Quantum Gaussian process model of potential energy surface for a polyatomic molecule

Predicting properties of complex molecules from first principles is considered to be one of the most promising applications of quantum computing. A computation of molecular properties within the Born-Oppenheimer approximation requires solving the electronic structure problem, fitting the results of potential energy calculations to produce global PES and solving the nuclear dynamics problem with the PES thus obtained. Several algorithms have been recently developed for solving electronic structure electronic-structure ; electronic-structure-1 ; electronic-structure-2 ; electronic-structure-3 ; electronic-structure-4 ; electronic-structure-5 ; electronic-structure-6 ; electronic-structure-7 ; electronic-structure-8 ; electronic-structure-9 ; electronic-structure-10 ; electronic-structure-11 ; electronic-structure-12 ; electronic-structure-13 ; electronic-structure-14 ; electronic-structure-15 and nuclear dynamics nuclear-dynamics ; nuclear-dynamics-1 ; nuclear-dynamics-2 ; nuclear-dynamics-3 ; nuclear-dynamics-4 ; nuclear-dynamics-5 problems on noisy intermediate-scale quantum (NISQ) computers. However, quantum algorithms for producing global PES of polyatomic molecules have not yet been demonstrated. The present work builds a quantum regression model of a six-dimensional PES for the molecular ion H₃O⁺. Our results demonstrate a comparison with the corresponding classical models and illustrate the role of entanglement of the qubits used for the quantum algorithm of constructing PES.

Recent work has demonstrated that PES of polyatomic molecules can be accurately represented by machine learning (ML) regression models, based on neural networks ML-for-PES2 ; ML-for-PES3 ; NNs-for-PES ; NNs-for-PESa ; NNs-for-PES-1a ; NNs-for-PES-1b ; NNs-for-PES-1c ; NNs-for-PES-2 ; NNs-for-PES-3 ; NNs-for-PES-4 ; NNs-for-PES-5 ; NNs-for-PES-6 ; carrington ; meuwly or kernel methods meuwly ; gp-1 ; gp-2 ; gp-3 ; jie-jpb ; gp-for-PES-2 ; gp-for-PES-3 ; gp-for-PES-4 ; gp-for-PES-5 ; gp-for-PES-6 ; gp-for-PES-7 ; gp-for-PES-8 ; gp-for-PES-9 ; gp-for-PES-10 ; gp-for-PES-11 ; gp-for-PES-12 ; kernel-for-PES ; rabitz-1 ; rabitz-2 ; rabitz-3 ; unke . Quantum computers have opened the possibility to research the quantum analogues of ML algorithms qml ; qml1 ; vcc ; vcc-1 ; vcc-2 ; qK-SVM ; qK-SVM-1 ; qK-SVM-2 ; qK-SVM-3 ; qK-SVM-4 ; qK-SVM-5 ; qK-SVM-6 ; quantum-regression ; quantum-regression-1 ; quantum-regression-2 ; quantum-regression-3 ; qgp . It has been shown that gate-based quantum devices can be used to build quantum kernels for kernel ML models qml ; qml1 ; qK-SVM ; qK-SVM-1 ; qK-SVM-2 ; qK-SVM-3 ; qK-SVM-4 ; qK-SVM-5 ; qK-SVM-6 ; qgp . While most applications of quantum kernels have been for support vector classification of low-dimensional data qml1 ; qK-SVM ; qK-SVM-1 ; qK-SVM-2 ; qK-SVM-3 ; qK-SVM-4 ; qK-SVM-5 ; qK-SVM-6 , several studies have considered quantum algorithms for regression quantum-regression ; quantum-regression-1 ; quantum-regression-2 ; quantum-regression-2 ; quantum-regression-3 ; qgp . Particularly relevant for the present work is Ref. qgp that applied Gaussian process regression to several model applications, such as regression of the one-dimensional function $x\sin x$. The goal of Ref. qgp was to simulate classical kernels using coherent states, or truncations of coherent states. In order to extend this work to regression problems for fitting PES, it is necessary to overcome several challenges. First, there is no general quantum circuit ansatz for building performant quantum kernels for PES interpolation. It is not known how to build a sequence of quantum gates in order to build the best quantum kernel for accurate models of PES. Second, accurate kernel regression models for complex problems with sparse data require optimization of kernel parameters. However, quantum kernel estimation is expensive, requiring many quantum measurements for each pair of training points. In addition, as will be illustrated in this work, the cost function used to train regression models with quantum kernels can be very sensitive to quantum circuit parameters. This makes kernel parameter optimization difficult. Third, the number of quantum kernel parameters grows quickly with the number of qubits and gates in the corresponding quantum circuit. This precludes grid search of optimal kernel parameters, often used for building classical kernel ridge regression models. This also makes search of optimal quantum circuit ansatz difficult.

