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The variational quantum eigensolver (VQE) is a hybrid quantum--classicalvariational algorithm that produces an upper-bound estimate of the ground-stateenergy of a Hamiltonian. As quantum computers become more powerful and gobeyond the reach of classical brute-force simulation, it is important to assessthe quality of solutions produced by them. Here we propose a dual variationalquantum eigensolver (dual-VQE) that produces a lower-bound estimate of theground-state energy. As such, VQE and dual-VQE can serve as quality checks ontheir solutions; in the ideal case, the VQE upper bound and the dual-VQE lowerbound form an interval containing the true optimal value of the ground-stateenergy. The idea behind dual-VQE is to employ semi-definite programming dualityto rewrite the ground-state optimization problem as a constrained maximizationproblem, which itself can be bounded from below by an unconstrainedoptimization problem to be solved by a variational quantum algorithm. Whenusing a convex combination ansatz in conjunction with a classical generativemodel, the quantum computational resources needed to evaluate the objectivefunction of dual-VQE are no greater than those needed for that of VQE. Wesimulated the performance of dual-VQE on the transverse-field Ising model, andfound that, for the example considered, while dual-VQE training is slower andnoisier than VQE, it approaches the true value with error of order $10^{-2}$.

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