Mirror subspace diagonalization: A quantum Krylov algorithm with near-optimal sampling cost

TL;DR

This paper introduces mirror subspace diagonalization (MSD), a quantum Krylov algorithm that significantly reduces sampling costs for ground-state energy estimation by optimizing finite-difference formulas and leveraging classical post-processing.

Contribution

MSD approaches the theoretical lower bound of sampling cost in quantum Krylov algorithms, offering a practical method with substantial reductions in measurement overhead for large-scale electronic structure problems.

Findings

MSD achieves up to 10,000 times reduction in sampling cost.

The method is particularly effective when the Hamiltonian's spectral norm is small.

Numerical results demonstrate significant efficiency improvements over traditional algorithms.

Abstract

Quantum Krylov algorithms have emerged as a promising approach for ground-state energy estimation in the near-term quantum computing era. A major challenge, however, lies in their inherently substantial sampling cost, primarily due to the individual measurement of each term in the Hamiltonian. While various techniques have been proposed to mitigate this issue, the sampling overhead remains a significant bottleneck, especially for practical large-scale electronic structure problems. In this work, we introduce an alternative method, dubbed mirror subspace diagonalization (MSD), which approaches the theoretical lower bound of the sampling cost for quantum Krylov algorithms. MSD leverages a finite-difference formula to express the Hamiltonian operator as a linear combination of time-evolution unitaries with symmetrically shifted timesteps, enabling efficient estimation of the Hamiltonian matrix within the Krylov subspace. In this scheme, the finite difference and statistical errors are simultaneously minimized by optimizing the timestep parameter and shifting the energy spectrum. Consequently, MSD attains the lower bound of the sampling cost of the quantum Krylov algorithms up to a logarithmic factor. Furthermore, we employ classical post-processing to infer Hamiltonian moments, which are used to mitigate the ground state energy error based on the Lanczos scheme. Through theoretical analysis of the sampling cost, we demonstrate that MSD is particularly effective when the spectral norm of the Hamiltonian is significantly smaller than its 1-norm. Such a situation arises, for example, in high-accuracy simulations of molecules using large basis sets that incorporate strong electronic correlations. Numerical results for various molecular models reveal that MSD can achieve sampling cost reductions ranging from approximately 10 to 10,000 times compared to the conventional quantum Krylov algorithm.