  We present an iterative generalisation of the quantum subspace expansion
algorithm used with a Krylov basis. The iterative construction connects a
sequence of subspaces via their lowest energy states. Diagonalising a
Hamiltonian in a given Krylov subspace requires the same quantum resources in
both the single step and sequential cases. We propose a variance-based
criterion for determining a good iterative sequence and provide numerical
evidence that these good sequences display improved numerical stability over a
single step in the presence of finite sampling noise. Implementing the
generalisation requires additional classical processing with a polynomial
overhead in the subspace dimension. By exchanging quantum circuit depth for
additional measurements the quantum subspace expansion algorithm appears to be
an approach suited to near term or early error-corrected quantum hardware. Our
work suggests that the numerical instability limiting the accuracy of this
approach can be substantially alleviated beyond the current state of the art.
