Amolak Ratan Kalra, PhD candidate
David R. Cheriton School of Computer Science
Supervisor: Professor Michele Mosca
This thesis revolves around two major themes. The first is the use of weight enumerators to study two different problems in quantum coding theory: magic state distillation and quantum channel capacity. The second is the study of the interaction between quantum circuits and number theory and the application of these techniques to the problems of circuit synthesis and stabilizer ranks. The following results are established:
1. Applications of Weight Enumerators: We study the applications of weight enumerators to magic state distillation and establish that many properties of T-state distillation protocols are directly determined by the code’s simple weight enumerator. By enforcing the physical consistency of the distillation process, we derive a new set of constraints on weight enumerators, yielding new upper bounds on the minimum distance of certain classical and quantum codes. Secondly, we use the coset weight enumerator formalism of DiVincenzo, Shor and Smolin to study noise thresholds of Pauli channels for stabilizer codes, in the context of quantum capacity. We report several new concatenated stabilizer codes of small length that show significant non-additivity. We also give a new closed form expression of coset weight enumerators of concatenated phase and bit flip repetition codes.
2. Number Theory and Quantum Circuits: We use number-theoretic tools, in particular the Barnes Wall lattice, to study the problem of stabilizer rank. In this context, we give the first quantitative lower bound on stabilizer fidelity in terms of stabilizer rank. We introduce a new magic measure called the Barnes Wall norm, and use it to give a lower bound on approximate stabilizer rank. Lastly, we use the machinery of Bruhat-Tits buildings to give a proof of the arithmetic nature of the single-qutrit Clifford+R gate set.
To attend this PhD defence in person, please go to QNC 2101. You can also attend virtually on MS Teams.
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