https://doi.org/10.1140/epjd/s10053-023-00780-9
Regular Article – Quantum Information
Closed-form expressions for the probability distribution of quantum walk on a line
Faculty of Engineering and the Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, 5290002, Ramat Gan, Israel
a
jadmn.jayakody@gmail.com
b
eliahu.cohen@biu.ac.il
Received:
17
June
2023
Accepted:
31
October
2023
Published online:
10
November
2023
Theoretical and applied studies of quantum walks are abundant in quantum science and technology thanks to their relative simplicity and versatility. Here we derive closed-form expressions for the probability distribution of quantum walks on a line. The most general two-state coin operator and the most general (pure) initial state are considered in the derivation. The general coin operator includes the common choices of Hadamard, Grover, and Fourier coins. The method of Fibonacci-Horner basis for the power decomposition of a matrix is employed in the analysis. Moreover, we also consider mixed initial states and derive closed-form expression for the probability distribution of the Quantum walk on a line. To prove the accuracy of our derivations, we retrieve the simulated probability distribution of Hadamard walk on a line using our closed-form expressions. With a broader perspective in mind, we argue that our approach has the potential to serve as a helpful mathematical tool in obtaining precise analytical expressions for the time evolution of qubit-based systems in a general context.
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© The Author(s), under exclusive licence to EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
