*IBM =
Journal of=20
Research and Development* **32**(1):16-23 (January=20
1988).

=20

=A9 Copyright 1988 by International = Business=20 Machines Corporation. Copying in printed form for private use is = permitted=20 without payment of royalty provided that (1) each reproduction is done = without=20 alteration and (2) the Journal reference and IBM copyright notice are = included=20 on the first page. The title and abstract, but no other portions, of = this paper=20 may be copied or distributed royalty free without further permission by=20 computer-based and other information-service systems. Permission to = republish=20 any other portion of this paper must be obtained from the Editor.

Concern with the thermodynamic limits of computation was preceded=20
historically by the paradox of Maxwell's=20
demon [1] and the realization that one bit of information is somehow =
equivalent to k ln 2 units of entropy, or about 2.3 X 10^{-24}=20
cal/Kelvin. This equivalence was implicit in the work of Szilard=20
[2] and became explicit in Shannon's=20
use [3] of the term "entropy" and the formula

=20

to describe the self-information of a message source.=20

The history of this subject is noteworthy because it offers an = example of how=20 ideas that are strikingly successful in one area of science (in this = case the=20 uncertainty principle and the theory of black-body radiation) can = stimulate=20 unconscious false analogies, and so impede progress in other areas of = science=20 (thermodynamics of measurement and computation).=20

In the nineteenth century, despite the vision of Babbage, computation = was=20 thought of as a mental process, not a mechanical one. Accordingly, the=20 thermodynamics of computation. if anyone had stopped to wonder about it, = would=20 probably have seemed no more urgent as a topic of scientific inquiry = than, say,=20 the thermodynamics of love. However, the need to think seriously about = the=20 thermodynamics of perceptual and mental processes was thrust upon = science by the=20 famous paradox of "Maxwell's demon," described as follows by its = inventor, in a=20 passage of admirable clarity and foresight [1]:=20

"This is only one of the instances in which conclusions we have drawn = from=20 our experience of bodies consisting of an immense number of molecules = may be=20 found not to be applicable to the more delicate observations and = experiments=20 which we may suppose made by one who can perceive and handle the = individual=20 molecules which we deal with only in large masses."=20

"In dealing with masses of matter, while we do not perceive the = individual=20 molecules, we are compelled to adopt what I have described as the = statistical=20 method of calculation, and to abandon the strict dynamical method, in = which we=20 follow every motion by the calculus."=20

"It would be interesting to enquire how far those ideas about the = nature and=20 methods of science which have been derived from examples of scientific=20 investigation in which the dynamical method is followed are applicable = to our=20 actual knowledge of concrete things, which, as we have seen, is of an=20 essentially statistical nature, because no one has yet discovered any = practical=20 method of tracing the path of a molecule, or of identifying it at = different=20 times."=20

"I do not think, however, that the perfect identity which we observe = between=20 different portions of the same kind of matter can be explained on the=20 statistical principle of the stability of averages of large numbers of=20 quantities each of which may differ from the mean. For if of the = molecules of=20 some substance such as hydrogen, some were of sensibly greater mass than = others,=20 we have the means of producing a separation between molecules of = different=20 masses, and in this way we should be able to produce two kinds of = hydrogen, one=20 of which would be somewhat denser than the other. As this cannot be = done, we=20 must admit that the equality which we assert to exist between the = molecules of=20 hydrogen applies to each individual molecule, and not merely to the = average of=20 groups of millions of molecules."

Ironically, Szilard came quite close to understanding the =
thermodynamic cost=20
of information destruction. At the end of his paper, where he followed =
one=20
version of his demon apparatus through a complete cycle of operation, he =
found=20
that resetting the demon in preparation for the next measurement =
generated=20
*k* ln 2 of entropy. Unfortunately, he did not pursue this finding =
to the=20
point of recognizing that information destruction is always =
thermodynamically=20
costly, and that therefore no thermodynamic cost need be postulated for=20
information acquisition.=20

Szilard's partial insight was lost as subsequent workers neglected =
resetting,=20
and instead attempted to prove in detail the irreversibility of various=20
measurement processes, particularly those in which the demon observes =
the=20
molecule with light. The emphasis on measurement and neglect of =
resetting=20
probably represented unconscious biases from everyday experience, where=20
information is thought of as valuable or at worst neutral, and from =
quantum=20
mechanics, which strikingly demonstrated the nontriviality of the =
measurement=20
process. The influence of quantum mechanics. particularly the quantum =
theory of=20
black-body radiation. can be seen in a discussion of Maxwell's demon in=20
Brillouin's influential 1956 book *Science=20
and Information Theory* [10]:=20

"It is not surprising that Maxwell did not think of including =
radiation in=20
the system in equilibrium at temperature *T*. Black body radiation =
was=20
hardly known in 1871, and it was thirty years before the thermodynamics =
of=20
radiation was clearly understood and Planck's theory was=20
developed."

