<s>
Reversible	B-Application
computing	I-Application
is	O
any	O
model	O
of	O
computation	O
where	O
the	O
computational	O
process	O
,	O
to	O
some	O
extent	O
,	O
is	O
time-reversible	O
.	O
</s>
<s>
In	O
a	O
model	O
of	O
computation	O
that	O
uses	O
deterministic	O
transitions	B-Application
from	O
one	O
state	O
of	O
the	O
abstract	O
machine	O
to	O
another	O
,	O
a	O
necessary	O
condition	O
for	O
reversibility	O
is	O
that	O
the	O
relation	O
of	O
the	O
mapping	B-Algorithm
from	O
states	O
to	O
their	O
successors	O
must	O
be	O
one-to-one	O
.	O
</s>
<s>
Reversible	B-Application
computing	I-Application
is	O
a	O
form	O
of	O
unconventional	B-Device
computing	I-Device
.	O
</s>
<s>
Due	O
to	O
the	O
unitarity	O
of	O
quantum	O
mechanics	O
,	O
quantum	B-Application
circuits	I-Application
are	O
reversible	O
,	O
as	O
long	O
as	O
they	O
do	O
not	O
"	O
collapse	O
"	O
the	O
quantum	O
states	O
they	O
operate	O
on	O
.	O
</s>
<s>
A	O
process	O
is	O
said	O
to	O
be	O
physically	O
reversible	O
if	O
it	O
results	O
in	O
no	O
increase	O
in	O
physical	O
entropy	B-Algorithm
;	O
it	O
is	O
isentropic	O
.	O
</s>
<s>
A	O
motivation	O
for	O
the	O
study	O
of	O
technologies	O
aimed	O
at	O
implementing	O
reversible	B-Application
computing	I-Application
is	O
that	O
they	O
offer	O
what	O
is	O
predicted	O
to	O
be	O
the	O
only	O
potential	O
way	O
to	O
improve	O
the	O
computational	O
energy	O
efficiency	O
of	O
computers	O
beyond	O
the	O
fundamental	O
von	O
Neumann	O
–	O
Landauer	O
limit	O
of	O
energy	O
dissipated	O
per	O
irreversible	O
bit	O
operation	O
.	O
</s>
<s>
Although	O
the	O
Landauer	O
limit	O
was	O
millions	O
of	O
times	O
below	O
the	O
energy	O
consumption	O
of	O
computers	O
in	O
the	O
2000s	O
and	O
thousands	O
of	O
times	O
less	O
in	O
the	O
2010s	O
,	O
proponents	O
of	O
reversible	B-Application
computing	I-Application
argue	O
that	O
this	O
can	O
be	O
attributed	O
largely	O
to	O
architectural	O
overheads	O
which	O
effectively	O
magnify	O
the	O
impact	O
of	O
Landauer	O
's	O
limit	O
in	O
practical	O
circuit	O
designs	O
,	O
so	O
that	O
it	O
may	O
prove	O
difficult	O
for	O
practical	O
technology	O
to	O
progress	O
very	O
far	O
beyond	O
current	O
levels	O
of	O
energy	O
efficiency	O
if	O
reversible	B-Application
computing	I-Application
principles	O
are	O
not	O
used	O
.	O
</s>
<s>
Landauer	O
's	O
principle	O
is	O
the	O
rigorously	O
valid	O
observation	O
that	O
the	O
oblivious	O
erasure	O
of	O
n	O
bits	O
of	O
known	O
information	O
must	O
always	O
incur	O
a	O
cost	O
of	O
in	O
thermodynamic	O
entropy	B-Algorithm
.	O
</s>
<s>
For	O
computational	O
processes	O
that	O
are	O
nondeterministic	O
(	O
in	O
the	O
sense	O
of	O
being	O
probabilistic	O
or	O
random	O
)	O
,	O
the	O
relation	O
between	O
old	O
and	O
new	O
states	O
is	O
not	O
a	O
single-valued	B-Algorithm
function	I-Algorithm
,	O
and	O
the	O
requirement	O
needed	O
to	O
obtain	O
physical	O
reversibility	O
becomes	O
a	O
slightly	O
weaker	O
condition	O
,	O
namely	O
that	O
the	O
size	O
of	O
a	O
given	O
ensemble	O
of	O
possible	O
initial	O
computational	O
states	O
does	O
not	O
decrease	O
,	O
on	O
average	O
,	O
as	O
the	O
computation	O
proceeds	O
forwards	O
.	O
</s>
<s>
The	O
implementation	O
of	O
reversible	B-Application
computing	I-Application
thus	O
amounts	O
to	O
learning	O
how	O
to	O
characterize	O
and	O
control	O
the	O
physical	O
dynamics	O
of	O
mechanisms	O
to	O
carry	O
out	O
desired	O
computational	O
operations	O
so	O
precisely	O
that	O
we	O
can	O
accumulate	O
a	O
negligible	O
total	O
amount	O
of	O
uncertainty	O
regarding	O
the	O
complete	O
physical	O
state	O
of	O
the	O
mechanism	O
,	O
per	O
each	O
logic	O
operation	O
that	O
is	O
performed	O
.	O
</s>
<s>
Although	O
achieving	O
this	O
goal	O
presents	O
a	O
significant	O
challenge	O
for	O
the	O
design	O
,	O
manufacturing	O
,	O
and	O
characterization	O
of	O
ultra-precise	O
new	O
physical	O
mechanisms	O
for	O
computing	O
,	O
there	O
is	O
at	O
present	O
no	O
fundamental	O
reason	O
to	O
think	O
that	O
this	O
goal	O
