Polyander visualization of quantum walks
Steven Duplij, Raimund Vogl
TL;DR
The paper addresses detailed joint visualization and analytic description of discrete-time quantum walks on a line by introducing polyander visualizations that track time evolution of fixed walker or coin states. It develops both Schrödinger/Fourier and combinatorial methods to obtain the final state $|\Psi_{tot}(t)\rangle$ and provides analytic Fourier-domain expressions for amplitudes, enabling explicit computation of $\varphi_{j,\ell}(t)$. Key contributions include the polyander visualization framework, explicit walker and coin probability distributions for the Hadamard walk, and a discussion of generalizations to Grover coins and braid/fusion extensions. The approach enhances visualization, analytic tractability, and broadens the scope for applying quantum-walk dynamics to complex coin-walker-topologies with potential implications for quantum information and data-visualization analogies.
Abstract
We investigate quantum walks which play an important role in the modelling of many phenomena. The detailed and thorough description is given to the discrete quantum walks on a line, where the total quantum state consists of quantum states of the walker and the coin. In addition to the standard walker probability distribution, we introduce the coin probability distribution which gives more complete quantum walk description and novel visualization in terms of the so called polyanders (analogs of trianders in DNA visualization). The methods of final states computation and the Fourier transform are presented for the Hadamard quantum walk.