Abstract
Considering recent advancements and successes in the development of efficient quantum algorithms for electronic structure calculations—alongside impressive results using machine learning techniques for computation—hybridizing quantum computing with machine learning for the intent of performing electronic structure calculations is a natural progression. Here we report a hybrid quantum algorithm employing a restricted Boltzmann machine to obtain accurate molecular potential energy surfaces. By exploiting a quantum algorithm to help optimize the underlying objective function, we obtained an efficient procedure for the calculation of the electronic ground state energy for a small molecule system. Our approach achieves high accuracy for the ground state energy for H_{2}, LiH, H_{2}O at a specific location on its potential energy surface with a finite basis set. With the future availability of largerscale quantum computers, quantum machine learning techniques are set to become powerful tools to obtain accurate values for electronic structures.
Introduction
Machine learning techniques are demonstrably powerful tools displaying remarkable success in compressing high dimensional data^{1,2}. These methods have been applied to a variety of fields in both science and engineering, from computing excitonic dynamics^{3}, energy transfer in lightharvesting systems^{4}, molecular electronic properties^{5}, surface reaction network^{6}, learning density functional models^{7} to classify phases of matter, and the simulation of classical and complex quantum systems^{8,9,10,11,12,13,14}. Modern machine learning techniques have been used in the state space of complex condensedmatter systems for their abilities to analyze and interpret exponentially large data sets^{9} and to speedup searches for novel energy generation/storage materials^{15,16}.
Quantum machine learning^{17}—hybridization of classical machine learning techniques with quantum computation—is emerging as a powerful approach allowing quantum speedups and improving classical machine learning algorithms^{18,19,20,21,22}. Recently, Wiebe et al.^{23} have shown that quantum computing is capable of reducing the time required to train a restricted Boltzmann machine (RBM), while also providing a richer framework for deep learning than its classical analog. The standard RBM models the probability of a given configuration of visible and hidden units by the Gibbs distribution with interactions restricted between different layers. Here, we focus on an RBM where the visible and hidden units assume {+1,−1} forms^{24,25}.
Accurate electronic structure calculations for large systems continue to be a challenging problem in the field of chemistry and material science. Toward this goal—in addition to the impressive progress in developing classical algorithms based on ab initio and density functional methods—quantum computing based simulation have been explored^{26,27,28,29,30,31}. Recently, Kivlichan et al.^{32} show that using a particular arrangement of gates (a fermionic swap network) is possible to simulate electronic structure Hamiltonian with linear depth and connectivity. These results present significant improvement on the cost of quantum simulation for both variational and phase estimation based quantum chemistry simulation methods.
Recently, Troyer and coworkers proposed using a restricted Boltzmann machine to solve quantum manybody problems, for both stationary states and time evolution of the quantum Ising and Heisenberg models^{24}. However, this simple approach has to be modified for cases where the wave function’s phase is required for accurate calculations^{25}.
Herein, we propose a threelayered RBM structure that includes the visible and hidden layers, plus a new layer correction for the signs of coefficients for basis functions of the wave function. We will show that this model has the potential to solve complex quantum manybody problems and to obtain very accurate results for simple molecules as compared with the results calculated by a finite minimal basis set, STO3G. We also employed a quantum algorithm to help the optimization of training procedure.
Results
Threelayers restricted Boltzmann machine
We will begin by briefly outlining the original RBM structure as described by^{24}. For a given Hamiltonian, H, and a trial state, \(\phi \rangle = \mathop {\sum}\nolimits_x \phi (x)x\rangle \), the expectation value can be written as:^{24}
where ϕ(x) = 〈xϕ〉 will be used throughout this letter to express the overlap of the complete wave function with the basis function x〉, \(\overline {\phi (x)} \) is the complex conjugate of ϕ(x).
We can map the above to a RBM model with visible layer units \(\sigma _1^z,\,\sigma _2^z...\,\sigma _n^z\) and hidden layer units h_{1}, h_{2}... h_{m} with \(\sigma _i^z\), h_{j} ∈ {−1, 1}. We use a visible unit \(\sigma _i^z\) to represent the state of a qubit i—when \(\sigma_{i}^{z}=1, {\hskip2pt}\sigma_{i}^{z}\rangle\) represents the qubit i in state \(1\rangle\) and when \(\sigma_{i}^{z}=1, {\hskip2pt}\sigma_{i}^{z}\rangle\) represents the qubit i in state \(0\rangle\). The total state of n qubits is represented by the basis \(x\rangle = \left {\sigma _1^z\sigma _2^z...\sigma _n^z} \right\rangle \). \(\phi (x) = \sqrt {P(x)} \) where P(x) is the probability for x from the distribution determined by the RBM. The probability of a specific set \(x = \{ \sigma _1^z,\sigma _2^z...\sigma _n^z\} \) is:
Within the above a_{i} and b_{j} are trainable weights for units \(\sigma _i^z\) and h_{j}. w_{ij} are trainable weights describing the connections between \(\sigma _i^z\) and h_{j} (see Fig. 1.)
