Tuesday, May 23, 2006

Seth Lloyd: counting quantum operations in spacetime

Sean Carroll has written a bizarre text praising Slavoj Žižek and comparing him to Seth Lloyd, a quantum computation expert from MIT. ;-)

If you don't know, Žižek is a Slovenian postmodern philosopher and a science-hater, the kind of "smart guys" who have provoked Alan Sokal to write his hoax and submit it to Social Text. The real motivation for Alan Sokal's hoax was a political one: Sokal couldn't stand the fact that the postmodern philosophers and literary critics make it far too obvious that the core of the Left is intellectually pretentious, hypocritical, anti-scientific, and vacuous: Sokal himself is a leftist but a rational one.

On the other hand, Sean Carroll indicates that the only reason why he does not make the postmodern philosophy a starting point and punch line of his research - and why he only focuses his blog writing on these paradigms - is that it might be rejected by the referees.

The first catastrophe of science, according to the Slovenian thinker, was the Big Bang itself. Something that has created this "monstrous" Universe whose monstrosity proves that something had to go terribly wrong. The second catastrophe after the Big Bang was the birth of the global capitalism, of course. ;-) Eventually, all of science is going to be reduced to language and words, the only thing that the lit crit hordes will ever be willing and capable to comprehend. Also, the Slovenian philosopher considers quantum mechanics to be "the symbolic real" or a "signifier reduced to a meaningless formula".

If you want to understand why some people think that Žižek is an obnoxious far left-wing simpleton if not a doubleton, see this excerpt from a movie about him. The people who say or even think that this guy is charismatic must be kind of ill, too. They call it a different taste; I call it a deviation.

Because I find all of his "ideas" far too irritating, let me leave the celebration of these deep intellectual constructions ;-) to our postmodern colleagues at Cosmic Variance, and look at something more serious or at least slightly more serious.

Is the Universe a quantum computer?

Much like discrete physics, it is another "great idea" that many people believed to be geniuses by themselves and their fan clubs propose all the time. In fact, these two ideas - discrete physics and Universe as a quantum computer - are closely related. However, they have a somewhat different focus.

What is the primary reason why some people want to believe that the Universe is a quantum computer (or perhaps even a classical one)?

The main reason is the same illusion that makes many people believe in creationism: it is just hard for a certain kind of folks to believe that the complexity, richness, and beauties of the real world may be a consequence of simple and "dry" laws of physics. The creationists decide that there must exist God because the simple laws simply can't reproduce the complex world.

The Universe-as-quantum-computer people are less radical. In their opinion, the Universe without God could work after all. But it must be based on specifically and intelligently designed fundamental laws that allow or "support" complexity: the Universe must have a very active and large socialist government. A government that prescribes Al Gore Rhythms to everyone and to everything.

Indeed, it is a Universe where all of us are monkeys who are typing nonsense, but because we have a smart government equipped with Al Gore Rhythms, the random typing results in Hamlet if not Cosmic Variance. ;-) The laws of this Universe must be like the rule #110 cellular automaton of Stephen Wolfram - that is neither random nor repetitive - or an analogous great achievement of the human thought. ;-) I am convinced that Wolfram, Markopoulou, Lloyd, and many others share this kind of belief.

I find the belief unscientific, postmodern, socialist, and irrational. The physical laws are indeed able to create complex structures and to gradually expand intelligence, self-organization, and beauty, but they don't have to include any specific "policies" in the fundamental laws to achieve these goals. The ability of the Universe to create organized structures is an emergent feature of other, more fundamental laws of Nature. In the context of life, the necessary mechanisms include the reproduction of DNA, mutations, and natural selection. Neither of these processes is a fundamental law of Nature.

All of them are derived, emergent notions. If you're highly interested in these interesting features of the world, they can be more important for you than the fundamental laws. Nevertheless, they are not fundamental laws themselves. They are not directly connected with the mathematically accurate description of the real world. In fact, they do not depend on the mathematical details behind the real world. The existence of complexity is a fact, and it is a fact that is compatible with the Standard Model, with General Relativity, with String Theory, and with other theories we know to be relevant for the real world. If you study these fundamental theories carefully enough, you will be able to derive the creative power of these laws.

