I arrived in Leeds, West Yorkshire, UK in February from the summer climate of Australia. Being back in Europe after spending 12 months on the opposite side of the globe was pleasant but I was immediately reminded why I left the continent in the first place: the winters are cold and dark and people don't smile without a reason.

During the first 4 weeks I was accommodated in a student housing in the outskirts of Leeds. I had to walk about 10 minutes to get to the University and 20 to get to the city. This is not much but Leeds is a compact city and I preferred to live in the heart of the action so I started to look for alternative ways of accommodation.

During my first year in Sydney I had concentrated on anyon walks and compared their behaviour to classical random walks and quantum walks. My supervisor Gavin and my supervisor in Leeds, Jiannis Pachos, recently made some considerable progress in the study of anyon walks by considering "Ising" type anyons which have some special properties. These properties make them more easily tractable analytically and we could perform longer simulations than were earlier possible. I worked closely with a Master's student Vaclav Zatloukal from the University of Prague and we developed codes both in Matlab and C which calculate the dynamics of Ising anyon walkers. It appears that only a small portion of the total Hilbert space is relevant to the probability distribution of the walker. Thus by restricting the calculations to this subspace only we could double the amount of time steps in the numerical calculations. This gives more information about the asymptotic behaviour of the Ising anyon walks.

Previously we have observed the algorithmic performance of the non-Abelian anyon walks to be somewhere between that of the random walks and quantum walks. The numerical calculations hint that the asymptotic behaviour, when the number of time steps approaches infinity that is, is the same as in random walks. This means that the variance of the spatial probability distribution depends linearly on the number of time steps, whereas the dependence is quadratic in quantum walks. This is interesting in the context of anyon statistics: the more complex exchange statistics of anyons seems to mix the quantum correlations and decrease the speedup which the correlations deliver.

This progress with the research has naturally been nice for me, but of course it is not the reason I am here. The main reason is of course to see England and travel around as much as possible... So I have already visited my precious home country Finland for two times, and went skiing in the Swiss Alps for four days. I have also visited York a couple of times and I can say it is a very nice city with a huge history. I also found a nice flat in the Leeds city centre which I now occupy with Vaclav. I can still walk to the university in 10 minutes and there are several pubs in a 100 metres radius so what more can I ask for.

With the summer approaching life in Leeds doesn't seem bad to me at all. On the other hand, life in Australia never was bad either. Greetings to all at QISS and see you ... not soon but soon enough!

A four-member squad of PhD students, consisting of myself, Aharon, Johann and Ressa, took an excursion to the University of Waterloo and the Fields Institute at the University of Toronto. We attended the Canadian Quantum Information Summer School and Student Conference, which I highly recommend to any beginning PhD students in the field of quantum information. In addition to that, we also had some other activities, which for me was the Workshop on Mathematics in Experimental Quantum Information Processing.

The Workshop was arranged at the IQC institute of the University of Waterloo. The main themes covered in this workshop were quantum state tomography, matrix numerical ranges of quantum observables and completely positive maps corresponding to quantum measurements. The topics were very technical and I have to say that I didn't understand many of the talks. But it was very interesting to see how rigorous mathematical machinery can be used to describe complex quantum mechanical systems. We also saw the experimental laboratories at IQC which was interesting to me. They have optical tables, liquid and solid state NMR devices and even a free air quantum key distributor. In addition, they are also building a lab to study quantum dots.

The second week we spent in Toronto in a summer school and a student conference. The summer school covered a wide range of topics in quantum information and it gave an idea of what is going on at different areas. The talks on implementing quantum information with light (Barry Sanders), quantum computational complexity (John Watrous), entanglement measures, entanglement in many-body systems and matrix product states (Guifre Vidal and Matthew Hastings) and the quantum threshold theorem (Panos Aliferis) appealed to me particularly. All in all, the topics were not directly related to my particular research interests but the school gave me a good perspective on quantum information and it might be beneficial to me later on.

