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International Journal of Theoretical Physics

Creating Very True Quantum Algorithms for Quantum Energy Based Computing

Koji Nagata, Tadao Nakamura, Han J.F. Geurdes, Josep Amengual i Batle, Soliman Abdalla, Ahmed Farouk, Do Ngoc Diep

Published Dec 11, 2017• 42 citations
Publisher
DOI
10.1007/s10773-017-3630-1

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Moore's law states that computing performance doubles every 18 months. While this has held true for 40 years, it is widely believed that this will soon come to an end. Quantum computation offers a potential solution to the eventual failure of Moore's law. Researchers have shown that efficient quantum algorithms exist and can perform some calculations significantly faster than classical computers. Quantum computers require very different algorithms than classical computers, so the challenge of quantum computation is to develop efficient quantum algorithms. Cybernet is working with the Air Force Research Laboratory (AFRL) to create image processing algorithms for quantum computers. We have shown that existing quantum algorithms (such as Grover's algorithm) are applicable to image processing tasks. We are continuing to identify other areas of image processing which can be improved through the application of quantum computing.
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 J 1 – J 2 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.