Noise-robust ground state energy estimates from deep quantum circuits

Harish J. Vallury1, Michael A. Jones1, Gregory A. L. White1, Floyd M. Creevey1, Charles D. Hill1,2, and Lloyd C. L. Hollenberg1

1School of Physics, University of Melbourne, Parkville, VIC 3010, Australia
2School of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia

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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.

Noise is the greatest challenge in present-day quantum computing. As the circuit depth increases for real-world world problems, the cumulative error in the quantum computation quickly overwhelms the results. Error correction and mitigation strategies exist but are either resource intensive or not powerful enough to compensate for such high levels of disruption — the question is, are there quantum algorithms which are inherently robust to noise which even the playing field? Variational quantum algorithms are a common approach to problems in chemistry and condensed matter physics, and involve preparing and measuring the energy of a trial state on a quantum computer. While noise typically disrupts this result, we have developed a technique whereby measuring additional higher weight observables (Hamiltonian moments) one can correct for noise induced imperfections in the trial state prepared on the quantum computer. In this work, we analyse the noise robustness of our method via a theoretical model, noisy simulations and ultimately through the implementation of deep quantum circuits on real hardware (upwards of 500 total CNOT gates). From the experimental results, we are able to determine the ground state energies of an ensemble of problems in quantum magnetism to a degree which, to be matched by conventional variational methods, would require some two orders of magnitude reduction in the device error rates.
Our results show that the remarkable filtering effect of the moments-based technique appears to circumvent the effects of noise at the core of present day quantum computing, and point the way to potentially achieving practical quantum advantage on hardware in the near-term.

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[3] Harish J. Vallury and Lloyd C. L. Hollenberg, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 295 (2023) ISBN:979-8-3503-4323-6.

[4] Floyd M. Creevey, Charles D. Hill, and Lloyd C. L. Hollenberg, "GASP: a genetic algorithm for state preparation on quantum computers", Scientific Reports 13 1, 11956 (2023).

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