Sampling Error Analysis in Quantum Krylov Subspace Diagonalization

Abstract

Quantum Krylov subspace diagonalization (QKSD) is an emerging method used in place of quantum phase estimation in the early fault-tolerant era, where limited quantum circuit depth is available. In contrast to the classical Krylov subspace diagonalization (KSD) or the Lanczos method, QKSD exploits the quantum computer to efficiently estimate the eigenvalues of large-size Hamiltonians through a faster Krylov projection. However, unlike classical KSD, which is solely concerned with machine precision, QKSD is inherently accompanied by errors originating from a finite number of samples. Moreover, due to difficulty establishing an artificial orthogonal basis, ill-conditioning problems are often encountered, rendering the solution vulnerable to noise. In this work, we present a nonasymptotic theoretical framework to assess the relationship between sampling noise and its effects on eigenvalues. We also propose an optimal solution to cope with large condition numbers by eliminating the ill-conditioned bases. Numerical simulations of the one-dimensional Hubbard model demonstrate that the error bound of finite samplings accurately predicts the experimental errors in well-conditioned regions.

24 pages, 8 figures

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• Molecular Properties from Quantum Krylov Subspace Diagonalization
• Partitioned Quantum Subspace Expansion
• Approximating dynamical correlation functions with constant depth quantum circuits
• Adiabatic state preparation from general initial states
• An Error Mitigated Non-Orthogonal Quantum Eigensolver via Shadow Tomography