Just got my beautiful Brompton wet in a sudden downpour on the way home. Yes, I dried it off, and now I’m sitting here with a cup of warm wet-chalkdust-tasting tea listening to the rain and waiting for last night’s chicken pilaf to warm up. It’s always even better the day after I make it! (Some of the things mentioned above will mean nothing to you if you did not read this earlier post.)
Yes, I’m still here at the Aspen Center for Physics, attending the SuperCosmology workshop. I’ve been attending some Cosmology discussions, but also doing some computations on another project (which I ought to tell you about some time) and thinking. This has been helped a lot by the Aspen Music Festival and School, since I’ve gone and sat in the nearby giant music tent in the mornings where the student orchestra is rehearsing pieces they’ll play in the concerts later in the evening. I love listening to orchestras rehearse. Especially large orchestral pieces (such as yesterday’s Shostakovich’s 1st Symphony) where the rehearsal entails deconstructing certain difficult passages by section. So you hear all the strands of a chord played separately by different bits of the orchestra and then put back together. You really appreciate a chord constructed by a master when you’ve heard it this way. Often more fun than going to the concert.
The Center is a wonderful place to do physics for so many reasons. One of them is the fact that there is a weekly colloquium given by one of the physicists from one of the workshops going on. You learn so much about what is going on in other fields.
(and they have really good cheese, wine, crackers and conversation after.)
So I’m supposed to sit here and write a second installment about stringy cosmology, following on from the first installment I gave here. Since there did not seem to be that much in the way of interest in it, as far as I can tell, I’ll instead tell you about this great colloquium I went to. “Topological Quantum Computation”, by Chetan Nayak.
Chetan told us about new ideas and approaches in quantum computers. So those of you who might know Chetan might wonder what on earth he’s doing talking about that stuff. Was he not working on matters to do with condensed matter physics, and topological quantum field theories showing up in strongly correlated electron systems? Yes, but that’s the point!
Let me back up (and turn off the pilaf).
First, what is a quantum computer? Well, such a thing does not exist, as far as we know. It is a dream that physicists would like to turn into a reality. The idea is often attributed to Feynman, and significant key refinements in the important concepts towards making it a reality were made by Deutch, and by Shor. You might start (as Feynman did) by wondering how well an ordinary computer will do in simulating a quantum system, and you quickly realize it would be highly inefficient. For example, to simulate N spins would require diagonalisations of matrices of size 2^N times 2^N. (That’s 2 to the power N, if some browsers miss the crucial character) This is very slow, and gets really bad as N grows. So you begin to think that maybe a computer that uses Quantum mechanics to do its actual computations is the way to go in simulating quantum processes. This is how it all began.
How might you get one to work? A “classical” computer (the one you’re reading this on now, unless you’re way in the future reading old historical records -Hi!) manipulates “bits”, which are realized and manipulated using transistor technology to do the various logical operations (NOT, OR, AND, etc) which build up everything your computer is doing. The basic bit takes two values, “0″ or “1″. (Or “up” and “down”, “pink” or “blue”, etc). The logical operations are then various manipulations you can do on a bit, or collection of bits, and then out comes an answer. A quantum computer uses instead a “q-bit”. A q-bit is rather different since it is inherently quantum mechanical in that it takes two quantum states, which I’ll call |0> and |1>, and forms a superposition of them. That’s your bit. Another example, (since you can’t have a QM discussion without it) is your basic Schrodinger’s Cat example. The q-bit there would be a superposition of the |dead> and |alive> states: |q-bit>=|dead> + a*|alive> where a is some complex number. So you build your computer out of these bits. You do manipulations on the bits with quantum mechanical operators which are in general some unitary operation. (In an N-dimensional Hilbert space, it would be U(N)). When you’re done, you read out your result. Since you’re working with a continuum of linear combinations of bits, it is rather like doing some humongously parallel computation, and this is (roughly) why this is such a potentially radical idea.
So, you ask, “why isn’t my Mac using this wonderful, truly innovative science?” No, it’s not because they’re not called “i-bits” but because there are problems making it a reality. Not just engineering problems, but physics ones. Just as with classical bits, you can implement q-bits and make a computer in a variety of ways (I hope there will be a pink-blue classical computer one day, perhaps based on pink and blue sugared almonds…). The trick is to do it in a way that allows you to do lots of basic computations in a fast and error-free way. And maybe not take up too much space. (There goes my almond computer, on all three counts.).
