[Paleopsych] Seth Lloyd: Ultimate physical limits to computation

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Seth Lloyd: Ultimate physical limits to computation

d'Arbeloff Laboratory for Information Systems and Technology, MIT 
Department of Mechanical Engineering, Massachusetts Institute of 
Technology 3-160, Cambridge, Massachusetts 02139, USA (slloyd at mit.edu)

NATURE 406 (2000 August 31):1047-56

An insight review article

Abstract: Computers are physical systems: the laws of physics dictate what 
they can and cannot do. In particular, the speed with which a physical 
device can process information is limited by its energy and the amount of 
information that it can process is limited by the number of degrees of 
freedom it possesses. Here I explore the physical limits of computation as 
determined by the speed of light c, the quantum scale h-bar and the 
gravitational constant G. As an example, I put quantitative bounds to the 
computational power of an 'ultimate laptop' with a mass of one kilogram 
confined to a volume of one litre.

[I did some different calculations, on an upper bound for the number of 
possible computations since the Big Bang. This was simply the time a 
photon can cross the Planck distance times the number of photons times the 
number of such instants since the Big Bang.

[a. The Plank length is sqrt(hG/c^3) = 1.62 x 10^-35 meters.

[b. The Plank time is the length divided by c, or sqrt(hG/c^5) = 5 x 
10^-44 seconds.

[c. The universe is 4 x 10^17 seconds old, or 10^61 Plank time units old.
[d. There are 10^89 photons in the universe.

[I cannot easily recall where I got these numbers, but I don't think they 
are in serious doubt. So the number of photons having moved across a 
Planck distance since the Big Bang is the product of b, c, and d and is 

[10^150 = (10^3)^50 ~ (2^10)^50 = 2^500.

[The number of actual calculations is vastly smaller than this, since 
particles do not communicate their movements instantaneously. How much 
vaster doesn't really matter right now. What does matter is that cracking 
a truth table puzzle that has 512 = 2^7 variables is impossible and that a 
512-bit encryption scheme is likewise impossible.

[All this sets sharp limits on the ability of anything whatever to 
calculate. It would not matter much, philosophically, if the figure were 
128 or 256 or 1024 or 2048 or even 1024^2. It is humanly-comprhensible 
number. We know from the results of Gödel, Church, Turing, and so on about 
the limitations of any finite calculators.^ But when we come to people the 
limits on reason are much sharper, since our brains weigh only three 

^[Recall that only finite sentences of finite length are admitted in the 
"formal systems" of the title of Gödel's 1931 paper, which not so by the 
way, means that finite has suddenly become an undefined term! This is true 
of Raymond Smullyan's _Formal Systems_ and all similar books I have seen. 
Of course, there are plenty of definitions of finite *inside* mathematics, 
logic, and set theory, definitions that are not equivalent, but every one 
of these theories is handled only by formal systems, with their undefined 
finite sentences of finite length. And so in mathematics, one of the core 
terms is left as a KNOW-HOW term and never actually gets defined. The same 
is true of cost in economics. I could go on.]

[The Enlightenment optimism of using reason to solve all problems is dead. 
No one in the Enlightenment thought about solving 512 variable truth 
tables and all, except maybe LaPlace, would have admitted to limits on 
reason. Socially, though, the weaking of this optimism took place on a 
different plane. World War I, with it senseless trench warfare, destroved 
much optimism about human nature, as far as governing itself goes. 
Relativity theory and quantum mechanics did their part, too, in the realm 
of science.

[The Enlightenment was killed for good during six seconds on the Dealey 

[Before the article here is a talk by Lloyd with Edge.]



"Something else has happened with computers. What's happened with society 
is that we have created these devices, computers, which already can 
register and process huge amounts of information, which is a significant 
fraction of the amount of information that human beings themselves, as a 
species, can process. When I think of all the information being processed 
there, all the information being communicated back and forth over the 
Internet, or even just all the information that you and I can communicate 
back and forth by talking, I start to look at the total amount of 
information being processed by human beings -- and their artifacts -- we 
are at a very interesting point of human history, which is at the stage 
where our artifacts will soon be processing more information than we 
physically will be able to process."



"Lloyd's Hypothesis" states that everything that's worth understanding 
about a complex system, can be understood in terms of how it processes 
information. This is a new revolution that's occurring in science.

Part of this revolution is being driven by the work and ideas of Seth 
Lloyd, a Professor of Mechanical Engineering at MIT. Last year, Lloyd 
published an article in the journal Nature -- "Ultimate Physical Limits to 
Computation" (vol. 406, no. 6788, 31 August 2000, pp. 1047-1054) -- in 
which he sought to determine the limits the laws of physics place on the 
power of computers. "Over the past half century," he wrote, "the amount of 
information that computers are capable of processing and the rate at which 
they process it has doubled every 18 months, a phenomenon known as Moore's 
law. A variety of technologies -- most recently, integrated circuits -- 
have enabled this exponential increase in information processing power. 
But there is no particular reason why Moore's law should continue to hold: 
it is a law of human ingenuity, not of nature. At some point, Moore's law 
will break down. The question is, when?"

His stunning conclusion?

"The amount of information that can be stored by the ultimate laptop, 10 
to the 31st bits, is much higher than the 10 to the 10th bits stored on 
current laptops. This is because conventional laptops use many degrees of 
freedom to store a bit whereas the ultimate laptop uses just one. There 
are considerable advantages to using many degrees of freedom to store 
information, stability and controllability being perhaps the most 
important. Indeed, as the above calculation indicates, to take full 
advantage of the memory space available, the ultimate laptop must turn all 
its matter into energy. A typical state of the ultimate laptop's memory 
looks like a plasma at a billion degrees Kelvin -- like a thermonuclear 
explosion or a little piece of the Big Bang! Clearly, packaging issues 
alone make it unlikely that this limit can be obtained, even setting aside 
the difficulties of stability and control."

Ask Lloyd why he is interested in building quantum computers and you will 
get a two part answer. The first, and obvious one, he says, is "because we 
can, and because it's a cool thing to do." The second concerns some 
interesting scientific implications. "First," he says, "there are 
implications in pure mathematics, which are really quite surprising, that 
is that you can use quantum mechanics to solve problems in pure math that 
are simply intractable on ordinary computers." The second scientific 
implication is a use for quantum computers was first suggested by Richard 
Feynman in 1982, that one quantum system could simulate another quantum 
system. Lloyd points out that "if you've ever tried to calculate Feynman 
diagrams and do quantum dynamics, simulating quantum systems is hard. It's 
hard for a good reason, which is that classical computers aren't good at 
simulating quantum systems."

Lloyd notes that Feynman suggested the possibility of making one quantum 
system simulate another. He conjectured that it might be possible to do 
this using something like a quantum computer. In 90s Lloyd showed that in 
fact Feynman's conjecture was correct, and that not only could you 
simulate virtually any other quantum system if you had a quantum computer, 
but you could do so remarkably efficiently. So by using quantum computers, 
even quite simple ones, once again you surpass the limits of classical 
computers when you get down to, say, 30 or 40 bits in your quantum 
computer. You don't need a large quantum computer to get a big huge 
speedup over classical simulations of physical systems.

"A salt crystal has around 10 to the 17 possible bits in it," he points 
out. "As an example, let's take your own brain. If I were to use every one 
of those spins, the nuclear spins, in your brain that are currently being 
wasted and not being used to store useful information, we could probably 
get about 10 to the 28 bits there."

Sitting with Lloyd in the Ritz Carlton Hotel in Boston, overlooking the 
tranquil Boston Public Gardens, I am suddenly flooded with fantasies of 
licensing arrangements regarding the nuclear spins of my brain. No doubt 
this would be a first in distributed computing

"You've got a heck of a lot of nuclear spins in your brain," Lloyd says. 
"If you've ever had magnetic resonance imaging, MRI, done on your brain, 
then they were in fact tickling those spins. What we're talking about in 
terms of quantum computing, is just sophisticated 'spin tickling'."

This leads me to wonder how "spin tickling" fits into intellectual 
property law. How about remote access? Can you in theory designate and 
exploit people who would have no idea that their brains were being used 
for quantum computation?

Lloyd points out that so far as we know, our brains don't pay any 
attention to these nuclear spins. "You could have a whole parallel 
computational universe going on inside your brain. This is, of course, 
fantasy. But hey, it might happen."

"But it's not a fantasy to explore this question about making computers 
that are much, much, more powerful than the kind that we have sitting 
around now -- in which a grain of salt has all the computational powers of 
all the computers in the world. Having the spins in your brain have all 
the computational power of all the computers in a billion worlds like ours 
raises another question which is related to the other part of the research 
that I do."

In the '80s, Lloyd began working on how large complex systems process 
information. How things process information at a very small scale, and how 
to make ordinary stuff, like a grain of salt or a cube of sugar, process 
information, relates to the complex systems work in his thesis that he did 
with the late physicist Heinz Pagels, his advisor at Rockefeller 
University. "Understanding how very large complex systems process 
information is the key to understanding how they behave, how they break 
down, how they work, what goes right and what goes wrong with them," he 

Science is being done in new an different ways, and the changes 
accelerates the exchange of ideas and the development of new ideas. Until 
a few years ago, it was very important for a young scientist to be to "in 
the know" -- that is, to know the right people, because results were 
distributed primarily by pre prints, and if you weren't on the right 
mailing list, then you weren't going to get the information, and you 
wouldn't be able to keep up with the field.

"Certainly in my field, and fundamental physics, and quantum mechanics, 
and physics of information," Lloyd notes, "results are distributed 
electronically, the electronic pre-print servers, and they're available to 
everybody via the World Wide Web. Anybody who wants to find out what's 
happening right now in the field can go to [13]http://xxx.lanl.gov and 
find out. So this is an amazing democratization of knowledge which I think 
most people aren't aware of, and its effects are only beginning to be 

"At the same time," he continues, "a more obvious way in which science has 
become public is that major newspapers such as The New York Times have all 
introduced weekly science sections in the last ten years. Now it's hard to 
find a newspaper that doesn't have a weekly section on science. People are 
becoming more and more interested in science, and that's because they 
realize that science impacts their daily lives in important ways."

A big change in science is taking place, and that's that science is 
becoming more public -- that is, belonging to the people. In some sense, 
it's a realization of the capacity of science. Science in some sense is 
fundamentally public.

"A scientific result is a result that can be duplicated by anybody who has 
the right background and the right equipment, be they a professor at 
M.I.T. like me," he points out, "or be they from an underdeveloped 
country, or be they an alien from another planet, or a robot, or an 
intelligent rat. Science consists exactly of those forms of knowledge that 
can be verified and duplicated by anybody. So science is basically, at it 
most fundamental level, a public form of knowledge, a form of knowledge 
that is in principle accessible to everybody. Of course, not everybody's 
willing to go out and do the experiments, but for the people who are 
willing to go out and do that, -- if the experiments don't work, then it 
means it's not science.

