It’s been recently demonstrated that quantum walks on graphs can solve certain computational problems faster than any classical algorithm. Therefore it is desirable to quantify those purely combinatorial properties of graphs which quantum walks take advantage of and try and separate them from those properties due to the encoding of the problem. In this paper we isolate the combinatorial propert...

We present explicit formulae for q-exponentials on quantum spaces which could be of particular importance in physics, i.e. the q-deformed Minkowski space and the q-deformed Euclidean space with two, three or four dimensions. Furthermore, these formulae can be viewed as 2-, 3or 4-dimensional analogues of the well-known q-exponential function.

Joining independent quantum searches provides novel collective modes of quantum search (merging) by utilizing the algorithm’s underlying algebraic structure. If n quantum searches, each targeting a single item, join the domains of their classical oracle functions and sum their Hilbert spaces (merging), instead of acting independently (concatenation), then they achieve a reduction of the search ...

Many naturally occurring phenomena require theoretical treatment utilizing complex analysis by methods such as the Cauchy-Riemann relations using hyper-geometrical spaces which treat inherently nonlinear, non-dispersive, collective nonlocal resonant states of a quantum system, so as to be consistent with the nonlinearity inherent in General Relativity. Typical quantum approaches form linear app...

We propose a proof of the recent observation due to Varagnolo and Vasserot that the q-deformed Fock spaces are modules of the quantum toroidal algebra U ′ q(sln,tor) (n ≥ 3) with the level (0, 1). The quantum toroidal action on the Fock space depends on a certain parameter κ. We find that with a specific choice of this parameter the action on the Fock spaces gives rise to the toroidal action on...

A construction of the noncommutative-geometric counterparts of classical classifying spaces is presented, for general compact matrix quantum structure groups. A quantum analogue of the classical concept of the classifying map is introduced and analyzed. Interrelations with the abstract algebraic theory of quantum characteristic classes are discussed. Various non-equivalent approaches to definin...

A clean geometric construction is presented for the recent generalization of the Quantum Hall Effect to higher dimensional spaces. A very explicit formula is derived for the Hamiltonian in the generalized Quantum Hall Effect. This explicit formula suggests that the quantum Hall effect in even dimensions larger than two turns out to be simply copies of the two-dimensional effect.

As a mathematical theory per se, noncommutative geometry (NCG) is by now well established. From the beginning, its progress has been crucially influenced by quantum physics: we briefly review this development in recent years. The Standard Model of fundamental interactions, with its central role for the Dirac operator, led to several formulations culminating in the concept of a real spectral tri...

This thesis aims at connecting the two research programs known as Categorical Quantum Mechanics and Dynamic Quantum Logic. This is achieved in three steps. First we define a procedure to extract a Modal Logic frame from a small category and a functor into the category of sets and relations. Second, we extend such methodology to locally small categories. Third, we apply it to the category of fin...

Quantum information provides fundamentally different computational resources than classical information. We prove that there is no unitary protocol able to add unknown quantum states belonging to different Hilbert spaces. This is an inherent restriction of quantum physics that is related to the impossibility of copying an arbitrary quantum state, i.e., the no-cloning theorem. Moreover, we demon...