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Re: This Week's Finds in Mathematical Physics (Week 235)

  1. Jul 19, 2006 #1
    Addenda: I thank Chris Weed for catching some errors, and thank Ben
    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

    >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

    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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    If you get stuck try my puzzle page at:

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  2. jcsd
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