Hamiltonian-reconstruction distance as a success metric for the variational quantum eigensolver

Abstract

The variational quantum eigensolver (VQE) is a hybrid quantum–classical algorithm for quantum simulation that can be run on near-term quantum hardware. A challenge in VQE—as well as any other heuristic algorithm for finding ground states of Hamiltonians—is to know how close the algorithm’s output solution is to the true ground state, when we do not know what the true ground state or ground-state energy is. This is especially important in iterative algorithms such as VQE, where we need to decide when to stop iterating. Stopping when the current solution energy no longer changes rapidly from one iteration to the next is a common choice but in many situations can lead to stopping too early and outputting an incorrect result. Recent developments in Hamiltonian reconstruction—the inference of a Hamiltonian given an eigenstate—give a tool that can be used to assess the quality of a variational solution to a Hamiltonian-eigensolving problem: a metric quantifying the distance between the problem Hamiltonian and the reconstructed Hamiltonian can give an indication of whether the variational solution is an eigenstate, which can (as a special case) indicate if the variational solution is not the ground state. Crucially, computing the Hamiltonian-reconstruction (HR) distance does not rely on knowing the true ground state or ground-state energy. We propose and study using the HR distance as a metric for assessing the success of VQE eigensolving. In numerical simulations and in demonstrations on a cloud-based trapped-ion quantum computer, we show that for examples of both one-dimensional transverse-field-Ising (11 qubits) and two-dimensional J1–J2 transverse-field-Ising (6 qubits) spin problems, the HR distance gives a helpful indication of whether VQE has yet found the ground state or not. Our demonstrations included cases where the energy plateaus as a function of the VQE iteration, which could have resulted in erroneous early stopping of the VQE algorithm, but where the HR distance correctly suggests to continue iterating. We find that HR distance has a useful correlation with the fidelity (i.e. state overlap) between the VQE solution and the true ground state, and with the energy difference between VQE solution’s energy and the true ground-state energy. Our work suggests that HR distance may be a useful tool for assessing success in VQE, including on noisy quantum processors in practice.