Discrete-Time Quantum Walk on a Line

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After 100 steps the distribution has two sharp peaks near

±𝑛/

, far from the origin where a classical walker would concentrate. Only same-parity sites are occupied at each step, so plot every other site:

In[1]:=

points[p_]:=Transpose[{Range[-100,100],p}]〚1;;;;2〛;distribution=Last[quantumWalk[100]];ListPlot[points[distribution],FillingAxis,PlotRangeAll,AxesLabel{"position","probability"}]

Out[1]=

The classical random walk of the same length is a binomial distribution: a single Gaussian peak at the origin. Overlaying the two shows ballistic versus diffusive transport:

In[2]:=

classical=Table[If[EvenQ[x],Binomial[100,(x+100)/2]/2^100,0],{x,-100,100}];ListPlot[{points[distribution],points[classical]},FillingAxis,PlotRangeAll,PlotLegends{"quantum","classical"}]

Out[2]=

	quantum
	classical

The spreading rates differ qualitatively: the quantum standard deviation grows linearly in the number of steps, while the classical one grows like

𝑡

In[3]:=

sites=Range[-100,100];spread[p_]:=Sqrt[p.sites^2-(p.sites)^2];evolution=quantumWalk[100];ListLinePlot[{Table[{t,spread[evolution〚t+1〛]},{t,0,100}],Table[{t,Sqrt[t]},{t,0,100}]},PlotLegends{"quantum (~ t)","classical (~ Sqrt[t])"},AxesLabel{"steps","std. dev."}]

Out[3]=

	quantum (~ t)
	classical (~ Sqrt[t])

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Discrete-Time Quantum Walk on a Line

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