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arXiv:1901.04715

Quantum chemistry on quantum annealers

Scott N. Genin, Ilya G. Ryabinkin, Artur F. Izmaylov

Published Jan 15, 2019• 20 citations
arXivHTMLPDF

Abstract

Quantum chemistry calculations for small molecules on quantum hardware have been demonstrated to date only on universal-gate quantum computers, not quantum annealers. The latter devices are limited to finding the lowest eigenstate of the Ising Hamiltonian whereas the electronic Hamiltonian could not be mapped to the Ising form without exponential growth of the Ising Hamiltonian with the size of the system [J. Phys. Chem. B 122, 3384 (2018)]. Here we propose a novel mixed discrete-continuous optimization algorithm, which finds the lowest eigenstate of the qubit coupled cluster (QCC) method using a quantum annealer for solving a discrete part of the problem. The QCC method is a potentially exact approach for constructing the electronic wave function in the qubit space. Therefore, our methodology allows for systematically improvable quantum chemistry calculations using quantum annealears. We illustrate capabilities of our approach by calculating QCC ground electronic states for the LiH, H$_2$O, and C$_6$H$_6$ molecules. C$_6$H$_6$ calculations involve 36 qubits and are the largest quantum chemistry calculations made on a quantum annealer (the D-Wave 2000Q system) to date. Our findings opens up a new perspective for use quantum annealers in high-throughput material discovery.

