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Quantum random walks: Quantum to classical phase transitions

Symeonakis Aristotelis

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Summary

Quantum walks (QW) are systems consisting of parts identified as the quantum coin, the quantum walker and procedures simulating coin tossing and spreading along a space. They constitute quantum versions of the proverbial classical random walk (CRW) which now is reformulated to admit its quantization to a QW. Quantization rules for the CRW to QW passage, the novel effects of enhanced mobility of a QW, its demonstrated computational universality, as well as a wealth of scientific-technological applications and physical implementation scenarios constitute a vibrant subfield of quantum information science and technology. This work deals with an additional problem: the controlled and designed quantum-to-classical (Q – C) transition in random walks. Utilizing recent progress in that problem, employs classical randomness imposed on the quantum coin subsystem of a QW and studies the induced onset of Q – C transition. Guided by a crossover condition, (involving the number of evolution steps-time and the strength of imposed randomness), governing the existence of Q – C transition, two QW models are studied. These are models of QWs on the lattices of integer and natural numbers. Distributions of lattice cite occupation probabilities as well as standard deviation parameter vs. number of steps and/or strength of the imposed randomness are systematically investigated as quantitative measures of the Q – C transition. The transition is manifested as a passage from the quadratically enhanced diffusion rate (ballistic regime) to a decelerated diffusion rate (classical regime). Next, a switch like designed Q – C transition is introduced: in terms of a layered QW diffusion model, the operational and applied character of that switch is manifested. A novel figure of merit for the performance of the switch is finally introduced: the Inverse Participation Ratio (IPR) of the distribution of occupation probabilities. The IPR demonstrates that the Q – C transition is a phenomenon similar to Anderson localization that entails a randomness induced suppression of QW hyper-diffusion.

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