Deep Variational Quantum Eigensolver: a divide-and-conquer method for solving a larger problem with smaller size quantum computers

TL;DR

Deep VQE is a divide-and-conquer hybrid quantum-classical algorithm that enables solving large, strongly correlated quantum systems using small-scale quantum computers by iteratively reducing system size and solving effective Hamiltonians.

Contribution

It introduces deep VQE, a novel method combining system reduction and VQE to handle large quantum problems on small quantum devices.

Findings

Successfully simulated a 64-qubit system with ~3% error

Demonstrated applicability to large systems with >1000 qubits

Achieved good accuracy on quasi 1D and 2D Heisenberg models

Abstract

We propose a divide-and-conquer method for the quantum-classical hybrid algorithm to solve larger problems with small-scale quantum computers. Specifically, we concatenate a variational quantum eigensolver (VQE) with a reduction in the system dimension, where the interactions between divided subsystems are taken as an effective Hamiltonian expanded by the reduced basis. Then the effective Hamiltonian is further solved by VQE, which we call deep VQE. Deep VQE allows us to apply quantum-classical hybrid algorithms on small-scale quantum computers to large systems with strong intra-subsystem interactions and weak inter-subsystem interactions, or strongly correlated spin models on large regular lattices. As proof-of-principle numerical demonstrations, we use the proposed method for quasi one-dimensional models, including one-dimensionally coupled 12-qubit Heisenberg anti-ferromagnetic models on Kagome lattices as well as two-dimensional Heisenberg anti-ferromagnetic models on square lattices. The largest problem size of 64 qubits is solved by simulating 20-qubit quantum computers with a reasonably good accuracy ~ a few %. The proposed scheme enables us to handle the problems of >1000 qubits by concatenating VQEs with a few tens of qubits. While it is unclear how accurate ground state energy can be obtained for such a large system, our numerical results on a 64-qubit system suggest that deep VQE provides a good approximation (discrepancy within a few percent) and has a room for further improvement. Therefore, deep VQE provides us a promising pathway to solve practically important problems on noisy intermediate-scale quantum computers.