![Quantum circuit on three qubit system and equivalent quantum walk scheme to implement same circuit is illustrated. Red circle represents |0⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert 0\rangle $$\end{document} of the real particle and green circle represents |1⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert 1\rangle $$\end{document} of the real particle. The input state is |Ψ⟩in=|000⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert \Psi \rangle _{in} = \vert 000\rangle $$\end{document} and output state is a superposition of four states |Ψ⟩out=12(|000⟩+|011⟩+|100⟩+|111⟩)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert \Psi \rangle _{out} = \frac{1}{2}\Big ( \vert 000\rangle + \vert 011\rangle + \vert 100\rangle + \vert 111\rangle \Big )$$\end{document}. CNOT23\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CNOT_{23}$$\end{document} is a position dependent shift operation given by S^-0S^-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{S}}^{0}_{-}{\hat{S}}^{1}_{-}$$\end{document} at position |01⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert 01\rangle $$\end{document} and identity at other states.](https://www.researchgate.net/publication/352074226/figure/fig4/AS:1030486596014081@1622698558190/Quantum-circuit-on-three-qubit-system-and-equivalent-quantum-walk-scheme-to-implement_Q320.jpg)
![Quantum circuit to create GHZ-state on three qubit system and equivalent quantum walk scheme to obtain GHZ-state is illustrated. Red circle represents |0⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert 0\rangle $$\end{document} of the real particle and green circle represents |1⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert 1\rangle $$\end{document} of the real particle. Here the QW-based scheme is more simplified compared to quantum circuit implementation. The input state is |Ψ⟩in=|000⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert \Psi \rangle _{in} = \vert 000\rangle $$\end{document} and output state is a GHZ-state |Ψ⟩out=12(|000⟩+|111⟩)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert \Psi \rangle _{out} = \frac{1}{\sqrt{2}}\Big ( \vert 000\rangle + \vert 111\rangle \Big )$$\end{document}.](https://www.researchgate.net/publication/352074226/figure/fig5/AS:1030486596014082@1622698558347/Quantum-circuit-to-create-GHZ-state-on-three-qubit-system-and-equivalent-quantum-walk_Q320.jpg)
June 2021
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550 Reads
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50 Citations
Quantum walk has been regarded as a primitive to universal quantum computation. In this paper, we demonstrate the realization of the universal set of quantum gates on two- and three-qubit systems by using the operations required to describe the single particle discrete-time quantum walk on a position space. The idea is to utilize the effective Hilbert space of the single qubit and the position space on which it evolves in order to realize multi-qubit states and universal set of quantum gates on them. Realization of many non-trivial gates and engineering arbitrary states is simpler in the proposed quantum walk model when compared to the circuit based model of computation. We will also discuss the scalability of the model and some propositions for using lesser number of qubits in realizing larger qubit systems.
![Discrete-time quantum walk scheme
Each step is composed of a quantum coin operation, Ĉθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{C}}_{\theta }$$\end{document}, with tunable effective coin bias parameters, θi, followed by a shift operation, Ŝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{S}$$\end{document}.](https://www.researchgate.net/publication/343188787/figure/fig1/AS:959173042917391@1605696082875/Discrete-time-quantum-walk-scheme-Each-step-is-composed-of-a-quantum-coin-operation_Q320.jpg)
![Mapping of multi-qubit states to position states
Multi-qubit states are re-ordered in such a way that the state 0(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left|0\right\rangle \,(\left|1\right\rangle )$$\end{document} of the last qubit corresponds to even (odd) position numbers and its correspondence in the position space.](https://www.researchgate.net/publication/343188787/figure/fig2/AS:959173051285504@1605696084269/Mapping-of-multi-qubit-states-to-position-states-Multi-qubit-statesare-re-ordered-in-such_Q320.jpg)
![Circuit implementation of quantum walks on a trapped-ion processor and its time evolution
a Circuit diagram for a DQW and DCA. Each dashed block describes one step in the quantum walk. b Discrete-time Quantum Walk. Comparison of the experimental results (left) and the theoretical quantum-walk probability distribution (right) for the first five steps with initial particle state b i and b iv ψc=0c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left|\psi \right\rangle }_{{\rm{c}}}={\left|0\right\rangle }_{{\rm{c}}}$$\end{document}, b ii and b iv ψc=1c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left|\psi \right\rangle }_{{\rm{c}}}={\left|1\right\rangle }_{{\rm{c}}}$$\end{document}, b iii and b vi ψc=0c+i1c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left|\psi \right\rangle }_{{\rm{c}}}={\left|0\right\rangle }_{{\rm{c}}}+i{\left|1\right\rangle }_{{\rm{c}}}$$\end{document}, and position state x=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left|x=0\right\rangle$$\end{document}. c Output of a step-5 Dirac Cellular Automaton for θ1 = 0 and, c i and c iv θ2 = π/4, c ii and c v θ2 = π/10 and c iii and c vi θ2 = π/20 with the initial state Ψin=(0c+i1c)⊗x=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left|{\Psi }_{{\rm{in}}}\right\rangle =({\left|0\right\rangle }_{{\rm{c}}}+i{\left|1\right\rangle }_{{\rm{c}}})\otimes \left|x=0\right\rangle$$\end{document}.](https://www.researchgate.net/publication/343188787/figure/fig3/AS:959173051297802@1605696084407/Circuit-implementation-of-quantum-walks-on-a-trapped-ion-processor-and-its-time_Q320.jpg)
