Unformatted text preview: ce. C. REVERSIBLE COMPUTING 63 Figure 14 Billiard model realization of the interaction gate.” (FredFigure II.22: “Billiard ball ball model realization of theinteractiongate.
kin & To↵oli, 1982)
All of the above requirements are met by introducing, in addition to collisions between two balls, collisions
between a ball and a fixed plane mirror. In this way, one can easily deflect the trajectory of a ball (Figure
15a), shift it sideways (Figure 15b), introduce a delay of an arbitrary number of time steps (Figure 1 Sc), and
guarantee correct signalis provided by simultaneously ﬁring balls into the input ports for for trivial
¶5. Input crossover (Figure 15d). Of course, no special precautions need be taken
crossover, where the1s in the the timing are such that two balls cannot possibly be present at the same
the logic or argument.
moment at the crossover point (cf. Figure 18 or 12a). Thus, in the billiard ball model a conservativelogic
wire is realized as a potential ball path, as determined by the mirrors. ¶6. Inside the box the balls ricochet o↵ each other and ﬁxed reﬂectors, Note that, since whichhave finite the computation. and wires require a certain clearance in order to
balls performs diameter, both gates
function properly. As a consequence, the metric of the space in which the circuit is embedded (here, we are
considering¶the After a ﬁxed time delay, the balls emerging (or not) from the output
7. Euclidean plane) is reflected in certain circuitlayout constraints (cf. P8, Section 2).
Essentially, withports deﬁne the output.
polynomial packing (corresponding to the Abeliangroup connectivity of Euclidean space)
some wires may have to be made longer than with exponential packing (corresponding to an abstract space
with freegroup connectivity)the number of 1s (balls) is conserved.
¶8. Obviously (Toffoli, 1977). ¶9. The computation is reversible because the laws of motion are reversible.
¶10. Interaction gate: Fig. II.22 shows the realization of the computational pr...
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 Fall '13
 BruceMacLennan
 Logic gate, fredkin gate, Fredkin

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