  The task of estimating the ground state of Hamiltonians is an important
problem in physics with numerous applications ranging from solid-state physics
to combinatorial optimization. We provide a hybrid quantum-classical algorithm
for approximating the ground state of a Hamiltonian that builds on the powerful
Krylov subspace method in a way that is suitable for current quantum computers.
Our algorithm systematically constructs the Ansatz using any given choice of
the initial state and the unitaries describing the Hamiltonian. The only task
of the quantum computer is to measure overlaps and no feedback loops are
required. The measurements can be performed efficiently on current quantum
hardware without requiring any complicated measurements such as the Hadamard
test. Finally, a classical computer solves a well characterized quadratically
constrained optimization program. Our algorithm can reuse previous measurements
to calculate the ground state of a wide range of Hamiltonians without requiring
additional quantum resources. Further, we demonstrate our algorithm for solving
problems consisting of thousands of qubits. The algorithm works for almost
every random choice of the initial state and circumvents the barren plateau
problem.