"This democratization of science, this making it public, is in a sense the 
realization of a promise that science has held for a long time. Instead of 
having to be a member of the Royal Society to do science, the way you had 
to be in England in the 17th, 18th, centuries today pretty much anybody 
who wants to do it can, and the information that they need to do it is 
there. This is a great thing about science. That's why ideas about the 
third culture are particularly apropos right now, as you are concentrating 
on scientists trying to take their case directly to the public. Certainly, 
now is the time to do it."


SETH LLOYD is an Associate Professor of Mechanical Engineering at MIT and 
a principal investigator at the Research Laboratory of Electronics. He is 
also adjunct assistant professor at the Santa Fe Institute. He works on 
problems having to do with information and complex systems from the very 
small -- how do atoms process information, how can you make them compute, 
to the very large -- how does society process information? And how can we 
understand society in terms of its ability to process information?

[14]Click Here for Seth Lloyd's Bio Page[15] [7.23.01]

SETH LLOYD: Computation is pervading the sciences. I believe it began 
about 400 years ago, if you look at the first paragraph of Hobbes's famous 
book Leviathan. He says that just as we consider the human body to be like 
a machine, like a clock where you have sinews and muscles to move energy 
about, a pulse beat like a pendulum, and a heart that pumps energy in, 
similar to the way a weight supplies energy to a clock's pendulum, then we 
can consider the state to be analogous to the body, since the state has a 
prince at its head, people who form its individual portions, legislative 
bodies that form its organs, etc. In that case, Hobbes asked, couldn't we 
consider the state itself to have an artificial life?

To my knowledge that was the first use of the phrase artificial life in 
the form that we use it today. If we have a physical system that's 
evolving in a physical way, according to a set of rules, couldn't we 
consider it to be artificial and yet living? Hobbes wasn't talking about 
information processing explicitly, but the examples he used were, in fact, 
examples of information processing. He used the example of the clock as 
something that is designed to process information, as something that gives 
you information about time. Most pieces of the clock that he described are 
devices not only for transforming energy, but actually for providing 
information. For example, the pendulum gives you regular, temporal 
information. When he next discusses the state and imagines it having an 
artificial life, he first talks about the brain, the seat of the state's 
thought processes, and that analogy, in my mind, accomplishes two things.

First, Hobbes is implicitly interested in information. Second, he is 
constructing the fundamental metaphor of scientific and technological 
inquiry. When we think of a machine as possessing a kind of life in and of 
itself, and when we think of machines as doing the same kinds of things 
that we ourselves do, we are also thinking the corollary, that is, we are 
doing the same kinds of things that machines do. This metaphor, one of the 
most powerful of the Enlightenment, in some sense pervaded the popular 
culture of that time. Eventually, one could argue, that metaphor gave rise 
to Newton's notions of creating a dynamical picture of the world. The 
metaphor also gave rise to the great inquiries into thermodynamics and 
heat, which came 150 years later, and, in some ways, became the central 
mechanical metaphor that has informed all of science up to the 20th 

The real question is, when did people first start talking about 
information in such terms that information processing rather than 
clockwork became the central metaphor for our times? Because until the 
20th century, this Enlightenment mode of thinking of physical things such 
as mechanical objects with their own dynamics as being similar to the body 
or the state was really the central metaphor that informed much scientific 
and technological inquiry. People didn't start thinking about this 
mechanical metaphor until they began building machines, until they had 
some very good examples of machines, like clocks for instance. The 17th 
century was a fantastic century for clockmaking, and in fact, the 17th and 
18th centuries were fantastic centuries for building machines, period.

Just as people began conceiving of the world using mechanical metaphors 
only when they had themselves built machines, people began to conceive of 
the world in terms of information and information-processing, only when 
they began dealing with information and information processing. All the 
mathematical and theoretical materials for thinking of the world in terms 
of information, including all the basic formulas, were available at the 
end of the 19th century, because all these basic formulas had been created 
by Maxwell, Boltzmann and Gibbs for statistical mechanics. The formula for 
information was known back in the 1880s, but people didn't know that it 
dealt with information. Instead, because they were familiar with things 
like heat and mechanical systems that processed heat, they called 
information in its mechanical or thermodynamic manifestation, entropy. It 
wasn't until the 1930s, when people like Claude Shannon and Norbert 
Wiener, and before them Harry Nyquist, started to think about information 
processing for the primary purpose of communication, or for the purposes 
of controlling systems so that the role of information and feedback could 
be controlled. Then came the notion of constructing machines that actually 
processed information. Babbage tried to construct one back in the early 
19th century, which was a spectacular and expensive failure, and one which 
did not enter into the popular mainstream.

Another failure concerns the outgrowth of the wonderful work regarding 
Cybernetics in other fields such as control theory, back in the late 
1950s, early 1960s, when there was this notion that cybernetics was going 
to solve all our problems and allow us to figure out how social systems 
work, etc. That was a colossal failure -- not because that idea was 
necessarily wrong, but because the techniques for doing so didn't exist at 
that point -- and, if we're realistic, may in fact never exist. The 
applications of Cybernetics that were spectacularly successful are not 
even called Cybernetics because they're so ingrained in technology, in 
fields like control theory, and in the aerospace techniques that were used 
to put men on the moon. Those were the great successes of Cybernetics, 
remarkable successes, but in a more narrow technological field.

This brings us to the Internet, which in some sense is almost like Anti 
Cybernetics, the evil twin of Cybernetics. The word Cybernetics comes from 
the Greek word kybernotos which means a governor -- helmsman, actually, 
the kybernotos was the pilot of a ship. Cybernetics, as initially 
conceived, was about governing, or controlling, or guiding. The great 
thing about the Internet, as far as I'm concerned, is that it's completely 
out of control. In some sense the fact of the Internet goes way beyond and 
completely contradicts the Cybernetic ideal. But, in another sense -- the 
way in which the Internet and cybernetics are related, Cybernetics was 
fundamentally on the right track. As far as I'm concerned what's really 
going on in the world is that there's a physical world where things 
happen. I'm a physicist by training and I was taught to think of the world 
in terms of energy, momentum, pressure, entropy. You've got all this 
energy, things are happening, things are pushing on other things, things 
are bouncing around.

But that's only half the story. The other half of the story, its 
complementary half, is the story about information. In one way you can 
think about what's going on in the world as energy, stuff moving around, 
bouncing off each other -- that's the way people have thought about the 
world for over 400 years, since Galileo and Newton. But what was missing 
from that picture was what that stuff was doing: how, why, what? These are 
questions about information. What is going on? It's a question about 
information being processed. Thinking about the world in terms of 
information is complementary to thinking about it in terms of energy. To 
my mind, that is where the action is, not just thinking about the world as 
information on its own, or as energy on its own, but looking at the 
confluence of information and energy and how they play off against each 
other. That's exactly what Cybernetics was about. Wiener, who is the real 
father of the field of Cybernetics, conceived of Cybernetics in terms of 
information, things like feedback control. How much information, for 
example, do you need to make something happen?

The first physicists studying these problems were scientists who happened 
to be physicists, and the first person who was clearly aware of the 
connection between information, entropy, and physical mechanics and energy 
like quanta was Maxwell. Maxwell, in the 1850s and 60s, was the first 
person to write down formulas that related what we would now call 
information -- ideas of information -- to things like energy and entropy. 
He was also the first person to make such an explicit connection.

He also had this wonderfully evocative far-out, William Gibsonesque notion 
of a demon. "Maxwell's Demon" is this hypothetical being that was able to 
look very closely at the molecules of gas whipping around in a room, and 
then rearrange them. Maxwell even came up with a model in which the demon 
was sitting at a partition, a tiny door, between two rooms and he could 
open and shut this door very rapidly. If he saw fast molecules coming from 
the right and slow molecules coming from the left, then he'd open the door 
and let the fast molecules go in the lefthand side, and let the slow 
molecules go into the righthand side.

And since Maxwell already knew about this connection between the average 
speed of molecules and entropy, and he also knew that entropy had 
something to do with the total number of configurations, the total number 
of states a system can have, he pointed out, that if the demon continues 
to do this, the stuff on the lefthand side will get hot, and the stuff on 
the righthand side will get cold, because the molecules over on the left 
are fast, and the molecules on the right are slow.

He also pointed out that there is something screwy about this because the 
demon is doing something that shouldn't take a lot of effort since the 
door can be as light as you want, the demon can be as small as you want, 
the amount of energy you use to open and shut the door can be as small as 
you desire, and yet somehow the demon is managing to make something hot on 
one side. Maxwell pointed out that this was in violation of all the laws 
of thermodynamics -- in particular the second law of thermodynamics which 
says that if you've got a hot thing over here and a cold thing over there, 
then heat flows from the hot thing to the cold thing, and the hot thing 
gets cooler and the cold thing gets hotter, until eventually they end up 
the same temperature. And it never happens the opposite way. You never see 
something that's all the same temperature spontaneously. Maxwell pointed 
out that there was something funny going on, that there was this 
connection between entropy and this demon who was capable of processing 

To put it all in perspective, as far as I can tell, the main thing that 
separates humanity from most other living things, is the way that we deal 
with information. Somewhere along the line we developed methods, 
sophisticated mechanisms, for communicating via speech. Somewhere along 
the line we developed natural language, which is a universal method for 
processing information. Anything that you can imagine is processed with 
information and anything that could be said, can be said using language.

That probably happened around a hundred thousand years ago, and since 
then, the history of human beings has been the development of ever more 
sophisticated ways of registering, processing, transforming, and dealing 
with information. Society through this methodology creates an 
organizational formula that is totally wild compared with the 
organizational structures of most other species, which makes the human 
species distinctive, if there is something at all that makes us 
distinctive. In some sense we're just like any ordinary species out there. 
The extent to which we are different has to do with having more 
sophisticated methods for processing information.

Something else has happened with computers. What's happened with society 
is that we have created these devices, computers, which already can 
register and process huge amounts of information, which is a significant 
fraction of the amount of information that human beings themselves, as a 
species, can process. When I think of all the information being processed 
there, all the information being communicated back and forth over the 
Internet, or even just all the information that you and I can communicate 
back and forth by talking, I start to look at the total amount of 
information being processed by human beings -- and their artifacts -- we 
are at a very interesting point of human history, which is at the stage 
where our artifacts will soon be processing more information than we 
physically will be able to process. So I have to ask, how many bits am I 
processing per second in my head? I could estimate it, it's going to be 
around ten billion neurons, something like 10 to the 15 bits per second, 
around a million billion bits per second.

Hell if I know what it all means -- we're going to find out. That's the 
great thing. We're going to be around to find out some of what this means. 
If you think that information processing is where the action is, it may 
mean in fact that human beings are not going to be where the action is 
anymore. On the other hand, given that we are the people who created the 
devices that are doing this mass of information processing, we, as a 
species, are uniquely poised to make our lives interesting and fun in 
completely unforeseen ways.