By setting 〈H〉 as the objective function of this RBM, we can use the standard gradient decent method to update parameters, effectively minimizing 〈H〉 to obtain the ground state energy.
However, previous prescriptions considering the use of RBMs for electronic structure problems have found difficulty as ϕ(x) can only be nonnegative values. We have thus appended an additional layer to the neural network architecture to compensate for the lack of sign features specific to electronic structure problems.
We propose an RBM with three layers. The first layer, σ^{z}, describes the parameters building the wave function. The h's within the second layer are parameters for the coefficients for the wave functions and the third layer s, represents the signs associated x〉:
The s uses a nonlinear function tanh to classify whether the sign should be positive or negative. Because we have added another function for the coefficients, the distribution is not solely decided by RBM. We also need to add our sign function into the distribution. Within this scheme, c is a regulation and d_{i} are weights for \(\sigma _i^z\). (see Fig. 1). Our final objective function, now with \(\phi \rangle = \mathop {\sum}\nolimits_x \phi (x)s(x)x\rangle \), becomes:
After setting the objective function, the learning procedure is performed by sampling to get the distribution of ϕ(x) and calculating to get s(x). We then proceed to calculate the joint distribution determined by ϕ(x) and s(x). The gradients are determined by the joint distribution and we use gradient decent method to optimize 〈H〉 (see Supplementary Note 1). Calculating the the joint distribution is efficient because s(x) is only related to x.
Electronic structure Hamiltonian preparation
The electronic structure is represented by N singleparticle orbitals which can be empty or occupied by a spinless electron:^{33}
where h_{ij} and h_{ijkl} are one and twoelectron integrals. In this study we use the minimal basis (STO3G) to calculate them. \(a_j^\dagger \) and a_{j} are creation and annihilation operators for the orbital j.
Equation (5) is then transformed to Pauli matrices representation, which is achieved by the JordanWigner transformation^{34}. The final electronic structure Hamiltonian takes the general form with \(\sigma _\alpha ^i \in \left\{ {\sigma _x,\sigma _y,\sigma _z,I} \right\}\) where σ_{x}, σ_{y}, σ_{z} are Pauli matrices and I is the identity matrix:^{35}
Quantum algorithm to sample Gibbs distribution
We propose a quantum algorithm to sample the distribution determined by RBM. The probability for each combination y = {σ^{z}, h} can be written as:
Instead of P(y), we try to sample the distribution Q(y) as:
where k is an adjustable constant with different values for each iteration and is chosen to increase the probability of successful sampling. In our simulation, it is chosen as \(O\big( {\mathop {\sum}\nolimits_{i,j} w_{ij}} \big)\).
We employed a quantum algorithm to sample the Gibbs distribution from the quantum computer. This algorithm is based on sequential applications of controlledrotation operations, which tries to calculate a distribution Q′(y) ≥ Q(y) with an ancilla qubit showing whether the sampling for Q(y) is successful^{23}.
This twostep algorithm uses one system register (with n + m qubits in use) and one scratchpad register (with one qubit in use) as shown in Fig. 2.
All qubits are initialized as 0〉 at the beginning. The first step is to use R_{y} gates to get a superposition of all combinations of {σ^{z}, h} with \(\theta _i = 2arcsin\left( {\sqrt {\frac{{e^{a_i/k}}}{{e^{a_i/k} + e^{  a_i/k}}}} } \right)\) and \(\gamma _j = 2arcsin\left( {\sqrt {\frac{{e^{b_j/k}}}{{e^{b_j/k} + e^{  b_j/k}}}} } \right)\):
where \(O(y) = \frac{{e^{\mathop {\sum}\nolimits_i a_i\sigma _i^z/k + \mathop {\sum}\nolimits_j b_jh_j/k}}}{{\mathop {\sum}\nolimits_{y\prime } e^{\mathop {\sum}\nolimits_i a_i\sigma _i^{z\prime }/k + \mathop {\sum}\nolimits_j b_jh_j^\prime /k}}}\) and y〉 corresponds to the combination \(y\rangle = \sigma _1^z...\sigma _n^zh_1...h_m\rangle \) as before when \(h_{j}=1,{\hskip2pt}\mathrm{h}{{j}}\rangle\) represents the corresponding qubit in state \(1\rangle\) and when \(h_{j}=1,{\hskip2pt}\mathrm{h}{j}\rangle\) represents the corresponding qubit in state \(0\rangle\).
