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arXiv:2004.11416 · Physical Review Letters

Quantum Solver of Contracted Eigenvalue Equations for Scalable Molecular Simulations on Quantum Computing Devices

Scott E. Smart, David A. Mazziotti

Published Feb 18, 2021• 117 citations
arXivHTMLPDF

Abstract

The accurate computation of ground and excited states of many-fermion quantum systems is one of the most consequential, contemporary challenges in the physical and computational sciences whose solution stands to benefit significantly from the advent of quantum computing devices. Existing methodologies using phase estimation or variational algorithms have potential drawbacks such as deep circuits requiring substantial error correction or nontrivial high-dimensional classical optimization. Here, we introduce a quantum solver of contracted eigenvalue equations, the quantum analog of classical methods for the energies and reduced density matrices of ground and excited states. The solver does not require deep circuits or difficult classical optimization and achieves an exponential speed-up over its classical counterpart. We demonstrate the algorithm though computations on both a quantum simulator and two IBM quantum processing units.

DOI
10.1103/physrevlett.126.070504

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The accurate computation of ground and excited states of many-fermion quantum systems is one of the most consequential, contemporary challenges in the physical and computational sciences whose solution stands to benefit significantly from the advent of quantum computing devices. Existing methodologies using phase estimation or variational algorithms have potential drawbacks such as deep circuits requiring substantial error correction or nontrivial high-dimensional classical optimization. Here, we introduce a quantum solver of contracted eigenvalue equations, the quantum analog of classical methods for the energies and reduced density matrices of ground and excited states. The solver does not require deep circuits or difficult classical optimization and achieves an exponential speed-up over its classical counterpart. We demonstrate the algorithm though computations on both a quantum simulator and two IBM quantum processing units.
Accurate computation of multiple eigenvalues of quantum Hamiltonians is essential in quantum chemistry, materials science, and molecular spectroscopy. Estimating excited-state energies is challenging for classical algorithms due to exponential scaling with system size, posing an even harder problem than ground-state calculations. We present a quantum algorithm for estimating eigenvalues and singular values of parametrized matrix families, including solving generalized eigenvalue problems that frequently arise in quantum simulations. Our method uses quantum amplitude amplification and phase estimation to identify matrix eigenvalues by locating minima in the singular value spectrum. We demonstrate our algorithm by proposing a quantum computing formulation of the pseudospectral collocation method for the Schrödinger equation. We estimate fault-tolerant quantum resource requirements for the quantum collocation method, showing favorable scaling in the size of the problem N [up to O ̃ ( N ) ] compared to classical implementations with O ̃ ( N ) , for certain well-behaved potentials. Additionally, unlike the standard collocation method, which results in a generalized eigenvalue problem requiring matrix inversion, our algorithm circumvents the associated numerical instability by scanning a parametrized matrix family and detecting eigenvalues through singular value minimization. This approach is particularly effective when multiple eigenvalues are needed or when the generalized eigenvalue problem involves a high condition number. In the fault-tolerant era, our method may thus be useful for simulating high-dimensional molecular systems with dense spectra involving highly excited states, such as those encountered in molecular photodynamics or quasicontinuum regimes in many-body and solid-state systems.
Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. For quantum systems, where the physical dimension grows exponentially, finding the eigenvalues of certain operators is one such intractable problem and remains a fundamental challenge. The quantum phase estimation algorithm efficiently finds the eigenvalue of a given eigenvector but requires fully coherent evolution. Here we present an alternative approach that greatly reduces the requirements for coherent evolution and combine this method with a new approach to state preparation based on ansätze and classical optimization. We implement the algorithm by combining a highly reconfigurable photonic quantum processor with a conventional computer. We experimentally demonstrate the feasibility of this approach with an example from quantum chemistry—calculating the ground-state molecular energy for He–H+. The proposed approach drastically reduces the coherence time requirements, enhancing the potential of quantum resources available today and in the near future. Quantum computers promise to efficiently solve problems that would be practically impossible with a normal computer. Peruzzo et al. develop a variational computation approach that uses any available quantum resources and, with a photonic quantum processing unit, find the ground-state molecular energy of He–H+.
