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Developing Hybrid Quantum Monte Carlo Algorithms for Low Quantum Overheads and Improved Noise Resilience

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Quantum chemistry problems have recently become particularly interesting targets for quantum computing algorithms due to their exponentially scaling Hilbert space which can be efficiently mapped onto a linear number of qubits, promising significant memory improvements in quantum over classical algorithms. However, in the era of noisy intermediate-scale quantum (NISQ) devices and hybrid algorithms, the size of chemical systems that can be treated is still severely limited, with hardware noise and qubit decoherence precluding useful computation even for modest numbers of qubits. Aditionally, most current quantum algorithms aim to use quantum devices to measure physical quantities of interest effectively exactly. Lowering noise to acceptable levels therefore requires non-trivial repetitions of the preparation and measurement procedure, which is both time- and resource-consuming. Quantum Monte Carlo (QMC) algorithms[1,2] have proven effective at lowering the computa- tional overhead of challenging problems, both in classical3 and quantum settings.[4] In this presentation, we combine ideas from conventional QMC algorithms with quantum computation to devise less cumbersome hybrid quantum algorithms. First, we show that, by using a quantum processor to compute projective Monte Carlo (PMC) residuals, one avoids the issue of having to importance sample the wavefunction contributions to the residuals, which is one of the bottlenecks of many QMC algorithms. Secondly, the resulting Monte Carlo estimate of the wavefunction and its properties is resilient to noisy measurement of the residuals so very few shots are necessary for the algorithm to succeed. Going further, we use stochastic representations of the wavefunction and the Hamiltonian to further reduce quantum overhead. While truncating the wavefunction parametrisation or the Hamiltonian would introduce a systematic error in a deterministic approach, this is not the case for QMC , in which we find that biases average out over the course of the calculation. Finally, we explore the expansion of these methods to excited electronic states. [1] G. H. Booth, A. J. W. Thom, A. Alavi, Journal of Chemical Physics 131, 054106 (2009). [2] M.-A. Filip, A. J. W. Thom, Journal of Chemical Physics 153, 214106 (2020) [3] J. E. Deustua, J. Shen, P. Piecuch, Journal of Chemical Physics 154, 124103 (2021) [4] M.-A. Filip, N. Fitzpatrick, D. Muñoz-Ramo, A. J. W. Thom, Physical Review Research 4, 023243 (2022)

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