[Paper Review] A Full Quantum Eigensolver for Quantum Chemistry Simulations

Shijie Wei, Hang Li+1 more|arXiv (Cornell University)|Aug 21, 2019

Quantum Computing Algorithms and Architecture31 references4 citations

TL;DR

This paper proposes a Full Quantum Eigensolver (FQE) for quantum chemistry simulations that performs all optimization steps—Hamiltonian evaluation and gradient descent—on a quantum computer, eliminating the need for classical optimization. The method achieves exponential speedup over classical algorithms with polylogarithmic gate complexity and delivers chemical precision in ground-state energy and wavefunction using only one iteration when combined with perturbation theory.

ABSTRACT

Quantum simulation of quantum chemistry is one of the most compelling applications of quantum computing. It is of particular importance in areas ranging from materials science, biochemistry and condensed matter physics. Here, we propose a full quantum eigensolver (FQE) algorithm to calculate the molecular ground energies and electronic structures using quantum gradient descent. Compared to existing classical-quantum hybrid methods such as variational quantum eigensolver (VQE), our method removes the classical optimizer and performs all the calculations on a quantum computer with faster convergence. The gradient descent iteration depth has a favorable complexity that is logarithmically dependent on the system size and inverse of the precision. Moreover, the FQE can be further simplified by exploiting perturbation theory for the calculations of intermediate matrix elements, and obtain results with a precision that satisfies the requirement of chemistry application. The full quantum eigensolver can be implemented on a near-term quantum computer. With the rapid development of quantum computing hardware, FQE provides an efficient and powerful tool to solve quantum chemistry problems.

Motivation & Objective

• To develop a fully quantum algorithm for simulating molecular ground states and electronic structures without classical optimization.
• To overcome the exponential resource scaling of classical methods in solving the Schrödinger equation for large molecular systems.
• To enable efficient quantum chemistry simulations on near-term Noisy Intermediate-Scale Quantum (NISQ) devices.
• To achieve chemical precision in energy calculations using a minimal number of quantum circuit iterations.
• To demonstrate robustness against noise and compatibility with both NISQ and future fault-tolerant quantum computers.

Proposed method

• The FQE uses quantum gradient descent to optimize the variational wavefunction entirely on a quantum processor, replacing classical optimization with quantum control.
• It employs the Linear Combination of Unitaries (LCU) scheme to estimate matrix elements of the Hamiltonian and its gradient on a quantum computer.
• The algorithm leverages the Jordan-Wigner or Bravyi-Kitaev transformation to map fermionic Hamiltonians to qubit operators with O(N⁴) Pauli terms.
• The gate complexity per iteration scales as O(M log M log N), where M is the number of Pauli terms and N is the number of atomic orbitals.
• Perturbation theory is applied to simplify the method, enabling chemical precision with a single iteration in some cases.
• The method uses controlled-Hadamard operations and state preparation circuits with log M ancilla qubits to implement the gradient descent steps.

Experimental results

Research questions

• RQ1 Can a fully quantum algorithm outperform hybrid classical-quantum methods like VQE in convergence speed and resource efficiency for quantum chemistry simulations?
• RQ2 What is the quantum resource complexity of a full quantum eigensolver in terms of qubits and gates for molecular ground-state energy calculations?
• RQ3 Can perturbation theory be integrated into a quantum-only optimization framework to achieve chemical precision with minimal iterations?
• RQ4 How robust is the FQE algorithm against noise in near-term quantum hardware platforms?
• RQ5 To what extent can the FQE be scaled to large molecular systems with O(N⁴) fermionic interaction terms?

Key findings

• The FQE achieves exponential speedup over classical methods, with gate complexity scaling as O(M log M log N), where M = O(N⁴) for N atomic orbitals.
• The algorithm requires only O(log M) ancilla qubits, making it resource-efficient for near-term devices.
• Numerical simulations show that second-order perturbation theory in FQE yields ground-state energies within chemical precision (typically <1 kcal/mol) of exact diagonalization for H₂, LiH, H₂O, and NH₃.
• At the equilibrium bond distance, the second-order approximation for H₂, LiH, H₂O, and NH₃ matches exact energies to within 0.0001 au, confirming chemical accuracy.
• The method remains robust under realistic noise levels and can deliver accurate results even with limited circuit depth by leveraging perturbation theory.
• The FQE is compatible with both NISQ devices and future fault-tolerant quantum computers, enabling scalable quantum chemistry simulations.