The second step is to calculate \(e^{w_{ij}\sigma _i^zh_j}\). We use controlledrotation gates to achieve this. The idea of sequential controlledrotation gates is to check whether the target qubit is in state 0〉 or state 1〉 and then rotate the corresponding angle (Fig. 2). If qubits \(\sigma _i^z h_j\) are in 00〉 or 11〉, the ancilla qubit is rotated by R_{y}(θ_{ij,1}) and otherwise by R_{y}(θ_{ij,2}), with \(\theta _{ij,1} = 2arcsin\left( {\sqrt {\frac{{e^{w_{ij}/k}}}{{e^{\left {w_{ij}} \right/k}}}} } \right)\) and \(\theta _{ij,2} = 2arcsin\left( {\sqrt {\frac{{e^{  w_{ij}/k}}}{{e^{w_{ij}/k}}}} } \right)\). Each time after one \(e^{w_{ij}\sigma _i^zh_j}\) is calculated, we do a measurement on the ancilla qubit. If it is in 1〉 we continue with a new ancilla qubit initialized in 0〉, otherwise we start over from the beginning (details in Supplementary Note 2).
After we finish all measurements the final states of the first m + n qubits follow the distribution Q(y). We just measure the first n + m qubits of the system register to obtain the probability distribution. After we get the distribution, we calculate all probabilities to the power of k and normalize to get the Gibbs distribution (Fig. 3).
The complexity of gates comes to O(mn) for one sampling and the qubits requirement comes to O(mn). If considering the reuse of ancilla qubits, the qubits requirements reduce to O(m + n) (see Supplementary Note 4). The probability of one successful sampling has a lower bound \(e^{\frac{{  1}}{k}\mathop {\sum}\nolimits_{i,j} 2w_{ij}}\) and if k is set to \(O\left( {\mathop {\sum}\nolimits_{i,j} w_{ij}} \right)\) it has constant lower bound (see Supplementary Note 3). If N_{s} is the number of successful sampling to get the distribution, the complexity for one iteration should be O(N_{s}mn) due to the constant lower bound of successful sampling as well as processing distribution taking O(N_{s}). In the meantime, the exact calculation for the distribution has complexity as O(2^{m+n}). The only error comes from the error of sampling if not considering noise in the quantum computer.
Summary of numerical results
We now present the results derived from our RBM for H_{2}, LiH and H_{2}O molecules. It can clearly be seen from Fig. 4 that our three layer RBM yields very accurate results comparing to the disorganization of transformed Hamiltonian which is calculated by a finite minimal basis set, STO3G. Points deviating from the ideal curve are likely due to local minima trapping during the optimization procedure. This can be avoided in the future by implementing optimization methods which include momentum or excitation, increasing the escape probability from any local features of the potential energy surface.
Further discussion about our results should mention instances of transfer learning. Transfer learning is a unique facet of neural network machine learning algorithms describing an instance (engineered or otherwise) where the solution to a problem can inform or assist in the solution to another similar subsequent problem. Given a diatomic Hamiltonian at a specific intermolecular separation, the solution yielding the variational parameters—which are the weighting coefficients of the basis functions—are adequate first approximations to those parameters at a subsequent calculation where the intermolecular separation is a small perturbation to the previous value.
Except for the last point in the Fig. 4d, we use 1/40 of the iterations for the last point in calculations initiated with transferred parameters from previous iterations of each points and still achieve a good result. We also see that the local minimum is avoided if the starting point achieve global minimum.
