Quantum Algorithm for Simulating Hamiltonian Dynamics with an Off-diagonal Series Expansion

Amir Kalev1 and Itay Hen2,3

1Information Sciences Institute, University of Southern California, Arlington, VA 22203, USA
2Information Sciences Institute, University of Southern California, Marina del Rey, CA 90292, USA
3Department of Physics and Astronomy, and Center for Quantum Information Science & Technology,University of Southern California, Los Angeles, California 90089, USA

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We propose an efficient quantum algorithm for simulating the dynamics of general Hamiltonian systems. Our technique is based on a power series expansion of the time-evolution operator in its off-diagonal terms. The expansion decouples the dynamics due to the diagonal component of the Hamiltonian from the dynamics generated by its off-diagonal part, which we encode using the linear combination of unitaries technique. Our method has an optimal dependence on the desired precision and, as we illustrate, generally requires considerably fewer resources than the current state-of-the-art. We provide an analysis of resource costs for several sample models.

Simulating the dynamics of quantum many-body systems is a central challenge in Physics, Chemistry and the Material Sciences as well as in other areas of science and technology. While for classical algorithms this task is in general intractable, quantum circuits offer a way around the classical bottlenecks by way of `circuitizing' the time evolution of the system in question. However, present-day quantum computing devices allow for the programming of only small and noisy quantum circuits, a state of matters that places severe constraints on the types of applications these devices may be used for in practice. The qubit and gate costs of circuitization procedures have therefore rightfully become key factors in determining the feasibility of any potential application and increasingly more efficient algorithms are continuously being devised. We propose a novel approach to resource-efficient Hamiltonian dynamics simulations on quantum circuits that we argue offers certain advantages, which directly translate to a shorter algorithm runtime, over state-of-the-art quantum simulation algorithms. We accomplish this by utilizing a series expansion of the quantum time-evolution operator in its off-diagonal elements wherein the operator is expanded around its diagonal component. This expansion allows one to effectively integrate out the diagonal component of the evolution, thereby reducing the overall gate and qubit complexities of the algorithm as compared to existing methods.

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Cited by

[1] Amir Kalev and Itay Hen, "An integral-free representation of the Dyson series using divided differences", arXiv:2010.09888.

[2] Yi-Hsiang Chen, Amir Kalev, and Itay Hen, "A quantum algorithm for time-dependent Hamiltonian simulation by permutation expansion", arXiv:2103.15334.

[3] Kyle Sherbert, Frank Cerasoli, and Marco Buongiorno Nardelli, "A systematic variational approach to band theory in a quantum computer", arXiv:2104.03409.

[4] Efekan Kökcü, Thomas Steckmann, J. K. Freericks, Eugene F. Dumitrescu, and Alexander F. Kemper, "Fixed Depth Hamiltonian Simulation via Cartan Decomposition", arXiv:2104.00728.

[5] Zohreh Davoudi, Norbert M. Linke, and Guido Pagano, "Toward simulating quantum field theories with controlled phonon-ion dynamics: A hybrid analog-digital approach", arXiv:2104.09346.

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