[Paper Review] Analytical Ground- and Excited-State Gradients for Molecular Electronic Structure Theory from Hybrid Quantum/Classical Methods

Robert M. Parrish, Gian-Luca R. Anselmetti+1 more|arXiv (Cornell University)|Oct 11, 2021

Quantum Computing Algorithms and Architecture73 references6 citations

TL;DR

This paper presents a Lagrangian formalism for computing analytical ground- and excited-state energy gradients in hybrid quantum/classical multistate contracted VQE (MC-VQE), enabling efficient quantum-classical separation and independent quantum resource scaling from the number of nuclear coordinates. The method achieves full analytical gradients with quantum cost comparable to energy evaluation, allowing accurate gradients for systems with hundreds of atoms using active-space simulations on near-term quantum hardware.

ABSTRACT

We develop analytical gradients of ground- and excited-state energies with respect to system parameters including the nuclear coordinates for the hybrid quantum/classical multistate contracted variational quantum eigensolver (MC-VQE) applied to fermionic systems. We show how the resulting response contributions to the gradient can be evaluated with a quantum effort similar to that of obtaining the VQE energy and independent of the total number of derivative parameters (e.g. number of nuclear coordinates) by adopting a Lagrangian formalism for the evaluation of the total derivative. We also demonstrate that large-step-size finite-difference treatment of directional derivatives in concert with the parameter shift rule can significantly mitigate the complexity of dealing with the quantum parameter Hessian when solving the quantum response equations. This enables the computation of analytical derivative properties of systems with hundreds of atoms, while solving an active space of their most strongly correlated orbitals on a quantum computer. We numerically demonstrate the exactness the analytical gradients and discuss the magnitude of the quantum response contributions.

Motivation & Objective

• To develop a scalable, analytical method for computing nuclear gradients in hybrid quantum/classical electronic structure methods, particularly for non-variational VQE approaches.
• To address the challenge of chain-rule response terms from intermediate parameters (e.g., VQE entangler parameters) in non-variational methods such as MC-VQE and SA-VQE.
• To reduce quantum computational overhead for gradient evaluation, making it independent of the number of nuclear coordinates or derivative parameters.
• To enable accurate computation of analytical gradients for large molecular systems (hundreds of atoms) with active-space simulations on near-term quantum computers.
• To provide a generalizable framework applicable to other hybrid quantum/classical algorithms beyond MC-VQE, including those for polarizabilities and non-adiabatic coupling vectors.

Proposed method

• Adopt a Lagrangian formalism to systematically derive total derivatives of energy with respect to system parameters, separating quantum and classical response contributions.
• Use the Lagrangian approach to eliminate scaling dependence on the number of derivative parameters (e.g., nuclear coordinates), ensuring quantum resource cost remains constant regardless of system size.
• Implement an iterative matrix-vector product formalism for the SA-VQE Hessian using widely spaced finite differences, avoiding explicit Hessian computation and reducing quantum circuit complexity.
• Leverage the parameter shift rule in combination with large-step finite differences to approximate directional derivatives and solve quantum response equations efficiently.
• Integrate classical response terms from intermediate parameters (e.g., VQE entangler parameters) using chain rule formalism within the Lagrangian framework.
• Apply the method to compute analytical gradients for both ground and excited states in active-space MC-VQE, including state-averaged (SA-VQE) and non-averaged variants.

Experimental results

Research questions

• RQ1 How can analytical gradients of molecular electronic energies be efficiently computed in non-variational hybrid quantum/classical algorithms like MC-VQE, where response terms from intermediate parameters are non-zero?
• RQ2 Can the quantum computational cost of computing gradients be made independent of the number of nuclear coordinates or derivative parameters using a formalism like the Lagrangian approach?
• RQ3 What is the impact of SA-VQE response contributions on the total nuclear gradient, and can they be computed accurately without explicitly constructing the full Hessian?
• RQ4 To what extent do finite-difference approximations with large steps and the parameter shift rule enable robust and efficient solution of quantum response equations in near-term devices?
• RQ5 Can the proposed method be generalized to other hybrid quantum/classical algorithms and observables beyond energy gradients, such as polarizabilities or NMR shifts?

Key findings

• The Lagrangian formalism successfully decouples quantum and classical response contributions, enabling analytical gradients with quantum resource cost independent of the number of derivative parameters.
• The iterative matrix-vector product formalism using large-step finite differences for the SA-VQE Hessian avoids explicit Hessian computation and enables accurate solution of response equations with minimal quantum overhead.
• Numerical results confirm the exactness of the analytical gradients, with SA-VQE response terms contributing approximately 2% of the total gradient in the octatetraene@MeOH system.
• The method enables analytical gradient computation for systems with hundreds of atoms, provided only the most strongly correlated orbitals are treated on the quantum computer.
• The framework is generalizable to other non-stationary hybrid algorithms such as SS-VQE, QFD, VQPE, and NO-VQE, and can be extended to compute derivatives of observables like polarizabilities and non-adiabatic coupling vectors.
• In small systems with deep quantum circuits, response terms are small (~2%) and may be neglected due to dominant noise sources; however, response terms are expected to become essential in larger systems.