Quantum Krylov subspace algorithms for ground- and excited-state energy estimation

Journal Article · 13 February 2022 · Physical Review A

DOI: https://doi.org/10.1103/physreva.105.022417 · OSTI ID:1962060

Cortes, Cristian L.  ^[1]; Gray, Stephen K.  ^[1]

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1. Argonne National Lab. (ANL), Lemont, IL (United States)

Quantum Krylov subspace diagonalization (QKSD) algorithms provide a low-cost alternative to the conventional quantum phase estimation algorithm for estimating the ground- and excited-state energies of a quantum many-body system. While QKSD algorithms typically rely on using the Hadamard test for estimating Krylov subspace matrix elements of the form $$\langle \phi_i|e^{-\widehat{H}τ}|\phi_j\rangle$$, the associated quantum circuits require an ancilla qubit with controlled multiqubit gates that can be quite costly for near-term quantum hardware. In this paper, we show that a wide class of Hamiltonians relevant to condensed-matter physics and quantum chemistry contain symmetries that can be exploited to avoid the use of the Hadamard test. We propose a multifidelity estimation protocol that can be used to compute such quantities, showing that our approach, when combined with efficient single-fidelity estimation protocols, provides a substantial reduction in circuit depth. In addition, here we develop a unified theory of quantum Krylov subspace algorithms and present three quantum-classical algorithms for the ground- and excited-state energy estimation problems, where each algorithm provides various advantages and disadvantages in terms of total number of calls to the quantum computer, gate depth, classical complexity, and stability of the generalized eigenvalue problem within the Krylov subspace.

• Research Organization: Argonne National Laboratory (ANL), Lemont, IL (United States)
• Sponsoring Organization: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE Office of Science (SC), Basic Energy Sciences (BES)
• Grant/Contract Number: AC02-06CH11357
• OSTI ID: 1962060
• Journal Information: Physical Review A, Journal Name: Physical Review A Journal Issue: 2 Vol. 105; ISSN 2469-9926
• Publisher: American Physical Society (APS)Copyright Statement
• Country of Publication: United States
• Language: English