Every physical system, just by existing, can register information. And 
every physical system, just by evolving according to its own peculiar 
dynamics, can process that information. I'm interested in how the world 
registers information and how it processes it. Of course, one way of 
thinking about all of life and civilization is as being about how the 
world registers and processes information. Certainly that's what sex is 
about; that's what history is about. But since I'm a scientist who deals 
with the physics of how things process information, I'm actually 
interested in that notion in a more specific way. I want to figure out not 
only how the world processes information, but how much information it's 
processing. I've recently been working on methods to assign numerical 
values to how much information is being processed, just by ordinary 
physical dynamics. This is very exciting for me, because I've been working 
in this field for a long time trying to come up with mathematical 
techniques for characterizing how things process information, and how much 
information they're processing.

About a year or two ago, I got the idea of asking the question, given the 
fundamental limits on how the world is put together -- (1) the speed of 
light, which limits how fast information can get from one place to 
another, (2) Planck's constant, which tells you what the quantum scale is, 
how small things can actually get before they disappear altogether, and 
finally (3) the last fundamental constant of nature, which is the 
gravitational constant, which essentially tells you how large things can 
get before they collapse on themselves -- how much information can 
possibly be processed. It turned out that the difficult part of this 
question was thinking it up in the first place. Once I'd managed to pose 
the question, it only took me six months to a year to figure out how to 
answer it, because the basic physics involved was pretty straightforward. 
It involved quantum mechanics, gravitation, perhaps a bit of quantum 
gravity thrown in, but not enough to make things too difficult.

The other motivation for trying to answer this question was to analyze 
Moore's Law. Many of our society's prized objects are the products of this 
remarkable law of miniaturization -- people have been getting extremely 
good at making the components of systems extremely small. This is what's 
behind the incredible increase in the power of computers, what's behind 
the amazing increase in information technology and communications, such as 
the Internet, and it's what's behind pretty much every advance in 
technology you can possibly think of -- including fields like material 
science. I like to think of this as the most colossal land grab that's 
ever been done in the history of mankind.

From an engineering perspective, there are two ways to make something 
bigger: One is to make it physically bigger, (and human beings spent a lot 
of time making things physically bigger, working out ways to deliver more 
power to systems, working out ways to actually build bigger buildings, 
working out ways to expand territory, working out ways to invade other 
cultures and take over their territory, etc.) But there's another way to 
make things bigger, and that's to make things smaller. Because the real 
size of a system is not how big it actually is, the real size is the ratio 
between the biggest part of a system and the smallest part of a system. Or 
really the smallest part of a system that you can actually put to use in 
doing things. For instance, the reason that computers are so much more 
powerful today than they were ten years ago is that every year and a half 
or so, the basic components of computers, the basic wires, logic chips 
etc., have gone down in size by a factor of two. This is known as Moore's 
Law, which is just a historical fact about history of technology.

Every time something's size goes down by a factor of two, you can cram 
twice as many of them into a box, and so every two years or so, the power 
of computers doubles, and over the course of fifty years the power of 
computers has gone up by a factor of a million or more. The world has 
gotten a million times bigger because we've been able to make the smallest 
parts of the world a million times smaller. This makes this an exciting 
time to live in, but a reasonable question to ask is, where is all this 
going to end? Since Moore proposed it in the early 1960s, Moore's Law has 
been written off numerous times. It was written off in the early 1970s 
because people thought that fabrication techniques for integrated circuits 
were going to break down and you wouldn't be able to get things smaller 
than a scale size of ten microns.

Now Moore's Law is being written off again because people say that the 
insulating barriers between wires in computers are getting to be only a 
few atoms thick, and when you have an insulator that's only a few atoms 
thick then electrons can tunnel through them and it's not a very good 
insulator. Well, perhaps that will stop Moore's Law, but so far nothing 
has stopped it.

At some point Moore's Law has to stop? This question involves the ultimate 
physical limits to computation: you can't send signals faster than the 
speed of light, you can't make things smaller than the laws of quantum 
mechanics tell you that you can, and if you make things too big, then they 
just collapse into one giant black hole. As far as we know, it's 
impossible to fool Mother Nature. I thought it would be interesting to see 
what the basic laws of physics said about how fast, how small, and how 
powerful, computers can get. Actually these two questions: given the laws 
of physics, how powerful can computers be; and where must Moore's Law 
eventually stop -- turn out to be exactly the same, because they stop at 
the same place, which is where every available physical resource is used 
to perform computation. So every little subatomic particle, every ounce of 
energy, every photon in your system -- everything is being devoted towards 
performing a computation. The question is, how much computation is that? 
So in order to investigate this, I thought that a reasonable form of 
comparison would be to look at what I call the ultimate laptop. Let's ask 
just how powerful this computer could be.

The idea here is that we can actually relate the laws of physics and the 
fundamental limits of computation to something that we are familiar with 
-- something of human scale that has a mass of about a kilogram, like a 
nice laptop computer, and has about a liter in volume, because kilograms 
and liters are pretty good to hold in your lap, are a reasonable size to 
look at, and you can put it in your briefcase, et cetera. After working on 
this for nearly a year what I was able to show was that the laws of 
physics give absolute answers to how much information you could process 
with a kilogram of matter confined to a volume of one liter. Not only 
that, surprisingly, or perhaps not so surprisingly, the amount of 
information that can be processed, the number of bits that you could 
register in the computer, and the number of operations per second that you 
could perform on these bits are related to basic physical quantities, and 
to the aforementioned constants of nature, the speed of light, Planck's 
constant, and the gravitational constant. In particular you can show 
without much trouble that the number of ops per second -- the number of 
basic logical operations per second that you can perform using a certain 
amount of matter is proportional to the energy of this matter.

For those readers who are technically-minded, it's not very difficult to 
whip out the famous formula E = MC2 and show, using work of Norm Margolus 
and Lev Levitin here in Boston that the total number of elementary logical 
operations that you can perform per second using a kilogram of matter is 
the amount of energy, MC2, times two, divided by H-bar Planck's constant, 
times pi. Well, you don't have to be Einstein to do the calculation; the 
mass is one kilogram, the speed of light is 3 times ten to the eighth 
meters per second, so MC2 is about ten to the 17th joules, quite a lot of 
energy (I believe it's roughly the amount of energy used by all the 
world's nuclear power plants in the course of a week or so), a lot of 
energy, but let's suppose you could use it to do a computation. So you've 
got ten to the 17th joules, and H-bar, the quantum scale, is ten to the 
minus 34 joules per second, roughly. So there you go. I have ten to the 
17th joules, I divide by ten to the minus 34 joules-seconds, and I have 
the number of ops: ten to the 51 ops per second. So you can perform 10 to 
the 51 operations per second, and ten to the 51 is about a billion billion 
billion billion billion billion billion ops per second -- a lot faster 
than conventional laptops. And this is the answer. You can't do any better 
than that, so far as the laws of physics are concerned.

Of course, since publication of the Nature article, people keep calling me 
up to order one of these laptops; unfortunately the fabrication plant to 
build it has not yet been constructed. You might then actually ask why it 
is that our conventional laptops, are so slow by comparison when we've 
been on this Moore's Law track for 50 years now? The answer is that they 
make the mistake, which could be regarded as a safety feature of the 
laptop, of locking up most of their energy in the form of matter, so that 
rather than using that energy to manipulate information and transform it, 
most of it goes into making the laptop sit around and be a laptop.

As you can tell, if I were to take a week's energy output of all the 
world's nuclear power plants and liberate it all at once, I would have 
something that looked a lot like a thermonuclear explosion, because a 
thermonuclear explosion is essentially taking roughly a kilogram of matter 
and turning it into energy. So you can see right away that the ultimate 
laptop would have some relatively severe packaging problems. Since I am a 
professor of mechanical engineering at MIT, I think packaging problems is 
where it's at. We're talking about some very severe material and 
fabrication problems to prevent this thing from taking not only you but 
the entire city of Boston out with it when you boot it up the first time.

Needless to say, we didn't actually figure out how we were going to put 
this thing into a package, but that's part of the fun of doing 
calculations according to the ultimate laws of physics. We decided to 
figure out how many ops per second we could perform, and to worry about 
the packaging afterwards. Now that we've got 10 to the 51 ops per second 
the next question is: what's the memory space of this laptop.

When I go out to buy a new laptop, I first ask how many ops per second can 
it perform? If it's something like a hundred megahertz, it's pretty slow 
by current standards; if it's a gigahertz, that's pretty fast though we're 
still very far away from the 10 to the 51 ops per second. With a 
gigahertz, we're approaching 10 to the 10th, 10 to the 11th, 10 to the 
12th, depending how ops per second are currently counted. Next, how many 
bits do I have -- how big is the hard drive for this computer, or how big 
is its RAM? We can also use the laws of physics to calculate that figure. 
And computing memory capability is something that people could have done 
back in the early decades of this century.

We know how to count bits. We take the number of states, and the number of 
states is two raised to the power of the number of bits. Ten bits, two to 
the tenth states, 1024 states. Twenty bits, two to the 20 bits, roughly a 
million states. You keep on going and you find that with about 300 bits, 
two to the 300, well, it's about ten to the one hundred, which is 
essentially a bit greater than the number of particles in the universe. If 
you had 300 bits, you could assign every particle in the universe a serial 
number, which is a powerful use of information. You can use a very small 
number of bits to label a huge number of bits.

How many bits does this ultimate laptop have?

I have a kilogram of matter confined to the volume of a liter; how many 
states, how many possible states for matter confined to the volume of a 
liter can there possibly be?

This happened to be a calculation that I knew how to do, because I had 
studied cosmology, and in cosmology there's this event, called the Big 
Bang, which happened a long time ago, about 13 billion years ago, and 
during the Big Bang, matter was at extremely high densities and pressures.

I learned from cosmology how to calculate the number of states for matter 
of very high densities and pressures. In actuality, the density is not 
that great. I have a kilogram of matter in a liter. The density is similar 
to what we might normally expect today. However, if you want to ask what 
the number of states is for this matter in a liter, I've got to calculate 
every possible configuration, every possible elementary quantum state for 
this kilogram of matter in a liter of volume. It turns out, when you count 
most of these states, that this matter looks like it's in the midst of a 
thermonuclear explosion. Like a little piece of the Big Bang -- a few 
seconds after the universe was born -- when the temperature was around a 
billion degrees. At a billion degrees, if you ask what most states for 
matter are if it's completely liberated and able to do whatever it wants, 
you'll find that it looks like a lot like a plasma at a billion degrees 
Kelvin. Electrons and positrons are forming out of nothing, going back 
into photons again, there's a lot of elementary particles whizzing about 
and it's very hot. Lots of stuff is happening and you can still count the 
number of possible states using the conventional methods that people use 
to count states in the early universe; you take the logarithm of the 
number of states, get a quantity that's normally thought of as being the 
entropy of the system (the entropy is simply the logarithm of the number 
of states which also gives you the number of bits, because the logarithm 
of the number of states, the base 2, is the number of bits -- because the 
number of bits raised to the power of -- 2 to the power of the number of 
bits is the number of states. What more do I need to say?)

