In the latest (April) issue of Scientifical American, Graham Collins has written an article on the theoretical possibility of a topological quantum computer, which can simulate any computation of a usual quantum computer within any desired non-zero approximation. Also, the operation of a topological quantum computer cn be simulated efficiently to arbitrary accuracy on a conventional quantum computer. In some sense, a topological quantum computer, is, to some approximation, as good as a usual quantum computer in computing, and vice versa.

This is good. Now, the attraction of a topological quantum computer is that it employs the topology of space-time paths in 2 + 1 space, or, more in more picturesque language, the topology of knots (and links) in 3-space. The interesting and useful thing about this is that by definition, the topology of a mathematical knot in 3-space is not unstable if you perturb the knot a little bit; by definition, the topology of the knot is stable.

The topological quantum computer uses two-dimensional quasiparticles called anyons, which are so called as they don't have to be bosons or fermions – their phase can change by some complex number that is not 1 or -1 if you interchange two anyon's positions (this is from the article – I'm not a physicist so excuse any misconceptions here!).

The article says something which struck me as very interesting:

Michael Freedman, now at Microsoft, lectured at Harvard University in the fall of 1988 on the possibility of using quantum topology for a computation. These ideas, published in a research paper in 1998, built on the discovery that certain mathematical quantities known as knot invariants were associated with the quantum physics of a two-dimensional surface evolving in time. If an instance of the physical system could be created and an appropriate measurement carried out, the knot invariant would be approximately computed automatically instead of via an inconveniently long calculation on a conventional computer. Equally difficult problems of more real-world importance would have similar shortcuts.

Now, if a knot invariant, which can be easily calculated from (potentially very large) square matrices is all you need to construct a computation of a topological quantum computer, and if a topological quantum computer can "approximately" do any computation that a usual quantum computer can do, does this mean that a normal boring everyday computer running maple or mathematica or even matlab can "approximately" simulate a quantum computer?

In the article, they give a topological quantum computer version of a CNOT gate (CNOT stands for controlled not) – which is just a tangle on six piece of string. You can write down a knot invariant for the closed version of this tangle by taking the quantum (super)trace of an at most 18 by 18 matrix (employing the 2-dimensional irreducible rep of the quantum (super) algebras related to sl(2) or osp(1|2)). This is not very hard to do.

The article says the following:

Acting on qubits, any computation can be built from a network of CNOT gates and one other operation – the multiplication of individual qubits by a complex phase.

Of course, the details are no doubt more subtle.

Links from the Scientific American article:

Topological Quantum Computation – Lecture notes by John Preskill – http://www.theory.caltech.edu/~preskill/ph219/topological.pdf

Anyon There? David Lindley in Physical Review Focus, Vol. 16, Story 14; Nov 2, 2005. http://focus.aps.org/story/v16/st14

Hi Sach,

I went to a talk on this last year, during the quantum information mini-workshop organised by the folks over at physics. I found it quite interesting because the quantum double turns up in topological quantum computing. Preskill says at the end of his notes that what he calls the “nonabelian superconductor” is usually called the quantum double in the literature.

Tel

I downloaded some lecture notes that I think are Preskill’s. Do quantum doubles pop up in topological quantum computing because they want an R-matrix? (I’ll say that that’s my guess)

Wouldn’t it be odd if knot invariants turned out to have some really applied aspect?! None of us would have thought it.

Hi Sach,

I’d say you’re spot on – they want (non-trivia)l R-matrices to model the braidings.

Preskill’s paper is quite good, I’ve been reading it on my laptop but the time has come to print it out and scribble all over it đź™‚

Knot invariants – real applications – spooooky

Tel

Hey Tel,

It’s a nice paper to read – I quickly looked at it a few weeks back – it’s not a hard core pure maths paper and it’s relatively easy to understand (apart from the physicsy stuff which I don’t know) – hooray!

Is it true that you can take any Hopf algebra and apply the “quantum double construction” to it to obtain a quasitriangular Hopf algebra? If this is true, then are the quantum groups and quantum supergroups a proper subset of the quasitriangular Hopf algebras? If so, then the topological quantum computing people could have a huge variety of non-trivial matrix reps of the braid group to play with.

It’d be very nice if the stuff we’ve done with knots, quantum (super)groups and quandum doubles had meaning/relevance/application outside of pure maths, especially (for me) if this applied to my PhD studies.

“Now, if a knot invariant, which can be easily calculated from (potentially very large) square matrices is all you need to construct a computation of a topological quantum computer, and if a topological quantum computer can “approximately” do any computation that a usual quantum computer can do, does this mean that a normal boring everyday computer running maple or mathematica or even matlab can “approximately” simulate a quantum computer?”

No, definitely not. These are very difficult calculations. Topological quantum computing is equivalent to standard (circuit-model) quantum computing. So you could factor efficiently, for example. An 18 x 18 matrix is easy, but the dimension grows exponentially with the number of qubits. To factor a 1024-bit number would require ~5000 qubits (see a paper by Preskill from the late 90s).

I should look at that paper. I reason I posed the sort-of question was that it is very easy to calculate a knot invariant – you just multiply some matrices together, take the trace of the resultant, and possibly multiply the result by an easily calculated number.

So, eg, to factor a, say, 1024-bit number, you might be taking the trace of a 15000 by 15000 matrix, or some other very large matrix… the time taken using this complicated process might take you much longer than other methods? I have to read up on this.

Watch for peer-to-peer enthusiasts to be quantum computing at 100.000.000-qubit mindspeed while the government-funded labs are still stuck in 64-bit institutional inertia. P2Ps will break with orthodoxy—quantum compute with NO transistors and NO clocks—-just take a shortcut through the grid with a streamline hyperfractal that is faster, easier, cleaner and more efficient—-by a order of powers. When you want a world-class problem solved in a fraction of the time, done right and under budget, it will be highly motivated, enthusiastic P2P collaborators who will have quantum computers up and running in their offices and residences while the government waffles and denies that anyone can challenge their 64-bit Pentiums. Carla Hein

Just searching on google and found your site. It was ranked fairly high on google to. Anyway just looking around to see why.

thanks

jamie

I havn’t kept up with developments in topological quantum computing in the last few months. I’ll be interesting to see if topological quantum computers are developed or if the study of top. quantum computing yields any practical application.