{1})|\phi_{0}\rangle),\end{split}$$

(16)

where the Hamiltonian expressed $H=\sum_{j}c_{j}h_{j}$. The $M_{ki}$ and $V_{k}$ are obtained in a quantum circuit via the Hadamard or Swap test. Note that the matrix elements in the first and second-order approximations i.e. Eq(2), Eq(3) and Eq(7) can be obtained by variational quantum algorithm.

In this section, we numerically test the performance of the previously described variational quantum algorithm on some of the systems. In the first example, consider $H_{2}$ molecule and find ground state energy $H_{2}$. We used a standard molecular basis set, the minimal STO-3G basis. Via JordanWigner or Bravyi-Kitaev transformation, the qubit-Hamiltonians of this molecule can be obtained. So,

$$H=g_{0}I+g_{1}Z_{0}+g_{2}Z_{1}+g_{3}Z_{0}Z_{1}+g_{4}Y_{0}Y_{1}+g_{5}X_{0}X_{1},$$

(17)

where the coefficients $g_{i}$ were all derived in this classical preprocessing step and $\{X_{i},Y_{i},Z_{i}\}$ are Puali matrixs. Whit using TIPT we can divid this Hamiltinain in two trem,

$$H_{0}=g_{0}I+g_{1}Z_{0}+g_{2}Z_{1}+g_{3}Z_{0}Z_{1}$$

$$H^{\prime}=g_{4}Y_{0}Y_{1}+g_{5}X_{0}X_{1}$$

$H_{0}$ is the unperturbed Hamiltonian and $H^{\prime}$ is the perturbation Hamiltonain. $H_{0}$ is diagonal matrix and we can obtain eigenstate and energy $H$ by using TIPT. For variational quantum algorithm, we consider an anstaz of the form

$$|\psi(\vec{\lambda})\rangle=e^{i\lambda_{1}X_{0}Y_{1}}e^{i\lambda_{2}Z_{0}I_{1}}|01\rangle$$

. Output of variational algorithm compared it to TIPT solution ground state energy as shown in figure 1. Also we implement a quantum circuitfor evaluating coefficients on quantum processor in figure 2

In second example let us consider a two-state system with

$$H_{0}=E_{1}|0\rangle\langle 0|+E_{2}|1\rangle\langle 1|$$

$$V(t)=\delta e^{i\omega t}|0\rangle\langle 1|+\delta e^{-i\omega t}|1\rangle\langle 0|.$$

Whit TDPT, we can obtain

$$i\mathchar 1406\relax\vec{\dot{c}}=\delta(e^{i(\omega-\omega_{21})t}|0\rangle\langle 1|+e^{-i(\omega-\omega_{21})t}|1\rangle\langle 0|)\vec{{c}},$$

where $\vec{{c}}$ is the two-component vector $\vec{{c}}=(c_{1}(t),\ c_{2}(t))$and $\omega_{21}=(E_{2}-E_{1})/\mathchar 1406\relax$. With the initial condition $c_{1}(0)=1$, and $c_{2}(0)=0$ this equation has the solution

$$|c_{2}(t)|^{2}=\frac{4\delta^{2}}{\delta^{2}+\mathchar 1406\relax^{2}(\omega-\omega_{21})^{2}}sin^{2}\Omega t\qquad|c_{1}(t)|^{2}=1-|c_{2}(t)|^{2}$$

(18)

where $\Omega=((\delta/\mathchar 1406\relax)^{2}+(\omega-\omega_{21})^{2}/4)^{1/2}$ is known as the Rabi frequency. The solution, which varies periodically in time, describes the transfer of probability from state 0 to state 1 and back. For variational quantum algorithm, we consider an anstaz of the form

$$|\psi(\vec{\lambda})\rangle=e^{i\lambda_{1}Z}e^{i\lambda_{2}X}|0\rangle$$

Output of variational algorithm compared it to TDPT solution the transfer of probability from states as shown in figure 3. Also We propose a quantum circuit for evaluating coefficients on quantum processor in figure 4

We have considered a variational quantum algorithm for investigation of perturbation theory. We considered perturbation theory as two categories, time-independent perturbation theory and time-dependent perturbation theory. We have used a variational quantum algorithm which, exact state is approximated by a parametrized state and variational principle. In the case TIPT and, TDPT using the time-dependent variational principle and McLachlan’s principle corresponding to the Schrödinger equation, respectively, we obtained the differential equations for simulating dynamics. We compare the analytical and numerical results obtained from the exact solution with the variational quantum algorithm which, are in good agreement with each other.