  Exploiting near-term quantum computers and achieving practical value is a
considerable and exciting challenge. Most prominent candidates as variational
algorithms typically aim to find the ground state of a Hamiltonian by
minimising a single classical (energy) surface which is sampled from by a
quantum computer. Here we introduce a method we call CoVaR, an alternative
means to exploit the power of variational circuits: We find eigenstates by
finding joint roots of a polynomially growing number of properties of the
quantum state as covariance functions between the Hamiltonian and an operator
pool of our choice. The most remarkable feature of our CoVaR approach is that
it allows us to fully exploit the extremely powerful classical shadow
techniques, i.e., we simultaneously estimate a very large number $>10^4-10^7$
of covariances. We randomly select covariances and estimate analytical
derivatives at each iteration applying a stochastic Levenberg-Marquardt step
via a large but tractable linear system of equations that we solve with a
classical computer. We prove that the cost in quantum resources per iteration
is comparable to a standard gradient estimation, however, we observe in
numerical simulations a very significant improvement by many orders of
magnitude in convergence speed. CoVaR is directly analogous to stochastic
gradient-based optimisations of paramount importance to classical machine
learning while we also offload significant but tractable work onto the
classical processor.

Quantum Physics