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arXiv:2205.01725 · Physical Review A

Many-fermion simulation from the contracted quantum eigensolver without fermionic encoding of the wave function

Scott E. Smart, David A. Mazziotti

Published Jun 15, 2022• 18 citations
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Abstract

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 ${\mathrm{O}}_{2}$, 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.

DOI
10.1103/physreva.105.062424

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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.
Abstract Molecular simulations generally require fermionic encoding in which fermion statistics are encoded into the qubit representation of the wave function. Recent calculations suggest that fermionic encoding of the wave function can be bypassed, leading to more efficient quantum computations. Here we show that the two-electron reduced density matrix (2-RDM) can be expressed as a unique functional of the unencoded N-qubit-particle wave function without approximation, and hence, the energy can be expressed as a functional of the 2-RDM without fermionic encoding of the wave function. In contrast to current hardware-efficient methods, the derived functional has a unique, one-to-one (and onto) mapping between the qubit-particle wave functions and 2-RDMs, which avoids the over-parametrization that can lead to optimization difficulties such as barren plateaus. An application to computing the ground-state energy and 2-RDM of H4 is presented.
Abstract Quantum computers are promising tools for simulating many-body quantum systems due to their potential scaling advantage over classical computers. While significant effort has been expended on many-fermion systems, here we simulate a model entangled many-boson system with the contracted quantum eigensolver (CQE). We generalize the CQE to many-boson systems by encoding the bosonic wavefunction on qubits. The CQE provides a compact ansatz for the bosonic wave function whose gradient is proportional to the residual of a contracted Schrödinger equation. We apply the CQE to a bosonic system, where N quantum harmonic oscillators are coupled through a pairwise quadratic repulsion. The model is relevant to the study of coupled vibrations in molecular systems on quantum devices. Results demonstrate the potential efficiency of the CQE in simulating bosonic processes such as molecular vibrations with good accuracy and convergence even in the presence of noise.
Quantum computers are promising tools for simulating many-body quantum systems due to their potential scaling advantage over classical computers. While significant effort has been expended on many-fermion systems, here we simulate a model entangled many-boson system with the contracted quantum eigensolver (CQE). We generalize the CQE to many-boson systems by encoding the bosonic wavefunction on qubits. The CQE provides a compact ansatz for the bosonic wave function whose gradient is proportional to the residual of a contracted Schrödinger equation. We apply the CQE to a bosonic system, where NNN quantum harmonic oscillators are coupled through a pairwise quadratic repulsion. The model is relevant to the study of coupled vibrations in molecular systems on quantum devices. Results demonstrate the potential efficiency of the CQE in simulating bosonic processes such as molecular vibrations with good accuracy and convergence even in the presence of noise.
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.
Electronic excited states of molecules are central to many physical and chemical processes, and yet they are typically more difficult to compute than ground states. In this paper we leverage the advantages of quantum computers to develop an algorithm for the highly accurate calculation of excited states. We solve a contracted Schr\"odinger equation (CSE)---a contraction (projection) of the Schr\"odinger equation onto the space of two electrons---whose solutions correspond identically to the ground and excited states of the Schr\"odinger equation. While recent quantum algorithms for solving the CSE, known as contracted quantum eigensolvers (CQEs), have focused on ground states, we develop a CQE based on the variance that is designed to optimize rapidly to a ground or excited state. We apply the algorithm to compute the ground and excited states of H2,H4{\mathrm{H}}_{2}, {\mathrm{H}}_{4}H2​,H4​, and BH.
Abstract Computing excited-state properties of molecules and solids is considered one of the most important near-term applications of quantum computers. While many of the current excited-state quantum algorithms differ in circuit architecture, specific exploitation of quantum advantage, or result quality, one common feature is their rooting in the Schrödinger equation. However, through contracting (or projecting) the eigenvalue equation, more efficient strategies can be designed for near-term quantum devices. Here we demonstrate that when combined with the Rayleigh–Ritz variational principle for mixed quantum states, the ground-state contracted quantum eigensolver (CQE) can be generalized to compute any number of quantum eigenstates simultaneously. We introduce two excited-state (anti-Hermitian) CQEs that perform the excited-state calculation while inheriting many of the remarkable features of the original ground-state version of the algorithm, such as its scalability. To showcase our approach, we study several model and chemical Hamiltonians and investigate the performance of different implementations.