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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