We introduce new hybrid algorithms, DBSCAN Solver and Solution Partitioning Solver, which use quantum annealing for solving Vehicle Routing Problem (VRP) and its practical variant: Capacitated Vehicle Routing Problem (CVRP). Both algorithms contain important classical components, but we also present two other algorithms, Full QUBO Solver and Average Partitioning Solver, which can be run only on a quantum processing unit (without CPU) and were prototypes which helped us develop better hybrid approaches. In order to validate our methods, we run comprehensive tests using D-Wave's Leap framework on well-established benchmark test cases as well as on our own test scenarios built based on realistic road networks. We also compared our new quantum and hybrid methods with classical algorithms - well-known meta-heuristics for solving VRP and CVRP. The experiments indicate that our hybrid methods give promising results and are able to find solutions of similar or even better quality than the tested classical algorithms.

We present a method for optimizing traffic signal settings which can be used for offline planning and realtime adaptive traffic management. The method is based on metaheuristics efficiently exploring space of possible settings and evaluating candidate solutions using a microscopic traffic simulation or metamodels of simulations built using machine learning algorithms (e.g., neural networks, LightGBM). We present results of extensive experiments and compare different algorithms and their configurations in order to find the best approach in our use case. Experiments were carried out on a realistic road network of Warsaw (maps originated from the OpenStreetMap service) and showed that LightGBM may outperform neural networks in terms of accuracy of approximations, time efficiency and optimality of traffic signal settings, which is a new and important result. We also show that in terms of traffic optimization genetic algorithms give the best results comparing to other metaheuristics.

We consider a product of two finite order quantum SU(2)-gates U-1, U-2 and ask when U-1 . U-2 has an infinite order. Using the fact that SU(2) is a double cover of SO(3) we actually study the product O(gamma, (k) over right arrow (12)) of two rotations O(phi, (k) over right arrow (12)) is an element of SO(3) and O(phi, (k) over right arrow (2)) is an element of SO(3) about axes (k) over right arrow (1), (k) over right arrow (2) is an element of R-3. In particular, we focus on the case when (k) over right arrow (1) . (k) over right arrow (2) = 0, and phi(1) = phi = phi(2) are rational multiples of pi and show that gamma is not a rational multiple of pi unless phi is an element of {k pi/2 : k is an element of Z}. The proof presented in this paper boils down to finding all pairs gamma, phi is an element of{a pi : a is an element of Q} that are solutions of cos gamma/2 = cos(2) phi/2.

We review a geometric approach to classification and examination of quantum correlations in composite systems. Since quantum information tasks are usually achieved by manipulating spin and alike systems or, in general, systems with a finite number of energy levels, classification problems are usually treated in frames of linear algebra. We proposed to shift the attention to a geometric description. Treating consistently quantum states as points of a projective space rather than as vectors in a Hilbert space we were able to apply powerful methods of differential, symplectic and algebraic geometry to attack the problem of equivalence of states with respect to the strength of correlations, or, in other words, to classify them from this point of view. Such classifications are interpreted as an identification of states with ’the same correlations properties‘, i.e. ones that can be used for the same information purposes, or, from yet another point of view, states that can be mutually transformed one to another by specific, experimentally accessible operations. It is clear that the latter characterization answers the fundamental question ’what can be transformed into what via available means?‘. Exactly such an interpretation, i.e. in terms of mutual transformability, can be clearly formulated in terms of actions of specific groups on the space of states and is the starting point for the proposed methods.

We consider the problem of deciding if a set of quantum one-qudit gates S = {U-1,...,U-n} is universal. We provide the compact-form criteria leading to a simple algorithm that allows deciding the universality of any given set of gates in a finite number of steps. Moreover, for a nonuniversal S our criteria indicate what types of gates can be added to S to turn it into a universal set.

We consider a product of two finite order quantum SU(2)-gates U_1, U_2 and ask when U_1· U_2 has an infinite order. Using the fact that SU(2) is a double cover of SO(3) we actually study the product O(γ,k⃗_12) of two rotations O(ϕ,k⃗_1)∈ SO(3) and O(ϕ,k⃗_2)∈ SO(3) about axes k⃗_1, k⃗_2∈ℝ^3. In particular we focus on the case when k⃗_1·k⃗_2=0, and ϕ_1=ϕ=ϕ_2 are rational multiple of π and show that γ is not a rational multiple of π unless ϕ∈{kπ/2:k∈ℤ}. The proof presented in this paper boils down to finding all pairs γ,ϕ∈{aπ : a∈ℚ} that are solutions of cosγ/2=cos^2ϕ/2.

We consider the problem of deciding if a set of quantum one-qudit gates \(\mathcal {S}=\{g_1,\ldots ,g_n\}\subset G\) is universal, i.e. if \({<}\mathcal {S}{>}\) is dense in G, where G is either the special unitary or the special orthogonal group. To every gate g in \(\mathcal {S}\) we assign the orthogonal matrix \(\mathrm {Ad}_g\) that is image of g under the adjoint representation \(\mathrm {Ad}:G\rightarrow SO(\mathfrak {g})\) and \(\mathfrak {g}\) is the Lie algebra of G. The necessary condition for the universality of \(\mathcal {S}\) is that the only matrices that commute with all \(\mathrm {Ad}_{g_i}\)’s are proportional to the identity. If in addition there is an element in \({<}\mathcal {S}{>}\) whose Hilbert–Schmidt distance from the centre of G belongs to \(]0,\frac{1}{\sqrt{2}}[\), then \(\mathcal {S}\) is universal. Using these we provide a simple algorithm that allows deciding the universality of any set of d-dimensional gates in a finite number of steps and formulate a general classification theorem.