Does causal dynamics imply local interactions?

Zoltán Zimborás1,2,3, Terry Farrelly4,5, Szilárd Farkas1, and Lluis Masanes6,7

1Wigner Research Centre for Physics, H-1121, Budapest, Hungary
2MTA-BME Lendület Quantum Information Theory Research Group, Budapest, Hungary
3Mathematical Institute, Budapest University of Technology and Economics, H-1111, Budapest, Hungary
4Institut für Theoretische Physik, Leibniz Universität, Appelstraße 2, 30167 Hannover, Germany
5ARC Centre for Engineered Quantum Systems, School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia
6Computer Science Department, University College London, United Kingdom.
7London Centre for Nanotechnology, University College London, United Kingdom.

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We consider quantum systems with causal dynamics in discrete spacetimes, also known as quantum cellular automata (QCA). Due to time-discreteness this type of dynamics is not characterized by a Hamiltonian but by a one-time-step unitary. This can be written as the exponential of a Hamiltonian but in a highly non-unique way. We ask if any of the Hamiltonians generating a QCA unitary is local in some sense, and we obtain two very different answers. On one hand, we present an example of QCA for which all generating Hamiltonians are fully non-local, in the sense that interactions do not decay with the distance. We expect this result to have relevant consequences for the classification of topological phases in Floquet systems, given that this relies on the effective Hamiltonian.

On the other hand, we show that all one-dimensional quasi-free fermionic QCAs have quasi-local generating Hamiltonians, with interactions decaying exponentially in the massive case and algebraically in the critical case. We also prove that some integrable systems do not have local, quasi-local nor low-weight constants of motion; a result that challenges the standard definition of integrability.

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Cited by

[1] Terry Farrelly, "A review of Quantum Cellular Automata", arXiv:1904.13318.

[2] Daniel Ranard, Michael Walter, and Freek Witteveen, "A converse to Lieb-Robinson bounds in one dimension using index theory", arXiv:2012.00741.

[3] Henrik Wilming and Albert H. Werner, "Lieb-Robinson bounds imply locality of interactions", arXiv:2006.10062.

[4] Tom Farshi, Daniele Toniolo, Carlos E. González-Guillén, Álvaro M. Alhambra, and Lluis Masanes, "Mixing and localisation in random time-periodic quantum circuits of Clifford unitaries", arXiv:2007.03339.

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