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This thesis explores one of the most prominent objects in quantum computing and fault-tolerance, the Clifford group, from a representation theory perspective. In particular, we look at the structure of the t tensor power oscillator representation of the symplectic group, intimately connected to the Clifford group, for multi-qudit systems -with odd local dimensions- described by the phase space formalism. The motivation of our work stems from the characterization of tensor power representations of the Clifford group. These representations appear in many applications of Quantum Information, from randomized benchmarking to quantum state distinction, as well as in quantum state tomography. Working with qudits, we are able to parametrize a Clifford unitary with the metaplectic representation of the symplectic group, which provides us with the capacity to investigate such representations in order to gain valuable information about the Clifford group. In parallel, the Clifford group and its related oscillator representation of the symplectic group are also important in representation theory, particularly in the study of the Howe duality, which involves a well-known notion in mathematical community, the so called theta correspondence. These representations are well understood in the case t < n in terms of self-orthogonal Calderbank-Shor-Steane (CSS) quantum codes and a correspondence termed "eta correspondence". The regime t > n was left unexplored. This thesis is an effort to investigate this regime. The objective is to decompose the t tensor power oscillator representation into irreducible ones, a goal that entailed two main levels. In the first, we decompose the representation into Sp irreducible ones, and the second, we restrict it to a special Abelian subgroup of the symplectic group, and we perform a further decomposition of the restricted oscillator representation into the subgroup’s irreducible ones (known as the N-spectrum in the literature). The analysis was carried out by implementing analytical formulas into the GAP System for Computational Discrete Algebra along with Sage Math and the proof techniques are based on mathematical tools from number theory as well as representation and character theory. The work done in this master thesis is a step forward in the characterization of the oscillator representation of the symplectic group, in an attempt to give relevant insights about the Clifford group.

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Anggelos Bampounis, THP
https://uni-koeln.zoom.us/j/94167440472?pwd=QlR3SGNHaFhQZkI1ZWJraE0rSTI0UT09
Contact: David Gross