  The variational principle of quantum mechanics is the backbone of hybrid
quantum computing for a range of applications. However, as the problem size
grows, quantum logic errors and the effect of barren plateaus overwhelm the
quality of the results. There is now a clear focus on strategies that require
fewer quantum circuit steps and are robust to device errors. Here we present an
approach in which problem complexity is transferred to dynamic quantities
computed on the quantum processor - Hamiltonian moments, $\langle H^n\rangle$.
From these quantum computed moments, estimates of the ground-state energy are
obtained using the "infinum" theorem from Lanczos cumulant expansions which
manifestly correct the associated variational calculation. With system dynamics
encoded in the moments the burden on the trial-state quantum circuit depth is
eased. The method is introduced and demonstrated on 2D quantum magnetism models
on lattices up to 5 $\times$ 5 (25 qubits) implemented on IBM Quantum
superconducting qubit devices. Moments were quantum computed to fourth order
with respect to a parameterised antiferromagnetic trial-state. A comprehensive
comparison with benchmark variational calculations was performed, including
over an ensemble of random coupling instances. The results showed that the
infinum estimate consistently outperformed the benchmark variational approach
for the same trial-state. These initial investigations suggest that the quantum
computed moments approach has a high degree of stability against trial-state
variation, quantum gate errors and shot noise, all of which bodes well for
further investigation and applications of the approach.
