Liberating the weights

Alexei S. Trifonov

Thomas G. Wong, Pascal Philipp

Continuous-time quantum walks are natural tools for spatial search, where one searches for a marked vertex in a graph. Sometimes the structure of the graph causes the walker to get trapped, such that the probability of finding the marked vertex is limited. We give an example with two linked cliques, proving that the captive probability can be liberated by increasing the weights of the links. This allows the search to succeed with probability 1 without increasing the energy scaling of the algorithm. Further increasing the weights, however, slows the runtime, so the optimal search requires weights that are neither too weak nor too strong.

Teodor Banica

The partial isometries of [Formula: see text] form compact semigroups [Formula: see text]. We discuss here the liberation question for these semigroups, and for their discrete versions [Formula: see text]. Our main results concern the construction of half-liberations [Formula: see text] and of liberations [Formula: see text]. We include a detailed algebraic and probabilistic study of all these objects, justifying our “half-liberation” and “liberation” claims.

Isaac L. Chuang

Climério Paulo da Silva Neto

Julien Bichon

We describe the quantum subspaces of Banica–Goswami’s half-liberated real spheres, showing in particular that there is a bijection between the symmetric ones and the conjugation stable closed subspaces of the complex projective spaces.