Subspace-search variational quantum eigensolver for excited states

Abstract

The variational quantum eigensolver (VQE), a variational algorithm to obtain an approximated ground state of a given Hamiltonian, is an appealing application of near-term quantum computers. The original work [A. Peruzzo et al.; \textit{Nat. Commun.}; \textbf{5}, 4213 (2014)] focused only on finding a ground state, whereas the excited states can also induce interesting phenomena in molecules and materials. Calculating excited states is, in general, a more difficult task than finding ground states for classical computers. To extend the framework to excited states, we here propose an algorithm, the subspace-search variational quantum eigensolver (SSVQE). This algorithm searches a low energy subspace by supplying orthogonal input states to the variational ansatz and relies on the unitarity of transformations to ensure the orthogonality of output states. The $k$-th excited state is obtained as the highest energy state in the low energy subspace. The proposed algorithm consists only of two parameter optimization procedures and does not employ any ancilla qubits. The disuse of the ancilla qubits is a great improvement from the existing proposals for excited states, which have utilized the swap test, making our proposal a truly near-term quantum algorithm. We further generalize the SSVQE to obtain all excited states up to the $k$-th by only a single optimization procedure. From numerical simulations, we verify the proposed algorithms. This work greatly extends the applicable domain of the VQE to excited states and their related properties like a transition amplitude without sacrificing any feasibility of it.

arXiv comment7 pages, 11 figures

References in corpus (10)

• Quantum Computing in the NISQ era and beyond
• A variational eigenvalue solver on a quantum processor
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• Characterizing Quantum Supremacy in Near-Term Devices
• Elucidating Reaction Mechanisms on Quantum Computers
• Variational Quantum Computation of Excited States
• Robust determination of molecular spectra on a quantum processor
• Variational quantum algorithms for discovering Hamiltonian spectra
• Hybrid quantum-classical approach to correlated materials
• Witnessing eigenstates for quantum simulation of Hamiltonian spectra

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