Quantum Krylov subspace algorithms for ground and excited state energy estimation

TL;DR

This paper introduces symmetry-exploiting quantum Krylov subspace algorithms that reduce circuit complexity for estimating ground and excited states, offering practical advantages for near-term quantum hardware.

Contribution

It develops a unified theory of quantum Krylov algorithms and proposes three new methods that improve efficiency by leveraging Hamiltonian symmetries and multi-fidelity estimation.

Findings

Significant reduction in circuit depth using symmetry-based methods

Three new algorithms with different trade-offs in quantum resource requirements

Enhanced stability and efficiency in energy estimation processes

Abstract

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, $\langle \phi_i|e^{-i\hat{H}\tau}|\phi_j \rangle$, the associated quantum circuits require an ancilla qubit with controlled multi-qubit gates that can be quite costly for near-term quantum hardware. In this work, 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 multi-fidelity 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, we develop a unified theory of quantum Krylov subspace algorithms and present three new quantum-classical algorithms for the ground and excited-state energy estimation problems, where each new 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.