Dual-VQE: A quantum algorithm to lower bound the ground-state energy

• URL: http://arxiv.org/abs/2312.03083v1
• Date: Tue, 5 Dec 2023 19:02:19 GMT
• Title: Dual-VQE: A quantum algorithm to lower bound the ground-state energy
• Authors: Hanna Westerheim, Jingxuan Chen, Zo\"e Holmes, Ivy Luo, Theshani Nuradha, Dhrumil Patel, Soorya Rethinasamy, Kathie Wang, and Mark M. Wilde
• Abstract summary: The variational quantum eigensolver (VQE) produces an upper-bound estimate of the ground-state energy of a Hamiltonian. We propose a dual variational quantum eigensolver (dual-VQE) that produces a lower-bound estimate of the ground-state energy.
• Score: 4.8746635005655286
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The variational quantum eigensolver (VQE) is a hybrid quantum--classical variational algorithm that produces an upper-bound estimate of the ground-state energy of a Hamiltonian. As quantum computers become more powerful and go beyond the reach of classical brute-force simulation, it is important to assess the quality of solutions produced by them. Here we propose a dual variational quantum eigensolver (dual-VQE) that produces a lower-bound estimate of the ground-state energy. As such, VQE and dual-VQE can serve as quality checks on their solutions; in the ideal case, the VQE upper bound and the dual-VQE lower bound form an interval containing the true optimal value of the ground-state energy. The idea behind dual-VQE is to employ semi-definite programming duality to rewrite the ground-state optimization problem as a constrained maximization problem, which itself can be bounded from below by an unconstrained optimization problem to be solved by a variational quantum algorithm. When using a convex combination ansatz in conjunction with a classical generative model, the quantum computational resources needed to evaluate the objective function of dual-VQE are no greater than those needed for that of VQE. We simulated the performance of dual-VQE on the transverse-field Ising model, and found that, for the example considered, while dual-VQE training is slower and noisier than VQE, it approaches the true value with error of order $10^{-2}$.

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