This is hard. The biggest problem people are worried about is errors. They basically kill your q-bit computation’s integrity way faster than for a classical analogue. There are several reasons for this, and among those are the fact that you have many more (an infinite set of: consider the complex number a) delicate numbers making up the q-bit. If you realized your bit as a tiny spin, say, it is really very sensitive to being knocked about by tiny environmental variations in the local magnetic field. Worse than that, you can’t peek into the computer’s computation half way and check to see if there are errors accumulating and then fix them: That would require you to read one of the intermediate quantum states which would (in the old language) collapse the wave-function and kill your superposition. So basically your mobile phone rings and your computer spits out nonsense. Not good. Well, it’s a lot worse than that, but you get the idea.
Well, there is a huge effort around the world in various universities trying to beat this error-correction problem (as it is called) both theoretically and experimentally. There actually are ways of introducing means of reading off errors and doing corrections (Shor ’95, Gottesman ’97 ….), by introducing a type of redundancy into the realizations of the q-bits, and people are trying to implement them in various
ways experimentally. Apparently, a tolerable error rate is about 1 in 10^5 operations. (That’s 10 to the power 5 if some browsers miss the crucial character) Current estimates of what is possible experimentally right now using “conventional” systems (Silicon, Gallium Arsenide, etc), come in at about 1 in 10^4 at best. (I’m sure there are those who would argue these numbers, but you get the idea.).
So far, in a colloquium about quantum computers, a physicist has been nodding and paying attention. Then the talk degenerates into (depending upon whether the speaker is a physicist, computer scientist, or engineer) rather dry and muddy discussion of various engineering solutions to the problem. All very interesting and important in fact, but if you don’t work on that stuff its dry, dry, dry as a….really dry thing. Instead, Chetan shifts gears and the talk gets *more* interesting.
What do we want? We want to have q-bits that are rather robust against local perturbations. This is where topology comes in. Topology is the study of properties of geometrical shapes which persist even after you do local deformations of it. The classic example is a donut (doughnut?) and a teacup. Imagine making them out of playdough or plasticine (do kids still play with that stuff? I hope so.) Well you could deform one into the other without ever tearing the playdough. You just do local deformations, squeezing and pushing here and there. The “hole” is the thing that is preserved. It is in the middle of the donut, and then it moves to the middle of the loop formed by the handle of the cup. Topology is all about the study of such persistent features. Another example is the study of knots, or how things are tangled or braided together. There are important features that stay the same about a knot tying together loops of string even if you locally waggle the strings a bit. You can classify different knots, or braids, or different surfaces (such as that of the cup and the pastry) according to what features are preserved under local deformations. It is a beautiful area of pure mathematics, and certain areas of physics (such as particle physics and string theory) But so what?
Ah! What if we represent our q-bits topologically instead?! Then they’d be less inclined to care about local environmental disturbances: They’d be really really robust. So how to do that? This is why Chetan is giving this talk, and not an engineer (not that there’s anything wrong with being an engineer; some of my best friends and colleagues are engineers). What you want is a physically realizable system (because you want to build it, right?) where the basic degrees of freedom -things that you want to manipulate and form superpositions of, like we were doing with spins and cats earlier- are topological. Well such systems are known. An example is the Fractional Quantum Hall Effect (Nobel prize 1998 to Laughlin, StÃ¶rmer, and Tsui by the way ). In very high magnetic fields and a low temperature, you can get what’s called a Hall current of charge carriers flowing perpendicular to the applied potential and applied magnetic field (which are themselves mutually perpendicular). You can measure a Hall resistance associated to this effect, and it varies in a way which is proportional to the applied magnetic field. The quantum Hall effect occurs when the variation is no longer linear, but there are plateaux in the resistance-field curve, resulting in a sort of quantized resistance. At those plateaux, the resistance associated to the other current, the one flowing due to the
applied electric field, actually drops to zero. The unit of quantization is essentially set by the basic fundamental unit of charge of the basic charge carriers, e.g. electrons. The fractional Quantum Hall effect is like the Quantum Hall effect but the plateaux occur at steps which imply a fractional charge carrier. This is very puzzling indeed if you think your basic charge carriers are things like electrons. What Laughlin showed is that the effective theory is different for the FQHE plateaux, and the electrons interact so strongly that they form a phase of matter called a “quantum fluid”, whose basic degrees of freedom are now new localized particles of fractional charge called “quasiparticles”.
