The Krylov subspace methods, being one category of the most important
classical numerical methods for linear algebra problems, their quantum
generalisation can be much more powerful. However, quantum Krylov subspace
algorithms are prone to errors due to inevitable statistical fluctuations in
quantum measurements. To address this problem, we develop a general theoretical
framework to analyse the statistical error and measurement cost. Based on the
framework, we propose a quantum algorithm to construct the Hamiltonian-power
Krylov subspace that can minimise the measurement cost. In our algorithm, the
product of power and Gaussian functions of the Hamiltonian is expressed as an
integral of the real-time evolution, such that it can be evaluated on a quantum
computer. We compare our algorithm with other established quantum Krylov
subspace algorithms in solving two prominent examples. It is shown that the
measurement number in our algorithm is typically 104 to 1012 times
smaller than other algorithms. Such an improvement can be attributed to the
reduced cost of composing projectors onto the ground state. These results show
that our algorithm is exceptionally robust to statistical fluctuations and
promising for practical applications.Comment: 18 pages, 5 figure