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Quantum Machine Learning for Electronic Structure Calculations
Rongxin Xia, Sabre Kais
TL;DR
Accurate electronic structure calculations remain challenging, especially for larger systems. The paper combines a three-layer restricted Boltzmann machine with quantum-assisted optimization to calculate molecular electronic ground-state energies. It reports high accuracy for H2, LiH, and H2O in a finite minimal basis, while noting sampling limitations for larger k.
Problem
Accurate electronic structure calculations for large systems remain challenging, motivating quantum-computing and machine-learning approaches.
Method
The approach uses a three-layer RBM with a sign layer for wave-function coefficients and a quantum algorithm to help optimize training.
Results
The three-layer RBM yields very accurate results for H2, LiH, and H2O compared with finite-basis STO-3G calculations.
Takeaways & Limitations
Quantum machine-learning techniques are presented as potential tools for electronic structure calculations and new-material design as larger quantum computers become available.
Takeaways & Limitations
For large k, the quantum algorithm may require extensive sampling, potentially eliminating quantum speedup.
Abstract
from arXiv · showhide
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 larger-scale quantum computers, quantum machine learning techniques are set to become powerful tools to obtain accurate values for electronic structures.
Introduction
Accurate electronic structure calculations remain challenging, motivating quantum simulation and quantum machine learning approaches. This paper combines an RBM with quantum optimization and adds a sign-correction layer for molecular wave functions.
- Quantum machine learning combines classical machine learning with quantum computation and may reduce RBM training time.
- Accurate electronic structure calculations for large systems remain challenging in chemistry and materials science.
- Quantum computing has been explored alongside classical ab initio and density functional methods to address electronic structure calculations.
- Prior RBM approaches for quantum many-body problems require modification when accurate wave-function phases are needed.
- The proposed three-layer RBM adds a sign-correction layer and uses a quantum algorithm to help optimize training.
Results
The paper develops a three-layer restricted Boltzmann machine with a sign layer and a quantum sampling algorithm for electronic-structure calculations. The approach produces accurate small-molecule ground-state energies while using polynomially scaling quantum resources under the stated setting.
- Three-layers restricted Boltzmann machine: The objective function is the energy expectation value, which is minimized by gradient descent to obtain the ground-state energy.Training samples the joint distribution determined by the coefficient and sign functions before estimating gradients.
- Three-layers restricted Boltzmann machine: The three-layer RBM adds a sign layer to represent coefficient signs that standard RBMs cannot capture for electronic structure.The visible layer builds the wave function, the hidden layer parameterizes coefficients, and the sign layer classifies positive or negative signs.
- Electronic Structure Hamiltonian Preparation: The electronic-structure Hamiltonian uses STO-3G molecular integrals and is transformed into a Pauli-matrix representation with the Jordan-Wigner transformation.The Hamiltonian is built from one- and two-electron integrals before conversion to Pauli operators.
- Quantum algorithm to sample Gibbs distribution: The quantum sampler uses sequential controlled rotations and an ancilla qubit to sample a distribution Q′(y) ≥ Q(y) associated with the RBM Gibbs distribution.The adjustable parameter k changes across iterations to increase successful-sampling probability.
- Quantum algorithm to sample Gibbs distribution: With ancilla reuse, the sampler requires O(m + n) qubits and has O(mn) gate complexity for one sampling, while exact classical distribution calculation scales as O(2^(m+n)).For one optimization iteration, the reported sampling complexity is O(N_smn), where N_s is the number of successful samples.
Methods
The study prepares molecular Hamiltonians with Jordan–Wigner-transformed STO-3G representations and optimizes a three-layer RBM using sampled gradients. Simulations use small learning rates, random initialization, one reused ancilla qubit, and exclude noise and system error.
- Hamiltonian preparation: H2, LiH, and H2O are represented in the STO-3G basis and transformed into four-qubit Hamiltonians using Jordan–Wigner transformation.The passage explicitly reports four-qubit final Hamiltonians for each molecule.
- Gradient estimation: Gradients are estimated from the local energy and parameter derivatives using expectation values over the joint wave-function distribution.The two functions φ(x) and s(x) are treated as real functions in the gradient estimator.
