Quantum Computing

A brief introduction to quantum computing has recently appeared as a preprint on arxiv. Here are the details:

arXiv.org > quant-ph > arXiv:0712.1098

Quantum Computations: Fundamentals And Algorithms
Authors: Steven Duplij (Kharkov National University), Illia Shapoval (NSC Kharkov Institute of Physics and Technology)
(Submitted on 7 Dec 2007)

Abstract: Basic concepts of quantum theory of information, principles of quantum calculations and the possibility of creation on this basis unique on calculation power and functioning principle device, named quantum computer, are briefly reviewed. The main blocks of quantum logic, schemes of implementation of quantum calculations, as well as some known today effective quantum algorithms, called to realize advantages of quantum calculations upon classical, are concerned. Among them special place is taken by Shor's algorithm of number factorization, Grover's algorithm of unsorted database search and, finally, the most promising in application methods of quantum phenomena simulation, particularly quantum chaos. The most perspective methods of experimental realization of quantum computer, namely nuclear-magnetic resonance and trapped ions realizations, are discussed. Phenomena of decoherence, its influence on quantum computer stability and methods of quantum error correction are described.

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Another short and interesting paper I encountered recently was on the equivalence between the circuit model and adiabatic quantum computing, by Lidar, et al. It can be accessed here. If you are unfamiliar with adiabatic quantum computation, a nice nontechnical introduction by Chris Lee is available here. Then, there's another interesting paper entitled "How Powerful is Adiabatic Quantum Computing?" by van Dam, Mosca and U. Vazirani, from IEEE.

Visualizing representations of U(1), SU(2), SU(3)

Hello:

I had a fun discovery in February. I was working on software to visualize quaternions at quaternions.sf.net. The idea is that one generates a large number of quaternions somehow, and then sorts those quaternions by their time. One can then decide to plot those quaternions in a 10 second universe at 10 frames/second, or 100 frames total. Depending on the time, it will determine what frame a quaternion is drawn.

Where the quaternion is drawn is determined by the x, y, and z. Generate the 100 frames, use some tools from ImageMagick, and out comes a gif animation that can be viewed on the web. The frames can be drawn with ImageMagick (kind of flat), GnuPlot (kind of like a physics paper), or POVRay, a great 3D image creator, the clear winner.

Space and time reversal
Yellow is input, from txyz=(-5, -5, -5, -5) to (0, 0, 0, 0).

Blue is a spatial reversal, from txyz=(-5, 5, 5, 5) to (0, 0, 0, 0)

Green is a time reversal, from txyz=(5, -5, -5, -5) to (0, 0, 0, 0)

What was up/down	What is	What can be
What was near/far	What was left/right	What can be that is

I could have written the code is anything, but I want my stuff to last. The only user interface that has lasted 30 years is the command line, so I decided to use that approach: small programs that play with quaternions, and can pipe their results into other programs that can do a math operation on quaternions. I've got about 40 such programs, like q_add, q_sin, etc. A Perl program called q_graph can takes streams of quaternions and make the animations in one step. The process for number crunching is fast, but making animation is not.

Back to the physics. I decide to generate important groups, such as SU(2), which can be generated from the expression exp(q-q*). Take a thousand randomly generated quaternions, stick it in that expression, and plot.

What was up/down	What is	What can be
What was near/far	What was left/right	What can be that is

The image is kind of interesting. Time is almost always greater than zero. It does fill up space smoothly. The Lie algebra su(2) has 3 degrees of freedom, but a quaternion has 4. How to get the 4th? Why not q/|q| exp(q-q*)? Notice that because I am using the same darn quaternion throughout, this particular bit of quaternion multiplication is Abelian because q/|q| exp(q-q*) = exp(q-q*) q/|q|. 3 degrees of freedom go into exp(q-q*), leaving 1 for q/|q|, meaning q/|q| happens to be a representation of U(1)! So take a new 1000 random quaternions, plug into q/|q| exp(q-q*), and see U(1)xSU(2), electroweak symmetry. This is more like a 4D sphere, but is has a real for negative time.

So how does one fill up uniformly a unit volume of spacetime using quaternions? I took the conjugate of one U(1)xSU(2) times a different quaternion, q* q'. Why should this create a different group? Quaternion multiplication as Hamilton practiced it is associative. I call a product that tosses a conjugate in the middle a "Euclidean product". The Euclidean product is not associative, since (a)*bc != (ab)*c. The norms are the same, but the results point in a different direction. The norm of (q/|q| exp(q-q*))* q'/|q'| exp(q'-q'*) is one, and there are 8 independent numbers that go into the system, just like the Lie algebra su(3).

What was up/down	What is	What can be
What was near/far	What was left/right	What can be that is

A longer page with a few more examples is over here: standard model . Based on the algebra, and even more importantly, these analytic animations, we now have a reason for the symmetries that appear in the standard model: it is about having the ability to describe smoothly any possible way of generating events in spacetime using quaternions. If you want to know how to bring in gravity, image 2 of these sphere, with slightly different sizes. The group Diff(M) will help smoothly describe the changes in those sizes by taking small continous steps with a changing metric.

doug

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EDIT: This is from our good friend Tim John, who has done (quite a) few good deeds this new year...starting by taking this page more seriously ;-).