A Multireference Quantum Krylov Algorithm for Strongly Correlated Electrons

Abstract: We introduce a multireference selected quantum Krylov (MRSQK) algorithm suitable for quantum simulation of many-body problems. MRSQK is a low-cost alternative to the quantum phase estimation algorithm that generates a target state as a linear combination of non-orthogonal Krylov basis states. This basis is constructed from a set of reference states via real-time evolution avoiding the numerical optimization of parameters. An efficient algorithm for the evaluation of the off-diagonal matrix elements of the overlap and Hamiltonian matrices is discussed and a selection procedure is introduced to identify a basis of orthogonal references that ameliorates the linear dependency problem. Preliminary benchmarks on linear H$_6$, H$_8$, and BeH$_2$ indicate that MRSQK can predict the energy of these systems accurately using very compact Krylov bases.

Overview of the Multireference Quantum Krylov Algorithm for Strongly Correlated Electrons

The paper introduces a multireference selected quantum Krylov (MRSQK) algorithm, a novel approach designed to simulate strongly correlated electron systems on quantum computers. Specifically, it addresses the electron many-body problem by constructing a Krylov subspace via real-time evolution from a set of orthogonal reference states, thus generating a target quantum state as a non-orthogonal linear combination of these Krylov basis states. This approach offers a low-cost alternative to the well-established quantum phase estimation (QPE) algorithm, making it suitable for near-term quantum devices with limited coherence times and gate fidelities.

Main Contributions and Methodology

The paper delineates several key contributions:

1. Algorithm Design: The MRSQK algorithm is crafted to efficiently evaluate off-diagonal matrix elements of overlap and Hamiltonian matrices, integral to quantum subspace diagonalization (QSD) techniques. The algorithm circumvents the need for parameter optimization, which is a significant limitation in methods such as the variational quantum eigensolver (VQE).
2. Multireference Approach: By employing multiple orthogonal reference states, the algorithm mitigates linear dependency issues that plague other quantum Krylov and subspace diagonalization methods.
3. Real-Time Evolution: Utilization of real-time dynamics ensures that the generated basis spans the classical Krylov space, capturing essential many-body correlations. The authors discuss using a Trotter-Suzuki approximation for implementing these dynamics, balancing the trade-off between quantum circuit depth and accuracy.
4. Reference Selection: References are derived from dominant determinants in a trial wave function constructed via the quantum Krylov method, enhanced by quantum measurement techniques.

Numerical Benchmarks and Results

The paper presents numerical benchmarks on small strongly correlated systems, including linear chains like \ce{H6}, \ce{H8}, and \ce{BeH2}. These serve as prototypical models for testing the algorithm's efficacy in simulating correlated electron environments. The principal findings are as follows:

• Energy Accuracy: MRSQK unerringly approximates ground-state energies with a notably compact representation, using significantly fewer states than the full configuration interaction (FCI) space. Such efficiency is pivotal for practical applications on quantum processors.
• Comparative Stability: In contrast to single-reference Krylov methods, MRSQK demonstrates numerical stability, as evidenced by lower condition numbers for the overlap matrix, enhancing the reliability of the results.
• Trotter Approximation: The errors introduced by approximating real-time evolution via a Trotter decomposition are quantified, showing that moderate Trotter numbers (e.g., 4 or 8) suffice for chemical accuracy, potentially easing the hardware requirements.

Theoretical and Practical Implications

Theoretically, the MRSQK method offers insights into constructing efficient quantum representations for correlated electron systems, potentially impacting quantum chemistry and materials science. Practically, it highlights a path forward for utilizing quantum computers capable of reducing calculations' complexity and cost via hybrid quantum-classical strategies. In this context, MRSQK could be a compelling alternative to deeply exploring electron correlation in systems where classical approaches like CI and coupled cluster are computationally infeasible.

Future Directions

Future research could focus on enhancing the robustness of real-time evolution implementations, perhaps exploring other approximations or adaptive time-stepping methodologies. Another promising avenue is refining the selection mechanisms, which could stabilize potential energy surfaces and optimize computational resources. Moreover, quantifying MRSQK's scalability to larger, more complex systems on actual quantum devices would provide critical insights into its practical applicability.

In conclusion, the multireference quantum Krylov algorithm represents a meaningful stride in adapting quantum computing capabilities to tackle significant challenges in quantum chemistry, providing a balanced trade-off between complexity and accuracy suitable for near-term quantum technologies.