When we count them, we find that there are roughly 10 to the 31 bits 
available. That means that there's 2 to the 10 to the 31 possible states 
that this matter could be in. That's a lot of states - but we can count 
them. The interesting thing about that is that you notice we've got 10 to 
the 31 bits, we're performing 10 to the 51 ops per second, so each bit can 
perform about 10 to the 20 ops per second. What does this quantity mean?

It turns out that the quantity -- if you like, the number of ops per 
second per bit is essentially the temperature of this plasma. And I take 
this plasma, I multiply it by Bell's constant, divide by Planck's 
constant, and what I get is the energy per bit, essentially; that's what 
temperature is. It tells you the energy per bit. It tells you how much 
energy is available for a bit to perform a logical operation. Since I know 
if I have a certain amount of energy I could perform a certain number of 
operations per second, then the temperature tells me how many ops per bit 
per second I can perform.

Then I know not only the number of ops per second, and the number of bits, 
but also the number of ops per bit per second that can be performed by 
this ultimate laptop, a kilogram of matter in a liter volume; it's the 
number of ops per bit per second that could be performed by these 
elementary particles back at the beginning of time by the Big Bang; it's 
just the total number of ops that each bit can perform per second. The 
number of times it can flip, the number of times it can interact with its 
neighboring bits, the number of elementary logical operations. And it's a 
number, right? 10 to the 20. Just the way that the total number of bits, 
10 to the 31, is a number -- it's a physical parameter that characterizes 
a kilogram of matter and a liter of volume. Similarly, 10 to the 51 ops 
per second is the number of ops per second that characterize a kilogram of 
matter, whether it's in a liter volume or not.

We've gone a long way down this road, so there's no point in stopping -- 
at least in these theoretical exercises where nobody gets hurt. So far all 
we've used are the elementary constants of nature, the speed of light, 
which tells us the rate of converting matter into energy or E = MC2. The 
speed of light tells us how much energy we get from a particular mass. 
Then we use the Planck scale, the quantum scale, because the quantum scale 
tells you both how many operations per second you can get from a certain 
amount of energy, and it also tells you how to count the number of states 
available for a certain amount of energy. So by taking the speed of light, 
and the quantum scale, we are able to calculate the number of ops per 
second that a certain amount of matter can perform, and we're able to 
calculate the amount of memory space that we have available for our 
ultimate computer.

Then we can also calculate all sorts of interesting issues, like what's 
the possible input-output rate for all these bits in a liter of volume. 
That can actually be calculated quite easily from what I've just 
described, because to get all this information into and out of a liter 
volume -- take a laptop computer -- you can say okay, here's all these 
bits, they're sitting in a liter volume, let's move this liter volume 
over, by its own distance, at the speed of light. You're not going to be 
able to get the information in or out faster than that.

We can find out how many bits per second we get into and out of our 
ultimate laptop. And we find we can get around 10 to the 40, or 10 to the 
41, or perhaps, in honor of Douglas Adams and his mystical number 42, even 
10 to the 42 bits per second in and out of our ultimate laptop. So you can 
calculate all these different parameters that you might think are 
interesting, and that tells you how good a modem you could possibly have 
for this ultimate laptop -- how many bits per second can you get in and 
out over the Ultimate Internet, whatever the ultimate Internet would be. I 
guess the Ultimate Internet is just space/time itself in this picture.

I noted that you can't possibly do better than this, right? These are the 
laws of physics. But you might be able to do better in other ways. For 
example, let's think about the architecture of this computer. I've got 
this computer that's doing 10 to the 51 ops per second, or 10 to the 31 
bits. Each bit can flip 10 to the 20 times per second. That's pretty fast. 
The next question is how long does it take a bit on this side of the 
computer to send a signal to a bit on that side of the computer in the 
course of time it takes it to do an operation.

As we've established, it has a liter volume, which is about ten 
centimeters on each side, so it takes about ten to the minus ten seconds 
for light to go from one side to another -- one ten billionth of a second 
for light to go from this side to the other. These bits are flipping 10 to 
the 20 times per second -- a hundred billion billion times per second. 
This bit is flipping ten billion times in the course of time it takes a 
signal to go from one side of the computer to the other. This is not a 
very serial computation. A lot of action is taking place over here in the 
time it takes to communicate when all the action is taking place over on 
this side of the computer. This is what's called a parallel computation.

You could say that in the kinds of densities of matter that we're familiar 
with, like a kilogram per liter volume, which is the density of water, we 
find that we can only perform a very parallel computation, if we operate 
at the ultimate limits of computation; lots of computational action takes 
place over here during the time it takes a signal to go from here to there 
and back again.

How can we do better? How could we make the computation more serial?

Let's suppose that we want our machine to do more serial computation, so 
in the time it takes to send a signal from one side of the computer to the 
other, there are fewer ops that are being done. The obvious solution is to 
make the computer smaller, because if I make the computer smaller by a 
factor of two, it only takes half the time for light, for a signal, for 
information, to go from one side of the computer to the other.

If I make it smaller by a factor of ten billion, it only takes one ten 
billionth of the time for its signal to go from one side of the computer 
to the other. You also find that when you make it smaller, these pieces of 
the computer tend to speed up, because you tend to have more energy per 
bit available in each case. If you go through the calculation you find out 
that as the computer gets smaller and smaller, as all the mass is 
compressed into a smaller and smaller volume, you can do a more serial 

When does this process stop? When can every bit in the computer talk with 
every other bit, in the course of time it takes for a bit to flip? When 
can everybody get to talk with everybody else in the same amount of time 
that it takes them to talk with their neighbors?

As you make the computer smaller and smaller, it gets denser and denser, 
until you have a kilogram of matter in an ever smaller volume. Eventually 
the matter assumes more and more interesting configurations, until it's 
actually going to take a very high pressure to keep this system down at 
this very small volume. The matter assumes stranger and stranger 
configurations, and tends to get hotter and hotter and hotter, until at a 
certain point a bad thing happens. The bad thing that happens is that it's 
no longer possible for light to escape from it -- it becomes a black hole.

What happens to our computation at this point. This is probably very bad 
for a computation, right? Or rather, it's going to be bad for 
input-output. Input is good, because stuff goes in, but output is bad 
because it doesn't come out since it's a black hole. Luckily, however, 
we're safe in this, because the very laws of quantum mechanics that we 
were using to calculate how much information a physical system can 
compute, how fast it can perform computations, and how much information it 
can register, actually hold.

Stephen Hawking showed, in the 1970s, that black holes, if you treat them 
quantum-mechanically, actually can radiate out information. There's an 
interesting controversy as to whether that information has anything to do 
with the information that went in. Stephen Hawking and John Preskill have 
a famous bet, where Preskill says yes -- the information that comes out of 
a black hole reflects the information that went in. Hawking says no -- the 
information that comes out of a black hole when it radiates doesn't have 
anything to do with the information that went in; the information that 
went in goes away. I don't know the answer to this.

But let's suppose for a moment that Hawking is wrong and Preskill is 
right. Let's suppose for a moment that in fact the information that comes 
out of a black hole when it evaporates, radiates information the wave 
length of the radiation coming out which is the radius of the black hole. 
This black hole, this kilogram black hole, is really radiating at a 
whopping rate; it's radiating out these photons with wave lengths of 10 to 
the minus 27 meters, this is not something you would actually wish to be 
close to -- it would be very dangerous. In fact it would look a lot like a 
huge explosion. But let's suppose that in fact that information that's 
being radiated out by the black hole is in fact the information that went 
in to construct it, but simply transformed in a particular way. What you 
then see is that the black hole can be thought of in some sense as 
performing a computation.

You take the information about the matter that's used to form the black 
hole, you program it in the sense that you give it a particular 
configuration, you put this electron here, you put that electron there, 
you make that thing vibrate like this, and then you collapse this into a 
black hole, 10 to the minus 27 seconds later, in one hundred billion 
billionth of a second, the thing goes cablooey, and you get all this 
information out again, but now the information has been transformed, by 
some dynamics, and we don't know what this dynamics is, into a new form.

In fact we would need to know something like string theory or quantum 
gravity to figure out how it's been transformed. But you can imagine that 
this could in fact function as a computer. We don't know how to make it 
compute, but indeed, it's taking in information, it's transforming it in a 
systematic form according to the laws of physics, all right, and then 
poop! It spits it out again.

It's a dangerous thing -- the Ultimate Laptop was already pretty 
dangerous, because it looked like a thermonuclear explosion inside of a 
liter bottle of coca cola. This is even worse, because in fact it looks 
like a thermonuclear explosion except that it started out at a radius of 
10 to the minus 27 meters, one billion billion billionth of a meter, so 
it's really radiating at a very massive rate. But suppose you could 
somehow read information coming out of the black hole. You would indeed 
have performed the ultimate computation that you could have performed 
using a kilogram of matter, in this case confining it to a volume of 10 to 
the minus 81 cubic meters. Pretty minuscule but we're allowed to imagine 
this happening.

Is there anything more to the story?

After writing my paper on the ultimate laptop in Nature, I realized this 
was insufficiently ambitious; that of course the obvious question to ask 
at this point is not what is the ultimate computational capacity of a 
kilogram of matter, but instead to ask what is the ultimate computational 
capacity of the universe as a whole? After all, the universe is processing 
information, right? Just by existing, all physical systems register 
information, just by evolving their own natural physical dynamics, they 
transform that information, they process it. So the question then is how 
much information has the universe processed since the Big Bang?


13. http://xxx.lanl.gov/ 14. 
http://www.edge.org/3rd_culture/bios/lloyd.html 15. 
http://www.edge.org/3rd_culture/bios/varela.html 16. 

[Now to the famous article by Seth Lloyd.]


Over the past half century, the amount of information that computers are 
capable of processing and the rate at which they process it has doubled 
every 18 months, a phenomenon known as Moore's law. A variety of 
technologies--most recently, integrated circuits-- have enabled this 
exponential increase in information processing power. But there is no 
particular reason why Moore's law should continue to hold: it is a law of 
human ingenuity, not of nature. At some point, Moore's law will break 
down. The question is, when?

The answer to this question will be found by applying the laws of physics 
to the process of computation (1-85). Extrapolation of current exponential 
improvements over two more decades would result in computers that process 
information at the scale of individual atoms. Although an Avogadro-scale 
computer that can act on 10^23 bits might seem implausible, prototype 
quantum computers that store and process information on individual atoms 
have already been demonstrated (64,65,76-80). Existing quantum computers 
may be small and simple, and able to perform only a few hundred operations 
on fewer than ten quantum bits or 'qubits', but the fact that they work at 
all indicates that there is nothing in the laws of physics that forbids 
the construction of an Avogadro-scale computer.

