Although the above gates are sufficient for the mathematics of logic, they are not sufficient to build a practical machine. A useful computer will also require the FANOUT and ERASE gates (Fig. 2).

**Fig. 2** Two non-standard gates that are required to build a computer,
in addition to a universal set of logic gates, are: (a) the FANOUT gate
which duplicates an input **A** and (b) the ERASE gate which deletes its
input.

First consider the FANOUT gate: Is it reversible? Certainly
no information has been destroyed so it is at least logically reversible.
Landauer showed that it could also be physically reversible [8].
Let us describe a simple model for FANOUT based on Bennett's scheme for a
reversible measurement (Fig. 3) [9]. Here a dark ball
is used to determine the presence or absence of a second (light) ball inside
a trap. The trap consists of a set of mirrors and may be
thought of as a one-bit memory register. If the trap is occupied then the
dark ball is reflected and leaves along direction **M** (with the light ball
continuing along its original trajectory); otherwise it passes unhindered
towards **N**. Upon leaving the trap, the dark ball's direction is used to
populate, or not, another trap.

**Fig. 3** A reversible measurement of the existence of a (light) ball
in a trap of mirrors (dark rectangles) [9]. A
(dark) ball enters the trap from **Y**. In the absence of a light ball in
the trap the dark ball will follow the path **HN**. In presence of a
light ball (timed to start at **X**) the dark ball will deflect the light one
from its unhindered trajectory **ABCDEF** to **ABGDEF** and will follow
the path **HIJKLM** itself. (Copyright 1988 by International Business
Machines Corporation, reprinted with permission.)

Let us now consider the ERASE operation which is required to ``clean out'' the computer's memory periodically. One type of erasure can be performed reversibly: If we have a backup copy of some information, we can erase further copies by uncomputing the FANOUT gate. The difficulty arises when we wish to erase our last copy, referred to here as the primitive ERASE.

Consider a single bit represented by a pair of equally probable
classical states of some particle. To erase the information about the
particle's state we must irreversibly compress phase-space by a factor
of two. If we allowed
this compressed phase-space to adiabatically expand, at temperature **T**,
to its original size, we could obtain an amount of work equal to
(where is Boltzmann's constant). Landauer concluded,
based on simple models and more general arguments about the compression
of phase-space, that the erasure of a bit of information at temperature
**T** requires the dissipation of at least heat (a result
known as Landauer's principle) [8].

Wed Aug 23 11:54:31 IDT 1995