DOI
10.48550/arxiv.1901.04715

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Quantum chemistry calculations for small molecules on quantum hardware have been demonstrated to date only on universal-gate quantum computers, not quantum annealers. The latter devices are limited to finding the lowest eigenstate of the Ising Hamiltonian whereas the electronic Hamiltonian could not be mapped to the Ising form without exponential growth of the Ising Hamiltonian with the size of the system [J. Phys. Chem. B 122, 3384 (2018)]. Here we propose a novel mixed discrete-continuous optimization algorithm, which finds the lowest eigenstate of the qubit coupled cluster (QCC) method using a quantum annealer for solving a discrete part of the problem. The QCC method is a potentially exact approach for constructing the electronic wave function in the qubit space. Therefore, our methodology allows for systematically improvable quantum chemistry calculations using quantum annealears. We illustrate capabilities of our approach by calculating QCC ground electronic states for the LiH, H2_22​O, and C6_66​H6_66​ molecules. C6_66​H6_66​ calculations involve 36 qubits and are the largest quantum chemistry calculations made on a quantum annealer (the D-Wave 2000Q system) to date. Our findings opens up a new perspective for use quantum annealers in high-throughput material discovery.
Quantum annealers are an alternative approach to quantum computing, which make use of the adiabatic theorem to efficiently find the ground state of a physically realizable Hamiltonian. Such devices are currently commercially available and have been successfully applied to several combinatorial and discrete optimization problems. However, the application of quantum annealers to problems in chemistry remains a relatively sparse area of research due to the difficulty in mapping molecular systems to the Ising model Hamiltonian. In this paper, we review two different methods for finding the ground state of molecular Hamiltonians using Ising model-based quantum annealers. In addition, we compare the relative effectiveness of each method by calculating the binding energies, bond lengths, and bond angles of the H3+ and H2O molecules and mapping their potential energy curves. We also assess the resource requirements of each method by determining the number of qubits and computation time required to simulate each molecule using various parameter values. While each of these methods is capable of accurately predicting the ground state properties of small molecules, we find that they are still outperformed by modern classical algorithms and that the scaling of the resource requirements remains a challenge.
Calculating the molecular ground-state energy is a central challenge in computational chemistry. Conventional methods such as the Complete Active Space Configuration Interaction scale exponentially with molecular size, limiting their applicability to large molecules. Quantum computing offers a promising alternative by mapping molecular Hamiltonians by qubits, enabling cheaper computational scaling. Previous studies have shown that it is possible to formulate molecular ground state calculations as discrete optimization problems, addressable by quantum annealing. However, these efforts have been limited by previous generations of hardware and suboptimal annealing techniques. Here, the H2OH_{2}OH2​O ground-state problem is mapped to an Ising Hamiltonian using the Xian-Bias-Kas (XBK) method. By taking advantage of enhanced qubit connectivity and shorter embedding chains, it is solved with a more than doubled probability of achieving Hartree-Fock-level solutions with respect to the most advanced predecessor. Advanced annealing strategies extend Hartree-Fock-level accuracy to significantly larger problem instances, enabling solutions that use nearly 2.5 times more physically embedded qubits than the largest cases previously reported and allowing to improve annealing results by two orders of magnitude, reaching an energy difference of 0.120~Hartree relative to Hartree-Fock. These results show tangible progress toward practical quantum annealing applications in NISQ era.
A unitary coupled cluster (UCC) form for the wave function in the variational quantum eigensolver has been suggested as a systematic way to go beyond the mean-field approximation and include electron correlation in solving quantum chemistry problems on a quantum computer. Although being exact in the limit of including all possible coupled cluster excitations, practically, the accuracy of this approach depends on the number and type of terms are included in the wave function parametrization. Another difficulty of UCC is a growth of the number of simultaneously entangled qubits even at the fixed Fermionic excitation rank. Not all quantum computing architectures can cope with this growth. To address both problems, we introduce a qubit coupled cluster (QCC) method that starts directly in the qubit space and uses energy response estimates for ranking the importance of individual entanglers for the variational energy minimization. Also, we provide an exact factorization of a unitary rotation of more than two qubits to a product of two-qubit unitary rotations. Thus, the QCC method with the factorization technique can be limited to only two-qubit entanglement gates and allows for very efficient use of quantum resources in terms of the number of coupled cluster operators. The method performance is illustrated by calculating ground-state potential energy curves of H 2 and LiH molecules with chemical accuracy, ≤1 kcal/mol, and a symmetric water dissociation curve.
Quantum chemistry is regarded to be one of the first disciplines that will be revolutionized by quantum computing. Although universal quantum computers of practical scale may be years away, various approaches are currently being pursued to solve quantum chemistry problems on near-term gate-based quantum computers and quantum annealers by developing the appropriate algorithm and software base. This work implements the general Quantum Annealer Eigensolver (QAE) algorithm to solve the molecular electronic Hamiltonian eigenvalue-eigenvector problem on a D-Wave 2000Q quantum annealer. The approach is based on the matrix formulation, efficiently uses qubit resources based on a power-of-two encoding scheme and is hardware-dominant relying on only one classically optimized parameter. We demonstrate the use of D-Wave hardware for obtaining ground and excited electronic states across a variety of small molecular systems. The approach can be adapted for use by a vast majority of electronic structure methods currently implemented in conventional quantum-chemical packages. The results of this work will encourage further development of software such as qbsolv which has promising applications in emerging quantum information processing hardware and has expectation to address large and complex optimization problems intractable for classical computers.
Abstract Until recently molecular energy calculations using quantum computing hardware have been limited to gate-based quantum computers. In this paper, a new methodology is presented to calculate the vibrational spectrum of a molecule on a quantum annealer. The key idea of the method is a mapping of the ground state variational problem onto an Ising or quadratic unconstrained binary optimization (QUBO) problem by expressing the expansion coefficients using spins or qubits. The algorithm is general and represents a new revolutionary approach for solving the real symmetric eigenvalue problem on a quantum annealer. The method is applied to two chemically important molecules: O2 (oxygen) and O3 (ozone). The lowest two vibrational states of these molecules are computed using both a hardware quantum annealer and a software based classical annealer. Extension of the algorithm to higher dimensions is explicitly demonstrated for an N-dimensional harmonic oscillator (N ≤ 5). The algorithm scales exponentially with dimensionality if a direct product basis is used but will exhibit polynomial scaling for a nondirect product basis.
An iterative version of the qubit coupled cluster (QCC) method [I. G. Ryabinkin et al., J. Chem. Theory Comput. 2019, 14, 6317] is proposed. The new method seeks to find ground electronic energies of molecules on noisy intermediate-scale quantum devices. Each iteration involves a canonical transformation of the Hamiltonian and employs constant-size quantum circuits at the expense of increasing the Hamiltonian size. We numerically studied the convergence of the method on ground-state calculations for LiH, H 2 O, and N 2 molecules and found that the exact ground-state energies can be systematically approached only if the generators of the QCC ansatz are sampled from a specific set of operators. We report an algorithm for constructing this set that scales linearly with the size of the Hamiltonian.