Here, we demonstrate that quantum kernel regression models with a fixed quantum circuit ansatz readily deployable on current gate-based quantum computers can yield comparable accuracy with classical ML models. Our focus is on building accurate models with a small number of training points, aiming to produce global PES with a small number of ab initio potential energy calculations. To achieve this, we employ Bayesian optimization for tuning the parameters of quantum gates and optimize kernels by maximizing a modified version of log marginal likelihood. We consider two problems: interpolation of PES in a six-dimensional (6D) configuration space and extrapolation of PES in the energy domain. We show that the quantum models may exhibit better extrapolation accuracy than classical models with radial basis function kernels universality-of-RBF-kernels , when trained by the same number and distribution of potential energy points. We also show that the accuracy of quantum models is significantly enhanced by two-qubit gates, which illustrates a critical role of qubit entanglement in quantum kernels for regression problems. By demonstrating Bayesian regression models with quantum kernels, our work complements Ref. qgp to set the stage for the quantum analogue of Bayesian optimization on quantum computing devices.

We use Gaussian process (GP) models to represent PES of the molecule H₃O⁺. The molecular geometry is described by the six-dimensional (6D) vector $\bm{x}$, as in our previous work jun-paper , where we built classical GP models of PES for H₃O⁺. A GP model is trained by $n$ input - output pairs, with inputs represented by $n$ molecular geometries $\bm{x}_{i}$ and outputs by $n$ corresponding values of the potential energy, collected into a column vector $\bm{y}$. The prediction of potential energy at an arbitrary point $\bm{x}^{\ast}$ in the 6D input space is given by gp-book :

where $\sigma^{2}$ is a hyperparameter representing variance of data noise, $\bf I$ is the identity matrix, $\bf K$ is an $n\times n$ kernel matrix with entries $k(\bm{x}_{i},\bm{x}_{j})$, ${\bm{k}}^{\top}(\bm{x}^{\ast})$ is the transpose of a column vector with $n$ entries $k(\bm{x}^{\ast},\bm{x}_{i})$, and $\bm{x}_{i}$ and $\bm{x}_{j}$ represent the molecular geometries for the training points $i$ and $j$. Because PESs are noiseless, we set $\sigma^{2}$ to zero.

The function $k(\bm{x},\bm{x}^{\prime})$ yielding the elements of the kernel matrix is the covariance function of the GP gp-book . It must satisfy the properties of a kernel function of a reproducing kernel Hilbert space (RKHS). Specifically, $k(\bm{x},\bm{x}^{\prime})$ must be positive-definite and symmetric to interchange of $\bm{x}$ and $\bm{x}^{\prime}$. In the present work, we build GP models with classical and quantum kernels. In both cases, the prediction of the model is given by Eq. (1). The difference is in the kernel matrix $\bf K$. For classical models, we use radial basis functions (RBF) as the kernel function,

RBF kernels are known to be universal universality-of-RBF-kernels and provide benchmark results for quantum models developed in this work.

For quantum kernels, we consider a quantum computer with $m$ qubits, initially in state $|0^{m}\rangle$. A sequence of gates operating on these qubits produces a quantum state ${\cal U}(\bm{x})|0^{m}\rangle$. The measurable square of the inner product

satisfies all the properties of a kernel of an RKHS. In order to build such quantum kernels, one must encode information about input vectors into parameters of the quantum gates of a quantum computer.