The purpose of this article is to determine just what limits the laws of 
physics place on the power of computers. At first, this might seem a 
futile task: because we do not know the technologies by which computers 
1000, 100, or even 10 years in the future will be constructed, how can we 
determine the physical limits of those technologies? In fact, I will show 
that a great deal can be determined concerning the ultimate physical 
limits of computation simply from knowledge of the speed of light, c = 
2.9979 x 10^8 ms^-1, Planck's reduced constant, h-bar= h/2pi = 1.0545 x 
10^-34 Js, and the gravitational constant, G = 6.673 x 10^-11 m^3 kg^-1 
s^-2. Boltzmann's constant, k-sub-B = 1.3805 x10^-23 J K^-1, will also be 
crucial in translating between computational quantities such as memory 
space and operations per bit per second, and thermodynamic quantities such 
as entropy and temperature. In addition to reviewing previous work on how 
physics limits the speed and memory of computers, I present results--which 
are new except as noted--of the derivation of the ultimate speed limit to 
computation, of trade-offs between memory and speed, and of the analysis 
of the behaviour of computers at physical extremes of high temperatures 
and densities.

Before presenting methods for calculating these limits, it is important to 
note that there is no guarantee that these limits will ever be attained, 
no matter how ingenious computer designers become. Some extreme cases such 
as the black-hole computer described below are likely to prove extremely 
difficult or impossible to realize. Human ingenuity has proved great in 
the past, however, and before writing off physical limits as unattainable, 
we should realize that certain of these limits have already been attained 
within a circumscribed context in the construction of working quantum 
computers. The discussion below will note obstacles that must be 
sidestepped or overcome before various limits can be attained.

Energy limits speed of computation

To explore the physical limits of computation, let us calculate the 
ultimate computational capacity of a computer with a mass of 1 kg 
occupying a volume of 1 litre, which is roughly the size of a conventional 
laptop computer. Such a computer, operating at the limits of speed and 
memory space allowed by physics, will be called the 'ultimate laptop' 
(Fig. 1).

[Figure 1 The ultimate laptop. The 'ultimate laptop' is a computer with a 
mass of 1 kg and a volume of 1 liter operating at the fundamental limits 
of speed and memory capacity fixed by physics. The ultimate laptop 
performs 2mc^2/pi h-bar = 5.4258 x 10^50 logical operations per second on 
~10^31 bits. Although its computational machinery is in fact in a highly 
specified physical state with zero entropy, while it performs a 
computation that uses all its resources of energy and memory space it 
appears to an outside observer to be in a thermal state at ~10^9 degrees 
Kelvin. The ultimate laptop looks like a small piece of the Big Bang.]

First, ask what limits the laws of physics place on the speed of such a 
device. As I will now show, to perform an elementary logical operation in 
time Delta-t requires an average amount of energy E greater than or equal 
to pi h-bar/2Delta-t. As a consequence, a system with average energy E can 
perform a maximum of 2E/pi h-bar logical operations per second. A 1-kg 
computer has average energy E = mc^2 = 8.9874 x 10^16 J. Accordingly, the 
ultimate laptop can perform a maximum of 5.4258 x 10^50 operations per 

Maximum speed per logical operation

For the sake of convenience, the ultimate laptop will be taken to be a 
digital computer. Computers that operate on nonbinary or continuous 
variables obey similar limits to those that will be derived here. A 
digital computer performs computation by representing information in the 
terms of binary digits or bits, which can take the value 0 or 1, and then 
processes that information by performing simple logical operations such as 
AND, NOT and FANOUT. The operation, AND, for instance, takes two binary 
inputs X and Y and returns the output 1 if and only if both X and Y are 1; 
otherwise it returns the output 0. Similarly, NOT takes a single binary 
input X and returns the output 1 if X = 0 and 0 if X = 1. FANOUT takes a 
single binary input X and returns two binary outputs, each equal to X. Any 
boolean function can be constructed by repeated application of AND, NOT 
and FANOUT. A set of operations that allows the construction of arbitrary 
boolean functions is called universal. The actual physical device that 
performs a logical operation is called a logic gate.

How fast can a digital computer perform a logical operation? During such 
an operation, the bits in the computer on which the operation is performed 
go from one state to another. The problem of how much energy is required 
for information processing was first investigated in the context of 
communications theory by Levitin (11-16), Bremermann (17-19), Beckenstein 
(20-22) and others, who showed that the laws of quantum mechanics 
determine the maximum rate at which a system with spread in energy Delta-E 
can move from one distinguishable state to another. In particular, the 
correct interpretation of the time-energy Heisenberg uncertainty principle 
Delta-E Delta-t is greater than or equal to h-bar is not that it takes 
time Delta-t to measure energy to an accuracy Delta-E (a fallacy that was 
put to rest by Aharonov and Bohm (23,24)), but rather that a quantum state 
with spread in energy Delta-E takes time at least Deltat = pi 
h-bar/2Delta-E to evolve to an orthogonal (and hence distinguishable) 
state(23-26). More recently, Margolus and Levitin (15,16) extended this 
result to show that a quantum system with average energy E takes time at 
least Delta-t = pi h-bar/2E to evolve to an orthogonal state.

Performing quantum logic operations

As an example, consider the operation NOT performed on a qubit with 
logical states |0> and |1>. (For readers unfamiliar with quantum 
mechanics, the 'bracket' notation |> signifies that whatever is contained 
in the bracket is a quantum-mechanical variable; |0> and |1> are vectors 
in a two-dimensional vector space over the complex numbers.) To flip the 
qubit, one can apply a potential H = E-sub-0|E-sub-0><E-sub-0|+ 
E-sub-1|E-sub-1><E-sub-1| with energy eigenstates |E-sub-0> = 
(1/sqrt2)(|0> + |1>) and |E-sub-1> = (1/sqrt2)(|0> - |1>). Because |0> = 
(1/sqrt2)(|E-sub-0> + |E-sub-1>) and |1> = (1/sqrt2)(|E-sub-0>- 
|E-sub-1>), each logical state |0>, |1>has spread in energy Delta-E = 
(E-sub-1 - E-sub-0)/2. It is easy to verify that after a length of time 
Delta-t = pi h-bar/2Delta-E the qubit evolves so that |0> -> |1> and |1> 
-> |0>. That is, applying the potential effects a NOT operation in a time 
that attains the limit given by quantum mechanics. Note that the average 
energy E of the qubit in the course of the logical operation is <0|H|0> = 
<1|H|1> = (E-sub-0+ E-sub-1)/2 = E-sub-0+ Delta-E. Taking the ground-state 
energy E-sub-0 = 0 gives E = Delta-E. So the amount of time it takes to 
perform a NOT operation can also be written as Delta-t = pi h-bar/2E. It 
is straightforward to show (15,16) that no quantum system with average 
energy E can move to an orthogonal state in a time less than Delta-t. That 
is, the speed with which a logical operation can be performed is limited 
not only by the spread in energy, but also by the average energy. This 
result will prove to be a key component in deriving the speed limit for 
the ultimate laptop.

AND and FANOUT can be enacted in a way that is analogous to the NOT 
operation. A simple way to perform these operations in a 
quantum-mechanical context is to enact a so-called Toffoli or 
controlled-controlled-NOT operation (31). This operation takes three 
binary inputs, X, Y and Z, and returns three outputs, X', Y' and Z'.

The first two inputs pass through unchanged, that is, X' = X, Y' = Y. The 
third input passes through unchanged unless both X and Y are 1, in which 
case it is flipped. This is universal in the sense that suitable choices 
of inputs allow the construction of AND, NOT and FANOUT. When the third 
input is set to zero, Z = 0, then the third output is the AND of the first 
two: Z' = X AND Y. So AND can be constructed. When the first two inputs 
are 1, X = Y = 1, the third output is the NOT of the third input, Z' = NOT 
Z. Finally, when the second input is set to 1, Y = 1, and the third to 
zero, Z = 0, the first and third output are the FANOUT of the first input, 
X' = X, Z' = X. So arbitrary boolean functions can be constructed from the 
Toffoli operation alone.

By embedding a controlled-controlled-NOT gate in a quantum context, it is 
straightforward to see that AND and FANOUT, like NOT, can be performed at 
a rate 2E/pi h-bar times per second, where E is the average energy of the 
logic gate that performs the operation. More complicated logic operations 
that cycle through a larger number of quantum states (such as those on 
non-binary or continuous quantum variables) can be performed at a rate 
E/pi h-bar--half as fast as the simpler operations (15,16). Existing 
quantum logic gates in optical-atomic and nuclear magnetic resonance (NMR) 
quantum computers actually attain this limit. In the case of NOT, E is the 
average energy of interaction of the qubit's dipole moment (electric 
dipole for optic-atomic qubits and nuclear magnetic dipole for NMR qubits) 
with the applied electromagnetic field. In the case of multiqubit 
operations such as the Toffoli operation, or the simpler two-bit 
controlled-NOT operation, which flips the second bit if and only if the 
first bit is 1, E is the average energy in the interaction between the 
physical systems that register the qubits.

Ultimate limits to speed of computation

We are now in a position to derive the first physical limit to 
computation, that of energy. Suppose that one has a certain amount of 
energy E to allocate to the logic gates of a computer. The more energy one 
allocates to a gate, the faster it can perform a logic operation. The 
total number of logic operations performed per second is equal to the sum 
over all logic gates of the operations per second per gate. That is, a 
computer can perform no more than

BigSigma-over-l 1/Delta-t-sub-l is less than or equal to

BigSigma-over- l underneath 2E-sub-l/pi h-bar= 2E/pi h-bar

operations per second. In other words, the rate at which a computer can 
compute is limited by its energy. (Similar limits have been proposed by 
Bremmerman in the context of the minimum energy required to communicate a 
bit (17-19), although these limits have been criticized for 
misinterpreting the energy-time uncertainty relation(21), and for failing 
to take into account the role of degeneracy of energy eigenvalues13,14 and 
the role of nonlinearity in communications7-9.) Applying this result to a 
1-kg computer with energy E = mc^2 = 8.9874 2 1016 J shows that the 
ultimate laptop can perform a maximum of 5.4258 x 10^50 operations per 

Parallel and serial operation

An interesting feature of this limit is that it is independent of computer 
architecture. It might have been thought that a computer could be speeded 
up by parallelization, that is, by taking the energy and dividing it up 
among many subsystems computing in parallel. But this is not the case. If 
the energy E is spread among N logic gates, each one operates at a rate 
2E/pi h-bar N, and the total number of operations per second, N 2E/pi 
h-barN = 2E/pi h-bar, remains the same. then the rate at which they 
operate and the spread in energy per gate decrease. Note that in this 
parallel case, the overall spread in energy of the computer as a whole is 
considerably smaller than the average energy: in general Delta-E = 
sqrt(BigSigma-over-l Delta-E-sub-l^2) ~ sqrt(N Delta-E-sub-l) whereas E = 
BigSigma-over-l E-sub-l = NE-sub-l. efficiently, but it does not alter the 
total number of operations per second. As I will show below, the degree of 
parallelizability of the computation to be performed determines the most 
efficient distribution of energy among the parts of the computer. 
Computers in which energy is relatively evenly distributed over a larger 
volume are better suited for performing parallel computations. More 
compact computers and computers with an uneven distribution of energy are 
performing parallel computations,

Comparison with existing computers

Conventional laptops operate much more slowly than the ultimate laptop. 
There are two reasons for this inefficiency. First, mot of the energy is 
locked up in the mass of the particles of which the computer is 
constructed, leaving only an infintesimal fraction for performing logic. 
Second, a conventional computer uses many degrees of freedom (billions and 
billions of electrons) for registering a single bit. From the physical 
perspective, such a computer operates in a highly redundant fashion. There 
are, however, good technological reasons for such redundancy, with 
conventional designs depending on it for reliability and 
manufacturability. But in the present discussion, the subject is not what 
computers are but what they might be, and in this context the laws of 
physics do not require redundancy to perform logical operations--recently 
constructed quantum microcomputers use one quantum degree of freedom for 
each bit and operate at the Heisenberg limit De;ta-t = pi h-bar/2 Delta-E 
for the time needed to flip a bit (64,65,76-80). Redundancy is, however, 
required for error correction, as will be discussed below.