cannot	O
eventually	O
be	O
accomplished	O
,	O
allowing	O
us	O
to	O
someday	O
build	O
computers	O
that	O
generate	O
much	O
less	O
than	O
1	O
bit	O
's	O
worth	O
of	O
physical	O
entropy	B-Algorithm
(	O
and	O
dissipate	O
much	O
less	O
than	O
kT	O
ln	O
2	O
energy	O
to	O
heat	O
)	O
for	O
each	O
useful	O
logical	O
operation	O
that	O
they	O
carry	O
out	O
internally	O
.	O
</s>
<s>
This	O
field	O
of	O
research	O
awaits	O
the	O
detailed	O
development	O
of	O
a	O
high-quality	O
,	O
cost-effective	O
,	O
nearly	O
reversible	B-Application
logic	I-Application
device	O
technology	O
,	O
one	O
that	O
includes	O
highly	O
energy-efficient	O
clocking	O
and	O
synchronization	O
mechanisms	O
,	O
or	O
avoids	O
the	O
need	O
for	O
these	O
through	O
asynchronous	O
design	O
.	O
</s>
<s>
This	O
sort	O
of	O
solid	O
engineering	O
progress	O
will	O
be	O
needed	O
before	O
the	O
large	O
body	O
of	O
theoretical	O
research	O
on	O
reversible	B-Application
computing	I-Application
can	O
find	O
practical	O
application	O
in	O
enabling	O
real	O
computer	O
technology	O
to	O
circumvent	O
the	O
various	O
near-term	O
barriers	O
to	O
its	O
energy	O
efficiency	O
,	O
including	O
the	O
von	O
Neumann	O
–	O
Landauer	O
bound	O
.	O
</s>
<s>
This	O
may	O
only	O
be	O
circumvented	O
by	O
the	O
use	O
of	O
logically	O
reversible	B-Application
computing	I-Application
,	O
due	O
to	O
the	O
second	O
law	O
of	O
thermodynamics	O
.	O
</s>
<s>
Reversible	O
functions	O
are	O
bijective	B-Algorithm
.	O
</s>
<s>
Similarly	O
,	O
in	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
model	O
of	O
computation	O
,	O
a	O
reversible	O
Turing	B-Architecture
machine	I-Architecture
is	O
one	O
whose	O
transition	O
function	O
is	O
invertible	O
,	O
so	O
that	O
each	O
machine	O
state	O
has	O
at	O
most	O
one	O
predecessor	O
.	O
</s>
<s>
Yves	O
Lecerf	O
proposed	O
a	O
reversible	O
Turing	B-Architecture
machine	I-Architecture
in	O
a	O
1963	O
paper	O
,	O
but	O
apparently	O
unaware	O
of	O
Landauer	O
's	O
principle	O
,	O
did	O
not	O
pursue	O
the	O
subject	O
further	O
,	O
devoting	O
most	O
of	O
the	O
rest	O
of	O
his	O
career	O
to	O
ethnolinguistics	O
.	O
</s>
<s>
In	O
1973	O
Charles	O
H	O
.	O
Bennett	O
,	O
at	O
IBM	O
Research	O
,	O
showed	O
that	O
a	O
universal	O
Turing	B-Architecture
machine	I-Architecture
could	O
be	O
made	O
both	O
logically	O
and	O
thermodynamically	O
reversible	O
,	O
and	O
therefore	O
able	O
in	O
principle	O
to	O
perform	O
an	O
arbitrarily	O
large	O
number	O
of	O
computation	O
steps	O
per	O
unit	O
of	O
physical	O
energy	O
dissipated	O
,	O
if	O
operated	O
sufficiently	O
slowly	O
.	O
</s>
<s>
In	O
1982	O
Edward	O
Fredkin	O
and	O
Tommaso	O
Toffoli	O
proposed	O
the	O
Billiard	B-Application
ball	I-Application
computer	I-Application
,	O
a	O
mechanism	O
using	O
classical	O
hard	O
spheres	O
to	O
do	O
reversible	B-Application
computations	I-Application
at	O
finite	O
speed	O
with	O
zero	O
dissipation	O
,	O
but	O
requiring	O
perfect	O
initial	O
alignment	O
of	O
the	O
balls	O
 '	O
trajectories	O
,	O
and	O
Bennett	O
's	O
review	O
compared	O
these	O
"	O
Brownian	O
"	O
and	O
"	O
ballistic	O
"	O
paradigms	O
for	O
reversible	B-Application
computation	I-Application
.	O
</s>
<s>
Aside	O
from	O
the	O
motivation	O
of	O
energy-efficient	O
computation	O
,	O
reversible	B-Application
logic	I-Application
gates	O
offered	O
practical	O
improvements	O
of	O
bit-manipulation	B-Algorithm
transforms	O
in	O
cryptography	O
and	O
computer	O
graphics	O
.	O
</s>
<s>
Since	O
the	O
1980s	O
,	O
reversible	B-Application
circuits	I-Application
have	O
attracted	O
interest	O
as	O
components	O
of	O
quantum	B-Device
algorithms	I-Device
,	O
and	O
more	O
recently	O
in	O
photonic	O
and	O
nano-computing	O
technologies	O
where	O
some	O
switching	O
devices	O
offer	O
no	O
signal	O
gain	O
.	O
</s>
<s>
Surveys	O
of	O
reversible	B-Application
circuits	I-Application
,	O
their	O
construction	O
and	O
optimization	O
,	O
as	O
well	O
as	O
recent	O
research	O
challenges	O
,	O
are	O
available	O
.	O
</s>
