The following are examples of successful projects carried out at QuSoft. Please get in touch with one of the QuSoft senior researchers in case you are interested to carry out your own bachelor or master thesis at QuSoft.

**BACHELOR PROJECTS**

## Kaustavdibya Phukan: Fractal properties and state evolution of magic state distillation

Quantum computers suffer from errors induced by the environment. Quantum error correction offers ways to deal with this problem. However, most such schemes require so called ‘magic states’. These magic states are obtained from a starting state by an iterative procedure, called ‘magic state distillation’. The question is which initial states actually end up as magic states after the iterative procedure. In other words: which initial states can be distilled to a magic state? The shape of the part of the space of one-qubit states that distil to a magic state turns out to be a fractal. In his Bachelor thesis supervised by Joris Kattemölle and Jasper van Wezel, Kaustav investigated these fractals. In specific, instead of distinguishing states that do from states that do not distill to a magic state, he resolved *how many* distillation steps it takes to approach a magic states sufficiently closely, enriching the already present fractal structure of magic state distillation.

## Stanley Kelder: Minimal noise subsystems. A numerical way to minimise decoherence for a qubit.

Qubits are affected by noise from their environment. To avoid this, the qubits are encoded. If the noise possesses a certain symmetry, the qubit can be completely protected by correct encoding. The qubit is then encoded in a noiseless subsystem. If this symmetry is perturbed, such a noiseless subsystem does not exist. The effects of noise can still be mitigated significantly by encoding the qubits in a *minimal noise subsystem*. In his Bachelor thesis supervised by Joris Kattemölle and Ben Freivogel, Stanley used numerical methods to find and study such subsystems in three physically relevant situations.

## Jorran de Wit: Teleportation of quantum states and entangled state pairs of arbitrary dimensions

In his thesis carried out under supervision of Kareljan Schoutens, Jorran explores generalizations of the original teleportation paper. Multiple qudit teleportation is described and furthermore it is worked out how higher dimensional qubits are teleported. Beyond this, it is shown that qudit teleportation is not the only variation to be done. As there are many variations, one variation is shown where an entangled Bell state pair shared between two parties Alice and Bob is teleported to two different parties, say Charlie and Dan. From here on, many more variations are imaginable.

Jorran is now master student in theoretical physics at the UvA.

## Jasper Dingerink: Quantum walks of an atom in an optical lattice

A quantum walk is the quantum version of a random walk. In his thesis supervised by Robert Spreeuw and Kareljan Schoutens, Jasper analysed one dimensional unbiased random walks and quantum walks. The probability distribution of a quantum walk looks completely different than the probability distribution of a classical random walk. Also the root mean square distance σ differs. For the quantum walk σ ∝ N , while for the classical random walk σ = sqrt(N). A quantum walk could be implemented with an atom in an optical lattice. However, by implementing the quantum walk it suffers from decoherence. By decoherence the probability distribution becomes more classical. Most of the research has been done by literature research, the simulations have been done with self-programmed code in python.

Jasper is now master student in theoretical physics at the UvA.

## Abel Sagodi: The qudit in the Fibonacci anyon model

Topological Quantum Computation (TQC) is a novel proposal for performing faster computation than on regular computers without the vulnerability to errors due to fluctuations of the environment as in the circuit model. The building blocks for a topological quantum computer are anyons that arise as excitations in a superconductor. In his double-bachelor thesis in mathematics and physics supervised by Jasper Stokman and Kareljan Schoutens, Abel gives an introduction to the underlying modular category framework of the anyon model in TQC. The underlying structure of modular categories are tensor categories with fusion and braiding rules. The anyons can be described as simple elements in such a category, while fusion is achieved as taking the tensor product between elements. By braiding anyons we can achieve any single-qubit operation (i.e. unitary transformation) with arbitrary precision. By applying morphisms on the objects in the category, described by the universal R-matrix, we can perform braidings. The properties of the Fibonacci model, a non-abelian anyon model based on SU(3)_k were investigated. Further, the representations of braidings of four anyons were calculated in order to define unitary transformations for the qutrit. Abel also presents an algorithm that calculates the representations of the braiding for arbitrary qudits.

**MASTER PROJECTS**

## Yfke Dulek: Quantum Homomorphic Encryption for Polynomial-Sized Circuits

In her thesis under the supervision of Christian Schaffner, Yfke has developed a new cryptographic encryption scheme which allows to carry out quantum computations on encrypted data. This can be very useful in the near-term setting of a “quantum cloud” where one would like to securely delegate a quantum computation to an untrusted quantum server.

Yfke’s master thesis has led to a scientific article published at the CRYPTO 2016 conference as well as in the Theory of Computing journal. Yfke has won the Ngi-NGN Informatie Scriptieprijs and presented the work at various scientific conferences such as QCrypt and QIP. Yfke is now a PhD candidate at QuSoft.

## Joris Kattemölle: Entanglement in the vacuum and the firewall paradox

In his thesis supervised by Ben Freivogel and Kareljan Schoutens, Joris calculated the amount of correlation between particles that are only there fore some observers. The amount of correlation forms a very important ingredient of the infamous firewall paradox: by a fundamental principle of the theory of really small things (quantum field theory), an observer who jumps into a black hole is incinerated by a wall of particles of very high energy. This, however, is in conflict with a fundamental principle of the theory of really big things (Einstein’s theory of general relativity), from which it follows that the observer should actually see the vacuum at the very same place. It is likely that the paradox only arises because physicists are using two separate theories next to each other. If we would have used a unified theory that describes both big and small things, there would probably be no paradox at all. However, this theory does not exist yet, and indeed there is a big effort to cook one up. Gaining a better understanding of the paradox, as we have contributed to by a little with our research, could reveal important clues about this ‘theory of everything’.

The scientific article forthcoming from Joris’ thesis was published in the Journal of High Energy Physics (JHEP). On the (Dutch) quantumuniverse blog, Joris has written about the firewall paradox and quantum computers.

Joris is now a PhD student at QuSoft.

## Tim Coopmans: Robust self-testing of (almost) all pure two-qubit states

Tim has written his MSc Logic thesis under the joint supervision of Christian Schaffner at QuSoft and Jed Kaniewski from QMATH at the University of Copenhagen. Tim has spent three months in Copenhagen to work on his thesis on the topic of robust self-testing where one would like to infer the (partial) presence of entanglement solely based on the observed violation of a Bell inequality. Such an inference is particularly challenging in the realistic setting of high noise levels where the observed Bell violation is not very large. Previous to Tim’s thesis, results that are robust to this kind of noise were only known for very special classes of states. A scientific article about the results is forthcoming.

Tim is now a PhD student at QuTech in Delft.