In sum, quantum mechanics provides a simple answer to the question of how 
fast information can be processed using a given amount of energy. Now it 
will be shown that thermodynamics and statistical mechanics provide a 
fundamental limit to how many bits of informatio can be processed using a 
given amount of energy confined to a given volume. Available energy 
necessarily limits the rate at which a computer can process information. 
Similarly, the maximum entropy of a physical system determines the amount 
of information it can process. Energy limits speed. Entropy limits memory.

Entropy limits memory space

The amount of information that a physical system can store and process is 
related to the number of distingt physical states that are accessible to 
the system. A collection of m two-state systems has 2^m accessible states 
and can register m bits of information. In general, a system with N 
accessible states can register log (base 2) N bits of information. But it 
has been known for more than a century that the number of accessible 
states of a physical system, W, is related to its thermodynamic entropy by 
the formula S = k-sub-B lnW, where kB is Boltzmann's constant. (Although 
this formula is inscribed on Boltzmann's tomb, it is attributed originally 
to Planck; before the turn of the century, kB was often known as Planck's 

The amount of information that can be registered by a physical system is I 
= S(E)/k-sub-B ln2 where S(E) is the thermodynamic entropy of a system 
with expectation value for the energy E. Combining this formula with the 
formula 2E/pi h-bar for the number of logical operations that can be 
performed per second, we see that when it is using all its memory, the 
number of operations per bit per second that our ultimate laptop can 
perform is k-sub-B x 2ln(2)E/pi h-barS = k-sub-B T/h-bar, where T = 
(partialderivativeS/partialderivativeE)^-1 is the temperature of 1 kg of 
matter in a maximum entropy in a volume of 1 liter. The entropy governs 
the amount of information the system can register and the temperature 
governs the number of operations per bit per second that it can perform.

Because thermodynamic entropy effectively counts the number of bits 
available to a physical system, the following derivation of the memory 
space available to the ultimate laptop is based on a thermo dynamic 
treatment of 1 kg of matter confined to a volume of 1 l in a maximum 
entropy state. Throughout this derivation, it is important to remember 
that although the memory space available to the computer is given by the 
entropy of its thermal equilibrium state, the actual state of the ultimate 
laptop as it performs a computation is completely determined, so that its 
entropy remains always equal to zero. As above, I assume that we have 
complete control over the actual state of the ultimate laptop, and are 
able to guide it through its logical steps while insulating it from all 
uncontrolled degrees of freedom. But as the following discussion will make 
clear, such complete control will be difficult to attain (see Box 1).

[Box 1: The role of thermodynamics in computation

[The fact that entropy and information are intimately linked has been 
known since Maxwell introduced his famous 'demon' well over a century ago 
(1). Maxwell's demon is a hypothetical being that uses its 
information-processing ability to reduce the entropy of a gas. The first 
results in the physics of information processing were derived in attempts 
to understand how Maxwell's demon could function (1-4. The role of 
thermodynamics in computation has been examined repeatedly over the past 
half century. In the 1950s, von Neumann (10) speculated that each logical 
operation performed in a computer at temperature T must dissipate energy 
k-sub-B T ln2, thereby increasing entropy by k-sub-B ln2. This speculation 
proved to be false. The precise, correct statement of the role of entropy 
in computation was attributed to Landauer (5), who showed that reversible, 
that is, one- to-one, logical operations such as NOT can be performed, in 
principle, without dissipation, but that irreversible, many-to-one 
operations such as AND or ERASE require dissipation of at least k-sub-B 
ln2 for each bit of information lost. (ERASE is a one-bit logical 
operation that takes a bit, 0 or 1, and restores it to 0.) The argument 
behind Landauer's principle can be readily understood (37). Essentially, 
the one-to-one dynamics of hamiltonian systems implies that when a bit is 
erased the information that it contains has to go somewhere. If the 
information goes into observable degrees of freedom of the computer, such 
as another bit, then it has not been erased but merely moved; but if it 
goes into unobservable degrees of freedom such as the microscopic motion 
of molecules it results in an increase of entropy of at least k-sub-B ln2.

[In 1973, Bennett (28-30) showed that all computations could be performed 
using only reversible logical operations. Consequently, by Landauer's 
principle, computation does not require dissipation. (Earlier work by 
Lecerf (27) had anticipated the possibility of reversible computation, but 
not its physical implications. Reversible computation was discovered 
independently by Fredkin and Toffoli(31).) The energy used to perform a 
logical operation can be 'borrowed' from a store of free energy such as a 
battery, 'invested' in the logic gate that performs the operation, and 
returned to storage after the operation has been performed, with a net 
'profit' in the form of processed information. Electronic circuits based 
on reversible logic have been built and exhibit considerable reductions in 
dissipation over conventional reversible circuits (33-35).

[Under many circumstances it may prove useful to perform irreversible 
operations such as erasure. If our ultimate laptop is subject to an error 
rate of epsilon bits per second, for example, then error-correcting codes 
can be used to detect those errors and reject them to the environment at a 
dissipative cost of epsilon k-sub-B T-sub-E ln2 J s^-1, where T-sub-E is 
the temperature of the environment. (k-sub-B T ln2 is the minimal amount 
of energy required to send a bit down an information channel with noise 
temperature T (ref. 14).) Such error-correcting routines in our ultimate 
computer function as working analogues of Maxwell's demon, getting 
information and using it to reduce entropy at an exchange rate of k-sub-B 
T ln2 joules per bit. In principle, computation does not require 
dissipation. In practice, however, any computer--even our ultimate 
laptop--will dissipate energy.

[The ultimate laptop must reject errors to the environment at a high rate 
to maintain reliable operation. To estimate the rate at which it can 
reject errors to the environment, assume that the computer encodes 
erroneous bits in the form of black-body radiation at the characteristic 
temperature 5.87 x 10^8 K of the computer's memory (21). The 
Stefan-Boltzmann law for black-body radiation then implies that the number 
of bits per unit area than can be sent out to the environment is B = pi^2 
k-sub-B^3 T^3/60ln(2)h-bar^3 c^2 = 7.195 x 10^42 bits per square meter per 
second. As the ultimate laptop has a surface area of 10^-2 m^2 and is 
performing ~10^50 operations per second, it must have an error rate of 
less than 10-^10 per operation in order to avoid over-heating. Even if it 
achieves such an error rate, it must have an energy throughput (free 
energy in and thermal energy out) of 4.04 x 10^26 W--turning over its own 
resting mass energy of mc^2 ~ 10^17 J in a nanosecond! The thermal load of 
correcting large numbers of errors clearly indicates the necessity of 
operating at a slower speed than the maximum allowed by the laws of 

[End of Box 1]

Entropy, energy and temperature

To calculate the number of operations per second that could be performed 
by our ultimate laptop, I assume that the expectation value of the energy 
is E. Accordingly, the total number of bits of memory space available to 
the computer is S(E,V)/k-sub-B ln2 where S(E,V) is the thermodynamic 
entropy of a system with expectation value of the energy E confined to 
volume V. The entropy of a closed system is usually given by the so-called 
microcanonical ensemble, which fixes both the average energy and the 
spread in energy DeltaE, and assigns equal probability to all states of 
the system within a range [E, E + Delta-E]. In the case of the ultimate 
laptop, however, I wish to fix only the average energy, while letting the 
spread in energy vary according to whether the computer is to be more 
serial (fewer, faster gates, with larger spread in energy) or parallel 
(more, slower gates, with smaller spread in energy). Accordingly, the 
ensemble that should be used to calculate the thermodynamic entropy and 
the memory space available is the canonical ensemble, which maximizes S 
for fixed average energy with no constraint on the spread in energy 
Delta-E. The canonical ensemble shows how many bits of memory are 
available for all possible ways of programming the computer while keeping 
its average energy equal to E. In any given computation with average 
energy E, the ultimate laptop will be in a pure state with some fixed 
spread of energy, and will explore only a small fraction of its memory 

In the canonical ensemble, a state with energy E-sub-i has probability 
p-sub-i = (1/Z(T) e^(-E-sub-ii/k-sub-BT) where Z(T) = BigSigma-over-i 
e^(E-sub-i/k-sub-BT) is the partition function, and the temperature T is 
chosen so that E = BigSigma-over-i p-sub-iE-sub-i. The entropy is S = 
-k-sub-bB BigSigma-over-i p-sub-i ln p-sub-i = E/T + k-sub-B lnZ. The 
number of bits of memory space available to the computer is S/k-sub-bB 
ln2. The difference between the entropy as calculated using the canonical 
ensemble and that calculated using the microcanonical ensemble is minimal. 
But there is some subtlety involved in using the canonical ensemble rather 
than the more traditional microcanonical ensemble. The canonical ensemble 
is normally used for open systems that interact with a thermal bath at 
temperature T. In the case of the ultimate laptop, however, it is applied 
to a closed system to find the maximum entropy given a fixed expectation 
value for the energy. As a result, the temperature T = 
(partialS/partialE)^-1 has a somewhat different role in the context of 
physical limits of computation than it does in the case of an ordinary 
thermodynamic system interacting with a thermal bath. Integrating the 
relationship T = (partialS/partialE)^-1 over E yields T = CE/S, where C is 
a constant of order unity (for example, C = 4/3 for black-body radiation, 
C = 3/2 for an ideal gas, and C = 1/2 for a black hole). Accordingly, the 
temperature governs the number of operations per bit per second, k-sub-B 
ln(2) E/h-barS ~ k-sub-B T/h-bar, that a system can perform. As I will 
show later, the relationship between temperature and operations per bit 
per second is useful in investigating computation under extreme physical 