In the present work, we use the quantum circuit depicted in Figure 1 to build quantum kernels. This quantum circuit was introduced in Ref. qK-SVM-2 for classification problems. We use one qubit to represent one dimension of the input space, resulting in a 6-qubit quantum circuit for the present problem. Each qubit is initialized in state $|0\rangle$. Following the initialization, quantum states are created by a sequence of gate operations ${\cal U}^{\dagger}(\bm{x}^{\prime}){\cal U}(\bm{x})$, as depicted in the upper panel of Figure 1. The values of the kernels are obtained by projecting the resulting quantum states onto the state $|0^{m}\rangle$.

The input vectors $\bm{x}$ are encoded into the quantum gates as follows:

where the superscripts in $\bm{x}^{i}$ and $\bm{x}^{j}$ denote the $i$-th and $j$-th components of the 6D vector $\bm{x}^{\top}=\left[\bm{x}^{1},\dots,\bm{x}^{6}\right]$, and $\theta_{i}$ and $\theta_{ij}$ are parameters of the quantum circuit to be optimized. As shown in Figure 1, the unitary transformation $\cal U$ is built as

with

where $\sigma_{Z,i}$ is the Pauli $Z$-gate acting on qubit $i$, and the second term in the exponent correspond to the ${R}_{ZZ}$ gate. This ansatz includes a sequence of two-qubit rotations, entangling each pair of the qubits in the circuit. The order of the individual $R_{ZZ}$ gates in Eq. (10) is arbitrary, because the $\sigma_{Z,i}$ operators commute. The parameters $\theta_{i}$ are independent for each one-qubit rotation gate in $\bf U$. In order to simplify the optimization of the kernel parameters, we require that the parameters of all two-qubit gates $\theta_{ij}$ be the same and set them equal to a single variable parameter $\theta_{12}$. The number of free parameters in the quantum kernel is thus equal to the number of $R_{Z}$ gates plus one, for a total of 7 parameters in $\bf U$.

The covariance functions of the GP models are thus parametrized by $\theta$ in the classical models and $\theta_{i=1,\dots,6}$ and $\theta_{12}$, hereafter represented collectively by $\bm{\theta}$, in the quantum models. GP models are trained by maximizing the logarithm of marginal likelihood (LML), which yields optimal parameters of the kernels gp-book . For GPs, LML can be written in closed form in terms of the kernel matrix $\bf K$ and its determinant as follows gp-book :

where the dependence on $\bm{\theta}$ is through the elements of the kernel matrix. While it is straightforward to train classical GP models with the RBF kernel by maximizing LML, it will be illustrated in the next section that LML for quantum models is extremely sensitive to $\bm{\theta}$ in some parts of the parameter space, leading to rapid variation of LML and lack of convergence of LML optimization. In order to overcome this problem, we show that quantum models can be trained by optimizing the following objective function instead of the LML:

where $a$ is a hyperparameter, set to 1 in the present work. It will be shown that the constant $a$ stabilizes optimization of LML and improves convergence.

To build quantum kernels, we use simulated qubits as implemented in the IBM qiskit package, using Statevector qiskit . Quantum states are generated by the operation of gate sequences on qubits initially all in state $|0\rangle$. The gate operations are noiseless. The kernels as defined in Eq. (3) are computed from the corresponding probability amplitudes in the quantum states of $m$ qubits after the sequence of gate operations. In order to examine the role of qubit entanglement, we consider two types of kernels for the quantum models: (a) kernels constructed as described above with 7 parameters $\theta_{i}$ and $\theta_{12}$; (b) kernels constructed as described above, but with all two-qubit gates $R_{ZZ}$ replaced with identity matrices, yielding quantum circuits with 6 free parameters and no entanglement between qubits. We will refer to these kernels as entangled and unentangled kernels, respectively.

