Addenda: I thank Chris Weed for catching some errors, and thank Ben(adsbygoogle = window.adsbygoogle || []).push({});

Rubiak-Gould and Nathan Urban for some interesting comments. Here's

my reply to Ben Rudiak-Gould:

John Baez wrote:

>> In fact, Rolf Landauer showed back in 1961 that getting rid of

>> one bit of information requires putting out this much energy

>> in the form of heat:

>>

>> kT ln(2)

Ben Rubiak-Gould replied:

>It's easy to understand where this formula comes from. Getting rid of a bit

>means emitting one bit of entropy, which is k ln 2 in conventional units.

>The associated quantity of heat is ST = kT ln 2.

Thanks; I should have said that.

Landauer's analysis showing that "forgetting information" costs

energy is still interesting, and it was surprising at the time.

There had been a number of other analyses of why Maxwell's demon

can't get you something for nothing, by Szilard and others, but

none (I think) had focussed on the key importance of resetting

the demon's memory to its initial state.

>But it seems to me that you're conflating two different issues here.

>One is the cost of forgetting a bit, which only affects irreversible

>computation, and the other is the cost of keeping the computation on

>track, which affects reversible computation also. Landauer's formula

>tells you the former, but I don't think there's any lower bound on the

>latter.

Not in principle: with a perfectly tuned dynamics, an analogue

system can act perfectly digital, since each macrostate gets mapped

perfectly into another one with each click of the clock. But with

imperfect dynamics, dissipation is needed to squeeze each macrostate

down enough so it can get mapped into the next - and the dissipation

makes the dynamics irreversible, so we have to pay a thermodynamic cost.

If I were smarter I could prove an inequality relating the "imperfection

of the dynamics" (how to quantify that?) to the thermodynamic cost

of computation, piggybacking off Landauer's formula.

Here's what Nathan Urban wrote:

John Baez wrote:

>[quantum computation]

>So, maybe it can work. I need to catch up on what people

>have written about this, even though most of it speaks the

>language of "error correction" rather than thermodynamics.

A nice recent overview of some of this work can be found in the

latest Physics Today:

22) Sankar Das Sarma, Michael Freedman, and Chetan Nayak, Topological

quantum computation, Physics Today (July 2006).

In this approach, error-free computation is accomplished using

topological quantum field theories, as topological theories are robust

against local perturbations.

The article has some nice discussion of anyons, braidings, non-Abelian

topological phases of condensed matter systems, etc. It speculates

that the nu = 12/5 state of the fractional quantum Hall effect might

support universal topological quantum computation (meaning that its

braiding operators could realize any desired unitary transformation).

Here's my reply:

Long time no see, Nathan!

Nathan Urban wrote:

John Baez wrote:

>>[quantum computation]

>>So, maybe it can work. I need to catch up on what people

>>have written about this, even though most of it speaks the

>>language of "error correction" rather than thermodynamics.

>A nice recent overview of some of this work can be found in the

>latest Physics Today (July 2006), in the article "Topological

>quantum computation" by Sarma, Freedman, and Nayak. (Nayak is

>at UCLA if you ever get out that way.)

Thanks, I'll check that out.

I'm usually too lazy to drive into LA, but now that I'm in Shanghai,

I thought I'd take the chance to visit Zhenghan Wang in the nearby

city of Hangzhou and talk to him about topological quantum computation.

Wang and Freedman both work for "Project Q", aptly named after the

Star Trek villain - it's Microsoft's project to develop quantum

computers using nonabelian anyons:

24) Topological quantum computing at Indiana University,

http://www.tqc.iu.edu [Broken]

>The article has some nice discussion of anyons, braidings,

>non-Abelian topological phases of condensed matter systems,

>etc. It speculates that the nu=12/5 state of the fractional

>quantum Hall effect might support universal topological

>quantum computation (meaning that its braiding operators

>could realize any desired unitary transformation).

Freedman, Larsen and Wang have already proved that certain versions of

Chern-Simons theory support universal quantum computation:

25) Michael Freedman, Michael Larsen, and Zhenghan Wang,

A modular functor which is universal for quantum computation,

available as quant-ph/0001108.

The fractional quantum Hall effect is supposedly described by

Chern-Simons theory, so this is relevant. I don't know anything

about the "nu = 12/5 state" of the fractional quantum Hall effect,

but the folks at Project Q *do* want to use the fractional quantum

Hall effect for quantum computation, and some people are looking

for nonabelian anyons in the nu = 5/2 state:

26) Parsa Bonderson, Alexei Kitaev and Kirill Shtengel,

Detecting non-abelian statistics in the nu = 5/2 fractional quantum

Hall state, Phys. Rev. Lett. 96 (2006) 016803. Also available as

cond-mat/0508616.

Apparently there's just one lab in the world that has the capability

of producing these fractional quantum Hall states!

The article in the latest Physics Today isn't free for nonsubscribers,

but this is, and it seems to cover similar ground:

23) Charles Day, Devices based on the fractional quantum Hall effect may

fulfill the promise of quantum computing, Physics Today (October 2005),

also available at http://www.physicstoday.org/vol-58/iss-10/p21.html [Broken]

It discusses both the nu = 12/5 and nu = 5/2 states. The former is

computationally universal; the latter seems easier to achieve.

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