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arXiv:2311.05058 · New Journal of Physics

Quantum simulation of excited states from parallel contracted quantum eigensolvers

Carlos L. Benavides-Riveros, Yuchen Wang, Samuel Warren, David A. Mazziotti

Published Mar 1, 2024• 22 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/ad2d1d

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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.
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.
The computation of electronic excited states and real-time quantum dynamics of many-fermion systems is among the most promising applications of near-term quantum computing. In this work, we generalize the reinforcement learning contracted quantum eigensolver (RL-CQE), previously developed for ground-state problems, to electronic excited states and real-time quantum dynamics, in which a deep Q-network agent adaptively selects the two-body operators at each iteration, yielding more compact ansätze and improved robustness with respect to critical hyperparameters. A key feature of the algorithm is a scalable state representation based on the ACSE residuals, whose dimension grows with the one-particle basis but remains independent of the number of targeted excited states. We also verify the equivalence of sign-free qubit operators in the excited-state setting, extending a result previously established for ground-state problems. Our RL-CQE for time evolution derives from a constant-scaling ansatz that represents the wave function with a fixed number of unitary transformations independent of simulation time ttt, enabled by the shared unitary structure of the purified ensemble treatment of excited states. Benchmarks on chemical systems demonstrate chemical accuracy with minimal operator counts across a range of bond lengths.
Calculating ground and excited states is an exciting prospect for near-term quantum computing applications, and accurate and efficient algorithms are needed to assess viable directions. We develop an excited-state approach based on the contracted quantum eigensolver (ES-CQE), which iteratively attempts to find a solution to a contraction of the Schrödinger equation projected onto a subspace and does not require a priori information on the system. We focus on the anti-Hermitian portion of the equation, leading to a two-body unitary ansatz. We investigate the role of symmetries, initial states, constraints, and overall performance within the context of the model strongly correlated rectangular H 4 system. We show that the ES-CQE achieves near-exact accuracy across the majority of states, covering regions of strong and weak electron correlation, while also elucidating challenging instances for two-body unitary ansatz.
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.
Variational quantum eigensolver (VQE) is a promising method for ground-state calculations on current noisy intermediate-scale quantum computers. However, the research progress of excited-state calculations on quantum computers is relatively slow. In order to extend the framework of VQE for excited-state calculations, we propose a new algorithm, orthogonal state reduction variational eigensolver (OSRVE), to determine the energies of excited states. Theoretical derivations prove that the optimized state in the OSRVE method can ensure the energy minimum and orthogonality constraint simultaneously, and OSRVE is also applicable for the degenerate state. The performance of OSRVE is demonstrated by numerical calculations of the H 4 and H 2 O molecules. Compared with other excited-state calculation algorithms, OSRVE has obvious advantages in calculating lower-order excited states. This work extends the VQE algorithm to excited-state calculations, and OSRVE can be implemented on near-term quantum computers.