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Quantum Pascal's Triangle and Sierpinski's carpet | | Tom Bannink
; Harry Buhrman
; | | Date: |
24 Aug 2017 | | Abstract: | In this paper we consider a quantum version of Pascal’s triangle. Pascal’s
triangle is a well-known triangular array of numbers and when these numbers are
plotted modulo 2, a fractal known as the Sierpinski triangle appears. We first
prove the appearance of more general fractals when Pascal’s triangle is
considered modulo prime powers. The numbers in Pascal’s triangle can be
obtained by scaling the probabilities of the simple symmetric random walk on
the line. In this paper we consider a quantum version of Pascal’s triangle by
replacing the random walk by the quantum walk known as the Hadamard walk. We
show that when the amplitudes of the Hadamard walk are scaled to become
integers and plotted modulo three, a fractal known as the Sierpinski carpet
emerges and we provide a proof of this using Lucas’s theorem. We furthermore
give a general class of quantum walks for which this phenomenon occurs. | | Source: | arXiv, 1708.7429 | | Services: | Forum | Review | PDF | Favorites | |
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