On the interplay between Babai and Cerny's conjectures

Motivated by the Babai conjecture and the Cerny conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with $n$ states in this class, we prove that the reset thresholds are upper-bounded by $2n^2-6n+5$ and can attain the value $\tfrac{n(n-1)}{2}$. In addition, we study diameters of the pair digraphs of permutation automata and construct $n$-state permutation automata with diameter $\tfrac{n^2}{4} + o(n^2)$.Comment: 21 pages version with full proof

Language and Automata Theory and Applications

International audienc

On the interplay between Babai and Černý’s conjectures

Motivated by the Babai conjecture and the Černý conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with n states in this class, we prove that the reset thresholds are upperbounded by 2n2 -6n + 5 and can attain the value (Formula presented). In addition, we study diameters of the pair digraphs of permutation automata and construct n-state permutation automata with diameter (formula presented). © Springer International Publishing AG 2017

On the Interplay Between Černý and Babai’s Conjectures

Motivated by the Babai conjecture and the Černý conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with n states in this class, we prove that the reset threshold is upper-bounded by 2n2−6n+5 and can attain the value n(n−1)2. In addition, we study diameters of the pair digraphs of permutation automata and construct n-state permutation automata with diameter n24+o(n2)

Characterization and Control in Large Hilbert spaces.

Computational devices built on and exploiting quantum phenomena have the potential to revolutionize our understanding of computational complexity by being able to solve certain problems faster than the best known classical algorithms. Unfortunately, unlike the digital computers quantum information processing devices hope to replace, quantum information is fragile by nature and lacks the inherent robustness of digital logic. Indeed, for whatever we can do to control the evolution, nature can also do in some random and unknown fashion ruining the computation. This thesis explores the task of building the classical control architecture to control a large quantum system and how to go about characterizing the behaviour of the system to determine the level of control reached. Both these tasks appear to require an exponential amount of resources as the size of the system grows. The inability to efficiently control and characterize large scale quantum systems will certainly militate against their potential computational usefulness making these important problems to solve. The solutions presented in this thesis are all tested for their practical usefulness by implementing them in either liquid- or solid-state nuclear magnetic resonance

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