Discussion
In conclusion, we present a combined quantum machine learning approach to perform electronic structure calculations. Here, we have a proof of concept and show results for small molecular systems. Screening molecules to accelerate the discovery of new materials for specific application is demanding since the chemical space is very large! For example, it was reported that the total number of possible small organic molecules that populate the ‘chemical space’ exceed 10^{60} ^{36,37}. Such an enormous size makes a thorough exploration of chemical space using the traditional electronic structure methods impossible. Moreover, in a recent perspective^{38}in Nature Reviews Materials the potential of machine learning algorithms to accelerate the discovery of materials was pointed out. Machine learning algorithms have been used for material screening. For example, out of the GDB17 data base, consisting of about 166 billion molecular graphs, one can make organic and druglike molecules with up to 17 atoms and 134 thousand smallest molecules with up to 9 heavy atoms were calculated using hybrid density functional (B3LYP/631G(2df,p). Machine learning algorithms trained on these data, were found to predict molecular properties of subsets of these molecules^{38,39,40}.
In the current simulation, H_{2} requires 13 qubits with the number of visible units n = 4, the number of hidden units m = 8 and additional 1 reusing ancilla qubit. LiH requires 13 qubits with the number of visible units n = 4, the number of hidden units m = 8 and additional 1 reusing ancilla qubit. H_{2}O requires 13 qubits with the number of visible units n = 6, the number of hidden units m = 6 and additional 1 reusing ancilla qubit. The order of scaling of qubits for the system should be O(m + n) with reusing ancilla qubits. The number of visible units n is equal to the number of spin orbitals. The choice of the number of hidden units m is normally integer times of n which gives us a scaling of O(n) with reusing ancilla qubits. Thus, the scaling of the qubits increases polynomially with the number of spin orbitals. Also, the complexity of gates O(n^{2}) scales polynomially with the number of spin orbitals while the scaling of classical Machine Learning approaches calculating exact Gibbs distribution is exponential. With the rapid development of largerscale quantum computers and the possible training of some machine units with the simple dimensional scaling results for electronic structure, quantum machine learning techniques are set to become powerful tools to perform electronic structure calculations and assist in designing new materials for specific applications.
Methods
Preparation of the Hamiltonian of H_{2}, LiH and H_{2}O
We treat H_{2} molecule with 2electrons in a minimal basis STO3G and use the JordanWigner transformation^{34}. The final Hamiltonian is of 4 qubits. We treat LiH molecule with 4electrons in a minimal basis STO3G and use the JordanWigner transformation^{34}. We assumed the first two lowest orbitals are occupied by electrons and the the final Hamiltonian is of 4 qubits. We treat H_{2}O molecule with 10electrons in a minimal basis STO3G, we use JordanWigner transformation^{34}. We assume the first four lowest energy orbitals are occupied by electrons and first two highest energy orbitals are not occupied all time. We also use the spin symmetry in^{41,42} to reduce another two qubits. With the reduction of the number of qubits, finally we have 6 qubits Hamiltonian^{35,43}. All calculations of integrals in second quantization and transformations of electronic structure are done by OpenFermion^{44} and Psi4^{45}.
Gradient estimation
The two functions ϕ(x) and s(x) are both real function. Thus, the gradient for parameter p_{k} can be estimated as \(2\left( {\left\langle {E_{loc}D_{p_k}} \right\rangle  \left\langle {E_{loc}} \right\rangle \left\langle {D_{p_k}} \right\rangle } \right)\) where \(E_{loc}(x) = \frac{{\langle xH\phi \rangle }}{{\phi (x)s(x)}}\) is so called local energy, \(D_{p_k}(x) = \frac{{\partial _{p_k}(\phi (x)s(x))}}{{\phi (x)s(x)}}\). 〈...〉 represents the expectation value of joint distribution determined by ϕ(x) and s(x) (details in Supplementary Note 1).
Implementation details
In our simulation we choose small constant learning rate 0.01 to avoid trapping in local minimum. All parameter are initialized as a random number between (−0.02,0.02). The range of initial random parameter is to avoid gradient vanishing of tanh. For each calculation we just need 1 reusing ancilla qubit all the time. Thus, in the simulation, the number of required qubits is m + n + 1. All calculations do not consider the noise and system error (details in Supplementary Note 5).
Data availability
The data and codes that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We would like to thank Dr. Ross Hoehn, Dr. Zixuan Hu, and Teng Bian for critical reading and useful discussions. S.K. and R.X. are grateful for the support from Integrated Data Science Initiative Grants, Purdue University.
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S.K. designed the research. R.X. performed the calculations. Both discussed the results and wrote the paper.
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Xia, R., Kais, S. Quantum machine learning for electronic structure calculations. Nat Commun 9, 4195 (2018). https://doi.org/10.1038/s4146701806598z
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