Interest in mathematics, knowledge of matrices and elementary linear algebra, good programming skills, knowledge of probabilities

No

Why is quantum theory more efficient in describing the small-scale world? The answer may be hidden in the structure of the large data acquired in quantum experiments.

All we obtain in experimenting on a quantum system, e.g. quantum computers, is a large dataset of probabilities (called a MOP), p(k|M,P), showing the probability of outcome k when the measurement M is performed on the system P [1]. For example, when P=[1 0;0 0] and we measure the spin of an electron along the Z-axis, we obtain the outcome “up” with probability one and the outcome “down” with probability zero. Measuring the spin along the X-axis, however, results in the outcome “left” with a probability 1/2 and the outcome “right” with a probability 1/2.

If we were to model the probabilities in a classical way, we had to decompose the MOP into two matrices with special properties [1,2,3]. The size of these two matrices characterizes the *memory* cost of the classical model. If the MOP is an NxN table, we know that it has a *trivial* decomposition which lives in an N-dimensional space. Evidently, for large MOPs obtained in quantum experiments, this *trivial* *cost* is huge!

In this project, supported by the Quantum Computing and Simulation Hub, we ask, “What is the smallest non-trivial decomposition of an MOP?” and we aim to develop an algorithm that performs the decomposition on input MOPs. This problem resembles the famous Non-negative Matrix Factorization (NMF) problem [4], for which some algorithms exist, and we hope to use them to our advantage.

The algorithm developed in this project will be used to find the smallest classical model for the output of quantum processors. This will allow us to benchmark their performance by characterizing the memory cost of their classical simulation.

During the project, you will be working closely with my PhD student and will be supervised and supported by myself. You will also receive support in terms of computational resources from the group and the Quantum Computing and Simulation Hub.

[1] J. Barrett, Information processing in generalized probabilistic theories. Phys. Rev. A **75**, 1 (2007).

[2] M. J. Grabowecky, C. A. J. Pollack, A. R. Cameron, R. W. Spekkens, K. J. Resch, Experimentally bounding deviations from quantum theory for a photonic three-level system using theory-agnostic tomography. Phys. Rev. A. **105**, 1 (2022).

[3] F. Shahandeh, Contextuality of General Probabilistic Theories. PRX Quantum. **2**, 010330 (2021).

[4] Non-negative matrix factorization. (2023, November 7). In *Wikipedia*. https://en.wikipedia.org/wiki/Non-negative_matrix_factorization