Directed Percolation with Non-Unitary Quantum Cellular Automata

In classical physics and computer science, cellular automata (CA) provide a powerful framework for investigating the emergence of large-scale complex structures from local dynamical rules [1]. Similarly, complex dynamics of quantum many body systems have been studied using Quantum Cellular Automata (QCA) [2], in which the cellular update rules are implemented by localizable unitary maps.

In this work, we develop a non-unitary QCA model, where the unitary rules are replaced by local completely-positive (CP) trace-preserving maps. This generalization of QCA enables exploration of non-equilibrium dynamics of quantum many-body systems with dissipation and decoherence. We construct a non-unitary QCA that in the fully dissipative limit reduces to Domany-Kinzel Cellular Automaton (DKCA) [3] - a classic model of directed percolation in one dimension. We simulate the dynamics of the model using infinite Time Evolving Block Decimation (iTEBD) for mixed states [4] verifying that the evolution agrees with the classical probabilistic DK dynamics. We then add small unitary rotations on the cells conditional on the state of their neighbors, exploring the effect of quantum corrections to the phase diagram of the DKCA. The presence of unitary rotations generates quantum entanglement in the steady state of the model, which we quantify using N-concurrance and negativity.

In addition to providing a novel framework for exploring fundamental non-equilibrium phenomena in quantum many body systems, non-unitary QCA models can be used
for developing new applications for quantum devices operating in the presence of environmental noise. Non-unitary QCA is physically realizable in atomic physics platforms with regular lattice structures, such trapped ions, superconducting qubits and arrays of Rydberg ensembles [5]. The entanglement generated in the steady states of non-unitary QCAs can be exploited in quantum information processing and metrology applications.

ACKNOWLEDGEMENTS
This research was funded in part by the Australian Research Council Centre of Excellence for Engineered Quantum Systems (Project number CE170100009).

REFERENCES
[1] S. Wolfram. Statistical mechanics of cellular automata. Rev. Mod. Phys., 55,601-644, (1983).
[2] P. Arrighi. An overview of quantum cellular automata. Natural Computing 18, 885 (2019).
[3] E. Domany and W. Kinzel. Equivalence of Cellular Automata to Ising models and Directed Percolation. Phys. Rev. Lett., 53, 311–314, (1984).
[4] R. Orús and G. Vidal. Infinite time-evolving block decimation algorithm beyond unitary evolution. Phys. Rev. B 78, 155117 (2008)
[5] T. M. Wintermantel, Y. Wang, G. Lochead, S. Shevate, G. K. Brennen, and S. Whitlock. Unitary and nonunitary quantum cellular automata with Rydberg arrays. Phys. Rev. Lett., 124, 070503, (2020).

Συνεδρία: 
Authors: 
Ramil Nigmatullin and Gavin Brennen
Room: 
4
Date: 
Thursday, December 10, 2020 - 13:35 to 13:50

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