Nature of project: **theory**, software

Available to full-time physicists or joint students.

Building a quantum computer is one of the main technological goals driving the current development of quantum theory. The distinction between classical and quantum computer arises from the fundamental and intriguing difference regarding the concept of information.

Physically, elementary pieces of information are represented by answers to yes/no questions, obtained by making measurements. In a classical computer, such an answer is represented by a bit, a binary number 0 or 1. It can be stored in a memory cell, to be read by measurement at a later time. Then there is a small probability that the read value differs from the stored one, as the bit may have flipped due to e.g. electrical interference inside the computer - this is how random noise distorts the reading of a classical bit. In contrast, reading information from a noisy quantum bit ("qubit") is much more involved, with a rich variety of theoretical, conceptual, and philosophical aspects associated to quantum measurements, and a geometric structure similar to that of special relativity.

The overall goal: to understand that there are mutually incompatible ways of extracting classical bits from a qubit, and describe how this feature is lost in the presence of dynamical noise. This goes way beyond the textbook quantum mechanics, and is related to active research within my group. The project can be organised into the following objectives:

(O1) Understanding the physics (literature review):

- historical developments of two-state quantum systems (spin, Stern-Gerlach experiment etc)

- the modern context of quantum information and computing

- general idea of the loss of quantum features due to dynamical noise

- focus on a specific feature: incompatibility of measurements

(O2) Understanding the maths (technically easy - only requires 2x2-matrices):

- formalism for qubit states and measurements - density matrices and POVMs

- definition of incompatibility

- modelling the noisy evolution of a qubit - master equation in the Heisenberg picture

(O3) Using appropriate software tools to do numerics:

- Qutip - quantum toolbox in Python [optional]

- coding from scratch, preferably with Python

(O4) Deciding what to do, and putting together (1)-(3) to obtain, present & interpret the results.

- The (suggested) task: apply noisy dynamics to pairs of qubit measurements and calculate numerically / analytically the time(s) at which their incompatibility is lost. Roughly speaking, this tells you how long it takes for the noise to wipe out “quantumness” from the information reading process. There is no “universal” answer - it depends on the noise model. So this is where the student can be creative - they will choose which models to study and what parameter combinations to use. I can suggest some simple models to start with.

In order to just pass the project, it suffices to contribute to each of the objectives 1-4 at a basic level. That is, the student needs to (O1) show basic understanding of the physics, (O2) get at least the central formulas and definitions right, (O3) correctly use a piece of software to e.g. plot something in python, and (O4) produce a meaningful result, e.g. a plot with interpretation, for at least one noise model.

*A successful project will develop beyond the above in one/some of the following directions:*

If the student aims at a good / excellent mark, they should

(O1) provide an in-depth account of quantum incompatibility, integrated into the general context of quantum computation and historical development,

(O2) present the formalism correctly, evidencing, in particular, clear understanding of the distinction between states and measurements & Heisenberg vs Schrodinger pictures of dynamical evolution,

(O3) show actual programming skills, with e.g. Python, preferably including QuTip,

(O4) study and then select different noise models from literature, discussing and comparing the results, providing meaningful interpretations.

Some suggestions:

- Qutip provides a general master equation solver, opening up lots of possibilities for numerical study.

- In order to achieve best results it is usually good to write the code in a general form where one can easily change the model parameters to quickly generate new results. One would then select the most interesting ones for the final report.

When considering where to take your project, please bear in mind the time available. It is preferable to do fewer things well than to try many and not get conclusive results on any of them. However, sometimes it is useful to have a couple of strands of investigation in parallel to work on in case delays occur.

*Additional scope or challenge if taken as a Year-4 project:* The project can be extended in a variety of ways; here are a few possibilities:

-Incompatibility, as a quantum feature, can be quantitatively described by a certain measure of its robustness against classical noise. This quantity can be computed at each point in time during the evolution, to get a picture of how incompatibility behaves as a function of time (as opposed to only looking at times where it reaches zero). This provides additional conceptual, mathematical, and numerical challenges.

- It is interesting to compare the dynamics of incompatibility to that of other quantum features under the same noise model. In particular, dynamics of quantum entanglement has been studied extensively in the literature. Is it more resistant to noise than incompatibility?

- Theoretical construction of mutually incompatible ways of extracting classical bits from a given qubit is useful for quantum cryptography. The noise-sensitivity of such schemes is an interesting topic, to which the student can contribute through numerical studies involving different types of noise, or via more challenging analytical effort.

*Initial literature for students:*

- M.A. Nielsen, I.L. Chuang, Quantum computation and quantum information, Cambridge, 2000.
- P. Busch, H.-J. Schmidt, Coexistence of qubit effects, Quantum Information Processing 9 143-169 (2010); arXiv:0802.4167 [quant-ph]
- P. Stano, D. Reitzner, T. Heinosaari, Coexistence of qubit effects, Phys. Rev. A 78 012315 (2008); arXiv:0802.4248 [quant-ph].
- C. Addis, T. Heinosaari, J. Kiukas, E.-M. Laine, S. Maniscalco, Dynamics of incompatibility of quantum measurements in open systems, Phys. Rev. A 93 022114 (2016); arXiv:1501.04554

The required mathematical background is minimal, mainly consisting of 2x2-matrices (eigenvalues and vectors etc) and basic probability theory. However, the conceptual level is challenging:

- Basic quantum mechanics lectures often begin with wave functions, Schrodinger equations with potentials etc. Most of this is not relevant for the matrix formalism used in quantum computation.

- Noisy systems are open, requiring master equations instead of Schrodinger equations.

- Noisy qubit states are described by density matrices; state vectors are not sufficient.

- Noisy measurements likewise require a more general description than usual Hermitian operators.

- In this project, the dynamics will be applied to measurements (Heisenberg picture) rather than to states (Schrodinger picture).

I will be available to explain these concepts - motivation to learn them is essential. Previous knowledge of quantum information theory would be useful but not essential.

milestone | to be completed by |
---|---|

Understanding the aim, context and the literature - O1 | end of November |

Understanding the formalism of qubit measurements and noisy evolution - O2 | Christmas |

Preliminary numerical / analytical results on the dynamical loss of incompatibility - contributes to O4 | end of February |

Comparison of different noise models using appropriate software - O3 and O4 | Easter |