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

In this talk, I will present unified framework for quantum Krylov subspace methods which aim to solve the ground and excited state energy estimation problem using near-term quantum computing hardware. I will show that a wide class of Hamiltonians relevant to nuclear physics, condensed matter physics, and chemistry contain symmetries that can be exploited to avoid the use of the Hadamard test. The Hadamard test, which is a requisite for a wide variety of quantum Krylov algorithms, uses an ancilla qubit with controlled multi-qubit gates that can be quite costly for near-term hardware. I will introduce a multi-fidelity estimation protocol that replaces the Hadamard test and, when combined with efficient single-fidelity estimation protocols, provides a substantial reduction in circuit depth. I will also introduce several new quantum Krylov algorithms that provide various advantages and disadvantages in terms of the number of calls to the quantum computer, gate depth, and classical complexity. To test the efficacy of the proposed algorithms, I will present numerical experiments of the proposed algorithms for the problem of finding the ground and excited-state energies of various quantum chemistry Hamiltonians, highlighting their fast convergence with a small number of iterations.