Hi folks,

I thought I might share a few thoughts with you today on something that I’ve been thinking about for a while. It is a fairly standard problem in computational mathematics, and arises naturally when dealing with solutions to partial differential equations with geometric invariants involving tensors of relatively high order (4+, eg, elasticity tensors in materials science). Namely, the matter of solving these equations over a 2 to 4 dimensional domain using finite element methods, ie, numerically.

It is simple enough to solve Laplace’s equation or the heat equation over a square domain, but the problem comes up when one increases the number of dimensions – or when one introduces a non-trivial (read, non-flat) metric tensor, such as when exploring numerical solutions / computational simulations of solutions to the equations of general relativity. For the former problem, such is easily solved on a desktop computer; for the latter problem, one needs a relatively powerful supercomputer.

Why? The key problem is the fact that, as one increases the number of dimensions in the classical approach to solving a system of equations numerically, the time to converge to a solution (if a solution exists, but that’s another matter, entirely, involving the growth and control of singularities) increases exponentially. For instance, for a square domain 100 finite elements wide, one needs to perform operations on 100 x 100 units maybe 50 times until convergence. For a cubic domain, it is 100 x 100 x 100 units, another 50 times. For a quartic domain, it is 100^4 units, another 50 times. So the issue is clear – it is devilishly hard to brute force solve problems of this nature.

For this reason, contemporary approaches tend to use sparse matrix methods (for ‘almost flat’ tensors) or impose a high degree of symmetry on the problem domain under exploration. But this doesn’t really solve the underlying problem. Surely one must be able to find a better way?

Well, maybe with quantum computers one might be able to. A recent paper (2013), as well as this slightly older document (from 2010), suggest one can solve linear systems of equations exponentially faster using a computer of such nature. So, if true, this is encouraging, as it would render computational problems like this slightly more tractable. The developments at Microsoft Station Q on topological quantum computers, and the progress on more conventional quantum computers by researchers at other labs, such as IBM, who recently made a breakthrough in error correction (extending possible systems of qubits to the 13 to 17 qubit range), suggest that the era of quantum computing is not too far away – we may be less than 20 years away from desktop devices.

In a moderately more speculative vein, I am intrigued by the more futuristic possibility of going beyond quantum computing, into the domain of what one might call hyper computing. This would deal with systems of ‘hyper qubits’ that are ‘hyperentangled’. That’s probably a bit vague, but I sort of had in mind atomic components of the system that had multiple quantum numbers, each of which were entangled with the other atoms of the system, as well as entangled internally. So essentially one would have two ‘strata’ / levels of entanglement. The key idea is that it might be possible to scale computing power not linearly as with bits, or exponentially as with qubits, but as 2 tetrated by the number of ‘hyper-qubits’ hyper-entangled. That would be a stupendous achievement, and, yes, futuristic. At the very least, it would make problems such as what I have described above much, much easier to solve, if not outright trivial.

For a slightly better idea of how this might work, the general idea might be to entangle a system (such as a pair of photons) in N degrees of freedom, as opposed to merely one for a standard qubit, and then consider then entangling this system with M copies of same. So there would be two strata to the machine, ie, it would be an (N, M) hyper-entangled system. Then, if one could scale N and M to sufficiently high extent, potentially by increasing the complexity of the system’s constituent ‘atomic systems’, then I suspect one would essentially have a system that grew in terms of some number tetrated by some other number, the latter of which would grow as some monotonic increasing function of N and M.