Meta-Variational Quantum Eigensolver: Learning Energy Profiles of Parameterized Hamiltonians for Quantum Simulation

We present the meta-variational quantum eigensolver (VQE), an algorithm capable of learning the ground-state energy profile of a parameterized Hamiltonian. If the meta-VQE is trained with a few data points, it delivers an initial circuit parameterization that can be used to compute the ground-state energy of any parameterization of the Hamiltonian within a certain trust region. We test this algorithm with an XXZ spin chain, an electronic ${\mathrm{H}}_4$ Hamiltonian, and a single-transmon quantum simulation. In all cases, the meta-VQE is able to learn the shape of the energy functional and, in some cases, it results in improved accuracy in comparison with individual VQE optimization. The meta-VQE algorithm introduces both a gain in efficiency for parameterized Hamiltonians in terms of the number of optimizations and a good starting point for the quantum circuit parameters for individual optimizations. The proposed algorithm can be readily mixed with other improvements in the field of variational algorithms to shorten the distance between the current state of the art and applications with quantum advantage.

Popular Summary

One of the major areas of interest in condensed-matter physics and electronic structure problems is obtaining the ground-state energy of a Hamiltonian that represents a physical system. This energy may depend on a series of parameters of the system, for instance an intermolecular distance or an external magnetic field. Current quantum computing techniques propose to use a parameterized quantum circuit that delivers the ground-state energy through the variational theorem. The goal is then to find the parameters of that circuit by using classical optimization methods. However, one has still to find, optimize, and run a different quantum circuit for each of these Hamiltonian parameters, increasing the computational cost of finding interesting regions such as that of minimum energy. On top of that, these variational algorithms are very sensitive to the initial parameters used in the optimization subroutine, leading to local minima that do not correspond to the true ground-state energy.

In this work, we present a quantum algorithm capable of learning the ground-state energy profile as a function of the Hamiltonian parameters. By selecting a set of the Hamiltonian parameters (the training set), we train a quantum circuit that encodes these parameters, so that the same circuit can be used independently of the values of those parameters, saving precious computational time. We compare its performance with other well-known algorithms and obtain better accuracy. This algorithm constitutes a novel example of an application of quantum machine learning to Hamiltonian simulation for near-to-intermediate-scale quantum computation.

Figure 1

Schematic illustration of the meta-VQE algorithm. The top diagram represents the training part of the algorithm, where the Hamiltonian parameters $\overrightarrow{\lambda }$ are encoded together with the variational parameters $\overrightarrow{\mathrm{\Phi }}$ and $\overrightarrow{\mathrm{\Theta }}$. By computing the expected value of the Hamiltonian for multiple values of $\overrightarrow{\lambda }$, we design a cost function $\mathcal{L}oss$ to be minimized. Once the algorithm converges to ${\mathrm{\Phi }}_{\mathrm{opt}}$ and ${\mathrm{\Theta }}_{\mathrm{opt}}$, we can use these values to obtain any $⟨H(\overrightarrow{\lambda })⟩$ (meta-VQE test) or as initial values of a standard VQE algorithm (opt-meta-VQE).

Figure 2

Results for the ground-state energy (arbitrary units) in relation to the parameter $\mathrm{\Delta }$ for an $n=14$ XXZ spin chain with a transverse field $\lambda =0.75$. The total number of layers considered is four, divided into encoding and processing for the meta-VQE and opt-meta-VQE. The meta-VQE learns the energy profile, whereas the standard VQE achieves better precision. However, using the results of the meta-VQE as a starting point for a VQE (opt-meta-VQE and opt-GA-VQE) improves the performance notably and avoids local minima, as shown by the plot of the absolute error with respect to the exact solution.

Figure 3

Circuit ansatz corresponding to four qubits and two encoding and processing layers. This ansatz is used for the XXZ meta-VQE. Each $R(\overrightarrow{x})$ gate corresponds to $R_z(x^{(1)})R_y(x^{(2)})$. The function used for the encoding layer is $f(\mathrm{\Delta },\overrightarrow{\phi })=w\mathrm{\Delta }+\varphi$.

Figure 4

Relative errors with respect to the exact ground state of the 1D XXZ model with $\lambda =0.75$ for the four algorithms considered in this work in relation to the $\mathrm{\Delta }$ parameter and the number of qubits $n$. The errors for the VQE, opt-meta-VQE, and opt-GA-VQE decrease slightly with the number of qubits. For the meta-VQE, the error scaling with $n$ depends on the phase of the system, a trait that can be explained by the hardness of learning a critical system with a single global cost function.

Figure 5

Relative errors for the rectangular ${\mathrm{H}}_4$ molecule in eight spin orbitals (STO-3G basis set) in relation to the intermolecular bond distance $d$. The 2-UpCCGSD model is used for all VQEs. The VQE lines denote the standard optimized 2-UpCCGSD model starting from the Hartree-Fock configuration (VQE0). Linear encoding (meta-VQE) replaces the UpCCGSD angles with an encoding of the form $\theta =\alpha +d\beta$, while the nonlinear encoding (nl-meta-VQE) uses a floating Gaussian with $\theta =\alpha e^{\beta \left(\gamma -d\right)}+\delta$. The opt-meta-VQE line denotes the canonical VQE initialized with the angles obtained from the meta-VQE. All errors are with respect to the exact diagonalization of the Hamiltonian within this basis set. Training points are chosen from 0.5 to 2.5 Å with 0.5 Å distance between them.

Figure 6

Left: ground-state energy as a function of the normalized external magnetic flux $f$ for a QCAD [29] simulation of a single transmon. Right: absolute error with respect to the exact diagonalization of the Hamiltonian. We compare the meta-VQE and opt-meta-VQE with the VQE and improved versions of it, where we use the result of the previous optimization point to initialize the parameters of the next one (VQE-smart) and the VQE with encoding layers (VQE-enc). Both VQE-smart and VQE-enc show more consistent results in comparison with the VQE with random initialization, where the algorithm gets trapped in local minima for some points. The introduction of encoding layers in VQE-enc seems not to be beneficial in comparison with the standard VQE circuit (VQE-smart). This could be a consequence of the increase in the number of parameters, where the optimization can face the barren-plateau problem [19]. On the other hand, opt-meta-VQE shows the best performance, indicating that the use of the meta-VQE result as the initialization point better captures the parameter-space region of the ground state.