- Implementation: A learning rate of 0.01 and random parameters between −0.02 and 0.02 are used to reduce local-minimum trapping and avoid tanh gradient vanishing.The simulation uses one reused ancilla qubit, requiring m+n+1 qubits.
- Data availability: The supporting data and code are available from the corresponding author upon reasonable request.
Additional Information
Supplementary materials are available, and the authors declare no competing interest.
- Supplementary information: Supplementary materials are available.
- Competing interests: The authors declare no competing interest.
Supplementary Note 1
The supplementary derivation defines a gradient-based optimization procedure for the RBM energy using a trial wave function and sampled expectations. Parameters are updated iteratively with a learning rate until the optimization reaches its stopping condition.
- Gradient derivation: The electronic-structure Hamiltonian and trial wave function provide the inputs for the RBM energy-gradient derivation.
- Gradient derivation: Defining Φ(x)=φ(x)s(x) combines the coefficient and sign functions for gradient evaluation.
- Gradient derivation: The expectation values are taken over the distribution determined by Φ(x), with Hamiltonian symmetry supplied by the Jordan–Wigner representation.
- Parameter optimization: The optimized parameters are ai, bj, wij, di, and c at iteration k.
- Parameter optimization: Gradient descent is used to optimize the RBM toward the ground energy, while gradients are estimated from sampled distributions.
- Parameter optimization: The learning rate αk controls convergence, and iterations continue until the maximum iteration count is reached.
Supplementary Note 2
The quantum sampler prepares superpositions and applies sequential controlled rotations to generate the RBM-derived Gibbs distribution. Ancilla measurements accept successful samples, after which measured probabilities are powered by k and normalized.
- Distribution recovery: The probability for each visible-hidden configuration y={σz,h} is used as the target distribution for the sampling procedure.
- Sampling modification: The algorithm modifies P(y) by raising probabilities to a large power k to increase successful measurement probability.
- State preparation: The sampling circuit begins with Ry gates that place visible and hidden qubits, together with the ancilla, in the required initial superposition.
- Controlled rotations: Controlled rotations compute weight-dependent terms by selecting angles according to the visible and hidden qubit states.
- Controlled rotations: Each controlled rotation is expressed through the corresponding rotation-gate construction.
- Measurement and acceptance: An ancilla measurement accepts a sample only when the ancilla is |1⟩; otherwise, the procedure restarts.
- Distribution recovery: After successful measurements, the first n+m system qubits are measured, and the resulting probabilities are raised to k and normalized into the Gibbs distribution.
Supplementary Note 3
The controlled-rotation procedure regulates sampling to obtain a constant lower bound on successful measurements. Choosing k balances success probability against sampling error.
- P ≥ 1/e^O(1) establishes a constant lower bound for successful sampling.
- The simulation chooses k = max(1, …) because larger k introduces larger sampling errors.
- The selected k gives a lower bound of success as e^-4, while numerical simulations produce substantially higher probabilities.
Supplementary Note 4
The C2(U) gate is decomposed into elementary operations for controlled rotations. For U = Ry(θ), the construction uses V = Ry(θ/2) and scales with the RBM dimensions.
- For U = Ry(θ), choosing V = Ry(θ/2) realizes the required decomposition.
- Control conditions |00⟩, |10⟩, and |01⟩ are obtained by adding X gates to the controlling qubits.
- Each wij requires four C2(U) gates, giving gate and qubit complexity O(mn), reducible to O(m + n) with qubit reuse.
Supplementary Note 5
The supplementary simulations track successful-sampling probabilities and RBM energies during optimization for H2, LiH, and H2O at a bond length of 1.75 Angstrom. The regulation parameter improves sampling success but larger values increase sampling demands and fluctuations.
- The simulations report successful-sampling probabilities for H2, LiH, and H2O at bond length 1.75 Angstrom.
- The optimization-energy plots likewise cover H2, LiH, and H2O during the same calculations.
- With no regulation, k = 1 yields a success probability proportional to 1/(2|wij|), requiring an exponential number of measurements.
- Using regulation gives a probability of 1/e^4 and requires a constant number of measurements for sufficient successful sampling.
- At final optimization, k is around 5 for all three molecules, while large k causes fluctuation that can be reduced with more samples.