Variational Quantum Computation of Excited States (1805.08138v5)

Abstract: The calculation of excited state energies of electronic structure Hamiltonians has many important applications, such as the calculation of optical spectra and reaction rates. While low-depth quantum algorithms, such as the variational quantum eigenvalue solver (VQE), have been used to determine ground state energies, methods for calculating excited states currently involve the implementation of high-depth controlled-unitaries or a large number of additional samples. Here we show how overlap estimation can be used to deflate eigenstates once they are found, enabling the calculation of excited state energies and their degeneracies. We propose an implementation that requires the same number of qubits as VQE and at most twice the circuit depth. Our method is robust to control errors, is compatible with error-mitigation strategies and can be implemented on near-term quantum computers.

• The paper introduces VQD, a novel extension of VQE that computes excited state energies by enforcing orthogonality using deflation techniques.
• The method achieves chemical accuracy in molecular Hydrogen simulations while utilizing the same qubit count and only a modest increase in circuit depth.
• Its resource efficiency and applicability on near-term quantum devices make it highly impactful for advancing quantum chemistry and materials science.

Variational Quantum Computation of Excited States: An Expert Overview

The paper "Variational Quantum Computation of Excited States" by Oscar Higgott, Daochen Wang, and Stephen Brierley presents a novel quantum algorithm aimed at the computation of excited state energies for electronic structure Hamiltonians. This development serves as an incremental advancement over existing quantum algorithms like the Variational Quantum Eigensolver (VQE), which has been primarily utilized for estimating ground state energies.

Methodological Contributions

The crux of the proposed method, termed Variational Quantum Deflation (VQD), lies in its ability to extend the VQE to compute excited states efficiently. The necessity of calculating excited states extends to applications like predicting charge and energy transfer processes, which are critical in photovoltaic materials and understanding chemical reactions involving photodissociation. Traditional classical methodologies, such as Density Functional Theory, are often inadequate for these complex calculations.

The VQD approach utilizes overlap estimation to effectively deflate already computed eigenstates, thus making room for subsequent calculations of higher excited states. This deflation technique strategically extends the parameter optimization process intrinsic to VQE by incorporating additional terms into the objective function that help in maintaining orthogonality between the trial wavefunction and previously found states.

Resource Efficiency

A significant advantage of the VQD is its low resource requirements. The method requires the same number of qubits as the original VQE approach and at most doubles the circuit depth, which is economically feasible for near-term quantum devices. Additionally, the sampling overhead needed for the calculation is negligible in the context of precision requirements, as demonstrated by the authors' analysis. These features make VQD particularly appealing for near-future implementations on existing quantum hardware.

Numerical Results

The efficacy of the VQD is underscored through numerical simulations of molecular Hydrogen (H₂) in the STO-3G basis. The simulations yield impressively accurate results, with errors significantly below the threshold of chemical accuracy for all examined energy levels. Importantly, VQD also demonstrates robustness in identifying degenerate states, presenting a systematic solution to a long-standing challenge in quantum computations of excited states.

Practical Implications and Future Directions

The contribution of VQD has immediate practical implications for quantum chemistry and materials science, providing a feasible computational pathway for quantum-enhanced calculation of excited states. This is of fundamental importance for designing more efficient materials and understanding complex biochemical reactions in innovative fields like drug discovery.

Theoretically, this work opens multiple avenues for future exploration. The potential to integrate VQD with error mitigation techniques suggests an opportunity for further enhancement of computational precision. Additionally, the method's compatibility with alternate ansatz formulations and optimizers invites extensive experimentation to potentially improve convergence rates and sensitivity.

In conclusion, the paper presents a sophisticated yet resource-efficient method for computing excited states on near-term quantum devices. As the quantum computing landscape continues to evolve, methods like VQD pave the way for more ambitious and complex problem-solving, extending the frontier of quantum simulations and their practical applications. These developments facilitate a better understanding of fundamental processes, contributing to the progress in both theoretical developments and industrial applications.