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

Quantum Krylov algorithms offer promising ground-state energy estimation in the near-term quantum computing era, yet face substantial sampling costs. This work introduces mirror subspace diagonalization (MSD), which approaches the theoretical lower bound of the sampling cost for quantum Krylov methods. MSD leverages a finite-difference formula to express the Hamiltonian as a linear combination of time-evolution unitaries, allowing efficient Hamiltonian matrix estimation within the Krylov subspace. By optimizing timestep parameters and shifting the energy spectrum, MSD minimizes both finite difference and statistical errors. 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. Our analysis shows that MSD is particularly effective when the Hamiltonian's spectral norm is much smaller than its 1-norm, often occurring in high-accuracy molecular simulations with large basis sets. Numerical results for various molecular models demonstrate MSD's sampling cost reductions, ranging from 10 to 10,000 times lower than conventional quantum Krylov algorithms. These findings pave the way for more accurate and resource-efficient quantum simulations in chemical physics.