The weekend was dedicated to the student conference where graduate students gave talks on their most recent work. I gave a talk on making quantum walks with anyons. Anyons and topological quantum computing did not seem to be very familiar to the audience and there was not much discussion concerning my talk, but the purpose of the talks was more about getting practice and I'm glad I had that opportunity. After the conference me and Aharon took the bus from Toronto to New York to have a bit of vacation. It was my first time in NY but certainly not the last, because everything about it is certainly true and it is really a great city!

From August 10 to 23 I am attending the "Thematic Program on Mathematics in Quantum Information" at the Fields Institute (University of Toronto), Canada. The whole program runs from July to August, covering a mixture of mathematical subjects in quantum information. I am attending three components of the program, those being:

Workshop on Mathematics in Experimental Quantum Information Processing (5 days)

Canadian Quantum Information Summer School (5 days)

Canadian Quantum Information Student Conference (2 days)

Additionally, there is the Distinguished Lecture Series which consists of three lectures on fundamental topics in quantum information, held by Matthew B. Hastings from Microsoft Station Q.

The first workshop I am attending is not directly related to my current research interests, but it is more of general interest and for the purpose of widening my perspective. The Summer School consists of talks given by active researchers on various subjects. Particular areas of interest for me are fault tolerant quantum computation, quantum computational complexity and quantum many-body problems. At the student conference I am supposed to give a talk on my recent research. I am going to discuss quantum walks with anyons, the first research subject that I have been working on during the course of my PhD studies.

I am really looking forward to my first travel as a PhD student and I hope it helps in getting some new ideas and lines of thought. I am very keen on meeting students who are doing their PhD studies on the same subject that I am, that is topological quantum information. I think this trip comes at a good point of my studies, when I am still looking for the general guidelines for my PhD thesis.

 Thematic Program on Mathematics in Quantum Information

When particles are confined in two dimensions, they can exhibit phenomena that do not occur in conventional three-dimensional systems. Such systems are said to have topological order, because the physical properties of the system are completely defined by the topological arrangement of the quasiparticles that exist in the system. That is to say that the correlations between the quasiparticles are of such nature that the time evolution of the system is only determined by the specific paths that the quasiparticles take. As long as the quasiparticles don't get too close to each other, the time evolution of the system does not change if there are small variations in the quasiparticle path; if a path can be deformed into another without taking the path over other quasiparticles, they are essentially the same path and the final state of the system is the same for all paths that can be deformed into each other without taking the paths over other quasiparticles.

But why do we talk about quasiparticles instead of particles? That is because in topologically ordered systems, the quasiparticles are not necessarily conventional particles such as electrons or atoms, but they are excitations of the topologically ordered ground state. The topologically ordered ground states can be produced by confining electrons or atoms in two dimensions, but the quasiparticles in such a system are excitations of the mutual ground state of the actual particles, and the excitations can have strange properties such as an electric charge that is a fraction of the electric charge of the actual particles, the phenomenon that is observed in the fractional quantum Hall effect. Topological systems can exhibit different kinds of quasiparticles depending on the character of the ground state of the system, and different types of quasiparticles can exist simultaneously in a topological system.

It turns out that it is possible to store and manipulate information in topologically ordered systems. When two particles are exchanged in such a system, their mutual wave function acquires a phase in Abelian systems or experiences a transformation in the Hilbert space in non-Abelian systems. Quasiparticles that have this property are called anyons.

The nontrivial action on the wave function upon the exchange of quasiparticles gives rise to a so-called topological charge. Different types of elementary quasiparticles possess different topological charges, because the effect of exchanging them is different. More interestingly, when two quasiparticles with a specific topological charge are examined as a single particle, their total topological charge can have multiple values, just like the angular momentum of conventional particles can take multiple values when several particles with specific angular momenta are regarded as a joint particle. Effectively this means that when the joint quasiparticle is exchanged with another quasiparticle, the effect in the total wave function is not unique but the transformation can have several options which are only determined upon a measurement (this only holds for the non-Abelian models where the effect is an operation in the Hilbert space).

