Quantum algorithms for grid-based variational time evolution

Pauline J Ollitrault1, Sven Jandura1, Alexander Miessen1, Irene Burghardt2, Rocco Martinazzo3,4, Francesco Tacchino1, and Ivano Tavernelli1

1IBM Quantum, IBM Research – Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
2Institute of Physical and Theoretical Chemistry, Goethe University Frankfurt, Max-von-Laue-Str. 7, D-60438 Frankfurt/Main, Germany
3Department of Chemistry, Università degli Studi di Milano, Via Golgi 19, 20133 Milan, Italy
4Istituto di Scienze e Tecnologie Chimiche “Giulio Natta”, CNR, Via Golgi 19, 20133 Milan, Italy

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The simulation of quantum dynamics calls for quantum algorithms working in first quantized grid encodings. Here, we propose a variational quantum algorithm for performing quantum dynamics in first quantization. In addition to the usual reduction in circuit depth conferred by variational approaches, this algorithm also enjoys several advantages compared to previously proposed ones. For instance, variational approaches suffer from the need for a large number of measurements. However, the grid encoding of first quantized Hamiltonians only requires measuring in position and momentum bases, irrespective of the system size. Their combination with variational approaches is therefore particularly attractive. Moreover, heuristic variational forms can be employed to overcome the limitation of the hard decomposition of Trotterized first quantized Hamiltonians into quantum gates. We apply this quantum algorithm to the dynamics of several systems in one and two dimensions. Our simulations exhibit the previously observed numerical instabilities of variational time propagation approaches. We show how they can be significantly attenuated through subspace diagonalization at a cost of an additional $\mathcal{O}(MN^2)$ 2-qubit gates where $M$ is the number of dimensions and $N^M$ is the total number of grid points.

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[1] Daniel Bultrini and Oriol Vendrell, "Mixed quantum-classical dynamics for near term quantum computers", Communications Physics 6 1, 328 (2023).

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[4] Mostafizur Rahaman Laskar, Kalyan Dasgupta, and Atanu Bhattacharya, "A Proposed Quantum Hamiltonian Encoding Framework for Time Evolution Operator Design of Potential Energy Function", arXiv:2308.06491, (2023).

[5] Hans Hon Sang Chan, Richard Meister, Tyson Jones, David P. Tew, and Simon C. Benjamin, "Grid-based methods for chemistry simulations on a quantum computer", Science Advances 9 9, eabo7484 (2023).

[6] Daniel J. Egger, Chiara Capecci, Bibek Pokharel, Panagiotis Kl. Barkoutsos, Laurin E. Fischer, Leonardo Guidoni, and Ivano Tavernelli, "Pulse variational quantum eigensolver on cross-resonance-based hardware", Physical Review Research 5 3, 033159 (2023).

[7] Alistair Letcher, Stefan Woerner, and Christa Zoufal, "From Tight Gradient Bounds for Parameterized Quantum Circuits to the Absence of Barren Plateaus in QGANs", arXiv:2309.12681, (2023).

[8] Anton Nykänen, Aaron Miller, Walter Talarico, Stefan Knecht, Arseny Kovyrshin, Mårten Skogh, Lars Tornberg, Anders Broo, Stefano Mensa, Benjamin C. B. Symons, Emre Sahin, Jason Crain, Ivano Tavernelli, and Fabijan Pavošević, "Toward Accurate Post-Born-Oppenheimer Molecular Simulations on Quantum Computers: An Adaptive Variational Eigensolver with Nuclear-Electronic Frozen Natural Orbitals", arXiv:2310.01302, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2023-12-07 07:05:29) and SAO/NASA ADS (last updated successfully 2023-12-07 07:05:30). The list may be incomplete as not all publishers provide suitable and complete citation data.