Montealegre Mora, Felipe (2022). Stabilizer and Numerical Representation Theory: Dualities, Algorithms and Applications. PhD thesis, Universität zu Köln.

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Abstract
Symmetry arguments and representation theoretical methods are widely used in theoretical physics. Quantum information theory is no exception to this: Here represen tation theory has been used, for example, to guide the development of a variety of quantum computational protocols. In this thesis, I present a series of contributions of this sort—representation theory applied to quantum information. These contributions are divided into two parts. Part I deals with the representation theory arising from the stabilizer formalism. Having originated within the context of quantum error correction, the stabilizer for malism quickly found more applications and today it is a vital tool in many areas of quantum information theory. Two important objects that appear in this formalism are the Clifford group and the oscillator representation of the symplectic group. In par ticular, tensor powers of the Clifford group and of the oscillator representation have recently come to the spotlight due to their applications in, for instance, quantum device characterization and the simulation of quantum computing. My contributions in this first part, Chapters 2 and 3, are in the context of these representations. In Chapter 2, I study these representations through the lens of Howe duality. Here, I generalize the recently developed theory of the η correspondence to provide a full decomposition for tensor power representations (TPRs). This theory has been previously used to understand a certain “maximal rank” sector of oscillator TPRs. In other words, the η correspondence has been used to partially decompose oscillator tensor powers. I show that not only can this formalism be used to fully decompose oscillator TPRs, but that it can be generalized in order to fully decompose Clifford TPRs. In Chapter 3, I present a new efficient construction of approximate unitary t designs. Unitary tdesigns are probability distributions on the unitary group that emulate the first t moments of the Haar distribution. The circuits arising in the construction shown here are dominated by Clifford unitaries, with a number O(t^4) of nonClifford gates that does not depend on the system size (e.g. the number of qubits). The proofs in this chapter build heavily on recent results which characterize the commutant of Clifford TPRs. Part II focuses on algorithms for the numerical decomposition of representations. My motivation here comes from semidefinite programming, where such algorithms can be used to significantly reduce the dimension of the optimization problem. The working principle of the package RepLAB, which tackles this problem, is presented in Chapter 5. In that chapter, I moreover show that under certain simplifying assumptions, this algorithm is stable against numerical perturbations. While the results in Chapter 5 strongly suggest that the output of RepLAB is cor rect, they do not rigorously prove this statement. In Chapter 6, I address this issue. I provide an algorithm that certifies whether a numerical decomposition of a representation is close to exact. This certifying algorithm has rigorous performance guarantees. I have coded it into the Python package RepCert, which I present and benchmark in Chapter 7. Taken together, RepLAB and RepCert may be used to decompose representations of compact groups in such a way that the output is guaranteed to be correct. The runtime of this combination depends on several factors: the complexity of taking products in the group, the largest dimension d of an irreducible block in the decomposition, and, most importantly, the dimension n of the representation. In the realistic scenario where the group product is cheap to compute and d ≪ n, the runtime is dominated by its dependence on n, scaling as O(n^3).
Item Type:  Thesis (PhD thesis)  
Creators: 


URN:  urn:nbn:de:hbz:38561233  
Date:  30 March 2022  
Language:  English  
Faculty:  Faculty of Mathematics and Natural Sciences  
Divisions:  Faculty of Mathematics and Natural Sciences > Department of Physics > Institute for Theoretical Physics  
Subjects:  Physics  
Uncontrolled Keywords: 


Date of oral exam:  3 March 2022  
Referee: 


Refereed:  Yes  
URI:  http://kups.ub.unikoeln.de/id/eprint/56123 
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