Wednesday, July 06, 2005

Quantum Systems are Big

Just thinking about the blog entry about water over at ars mathematica.

One of the things that is underappreciated about quantum systems is how big they are. Consider a classical system made of two subsystems A and B. Then the space of configurations of the union of the systems is AxB. Typically this is a finite dimensional manifold. To describe the state of the combined system we just need to describe the A susbsystem and then the B subsystem. In this sense classical mechanics is additive.

In quantum mechanics we look at much larger spaces. Typically we're looking at wavefunctions, complex valued functions on the classical configuration space. When we combine two systems the state space is the tensor product of the state spaces of the individual systems. So in a sense combining quantum mechanical systems is multiplicative. When you start combining quantum systems they get very large.

Anyway, to get an idea of what I'm talking about see my comment. With what seems like a really simple system we're way beyond what is reasonable to simulate on a computer. It's not surprising then that it's hard to predict the behaviour of water. On the contrary, we should be happy that we can make any kinds of predictions at all about the bulk properties of matter.

Now, many years ago, I used to work in a computational chemistry group. People were simulating molecules all the time. Not just little ones - big ones with hundreds or even thousands of atoms. They were trying to understand biology at a molecular level in order to design drugs. But given that water is so hard to understand this seems like an insurmountable task. They either used simplified quantum models (eg. single electron models) or empirical classical models based on masses and springs. Typically what happened was post hoc fitting of data. The empirical models had many parameters that could be tweaked. The user would run sims with many different parameters, eventually see the real data (eg. from X-ray diffraction techniques or NMR) and then choose the model that best fit the experimental results claiming it explained the observed behaviour. It had next to zero predictive value. Occasionally my colleagues would have moments of lucidity and realise that they might as well predict molecular behaviour using yarrow stalks and the I Ching - and then a short while later they'd go back to work tweaking the parameters.

And you thought you were helping to cure cancer when you donated your cycles to folding@home!

Incidentally, the largeness of quantum systems is why quantum computers are potentially so powerful. The states of a 5 qubit computer form a 5D vector space. The states of a 10 qubit system form a 10D vector space. The latter is vastly larger than the former and has much more complex time evolution. The potential increase in power is much more than what you get from upgrading a 5 bit classical computer to a 10 bit one. (When I say 10 bits, I don't mean a computer with a 10 bit address bus, I mean 10 bits total memory.) But, alas, I suspect, for the same reason, that an (N+1)-qubit computer is much harder to build than an N-qubit computer, and so the cost of making a quantum computer of a given power will be much the same as the cost of making a classical computer of equivalent power. (I'll make an exception for specialised types of problem that you can't even begin to tackle classcially such as certain types of encryption.)



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