Quantum Krylov subspace algorithms for ground and excited state energy estimation (2109.06868v3)

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.

Quantum Krylov Subspace Algorithms for Ground and Excited State Energy Estimation

This paper presents an insightful examination of quantum Krylov subspace algorithms (QKSA) designed for the efficient estimation of ground and excited state energies in quantum many-body systems. Departing from conventional quantum phase estimation (QPE), these algorithms offer substantial computational benefits in terms of gate depth and resource efficiency, making them particularly relevant for the noisy intermediate-scale quantum (NISQ) era.

The authors introduce several variants of quantum Krylov subspace diagonalization methods, broadly categorized by the selection of Hamiltonian functions and non-orthogonal bases, which results in distinct generalized eigenvalue problems. An essential contribution is a multi-fidelity protocol that exploits inherent symmetries in specific Hamiltonians, circumventing the traditionally required Hadamard test. This approach, combined with efficient fidelity estimation protocols, significantly reduces circuit depth, presenting fewer multi-qubit gate demands—a crucial factor given current hardware limitations.

Theoretical Contributions

Two primary innovations are highlighted: new hybrid quantum-classical algorithms that estimate energy levels, and a framework for exploiting Hamiltonian symmetries to simplify quantum circuit operations. Three novel algorithms are proposed—two hinge on filter diagonalization methods (FDMs) with Fourier transform-based energy filtering, while the third aligns with the time dynamic analyses akin to quantum phase estimation proposed by Klymko et al.

1. Krylov Subspace Diagonalization Method (KDM): This approach draws on real-time dynamics for basis state formation, leveraging both Hamiltonian and unitary evolution operators. It provides flexibility across various Hamiltonian models.
2. Filter Diagonalization Method (FDM): Takes advantage of non-orthogonal states influenced by specific energy filtering, improving convergence reliability for selected quantum systems.

Each method underlines different trade-offs concerning computational complexity, measurement requirements, and algorithmic stability. Notably, methods utilizing unitary time evolution demand fewer quantum operations, substantially enhancing computational expedience.

Numerical Findings

The efficacy of the proposed QKSA is validated through numerical simulations focusing on molecular systems (e.g., water and hydrogen chains), illustrating rapid convergence to chemical accuracy within a minimal number of time steps. The large frequency windows applied within FDM demonstrated superior convergence characteristics, achieving chemical accuracy stages comparably with KDM while ensuring lower computational overheads. This strategic choice addresses condition number imbalances, establishing FDM as potentially more resilient over extended operations—critical for capturing interior excited states and achieving detailed spectral resolution in complex quantum systems.

Implications and Future Outlook

Quantum Krylov subspace algorithms depict a promising route for addressing quantum eigenpair problems in contemporary quantum computing frameworks, particularly significant for applications in quantum chemistry and material science. These innovations underscore the potential to further optimize algorithmic designs, reducing the dependence on quantum ancillae and multi-qubit gates while sustaining high fidelity in numerical simulations.

The paper invites future exploration into the integration of QKSA with variational frameworks and potential algorithmic adaptations under realistic noise models. An in-depth analysis of Trotter errors and implementation fidelity under constrained NISQ hardware could further refine these techniques, aligning them with practical quantum computation demands and enabling broader deployment across diverse scientific domains.