  Quantum natural gradient has emerged as a superior minimisation technique in
quantum variational algorithms. Classically simulating the algorithm running on
near-future quantum hardware is paramount in its study, as it is for all
variational algorithms. In this case, state-vector simulation of the
P-parameter/gate ansatz circuit does not dominate the runtime; instead,
calculation of the Fisher information matrix becomes the bottleneck, involving
O(P^3) gate evaluations, though this can be reduced to O(P^2) gates by using
O(P) temporary state-vectors. This is similar to the gradient calculation
subroutine dominating the simulation of quantum gradient descent, which has
attracted HPC strategies and bespoke simulation algorithms with asymptotic
speedups. We here present a novel simulation strategy to precisely calculate
the quantum natural gradient in O(P^2) gates and O(1) state-vectors. While more
complicated, our strategy is in the same spirit as that presented for gradients
in Reference 6, and involves iteratively evaluating recurrent forms of the
Fisher information matrix. Our strategy uses only "apply gate", "clone state"
and "inner product" operations which are present in practically all quantum
computing simulators. It is furthermore compatible with parallelisation
schemes, like hardware acceleration and distribution. Since our scheme
leverages a form of the Fisher information matrix for strictly unitary ansatz
circuits, it cannot be simply extended to density matrix simulation of quantum
natural gradient with non-unitary circuits.