By the 1950s the development of the theory of computation by Turing = and=20 others had made it commonplace to think of computation as a mechanical = process.=20 Meanwhile the development of electronic digital computers had naturally = raised=20 the question of the ultimate thermodynamic cost of computation, = especially since=20 heat removal has always been a major engineering consideration in the = design of=20 computers, limiting the density with which active components can be = packed.=20

The general folklore belief at this time, descended from Szilard's =
and=20
Brillouin's analyses, is expressed in a remark [11]=20
from a 1949 lecture by von Neumann, to the effect that a computer =
operating at=20
temperature *T* must dissipate at least *kT *ln 2 of energy =
"per=20
elementary act of information, that is, per elementary decision of a =
two-way=20
alternative and per elementary transmittal of one unit of information."=20

A major turning point in under-standing the thermodynamics of =
computation=20
took place when Landauer=20
[6] attempted to prove this folklore belief and found he couldn't. =
He was=20
able to prove a lower bound of order *kT* for some data operations, =
but not=20
for others. Specifically. he showed that "logically irreversible"=20
operations-those that throw away information about the previous logical =
state of=20
the computer-necessarily generate in the surroundings an amount of =
entropy equal=20
to the information thrown away. The essence of Landauer's argument was =
that such=20
operations compress the phase space spanned by the computer's=20
information-bearing degrees of freedom, and so, in order to occur =
spontaneously,=20
they must allow a corresponding expansion, in other words, an entropy =
increase,=20
in other degrees of freedom.=20

[This argument is not without its subtleties; for example, a =
many-to-one=20
operation such as erasure may be thermodynamically reversible or not, =
depending=20
on the data to which it is applied. When truly *random *data (e.g., =
a bit=20
equally likely to be 0 or 1) is erased, the entropy increase of the =
surroundings=20
is compensated by an entropy decrease of the data, so the operation as a =
whole=20
is thermodynamically reversible. This is the case in resetting Maxwell's =
demon.=20
where two equiprobable states of the demon's mind must be compressed =
onto one.=20
By contrast, in computations, logically irreversible operations are =
usually=20
applied to nonrandom data deterministically generated by the =
computation. When=20
erasure is applied to such data, the entropy increase of the environment =
is not=20
compensated by an entropy decrease of the data, and the operation is=20
thermodynamically irreversible [7].]=20

About 1970, having read Landauer's paper and heard him talk. I began = thinking=20 about the thermodynamics of computation. Initially I assumed, as he, = that at=20 least some logically irreversible operations were necessary to = nontrivial=20 computation. However, as a side project, I experimented with simple = computations=20 that could be done without them. For example, I wrote a reversible = program that=20 used repeated subtraction to test whether one integer is divisible by = another.=20 Such experiments revealed a common pattern: The computation consisted of = two=20 halves, the second of which almost exactly undid the work of the first. = The=20 first half would generate the desired answer (e.g.. divisible or not) as = well=20 as, typically, some other information (e.g., remainder and quotient). = The second=20 half would dispose of the extraneous information by reversing the = process that=20 generated it, but would keep the desired answer. This led me to realize = [12]=20 that any computation could be rendered into this reversible format by=20 accumulating a history of all information that would normally be thrown = away,=20 then disposing of this history by the reverse of the process that = created it. To=20 prevent the reverse stage from destroying the desired output along with = the=20 undesired history, it suffices, before beginning the reverse stage, to = copy the=20 output on blank tape. No history is recorded during this copying = operation. and=20 none needs to be, since copying onto blank tape is already logically = reversible:=20 the reverse stage of computation then destroys only the original of the = output,=20 leaving the copy intact. My technique for performing an arbitrary = computation=20 reversibly is illustrated in Table=20 1, with underbars indicating the positions of the tape heads.=20

