[Paper Review] Deep variational quantum eigensolver for excited states and its application to quantum chemistry calculation of periodic materials

Kaoru Mizuta, Mikiya Fujii+5 more|arXiv (Cornell University)|Apr 1, 2021

Quantum Computing Algorithms and ArchitectureComputer Science55 references26 citations

TL;DR

This paper extends the Deep Variational Quantum Eigensolver (Deep VQE) to compute excited states in quantum chemistry, particularly for periodic materials like hydrogen chains. By introducing a modified coarse-graining scheme that includes both intra- and inter-subsystem excitations and adding penalty terms to eliminate spurious eigenvalues, the method achieves ground and first-excited-state energies within O(1)% error using up to four fewer qubits than standard VQE, enabling near-term quantum advantage on NISQ devices.

ABSTRACT

A programmable quantum device that has a large number of qubits without fault-tolerance has emerged recently. Variational Quantum Eigensolver (VQE) is one of the most promising ways to utilize the computational power of such devices to solve problems in condensed matter physics and quantum chemistry. As the size of the current quantum devices is still not large for rivaling classical computers at solving practical problems, Fujii et al. proposed a method called "Deep VQE" which can provide the ground state of a given quantum system with the smaller number of qubits by combining the VQE and the technique of coarse-graining [K. Fujii, et al, arXiv:2007.10917]. In this paper, we extend the original proposal of Deep VQE to obtain the excited states and apply it to quantum chemistry calculation of a periodic material, which is one of the most impactful applications of the VQE. We first propose a modified scheme to construct quantum states for coarse-graining in Deep VQE to obtain the excited states. We also present a method to avoid a problem of meaningless eigenvalues in the original Deep VQE without restricting variational quantum states. Finally, we classically simulate our modified Deep VQE for quantum chemistry calculation of a periodic hydrogen chain as a typical periodic material. Our method reproduces the ground-state energy and the first-excited-state energy with the errors up to O(1)% despite the decrease in the number of qubits required for the calculation by two or four compared with the naive VQE. Our result will serve as a beacon for tackling quantum chemistry problems with classically-intractable sizes by smaller quantum devices in the near future.

Motivation & Objective

• To extend the Deep VQE framework to accurately compute low-lying excited states in quantum systems.
• To resolve the issue of meaningless eigenvalues in Deep VQE that restricts variational quantum state choices.
• To apply the modified Deep VQE to quantum chemistry calculations of periodic materials, such as hydrogen chains, to assess its feasibility for near-term quantum devices.
• To demonstrate that coarse-graining with higher-order excitations (e.g., double-particle) outperforms combining standard Deep VQE with Quantum Subspace Expansion (QSE).

Proposed method

• Proposes a modified local basis set that includes both intra-subsystem and inter-subsystem excitations to better capture low-energy excited states.
• Introduces penalty terms in the coarse-grained Hamiltonian to push spurious eigenvalues out of the low-energy spectrum, enabling the use of arbitrary variational quantum states.
• Provides a mathematically rigorous sufficient condition for penalty term magnitude to ensure validity of the method regardless of the chosen ansatz.
• Employs the Quantum Subspace Expansion (QSE) and perturbation theory to validate and improve the accuracy of low-energy eigenstates in the coarse-grained model.
• Uses a hardware-efficient ansatz with parametric single-qubit rotations and CZ entangling gates in classical simulations.
• Applies the SSVQE (State-Splitting VQE) cost function with weighted terms to target the first excited state in the effective model.

Experimental results

Research questions

• RQ1 Can Deep VQE be successfully extended to compute excited states while maintaining accuracy and reducing qubit requirements?
• RQ2 How can the problem of meaningless eigenvalues in Deep VQE be resolved without restricting the choice of variational quantum states?
• RQ3 Does including higher-order excitations (e.g., double-particle) in the coarse-graining process lead to better energy estimates than combining Deep VQE with QSE on lower-order excitations?
• RQ4 Can the modified Deep VQE accurately reproduce ground and excited-state energies for periodic quantum materials like hydrogen chains with significantly fewer qubits?
• RQ5 What is the impact of different coarse-graining schemes (e.g., single vs. double-particle excitations) on the accuracy of low-energy eigenstates?

Key findings

• The modified Deep VQE achieves ground-state energy and first-excited-state energy with errors up to O(1)% for a periodic hydrogen chain, despite reducing the number of qubits by two to four compared to standard VQE.
• Including both intra- and inter-subsystem excitations in the coarse-graining basis leads to more accurate low-energy eigenvalues than the original Deep VQE, which only considered inter-subsystem excitations.
• The addition of penalty terms to the coarse-grained Hamiltonian successfully removes meaningless eigenvalues from the low-energy spectrum, allowing the use of any variational quantum state without restriction.
• Classical simulations show that Deep VQE with double-particle excitations outperforms the combination of standard Deep VQE with QSE on single-particle excitations, due to a larger active subspace.
• The method maintains accuracy even with reduced qubit counts, demonstrating its potential for simulating classically intractable materials on near-term NISQ devices.
• The sufficient condition for penalty term magnitude is derived, ensuring the method's validity for arbitrary variational ansätze and providing a robust framework for future implementations.