# Can we make a Maxwell's Demon using Quantum Computers?

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:

(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:

(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:

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?

• 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. Commented Oct 10, 2012 at 5:42
• Look into this thought experiment called Szilard's engine. Commented Oct 10, 2012 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 Commented Oct 10, 2012 at 13:13
• 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. Commented Oct 11, 2012 at 2:10
• @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. Commented Oct 28, 2013 at 7:45

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.

• 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. Commented Oct 28, 2013 at 7:41
• The poster said “operation of Maxwell's demon cannot be reversible” and apparently inferred that they can’t be implemented with unitary operators. But the whole universe is irreversible, whereas an objective existence of non-unitary evolution of quantum states (so named state vector reduction a.k.a. wave function collapse) is a controversial topic in modern science. Please, improve this stuff: for which exactly reason any unitary evolution cannot effect a Maxwell demon behaviour? Commented Oct 20, 2014 at 17:47

If you make a Maxwell's Demon using a quantum computer, you may find that it is actually a heat engine operating on the difference in temperature between the system and the quantum computer's memory. (For you to analyze it this way, it might help assume a very small energy difference between a 0 bit in memory and a 1 bit.)

As soon as the memory fills up (in thermodynamic terms, reaches the same temperature as the system), the Maxwell's Demon will stop working.

The energy of quantum systems encodes redundant information. If a quantum computer were used to to emulate a vast number of physical systems this might be possible.