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

Noise-robust ground state energy estimates from deep quantum circuits