Key properties of physical systems can be described by the eigenvalues of matrices that represent the system. Computational algorithms that determine the eigenvalues of these matrices exist, but they generally suffer from a loss of performance as the matrix grows in size. This process can be expanded to quantum computation to find the eigenvalues with better performance than the classical algorithms. One application of such an eigenvalue solver is to determine energy levels of a molecule given a matrix representation of its Hamiltonian using the variational principle. Using a variational quantum eigensolver, we determine the ground state energies of different molecules. We focus on the choice of optimization strategy for a Qiskit simulator on low-end hardware. The benefits of several different optimizers were weighed in terms of accuracy in comparison to an analytic classical solution as well as code efficiency.
The simulation of strongly correlated many-electron systems is one of the most promising applications for near-term quantum devices. Here we use a class of eigenvalue solvers [presented in Smart and Mazziotti, Phys. Rev. Lett. 126, 070504 (2021)] in which a contraction of the Schr\"odinger equation is solved for the two-electron reduced density matrix (2-RDM) to resolve the energy splittings of the ortho-, meta-, and para-isomers of benzyne C6H4{\text{C}}_{6}{\text{H}}_{4}C6​H4​. In contrast to the traditional variational quantum eigensolver, the contracted quantum eigensolver can solve an integration (or contraction) of the many-electron Schr\"odinger equation onto the two-electron space. The quantum solution of the anti-Hermitian part of the contracted Schr\"odinger equation provides a scalable approach with few variational parameters that has its foundations in 2-RDM theory. Experimentally, a variety of error-mitigation strategies enable the calculation, including a linear shift in the 2-RDM targeting the iterative nature of the algorithm as well as a projection of the 2-RDM onto the convex set of approximately NNN-representable 2-RDMs defined by the 2-positive NNN-representability conditions. The relative energies exhibit single-digit millihartree errors, capturing a large part of the electron correlation energy, and the computed natural orbital occupations reflect the significant differences in the electron correlation of the isomers.
Quantum computers potentially have an exponential advantage over classical computers for the quantum simulation of many-fermion quantum systems. Nonetheless, fermions are more expensive to simulate than bosons due to the fermionic encoding---a mapping by which the qubits are encoded with fermion statistics. Here we generalize the contracted quantum eigensolver (CQE) to avoid fermionic encoding of the wave function. In contrast to the variational quantum eigensolver, the CQE solves for a many-fermion stationary state by minimizing the contraction (projection) of the Schr\"odinger equation onto two fermions. We avoid fermionic encoding of the wave function by contracting the Schr\"odinger equation onto an unencoded pair of particles. Solution of the resulting contracted equation by a series of unencoded two-body exponential transformations generates an unencoded wave function from which the energy and two-fermion reduced density matrix (2-RDM) can be computed. We apply the unencoded and the encoded CQE algorithms to the hydrogen fluoride molecule, the dissociation of oxygen O2{\mathrm{O}}_{2}O2​, and a series of hydrogen chains. Both algorithms show comparable convergence towards the exact ground-state energies and 2-RDMs, but the unencoded algorithm has computational advantages in terms of state preparation and tomography.
Key properties of physical systems can be described by the eigenvalues of matrices that represent the system. Computational algorithms that determine the eigenvalues of these matrices exist, but they generally suffer from a loss of performance as the matrix grows in size. This process can be expanded to quantum computation to find the eigenvalues with better performance than the classical algorithms. One application of such an eigenvalue solver is to determine energy levels of a molecule given a matrix representation of its Hamiltonian using the variational principle. Using a variational quantum eigensolver, we determine the ground state energies of different molecules. We focus on the choice of optimization strategy for a Qiskit simulator on low-end hardware. The benefits of several different optimizers were weighed in terms of accuracy in comparison to an analytic classical solution as well as code efficiency.