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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).

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).

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 .

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