=20

A proof of the thermodynamic reversibility of computation requires = not only=20 showing that logically irreversible operations can be avoided, but also = showing=20 that, once the computation has been rendered into the logically = reversible=20 format, some actual hardware. or some physically reasonable theoretical = model,=20 can perform the resulting chain of logically reversible operations in a=20 thermodynamically reversible fashion. Approaching the problem with a = background=20 of prior interests in biochemistry and computability theory. I saw an = analogy=20 between DNA and RNA and the tapes of a Turing machine. The notion of an=20 informational macromolecule. undergoing transitions of its logical state = by=20 highly specific (e.g., enzyme-catalyzed) reversible chemical reactions, = offered=20 a felicitous model within which thermodynamic questions about = information=20 processing could be asked and rigorously answered. Within this = theoretical=20 framework it is easy to design an "enzymatic Turing machine" [7,=20 12]=20 which would execute logically reversible computations with a dissipation = per=20 step proportional to the speed of computation. Near equilibrium, the = machine=20 would execute a slightly biased random walk, making backward steps = nearly as=20 often as forward ones. The backward steps would not result in errors, = since they=20 would be undone by subsequent forward steps. True errors -- transitions = to=20 logically unrelated states -- would also occur in any system with finite = potential energy barriers, but their rate could be made small (in = principle=20 arbitrarily small) compared to the rate of logically correct forward and = backward transitions. The enzymatic Turing machine is an example of a = "Brownian"=20 reversible computer, in which the non-information-bearing degrees of = freedom are=20 strongly coupled to, and exert a viscous drag on, the = information-bearing ones,=20 resulting in a dissipation per step proportional to the speed of = computation.=20

Although there are no known general-purpose (i.e., universal) =
enzymatic=20
Turing machines in nature, there are enzymes analogous to =
special-purpose Turing=20
machines, notably RNA polymerase. This enzyme, whose function is to make =
an RNA=20
transcript of the genetic information in one or more DNA genes, may be =
viewed as=20
a special-purpose tape-copying Turing machine. Under physiological =
conditions=20
the enzyme is driven hard forward, and dissipates about 20 *kT* per =
step.=20
however. the operation of RNA polymerase is both logically and =
thermodynamically=20
reversible, and it is routinely operated both forward and backward in =
the=20
laboratory by varying the relative concentrations of reactants =
(nucleoside=20
triphosphates) and product (pyrophosphate) [13,=20
14].=20
When operating backward the enzyme performs the logical inverse of =
copying: It=20
removes bases one by one from the RNA strand, checking each one for=20
complementarity with the DNA before removing it.=20

Edward=20 Fredkin, at MIT, independently = arrived at=20 similar conclusions concerning reversible computation. Fredkin was = motivated by=20 a conviction that computers and physics should be more like each other. = On one=20 hand he was dissatisfied with a theoretical physics based on partial=20 differential equations and continuous space-time. He felt it = unreasonable to=20 invoke an infinite number of bits of information to encode the state of = one=20 cubic centimeter of nature, and an infinite number of digital operations = to=20 exactly simulate one second of its evolution. By the same token he felt = it wrong=20 to base the theory of computation on irreversible primitives, not found = in=20 physics. To remedy this he found a reversible three-input three-output = logic=20 function, the "conservative logic gate" able to simulate all other logic = operations, including the standard ones AND, OR, and NOT [15,=20 16].=20 He showed that conservative logic circuits can perform arbitrary = computations by=20 essentially the same programming trick I had used with reversible Turing = machines: Do the computation, temporarily saving the extra information = generated=20 in the course of obtaining the desired answer, then dispose of this = information=20 by the reverse of the process by which it was created.=20

Fredkin's displeasure with continuum models resembles Landauer's =
well-known=20
displeasure [17]=20
with mathematical operations that have no physical way of being =
performed, e.g.,=20
calculating the 10^{100}th digit of pi. These doubts, however, =
led=20
Fredkin to pursue the radical goal of finding a fully discrete basis for =
physics, whereas in Landauer they merely inspired a certain aesthetic=20
indifference toward nonconstructive mathematics.=20

Fredkin was joined by T.=20 Toffoli (who in his doctoral thesis [18]=20 had refuted, by counterexample, an accepted but erroneous proof that = reversible=20 cellular automata cannot be computationally universal), and later by = Gerard=20 Vichniac and Norman=20 Margolus to form the Information=20 Mechanics group at MIT. The = activities of=20 this group are largely responsible for stimulating the current interest = in=20 reversible cellular automata with direct physical significance, notably=20 deterministic Ising models [19,=20 20,=20 21]=20 and momentum-conserving lattice gases that support a macroscopic = hydrodynamics=20 [22].=20

A major step toward Fredkin's goal of finding a reversible physical =
basis for=20
computation was his discovery of the billiard-ball model of computation =
[16].=20
This takes advantage of the fact that a collision between two classical =
hard=20
spheres ("balls") diverts each one from the path it would have followed =
had the=20
other been absent; thus a collision can be thought of as a two-input,=20
four-output logic function whose outputs, for inputs A and *B, =
*are,=20
respectively (cf. Figure=20
1),=20