Some essential features of Nature's "creativity" may be identified as a rather sharp consequence of some aspects of the fundamental laws - but the "creativity" can never become a complete fundamental law by itself. Simple assumptions such as sufficient space, viable building blocks of matter, meaningful microscopic laws, and freedom from any kind of macroscopic constraints are sufficient for Nature to develop very rich structures. At sufficient time scales, She is always brighter than we are, and our attempts to dictate Her how to make the world rich can only make things worse, not better. As you can see, I formulate these ideas generally enough so that they can be used in political philosophy, too.

Because so far I have not seen any specific connections of the proposed theories painting the Universe as a quantum computer with physics insights that I consider quantitatively tested, qualitatively motivated, or associated with the insights of the last 500 years, I treat all papers proposing that the Universe is a "computer" as religious texts. There is no science in them. It's a new kind of religion decorated with some inconsequential mathematics.

Other people may believe that the Universe is not a Matrix but rather a special kind of a feminist organization because feminism is necessary for a good Universe to work. These two theories are comparably justifiable.

Seth Lloyd may have something to say

But I am far from saying that there are absolutely no ideas associated with the papers whose basic paradigm is the religion described above. Seth Lloyd of MIT, the author of the first semi-realistic prototype of a quantum computer, has proposed slightly more concrete ways how the concepts from quantum computation could be relevant for physics, especially physics of quantum gravity. He has written a book. Instead, let me link two preprints:
Let us ignore the religious parts of these papers - the attempts to dictate Nature that She should have some special affirmative action policies to increase its complexity because unconstrained, free Nature without a socialist, complex government surely can't work, as the religion believes. ;-) Are there useful ideas that could actually be used in our analysis of the theories that describe the real world, as opposed to some religious fantasies about the world?

The answer is Maybe.

The main non-trivial concept that Lloyd proposes is to compute the number of operations (ops) that can be squeezed in a given spacetime volume. If the spacetime region has a spatial section of radius "R" and if it lasts for time "T", then Lloyd proposes the maximum number of operations in this volume to be
  • R T / pi Gnewton.
Note that he even conjectures the numerical coefficient (which I probably don't believe to be justified, even if I believe the general form). ;-) This should be viewed as a counterpart of the maximum amount of information that you can squeeze into a region of surface area "A", namely
  • A / 4 Gnewton
"bits" (divided by "ln(2)"), also known as the Bekenstein-Hawking entropy. Note that these two formulae are related. The first formula does not count bits but rather operations (another dimensionless quantity), and one spatial dimension is replaced by time. I view this analogy as a very natural one. The formulae are kind of Wick rotations of each other.

While the entropy is the best physics concept that can define a counterpart of the "number of bits" from computer science, it is harder to define "the number of operations" in physics. I would be thrilled if someone could write down a formula how to extract the "number of operations" that occured in a macroscopic system from the path integral. I say "path integral" because we clearly need a quantum, spacetime formalism to talk about the number of operations.

Note that the number of operations of a real computer - or a quantum computer - is much smaller than the (intuitively evaluated) actual number of atomic processes behind these operations. Classical computers still require a huge number of atoms to behave "collectively" if the resulting behavior can be interpreted as a reliable binary operation, and they require them to do "the same thing" for a certain period of time that is much longer than the atomic timescales. In principle, we can imagine that one "transistor" in the future computers may be composed of "N" atoms where "N" is going to be of order one, even though we are far from this point (but not astronomically far).

Also, quantum computers may be created at some point, and decades or centuries of refinement can make the elementary pieces of such computers as small as one atom, too. Another difficult task is to make the computers so fast that almost every "distinguishable change" is used as a real operation.

This means that in principle, we can estimate the number of operations that such computers can make - at least its order of magnitude. A challenge for the reader is to find a quantitative counterpart of the coarse-grained entropy formula
  • S = - Tr ( rho ln rho )
for the number of operations. One trivial definition that you might propose is simply the increase of the entropy ;-), but I want something better - and something that can be nonzero even in the equilibrium.

Such a formula, probably based on Feynman's approach to the quantum theory, could make Lloyd's "operation counting holographic bound" a well-defined conjecture. Without such a definition of the "number of operations" that can be used in theories that actually describe the real Universe, religion seems to dominate in Lloyd's papers, too. The papers talk about the Universe but they think about a computer which is a different thing.

But with some bright idea added to the picture, Lloyd's speculation could become a part of serious physics, unlike the ideas of the Slovenian postmodern philosopher. But you should realize that it is subtle to define the number of operations because the world is continuous and it is not as easy as a discrete classical computer or discrete quantum computer whose information is digitized.

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