Calculating the maximum memory space

To calculate exactly the maximum entropy for a kilogram of matter in a 
litre volume would require complete knowledge of the dynamics of 
elementary particles, quantum gravity, and so on. Although we do not 
possess such knowledge, the maximum entropy can readily be estimated by a 
method reminiscent of that used to calculate thermodynamic quantities in 
the early Universe (86). The idea is simple: model the volume occupied by 
the computer as a collection of modes of elementary particles with total 
average energy E. The maximum entropy is obtained by calculating the 
canonical ensemble over the modes. Here, I supply a simple derivation of 
the maximum memory space available to the ultimate laptop. A more detailed 
discussion of how to calculate the maximum amount of information that can 
be stored in a physical system can be found in the work of Bekenstein 

For this calculation, assume that the only conserved quantities other than 
the computer's energy are angular momentum and electric charge, which I 
take to be zero. (One might also ask that the number of baryons be 
conserved, but as will be seen below, one of the processes that could take 
place within the computer is black-hole formation and evaporation, which 
does not conserve baryon number.) At a particular temperature T, the 
entropy is dominated by the contributions from particles with mass less 
than k-sub-B T/2c^2. The lth such species of particle contributes energy E 
= r-sub-i<pi^2 V(k-sub-B T)^4/30 h-bar^3 c^3 and entropy S = 2r-sub-l 
k-sub-B pi^2 V(k-sub-B T)^3/ 45h-bar^3 c^3= 4E/3T, where r-sub-l is equal 
to the number of particles/antiparticles in the species (that is, 1 for 
photons, 2 for electrons/positrons) multiplied by the number of 
polarizations (2 for photons, 2 for electrons/positrons) multiplied by a 
factor that reflects particle statistics (1 for bosons, 7/8 for fermions). 
As the formula for S in terms of T shows, each species contributes (2pi)^5 
r-sub-l</90ln2 ~ 10^2 bits of memory space per cubic thermal wavelength 
lambda-sub-T^3, where lambda-sub-T = 2pi h-barc/k-sub-B T. Re-expressing 
the formula for entropy as a function of energy, the estimate for the 
maximum entropy is

S = (4/3)k-sub-B(pi^2rV/30h-bar^3 c^3)^(1/4) E^(3/4) = k-sub-B ln(2)I

where r = BigSigma-over-l r-sub-l. Note that S depends only insensitively 
on the total number of species with mass less than k-sub-bB T/2c^2.

A lower bound on the entropy can be obtained by assuming that energy and 
entropy are dominated by black-body radiation consisting of photons. In 
this case, r = 2, and for a 1-kg computer confined to a volume of a 1 
liter we have k-sub-B T = 8.10 x10^-15 J, or T = 5.87 x 10^8 K. The 
entropy is S = 2.04 x 10^8 J K^-1, which corresponds to an amount of 
available memory space I = S/k-sub-Bln2 = 2.13 x 10^31 bits. When the 
ultimate laptop is using all its memory space it can perform 
2ln(2)k-sub-BE/pi h-bar S = 3ln(2)k-sub-BT/2pi h-bar ~ 10^19 operations 
per bit per second. As the number of operations per second 2E/pi h-bar is 
independent of the number of bits available, the number of operations per 
bit per second can be increased by using a smaller number of bits. In 
keeping with the prescription that the ultimate laptop operates at the 
absolute limits given by physics, in what follows I assume that all 
available bits are used.

This estimate for the maximum entropy could be improved (and slightly 
increased) by adding more species of massless particles (neutrinos and 
gravitons) and by taking into effect the presence of electrons and 
positrons. Note that k-sub-BT/2c^2 = 4.51 x 10^-32 kg, compared with the 
electron mass of 9.1 x 10^31 kg. That is, the ultimate laptop is close to 
a phase transition at which electrons and positrons are produced 
thermally. A more exact estimate of the maximum entropy and hence the 
available memory space would be straightforward to perform, but the 
details of such a calculation would detract from my general exposition, 
and could serve to alter S only slightly. S depends insensitively on the 
number of species of effectively massless particles: a change of r by a 
factor of 10,000 serves to increase S by only a factor of 10.

Comparison with current computers

The amount of information that can be stored by the ultimate laptop, 
~10^31 bits, is much higher than the ~10^10 bits stored on current 
laptops. This is because conventional laptops use many degrees of freedom 
to store a bit whereas the ultimate laptop uses just one. There are 
considerable advantages to using many degrees of freedom to store 
information, stability and controllability being perhaps the most 
important. Indeed, as the above calculation indicates, to take full 
advantage of the memory space available, the ultimate laptop must turn all 
its matter into energy. A typical state of the ultimate laptop's memory 
looks like a plasma at a billion degrees Kelvin--like a thermonuclear 
explosion or a little piece of the Big Bang! Clearly, packaging issues 
alone make it unlikely that this limit can be obtained, even setting aside 
the difficulties of stability and control.

Even though the ultimate physical limit to how much information can be 
stored in a kilogram of matter in a litre volume is unlikely to be 
attained, it may nonetheless be possible to progress some way towards such 
bit densities. In other words, the ultimate limits to memory space may 
prove easier to approach than the ultimate limits to speed. Following 
Moore's law, the density of bits in a computer has gone down from 
approximately one per square centimetre 50 years ago to one per square 
micrometre today, an improvement of a factor of 10^8. It is not 
inconceivable that a similar improvement is possible over the course of 
the next 50 years. In particular, there is no physical reason why it 
should not be possible to store one bit of information per atom. Indeed, 
existing NMR and ion-trap quantum computers already store information on 
individual nuclei and atoms (typically in the states of individual nuclear 
spins or in hyperfine atomic states). Solid-state NMR with high gradient 
fields or quantum optical techniques such as spectral hole-burning provide 
potential technologies for storing large quantities of information at the 
atomic scale. A kilogram of ordinary matter holds on the order of 10^25 
nuclei. If a substantial fraction of these nuclei can be made to register 
a bit, then we could get close to the ultimate physical limit of memory 
without having to resort to thermonuclear explosions. If, in addition, we 
make use of the natural electromagnetic interactions between nuclei and 
electrons in the matter to perform logical operations, we are limited to a 
rate of ~10^15 operations per bit per second, yielding an overall 
information processing rate of ~10^40 operations per second in ordinary 
matter. Although less than the ~1051 operations per second in the ultimate 
laptop, the maximum information processing rate in 'ordinary matter' is 
still quite respectable. Of course, even though such an 'ordinary matter' 
ultimate computer need not operate at nuclear energy levels, other 
problems remain--for example, the high number of bits still indicates 
substantial input/output problems. At an input/output rate of 10^12 bits 
per second, an Avogadro-scale computer with 10^23 bits would take about 
10,000 years to perform a serial read/write operation on the entire 
memory. Higher throughput and parallel input/output schemes are clearly 
required to take advantage of the entire memory space that physics makes 

Size limits parallelization

Up until this point, I have assumed that the ultimate laptop occupies a 
volume of 1 liter. The previous discussion, however, indicates that 
benefits are to be obtained by varying the volume to which the computer is 
confined. Generally speaking, if the computation to be performed is highly 
parallelizable or requires many bits of memory, the volume of the computer 
should be greater and the energy available to perform the computation 
should be spread out evenly among the different parts of the computer. 
Conversely, if the computation to be performed is highly serial and 
requires fewer bits of memory, the energy should be concentrated in 
particular parts of the computer.

A good measure of the degree of parallelization in a computer is the ratio 
between the time it takes to communicate from one side of the computer to 
the other, and the average time it takes to perform a logical operation. 
The amount of time it takes to send a message from one side of a computer 
of radius R to the other is t-sub-com = 2R/c. The average time it takes a 
bit to flip in the ultimate laptop is the inverse of the number of 
operations per bit per second calculated above: t-sub-flip = pi 
h-barS/k-sub-B2ln(2)E. The measure of the degree of parallelization in the 
ultimate laptop is then

t-sub-com/t-sub-flip = k-sub-B 4ln(2) RE / pi h-bar cS which is 
equalivalent to k-sub-B RT / h-bar c = 2piR/lambda-sub-T

That is, the amount of time it takes to communicate from one side of the 
computer to the other, divided by the amount of time it takes to flip a 
bit, is approximately equal to the ratio between the size of the system 
and its thermal wavelength. For the ultimate laptop, with 2R = 10-^1 m, 
2E/pi h-bar ~ 10^51 operations per second, and S/k-sub-Bln2 ~ 10^31 bits, 
t/tflip ~ 10^10. The ultimate laptop is highly parallel. A greater degree 
of serial computation can be obtained at the cost of decreasing memory 
space by compressing the size of the computer or making the distribution 
of energy more uneven. As ordinary matter obeys the Beckenstein bound 
(20-22), k-sub-BRE/h-bar cS > 1/2pi, as the computer is compressed, 
t-sub-com/t-sub-flip ~ k-sub-BRE/h-bar cS will remain greater than one, 
that is, the operation will still be somewhat parallel. Only at the 
ultimate limit of compression--a black hole--is the computation entirely 

Compressing the computer allows more serial computation

Suppose that we want to perform a highly serial computation on a few bits. 
Then it is advantageous to compress the size of the computer so that it 
takes less time to send signals from one side of the computer to the other 
at the speed of light. As the computer gets smaller, keeping the energy 
fixed, the energy density inside the computer increases. As this happens, 
different regimes in high-energy physics are necessarily explored in the 
course of the computation. First the weak unification scale is reached, 
then the grand unification scale. Finally, as the linear size of the 
computer approaches its Schwarzchild radius, the Planck scale is reached 
(Fig. 2). (No known technology could possibly achieve such compression.) 
At the Planck scale, gravitational effects and quantum effects are both 
important: the Compton wavelength of a particle of mass m, lC = 2pi 
h-bar/mc, is on the order of its Schwarzschild radius, 2Gm/c 2. In other 
words, to describe behaviour at length scales of the size l-sub-P = 
sqrt(h-barwG/c^3) = 1.616 x 10^-35 m, timescales t-sub-P = sqrt(h-bar/c^5 
= 5.391 x 10^-44 s, and mass scales of m-sub-P = sqrt(h-barc/G)  ~ 2.177 x 
10^-8 kg, a unified theory of quantum gravity is required. We do not 
currently possess such a theory. Nonetheless, although we do not know the 
exact number of bits that can be registered by a 1-kg computer confined to 
a volume of 1 l, we do know the exact number of bits that can be 
registered by a 1-kg computer that has been compressed to the size of a 
black hole (87). This is because the entropy of a black hole has a 
well-defined value.

