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Locality of the Feynman Hamiltonian
The step operator T is local by construction, that is, a single computational step changes at most the qubit at the lattice position of the head. In order to provide a quantitative measure of locality of powers of H, we prepare a QTM state
q[i]:= InitQTM[2,b,Interval[{0,4}],{{1,b[0]},{3,b[0]}},9,h,i]
and calculate the set of functions 
defined as
measure the overlap of 
and 
for given 
. As a consequence of the locality of H we expect that the overlap for fixed n decreases fast with increasing i because the head positions of 
and 
become more spatially separated and we need more and more iterations of H to produce any significant overlap between 
and 
. As an example, we plot 
(see Figure 2).
If we take 
for fixed i and increase n, we expect to have a significant overlap when 
. As an example, we plot 
(see Figure 3).
From this we can see that the time translation operator 
is completely delocalized because infinitely many powers of H contribute, that is, Abs[QInner[q[0],U(t)[q[i]]]
0 for all 
.
Copyright © 2002 Wolfram Media, Inc. All rights reserved.