Hamiltonian-reconstruction distance as a success metric for the Variational Quantum Eigensolver

• URL: http://arxiv.org/abs/2403.11995v1
• Date: Mon, 18 Mar 2024 17:28:06 GMT
• Title: Hamiltonian-reconstruction distance as a success metric for the Variational Quantum Eigensolver
• Authors: Leo Joon Il Moon, Mandar M. Sohoni, Michael A. Shimizu, Praveen Viswanathan, Kevin Zhang, Eun-Ah Kim, Peter L. McMahon,
• Abstract summary: Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm for quantum simulation that can run on near-term quantum hardware. A challenge in VQE is to know how close the algorithm's output solution is to the true ground state, when the true ground state and ground-state energy are unknown. Recent developments in Hamiltonian reconstruction give a metric can be used to assess the quality of a variational solution to a Hamiltonian-eigensolving problem.
• Score: 1.0916270449935084
• License: http://creativecommons.org/licenses/by/4.0/
• 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 the true ground state and ground-state energy are unknown. This is especially important in iterative algorithms, such as VQE, where one wants to avoid erroneous early termination. Recent developments in Hamiltonian reconstruction -- the inference of a Hamiltonian given an eigenstate -- give a metric can be used to assess the quality of a variational solution to a Hamiltonian-eigensolving problem. This metric can assess the proximity of the variational solution to the ground state without any knowledge of the true ground state or ground-state energy. 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 Hamiltonian-reconstruction distance gives a helpful indication of whether VQE has yet found the ground state or not. Our experiments 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 Hamiltonian-reconstruction distance correctly suggests to continue iterating. We find that the Hamiltonian-reconstruction distance has a useful correlation with the fidelity between the VQE solution and the true ground state. Our work suggests that the Hamiltonian-reconstruction distance may be a useful tool for assessing success in VQE, including on noisy quantum processors in practice.

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