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Although I'm reasonably sure that quantum computing advances will not lead to the ability to construct a machine that globally violates the 2nd law of thermodynamics, it feels like a difficult position to defend when I see things like this:

http://www.flickr.com/photos/jurvetson/8054771535 Rose's Law

(summary: an alternative to Moore's law that predicts basically unbounded growth is computing ability)

There's a proviso with this, while quantum computers could solve problems that would take a classical computer almost forever, it can't solve all problems. Wikipedia helped clarify this for me:

problems quantum computing can solve

(BQP are problems solvable by quantum computers with polynomial time)

This image is to say, although quantum computers could solve every computing problem we have now and more, there are things it can't solve. But is THIS the reason we couldn't make a quantum computer into a Maxwell's Demon?

My biggest problem with a 10,000 Qubit computer is the internal memory. The Landauer limit dictates that any given memory transition must expend at least $k T \ln 2$ of energy to make the transition, otherwise it then violates the 2nd law. What is the "internal memory" analog to such a massively powerful quantum computer? Would it also be subject to this limit, or would the internal states, never actually existing in a way, circumvent it?

So let me narrow the question, looking at the idea of Maxwell' Demon:

Maxwell's Demon, Wikipedia

Let's formalize this to say that we have some observable vector $\vec{y}$, that we get from sensors in the $A$ gas. We then take that information, process it, and apply the information to a decision vector $\vec{u}$, which in the above image is a single Boolean open/shut decision. What is the reason that we can not build this with a quantum supercomputer? My guesses are:

  1. The internal memory switches would use energy, increasing global entropy (seems to violate the claims of quantum computing)
  2. The particular type of problem can't be solved by quantum computers with sufficient efficiency (implying there exists a set of problems that could violate the 2nd law if solved with polynomial time)
  3. The problem is irrelevant of both memory and processing capability, as the sensors and the control input themselves require too much energy (seems this would trash certain explanations I've heard for why various Maxwell's Demons wouldn't work, in fact, it would seem to flagrantly ignore the principle behind the Landauer limit)

As you can see, none of my explanations are consistent. Is the universe big enough for both superpowerful quantum computers and the 2nd law?

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You're imagining that quantum computers are "so smart that they can fool the system and universal laws of physics". But this ain't the case. A quantum computer is just another physical system. The reason why Maxwell's daemon fails to act as a perpetual motion of the second kind is that the energy needed to collect the information and open the door between the half-vessels exceeds the energy we gain, and this is true for a classical devil with a brain doing the job and imagined by Maxwell as well as a quantum computer or anyone else. –  Luboš Motl Oct 10 '12 at 5:42
    
Look into this thought experiment called Szilard's engine. –  Prathyush Oct 10 '12 at 8:57
    
@LubošMotl My issue is that I read your answer arguing two things 1. that the quantum computer is physical so it increasing entropy in its operation 2. the information collection alone destroys the proposal. Obviously we can't assume the computer has god-like knowledge, but may we assume it has god-like computational power? As Anixx points out, reading the output of the computer would cost energy. Energy consumption may scale logarithmically for some problems, and polynomial for others. Could a set of problem types in complexity theory violate the 2nd law?! ..or maybe not, I don't know –  AlanSE Oct 10 '12 at 13:13
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The way I always heard it was that it costs entropy when the demon wants to erase his memory of the past. He can't continue to observe the particles forever without eventually having to dump the old information to the environment. This goes for a classical as well as a quantum demon. –  Dan Stahlke Oct 11 '12 at 2:10
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@AlanSE I presume you're now aware of Shoichi Toyabe; Takahiro Sagawa; Masahito Ueda; Eiro Muneyuki; Masaki Sano (2010-09-29). "Information heat engine: converting information to energy by feedback control". Nature Physics 6 (12): 988–992. arXiv:1009.5287. Bibcode:2011NatPh...6..988T. doi:10.1038/nphys1821 They actually build and test a Maxwell Daemon in this experiment and show that it fulfils Landauer's principle. –  WetSavannaAnimal aka Rod Vance Oct 28 '13 at 7:45
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up vote 4 down vote accepted

Quantum computer performs unitary, reversible operation so it is not a subject to Landauer's principle until the reading the final result.

Still operation of Maxwell's demon cannot be reversible, as it needs to conduct measurements.

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Actually isn;t it the initialisation of a quantum computer that will be subject to Landauer's principle? When the machine is initialised, it must "forget" whatever random state the qubits begin in. I think this amounts to pretty much the same thing as what you say: either way, it implies some pretty hefty energy requirements for high numbers of qubits. –  WetSavannaAnimal aka Rod Vance Oct 28 '13 at 7:41
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