  In the lead up to fault tolerance, the utility of quantum computing will be
determined by how adequately the effects of noise can be circumvented in
quantum algorithms. Hybrid quantum-classical algorithms such as the variational
quantum eigensolver (VQE) have been designed for the short-term regime.
However, as problems scale, VQE results are generally scrambled by noise on
present-day hardware. While error mitigation techniques alleviate these issues
to some extent, there is a pressing need to develop algorithmic approaches with
higher robustness to noise. Here, we explore the robustness properties of the
recently introduced quantum computed moments (QCM) approach to ground state
energy problems, and show through an analytic example how the underlying energy
estimate explicitly filters out incoherent noise. Motivated by this
observation, we implement QCM for a model of quantum magnetism on IBM Quantum
hardware to examine the noise-filtering effect with increasing circuit depth.
We find that QCM maintains a remarkably high degree of error robustness where
VQE completely fails. On instances of the quantum magnetism model up to 20
qubits for ultra-deep trial state circuits of up to ~500 CNOTs, QCM is still
able to extract reasonable energy estimates. The observation is bolstered by an
extensive set of experimental results. To match these results, VQE would need
hardware improvement by some 2 orders of magnitude on error rates.