What is really interesting for our purposes is the topological facts hiding here. It turns out that the quasiparticles have very interesting properties when you interchange them, or if you take one and encircle another with it. Their wavefunctions come back multiplied by phases with fractional exponents (complex numbers arising from taking n-th roots of unity, for example). These phases are measurable (using e.g. the Bohm-Aharanov effect), but because they arise from taking various paths, or doing certain exchanges, they are tolopogical data. In fact, you can represent the basic degrees of freedom of the system in terms of braiding, or knot theory. (You can see the braids if you imagine little threads attached to the particles, and then you exchange the particles in various ways. You start braiding the threads.) These paths, these entanglements, are the quantum states to use as q-bits, and these are really things that you can make and study! It turns out that there are rather nice quantum field theories that you can write down which model the topological essence of these systems rather nicely. Moreover, they are inherently topological in their definition and rather clean to study. The Chern-Simons theory is an example.
Actually, there’s a more technical point: to make them useful q-bits you need to make sure that your have a full representation of the unitary action on the associated Hilbert space. Move on if you don’t care about the details, but the issue is that you need to have a more subtle braiding such that the monodromy matrices that you get by doing various exchanges are non-Abelian, and that they furnish a representation of the Unitary group U(N) for an N-dimensional Hilbert space. Then, your Chern-Simons effective theory is in fact a non-Abelian one. It is believed (but still being worked on that the 5/2 plateau has such an effective theory (as suggested by Moore and Reade in 91).
These theories are well-studied in the context of both formal field theory and string theory, and both fields have been enriched by work in this area. Also, the field of mathematics has been enriched in this area due to some of Witten’s Fields Medal work, on topological quantum field theory and Knots. Also, real condensed matter theorists like Chetan and his phd advisor of several years ago, Frank Wilczek (Nobel prize last year, by the way, for something else, with Gross and Politzer) have been applying these models to real physical systems. (There are unfortunately three different uses of the word “field” in this paragraph. Anybody care to point them out? Still with me?)
Ok, this post is way too long and my pilaf needs to start heating all over again. So I’d better get to the punchline. Story so far: It would be great to make a quantum computer as it will transform our world. It is hard because q-bit computations are really sensitive to little perturbations in the environment. Topological states are not. So make q-bits out of topological degrees of freedom of a physical system.
So what you’ve got to do is start looking out for ways to engineer it so that you can prepare, manipulate and read out the data from the topological degrees of freedom of a q-bit that you make in the lab. Chetan outlined a number of ways of doing this, involving very clever uses of Josephson junctions (another Nobel, 1973, with Esaki and Giaever) to have the various currents tunneling between various readouts to make gates, and the topological features are the paths the quasiparticles take around various trapped quasiparticles that have been prepared to make the q-bit.
I imagine that you get the idea now. Use topology in an essential way to furnish quantum states upon which to build q-bits from which to build a quantum computer. It turns out that the error in this realization of the system is better than 1 in 10^30 (10 to the power 30…it is controlled essentially by the resistance, which you’ll recall drops precipitously to zero at the plateau).
There’s a long way to go in this program, Chetan tells us, not the least because we’re not going to be making practical computers with devices that only work at about 5 milliKelvin with and applied magnetic field of 10 Tesla! But the point is that there are probably several other systems (for example in high temperature superconductors) that might have accessible topological degrees of freedom, and so this is a program that has a lot of exciting exploration ahead. Have a look at the paper of Freedman, Nayak and Shtengel, Phys. Rev. Lett. 94, 066401 (2005), and references therein for more information. (And note that these guys are working for Microsoft, so you Apple people should make sure Apple’s in on the act. I’ll cry bitterly to see the Windows operating system slowing a quantum computer down to classical speeds after years of innovative research. )
This is a really fascinating combination of very practical concerns with seemingly esoteric things that we string theories work on – like topological field theories – all done in a very satisfyingly clever way. It was also great to see several of my stringy colleagues suddenly start sitting at the edge of their seats during during the colloquium (some of them stopped playing -or whatever- on their wireless-web-connected laptops and sat up), just as I was.
Really, really great stuff. (Which reminds me….Chicken pilaf time!)