The values that the topological charges of the joint particles can have is determined by the so-called fusion rules. The fusion rules are different for different anyon models. It has been proven that some anyon models, even fairly simple ones, allow universal quantum computing by encoding qubits to the degenerate ground state of topologically ordered system, operating on the ground state by exchanging quasiparticles with each other, and measuring the outcome by taking an ancilla particle around the whole system or bringing the quasiparticles close to each other and observing the energy spectrum of the system.

What is great about topologically ordered systems is that they are ignorant of small local perturbations, such as interactions with phonons. This protects the qubits from errors because the information is stored nonlocally in the mutual state of the interacting particles, and there is no need for error correction in the usual sense. The only errors occur when a pair of a quasiparticle and an antiquasiparticle is created from vacuum and they make nontrivial loops around other quasiparticles before they get annihilated again. Ultimately there is also the possibility that a perturbation is of such measure that the whole system breaks down and the topological order of the system is lost. But the idea of topological quantum information is very neat because the information is protected from errors physically and if the system can be maintained in its ground state there are no sources of error. The greatest caveat is just that it is very challenging to produce practical physical systems that exhibit topological order that would allow universal quantum computing.

To get a more formal introduction to topological quantum information, the following sources are recommended:

Chapter 9 of the Lectures on quantum computing by John Preskill

"Topological quantum computation", Physics Today 59 (7): 32 (2006)

"Why should anyone care about computing with anyons?", Proc.R.Soc.A 464 (2089): 1 (2008)

"Non-Abelian anyons and topological quantum computation", Rev. Mod. Phys. 80: 1083 (2008)

With a bit of editing, it is possible to run C (or Fortran) programs as Matlab functions. (It should also be possible to do it the other way around.) What has to be done is just modifying the original C source file so that Matlab knows what to do with it, and then compile source inside Matlab.

Suppose you have a C program called example.c. First, you should include a couple of files in the beginning of the source:

#include "mex.h"
#include "matrix.h"

These files include some functions that Matlab needs to understand the C code. Next, inside the source file there is the "main" function which takes command line parameters as arguments:

int main(int argc, char **argv)
{
...
return 0;
}

When Matlab interprets the source file, the main function is not special in any way, so for the sake of clarity it could be named "examplemain". The return values of the main function are not really needed, so the "return 0" part in the end (and any other return points inside the main program) can be removed and the function can be declared as void. The main function will just be called from a new function which acts as a "gateway" between Matlab and C, so the input arguments could as well be called "N_arguments" and "arguments". Thus, the former main function would now look like

void examplemain(
int N_arguments,
char *arguments[])
{
...
}

(the second asterisk in argv can be lifted). Now what needs to be done is just defining the gateway function which is called "mexFunction" (this should be the last function in the source file):

void mexFunction(
int nlhs,
mxArray *plhs[],
int nrhs
const mxArray *prhs[]
)

{
char *matlabarguments[nrhs];
long i;
for (i = 0; i <= nrhs-1; i++)
{
matlabarguments[i+1] = mxArrayToString(prhs[i]);
}
examplemain(nrhs, matlabarguments);
}

The argument array "prhs" contains the arguments which are used when the program is called from Matlab: example(prhs[0],prhs[1],...), and the integer "nrhs" is just the number of the input arguments. The Matlab output is stored to the array "plhs". To get valid Matlab output, the main function has to be edited so that the output is stored to plhs. "nlhs" is the number of the Matlab output arguments. It is important that the argument declarations are in the exact form given above.

The only role of the mexFunction is to call the main program with the arguments passed from Matlab. The Matlab arguments are of the mxArray type, which is not recognized by C, so they have to be converted to C strings one by one (mxArrayToString). Notice that the subscripts of the mxArray objects start from 0 while those of a char array start from 1.

Once this is done, you can just go into Matlab and type

>> mex example.c

On Intel Macs, this generates a file example.mexmaci. The program can now be run as

>> [out1,out2,...] = example(in1,in2,...);

It took me quite a while to figure this out, so I thought sharing it might save someone's time. Hope it helps!