Although Ref. qK-SVM-2 illustrated that the quantum circuit ansatz described in the previous section can be used to build kernels for classification problems, this ansatz has not been used for regression models. Therefore, out first goal is to explore the possibility of using the quantum circuit depicted in Figure 1 for regression problems. Specifically, we aim to build accurate interpolation and extrapolation models with a limited number of training points (200 to 1500 for a 6D problem). In this limit, and especially for extrapolation problems, kernel regression models must be sensitive to kernels. We use a comparison with the models based on optimized RBF kernels to benchmark the performance of the quantum kernels. It should be noted that RBF kernels do not always represent the best classical kernels for kernel models of PES. As we demonstrated previously, classical GP models of PES can be improved by increasing the kernel complexity by combining different simple mathematical forms of kernels into composite kernels jun-paper ; kasra . However, RBF kernels are proven to be universal universality-of-RBF-kernels and represent one of the most frequently used type of kernels. Our goal is not to illustrate that quantum kernels can outperform classical kernels for small data regression problems. Rather, we aim to show that quantum kernels can produce regression models of similar accuracy as classical kernels.

Specifically, the present section illustrates:

• $\circ$

  how to optimize quantum circuits to build accurate quantum GP models;

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  the feasibility of building accurate GP regression models with quantum kernels using a fixed quantum circuit ansatz depicted in Figure 1;

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  comparison of quantum GP models for interpolation and extrapolation (in the energy domain) of PES with the classical models with optimized RBF kernels;

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  comparison of quantum GP models of PES with and without entanglement between qubits.

The ab initio results for the PES of H₃O⁺ are taken from Ref. h3o+ . There are a total of 31124 potential energy points, spanning the energy range $[0,21000]$ cm^-1. We construct global 6D PES by training GP models using $n$ ab initio points in a specific energy interval. The value of $n$ and the energy range for the training points is specified in the caption for each figure. The accuracy of the resulting models is quantified by computing the root mean squared error (RMSE)

where $\hat{f}$ are the GP model predictions given by Eq. (1), $y_{i}$ represent the ab initio potential energy points from Ref h3o+ , and the sum extends over all ab initio points that are not used for training the models. For models trained by potential energy points from a limited range of energies (e.g. at energies $\leq 10,000$ cm^-1), these RMSEs covering the entire energy range up to 21,000 cm^-1 quantify the ability of GP models to extrapolate in the energy domain.

We have demonstrated that quantum circuits of gate-based quantum computers can be used to build kernels for regression models of global PES for polyatomic molecules. Such kernels can be obtained by measuring the individual qubit states. We have shown that such kernels can be constructed with a fixed quantum circuit ansatz, previously used for classification problems, provided the quantum gate parameters are optimized to maximize $\log[{\cal L}+1]$, where ${\cal L}$ is marginal likelihood. This yields Gaussian process models of PES with quantum kernels. While the standard procedure for training Gaussian process models is to maximize $\log{\cal L}$, our results illustrate that $\log{\cal L}$ is very sensitive to variation of the circuit parameters, making the optimization challenging. However, we have shown that maximization of $\log[{\cal L}+1]$ can be performed with Bayesian optimization, yielding stable results that correspond to accurate regression models with quantum kernels.

We have compared the accuracy of Gaussian process models of PES with quantum kernels based on entangled qubits, quantum kernels with unentangled kernels and classical Gaussian process models with RBF kernels. In all cases considered, the accuracy of quantum models including two-qubit rotation gates is comparable with the accuracy of classical models with RBF kernels. The quantum models with entangled kernels outperform the classical models with optimized RBF kernels for the class of problems aiming to construct the 6D PES at high energies based on 1500 ab initio points at low energies. At the same time, the accuracy of all quantum models drops significantly, when the entangling two-qubit gates are omitted from the quantum circuits. This illustrates the critical role of qubit entanglement in the quantum kernel computation algorithm.

Our work demonstrates that quantum kernels obtained with a small number of qubits and quantum gates can be used for accurate regression models. This is important because finite fidelity of current NISQ devices is a major obstacle to increasing the size of quantum circuits. The quantum circuit used in the present work can be readily implemented on the current IBM quantum computer. Moreover, we have built quantum kernels for Gaussian process models, which themselves could be used as surrogate models underlying Bayesian optimization. Thus, our work complements Ref. qgp to pave the way for the development of the quantum analogue of Bayesian optimization. If quantum kernels prove to offer better inference for supervised learning tasks with a small number of training points than classical kernels, Bayesian optimization with quantum GPs may offer a useful application of quantum computing to optimization of functions that are exceedingly expensive to evaluate.

This work was supported by NSERC of Canada