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Unformatted text preview: n certain circuitlayout constraints (cf. P8, Section 2).
Essentially, with polynomial packing (corresponding to the Abeliangroup connectivity of Euclidean space)
6.2. The InteractionNote that the second gatefrom the bottom should be pq . defined by Figure
To↵oli, 1982) Gate. The interaction pq is the conservativelogic primitive
some wires may have to be made longer than with exponential packing (corresponding to an abstract space
13a, which also assigns its graphical representation.7
with freegroup connectivity) (Toffoli, 1977).
In the billiard ball model, the interaction gate is realized simply as the potential locus of collision of two
balls. With reference to Figure 14, let p, q be the values at a certain instant of the binary variables associated
with the two points P, Q, and consider the valuesfour time steps later in this particular exampleof the
variables associated with the four points A, B, C, D. It is clear that these values are, in the order shown in the
figure, pq, ¬pq, p¬q; and pq. In other words, there will be a ball at A if and only if there was a ball at P and
one at Q; similarly, there will be a ball at B if and only if there was a ball at Q and none at P; etc. 6.3. Interconnection; Timing and Crossover; The Mirror. Owing to its AND and NOT
capabilities, the interaction gate is clearly a universal logic primitive (as explained in Section 5, we assume
the availability of input constants). To verify that these capabilities are retained in the billiard ball model,
one FiguremakeThe mirrorone can realize solid dash) can be used to deflect a ball’s path (a), introduce a
must 15. sure that (indicated by a the appropriate interconnections, i.e., that one can suitably route
Figure II.24: “The mirror (indicated (c), and realize nontrivial crossover deﬂect
sideways shift (b), introduce delay by a solid dash) can be used to we
balls from one collision locus to another andamaintain proper timing. In particular, since(d). are considering
a ball’smust provide a way of performin...
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 Fall '13
 BruceMacLennan

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