[Figure 2. Computing at the black-hole limit. The rate at which the 
components of a computer can communicate is limited by the speed of light. 
In the ultimate laptop, each bit can flip ~10^19 times per second, whereas 
the time taken to communicate from one side of the 1-liter computer to the 
other is on the order of 10^9 s--the ultimate laptop is highly parallel. 
The computation can be speeded up and made more serial by compressing the 
computer. But no computer can be compressed to smaller than its 
Schwarzschild radius without becoming a black hole. A 1-kg computer that 
has been compressed to the black-hole limit of R-sub-S = 2Gm/c^2 = 1.485 x 
10^-27 m can perform 5.4258 x 10^50 operations per second on its I = 
4piGm^2/ln(2)h-barc = 3.827 x 10^16 bits. At the black-hole limit, 
computation is fully serial: the time it takes to flip a bit and the time 
it takes a signal to communicate around the horizon of the hole are the 

In the following discussion, I use the properties of black holes to place 
limits on the speed, memory space and degree of serial computation that 
could be approached by compressing a computer to the smallest possible 
size. Whether or not these limits could be attained, even in principle, is 
a question whose answer will have to await a unified theory of quantum 
gravity (see Box 2).

[Box 2: Can a black hole compute?

[No information can escape from a classical black hole: what goes in does 
not come out. But the quantum mechanical picture of a black hole is 
different. First of all, black holes are not quite black; they radiate at 
the Hawking temperature T given above. In addition, the well-known 
statement that 'a black hole has no hair'--that is, from a distance all 
black holes with the same charge and angular momentum look essentially 
alike--is now known to be not always true (89-91). Finally, research in 
string theory (92-94) indicates that black holes may not actually destroy 
the information about how they were formed, but instead process it and 
emit the processed information as part of the Hawking radiation as they 
evaporate: what goes in does come out, but in an altered form.

[If this picture is correct, then black holes could in principle be 
'programmed': one forms a black hole whose initial conditions encode the 
information to be processed, lets that information be processed by the 
planckian dynamics at the hole's horizon, and extracts the answer to the 
computation by examining the correlations in the Hawking radiation emitted 
when the hole evaporates. Despite our lack of knowledge of the precise 
details of what happens when a black hole forms and evaporates (a full 
account must await a more exact treatment using whatever theory of quantum 
gravity and matter turns out to be the correct one), we can still provide 
a rough estimate of how much information is processed during this 
computation (95-96). Using Page's results on the rate of evaporation of a 
black hole (95), we obtain a lifetime for the hole t-sub-life = G^2 m^3 / 
3C h-bar c^4, where C is a constant that depends on the number of species 
of particles with a mass less than k-sub-BT, where T is the temperature of 
the hole. For O (10^1 to 10^2) such species, C is on the order of 10^-3 to 
10^-2, leading to a lifetime for a black hole of mass 1 kg of ~10^-19 s, 
during which time the hole can perform ~10^32 operations on its ~10^16 
bits. As the actual number of effectively massless particles at the 
Hawking temperature of a 1-kg black hole is likely to be considerably 
larger than 10^2, this number should be regarded as an upper bound on the 
actual number of operations that could be performed by the hole. Although 
this hypothetical computation is performed at ultra-high densities and 
speeds, the total number of bits available to be processed is not far from 
the number available to current computers operating in more familiar 

[End of Box 2]

The Schwarzschild radius of a 1-kg computer is RS = 2Gm/c 2= 1.485 x 10^27 
m. The entropy of a black hole is Boltzmann's constant multiplied by its 
area divided by 4, as measured in Planck units. Accordingly, the amount of 
information that can be stored in a black hole is I = 4piGm^2/ln(2)h-bar c 
= 4pim^2/ln(2)m-sub-P^2. The amount of information that can be stored by 
the 1-kg computer in the black- hole limit is 3.827 x 10^16 bits. A 
computer compressed to the size of a black hole can perform 5.4258 x 10^50 
operations per second, the same as the 1-l computer.

In a computer that has been compressed to its Schwarzschild radius, the 
energy per bit is E/I = mc^2/I = ln(2)h-bar c^3 / 4pimG = ln(2)k-sub-B 
T/2, where T = (partialS/partialE)^-1 = h-bar c/4pi k-sub-B R-sub-S is the 
temperature of the Hawking radiation emitted by the hole. As a result, the 
time it takes to flip a bit on average is t-sub-flip = pi h-barI/2E = 
pi^2RS/c ln2. In other words, according to a distant observer, the amount 
of time it takes to flip a bit, t-sub-flip, is on the same order as the 
amount of time t-sub-com = piR-sub-S/c it takes to communicate from one 
side of the hole to the other by going around the horizon: 
t-sub-com/t-sub-flip = ln2/pi. In contrast to computation at lesser 
densities, which is highly parallel, computation at the horizon of a black 
hole is highly serial: every bit is essentially connected to every other 
bit over the course of a single logic operation. As noted above, the 
serial nature of computation at the black-hole limit can be deduced from 
the fact that black holes attain the Beckenstein bound (20-22), k-sub-B 
RE/h-barcS = 1/2pi.

Constructing ultimate computers

Throughout this entire discussion of the physical limits to computation, 
no mention has been made of how to construct a computer that operates at 
those limits. In fact, contemporary quantum 'microcomputers' such as those 
constructed using NMR (76-80) do indeed operate at the limits of speed and 
memory space described above. Information is stored on nuclear spins, with 
one spin registering one bit. The time it takes a bit to flip from a state 
|uparrow> to an orthogonal state |downarrow> is given by pi h-bar/2muB = 
pi h-bar/2E, where m is the spin's magnetic moment, B is the magnetic 
field, and E = muB is the average energy of interaction between the spin 
and the magnetic field. To perform a quantum logic operation between two 
spins takes a time pi h-bar/2E-sub-gammma, where E-sub-gamma is the energy 
of interaction between the two spins.

Although NMR quantum computers already operate at the limits to 
computation set by physics, they are nonetheless much slower and process 
much less information than the ultimate laptop described above. This is 
because their energy is locked up largely in mass, thereby limiting both 
their speed and their memory. Unlocking this energy is of course possible, 
as a thermonuclear explosion indicates. But controlling such an 'unlocked' 
system is another question. In discussing the computational power of 
physical systems in which all energy is put to use, I assumed that such 
control is possible in principle, although it is certainly not possible in 
current practice. All current designs for quantum computers operate at low 
energy levels and temperatures, exactly so that precise control can be 
exerted on their parts.

As the above discussion of error correction indicates, the rate at which 
errors can be detected and rejected to the environment by error-correction 
routines places a fundamental limit on the rate at which errors can be 
committed. Suppose that each logical operation performed by the ultimate 
computer has a probability e of being erroneous. The total number of 
errors committed by the ultimate computer per second is then 2epsilonE/pi 
h-bar. The maximum rate at which information can be rejected to the 
environment is, up to a geometric factor, ln(2)cS/R (all bits in the 
computer moving outward at the speed of light). Accordingly, the maximum 
error rate that the ultimate computer can tolerate is epsilon less than or 
equal to piln(2)h-bar cS/2ER = 2t-sub-flip/t-sub-com. That is, the maximum 
error rate that can be tolerated by the ultimate computer is the inverse 
of its degree of parallelization.

Suppose that control of highly energetic systems were to become possible. 
Then how might these systems be made to compute? As an example of a 
'computation' that might be performed at extreme conditions, consider a 
heavy-ion collision that takes place in the heavy-ion collider at 
Brookhaven (S. H. Kahana, personal communication). If one collides 100 
nucleons on 100 nucleons (that is, two nuclei with 100 nucleons each) at 
200 GeV per nucleon, the operation time is pi h-bar/2E ~ 10-29 s. The 
maximum entropy can be estimated to be ~4k-sub-B per relativistic pion (to 
within a factor of less than 2 associated with the overall entropy 
production rate per meson), and there are ~10^4 relativistic pions per 
collision. Accordingly, the total amount of memory space available is 
S/k-sub-B ln2 ~ 10^4 to 10^5 bits. The collision time is short: in the 
centre-of-mass frame the two nuclei are Lorentz-contracted to D/gamma 
where D = 12-13 fermi and gamma = 100, giving a total collision time of 
~10^-25 s. During the collision, then, there is time to perform 
approximately 10^4 operations on 10^4 bits--a relatively simple 
computation. (The fact that only one operation per bit is performed 
indicates that there is insufficient time to reach thermal equilibrium, an 
observation that is confirmed by detailed simulations.) The heavy-ion 
system could be programmed by manipulating and preparing the initial 
momenta and internal nuclear states of the ions. Of course, we would not 
expect to be able do word processing on such a 'computer'. Rather it would 
be used to uncover basic knowledge about nuclear collisions and 
quark-gluon plasmas: in the words of Heinz Pagels, the plasma 'computes 
itself ' (88).

At the greater extremes of a black-hole computer, I assumed that whatever 
theory (for example, string theory) turns out to be the correct theory of 
quantum matter and gravity, it is possible to prepare initial states of 
such systems that causes their natural time evolution to carry out a 
computation. What assurance do we have that such preparations exist, even 
in principle?

Physical systems that can be programmed to perform arbitrary digital 
computations are called computationally universal. Although computational 
universality might at first seem to be a stringent demand on a physical 
system, a wide variety of physical systems-- ranging from 
nearest-neighbour Ising models52 to quantum electrodynamics84 and 
conformal field theories (M. Freedman, unpublished results)--are known to 
be computationally universal (51-53,55-65). Indeed, computational 
universality seems to be the rule rather than the exception. Essentially 
any quantum system that admits controllable nonlinear interactions can be 
shown to be computationally universal (60,61). For example, the ordinary 
electrostatic interaction between two charged particles can be used to 
perform universal quantum logic operations between two quantum bits. A bit 
is registered by the presence or absence of a particle in a mode. The 
strength of the interaction between the particles, e^2/r, determines the 
amount of time t-sub-flip = pi h-bar r/2e^2 it takes to perform a quantum 
logic operation such as a controlled-NOT on the two particles. The time it 
takes to perform such an operation divided by the amount of time it takes 
to send a signal at the speed of light between the bits t-sub-com = r/c is 
a universal constant, t-sub-flip/t-sub-com= pi h-bar c/2e^2= p/2alpha, 
where alpha= e^2/h-bar c ~1/137 is the fine structure constant. This 
example shows the degree to which the laws of physics and the limits to 
computation are entwined.

In addition to the theoretical evidence that most systems are 
computationally universal, the computer on which I am writing this article 
provides strong experimental evidence that whatever the correct underlying 
theory of physics is, it supports universal computation. Whether or not it 
is possible to make computation take place in the extreme regimes 
envisaged in this paper is an open question. The answer to this question 
lies in future technological development, which is difficult to predict. 
If, as seems highly unlikely, it is possible to extrapolate the 
exponential progress of Moore's law into the future, then it will take 
only 250 years to make up the 40 orders of magnitude in performance 
between current computers that perform 10^10 operations per second on 
10^10 bits and our 1-kg ultimate laptop that performs 1051 operations per 
second on 10^31 bits.


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