  The Krylov subspace methods, being one category of the most important
classical numerical methods for linear algebra problems, can be much more
powerful when generalised to quantum computing. 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. To achieve an error comparable to that of the classical Lanczos
algorithm at the same subspace dimension, our algorithm typically requires
orders of magnitude fewer measurements than others. 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.