A and not =

Fredkin showed that, with the addition of "mirrors" to redirect the = balls,=20 such collisions can simulate any conservative logic function, and = therefore any=20 ordinary logic function. This implies that an infinite two-dimensional = hard=20 sphere gas, in an appropriate periodic potential (i.e., a periodic array = of=20 mirrors), is

The billiard-ball computer is the prime example of a ballistic = reversible=20 computer. In contrast to the Brownian computers described earlier, = ballistic=20 computers operate with zero dissipation at finite speed, but they depend = on=20 isolating the information-bearing degrees of freedom from all sources of = thermal=20 noise, such as internal degrees of freedom of the balls or mirrors. = Another way=20 of characterizing the difference between Brownian and ballistic = computers is to=20 say that the former work by creating a low-potential energy labyrinth in = configuration space, isomorphic to the desired computation, through = which the=20 system drifts despite thermal noise; the latter instead work by creating = a=20 dynamical trajectory isomorphic to the desired computation, which the = system=20 follows exactly in the absence of noise.=20

A number of other classical-mechanical models of reversible = computation can=20 be characterized as clocked Brownian models: The information -bearing = degrees of=20 freedom are locked to and driven by a master "clock" degree of freedom, = with=20 dissipation proportional to speed. These include the early=20 coupled-potential-well models of Landauer and Keyes [6,=20 23],=20 which were invented before the trick of reversible programming was = known, but=20 would function as Brownian reversible computers if reversibly = programmed; the=20 author's clockwork Turing machine [7],=20 which invokes infinitely hard potentials to achieve zero error in a = Brownian=20 setting: Likharev's reversible computer based on Josephson junctions [24],=20 which could probably be built and Landauer's ball-and-pipe model [15,=20 25].=20

Returning to the question of Maxwell's demon, we can now give a =
detailed=20
entropy accounting of the demon's cycle of operation. We refer to Szilard's=20
[2] version of the demon, which uses a gas consisting of a single =
molecule.=20
The demon first inserts a partition trapping the molecule on one side or =
the=20
other, next performs a measurement to learn which side the molecule is =
on, then=20
extracts *kT* ln 2 of work by allowing the molecule to expand =
isothermally=20
to fill the whole container again, and finally clears its mind in =
preparation=20
for the next measurement. The discussion below of the classical Szilard =
engine=20
follows [7];=20
an analogous quantum analysis has been given by Zurek [26].=20

According to our current understanding. each step of the cycle is=20 thermodynamically reversible if we make the usual idealization that = operations=20 are carried out quasistatically. In particular, the measurement is = reversible=20 and does not increase the entropy of the universe. What the measurement = does do,=20 however, is to establish a correlation between the state of the demon's = mind and=20 the position of the molecule. This correlation means that after the = measurement=20 the entropy of the combined system (demon + molecule) is no longer equal = to the=20 sum of the entropies of its parts. Adopting a convenient origin for the = entropy=20 scale, the entropy of the molecule is one bit (since it may be, = equiprobably, on=20 either side of the partition), the entropy of the demon's mind is one = bit (since=20 it may think, equiprobably, that the molecule is on either side of the=20 partition), but the entropy of the combined system is only one bit. = because the=20 system as a whole. owing to the correlation, has only two equiprobable = states,=20 not four.=20

The next phase of the cycle, the isothermal expansion, reduces the = entropy of=20 the environment by one bit while increasing the entropy of the demon + = molecule=20 system from one bit to two bits. Because the expansion destroys the = correlation=20 between demon and molecule (rendering the information obtained by the=20 measurement obsolete), the entropy of the demon + molecule system is now = equal=20 to the sum of the entropies of its parts, one bit each.=20

The last phase of the cycle, resetting the demon's mind, reduces the = entropy=20 of the demon from one bit to zero, and accordingly, by Landauer's = argument, must=20 increase the entropy of the environment by one bit. This increase = cancels the=20 decrease brought about during the expansion phase, bringing the cycle to = a close=20 with no net entropy change of demon, molecule, or environment.=20

One may wonder how, in view of the arguments of Brillouin and others, =
the=20
demon can make its measurement without dissipation. Though plausible, =
these=20
arguments only demonstrated the dissipativeness of certain particular =
mechanisms=20
of measurement, not of all measurements. In a sense, the existence of =
copying=20
mechanisms such as RNA polymerase demonstrates the reversibility of =
measurement,=20
if one is willing to call RNA synthesis a measurement of the DNA. More=20
traditional reversible-measurement schemes can also be devised which are =
ideal=20
in the sense of having no other effect than to establish the desired =
correlation=20
between the measuring apparatus and the system being measured. Such a=20
measurement begins with the measuring apparatus in a standard dynamical =
or=20
thermodynamic state and ends with it in one of several states depending =
on the=20
initial state of the system being measured, meanwhile having produced no =
change=20
either in the environment or in the system being measured. Figure=20
2, for example, shows a classical billiard-ball mechanism based on =
the ideas=20
of Fredkin that uses one billiard ball (dark) to test the presence of =
another=20
(light) without disturbing the dynamical state of the latter. The =
apparatus=20
consists of a number of fixed mirrors (dark rectangles) which reflect =
the=20
billiard balls. First assume that the dark ball is absent. Then a light =
ball=20
injected into the apparatus at *X* will follow the closed =
diamond-shaped=20
trajectory *ABCDEFA* forever, representing the value 1; conversely, =
the=20
absence of the light ball (i.e., no balls in the apparatus at all) =
represents=20
the value 0. The goal of the measurement is to inject another ball (dark =
color)=20
into the apparatus in such a way that it tests whether the light ball is =
present=20
without altering the light ball's state. By injecting the dark ball at Y =
at=20
the-appropriate time, the light ball (if present) is diverted from, but =
then=20
returned to, its original path (following *BGD *instead of *BCD), =
*while the dark ball leaves the apparatus at *M* if the light =
ball was=20
present and at *N* if it was absent.

**Figure 2.** Reversible measurement in the billiard-ball model of =
computation.

=20

One can design analogous mechanisms [7,=20 8]=20 for reversibly measuring which side of Szilard's engine the molecule is = on=20 without otherwise disturbing the thermodynamic state of the engine or = the=20 environment. Such reversible nondemolition measurement schemes in = general exist=20 for classical systems, and for quantum systems in which the goal of the=20 measurement is to distinguish among orthogonal states of the system, = since these=20 states may in principle be made eigenstates of an appropriate = observable. Of=20 course a quantum measurement cannot avoid disturbing a system which is = presented=20 to it in a superposition of eigenstates the measuring apparatus is = designed to=20 measure. The relation of irreversibility to quantum measurement has been = considered by many authors (cf. the concise discussion in [27]=20 and references therein).=20

An active research area recently has been the theory of quantum = reversible=20 computation. Chemical Brownian computers such as RNA polymerase are of = course=20 quantum systems, but because of the high temperature and short thermal = de=20 Broglie wavelength, quantum effects are subtle and quantitative (e.g.,=20 zero-point and tunneling corrections to reaction rates) rather than = qualitative.=20

More distinctively quantum models have been considered by a number of = authors=20 [28-35].=20 These models are somewhat abstract by comparison with classical models,=20 consisting typically of an array of two-state spins (each representing = one bit)=20 and a time evolution operator or Hamiltonian designed to make the spins = pass=20 through a sequence of states corresponding to the desired computation. = The=20 computationally relevant states are generally a subset of a set of = orthonormal=20 "basis states," in which each spin is either up or down.=20

Margolus=20 [33] and Benioff=20 [34] considered the problem of finding a universal quantum computer = (with=20 infinite memory) whose Feynman-type Hamiltonian nevertheless would have = a finite=20 range of interaction. For a serial computer such as a Turing machine, in = which=20 only one part is active at a time, this is not difficult; but when an = analogous=20 construction is attempted for a parallel machine such as a cellular = automaton,=20 Margolus found, on the one hand, that the finite range of interaction = forbade=20 synchronous updating of all the sites, and. on the other hand, that with = asynchronous updating the computation no longer proceeded ballistically. =

=20

- J. C. Maxwell,
*Theory of Heat. =*4th Ed.,=20 Longmans, Green & Co., London, 1875 (1st Ed. 1871), pp. 328-329.=20 - L. Szilard.
*Z. Phys.***53**, = 840-856=20 (1929).=20 - C. E. Shannon and W. Weaver,
*The = Mathematical=20 Theory of Communication.*University of Illinois Press, = Urbana-Champaign,=20 IL, 1949.=20 - M. von Smoluchowski,
*Z. Phys. =*(1912).=20 - M. von Smoluchowski, lecture = notes, Leipzig,=20 1914 (as quoted by Szilard [2]).=20
- R.=20
Landauer,
*IBM=20 J. Res. Develop.***3***,*183-191=20 (1961).=20

Created: November 22, 1998

Last Modified: November 23, 1998 =

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