Piotr Gawron (IITiS PAN)

• Ryszard Winiarczyk (IITiS PAN)
• Jan Sładkowski (University of Silesia)
• Tadeusz Czachórski (IITiS PAN and Silesian University of Technology)
• Jerzy Klamka (IITiS PAN and Silesian University of Technology)
• Jarosław Adam Miszczak (IITiS PAN)
• Zbigniew Puchała (IITiS PAN)

• [1] Z. Puchała, J.A. Miszczak, P. Gawron, C.F. Dunkl, J.A. Holbrook, K. Życzkowski, "Restricted numerical shadow and geometry of quantum entanglement", preprint (2012). arXiv:1201.2524.

  The restricted numerical range W_R(A) of an operator A acting on a D-dimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset R of the of set of pure quantum states of dimension D. One considers for instance the set of real states, or in the case of composite spaces, the set of product states and the set of maximally entangled states. Combining the operator theory with a probabilistic approach we introduce the restricted numerical shadow of A -- a normalized probability distribution on the complex plane supported in W_R(A). Its value at point z in C is equal to the probability that the expectation value <\psi|A|\psi> is equal to z, where |\psi> represents a random quantum state in subset R distributed according to the natural measure on this set, induced by the unitarily invariant Fubini--Study measure. Studying restricted shadows of operators of a fixed size D=N_A N_B we analyse the geometry of sets of separable and maximally entangled states of the N_A x N_B composite quantum system. Investigating trajectories formed by evolving quantum states projected into the plane of the shadow we study the dynamics of quantum entanglement. A similar analysis extended for operators on D=2^3 dimensional Hilbert space allows us to investigate the structure of the orbits of GHZ and W quantum states of a three--qubit system.

• [2] Gawron P., Klamka J., Winiarczyk R., "Noise effects in the quantum search algorithm from the computational complexity point of view", International Journal of Applied Mathematics and Computer Science, Vol. 22 (2012). arXiv:1108.1915.

  We analyse the resilience of the quantum search algorithm in the presence of quantum noise modelled as trace preserving completely positive maps. We study the influence of noise on computational complexity of the quantum search algorithm. We show that only for small amounts of noise the quantum search algorithm is still more efficient than any classical algorithm.

• [3] J.A. Miszczak, "Models of quantum computation and quantum programming languages", Bulletin of the Polish Academy of Sciences - Technical Sciences, Vol. 59 (2011): 305-324.

  The goal of this report is to provide an introduction to the basic computational models used in quantum information theory. We various review models of quantum Turing machine, quantum circuits and quantum random access machine (QRAM) along with their classical counterparts. We also provide an introduction to quantum programming languages, which are developed using the QRAM model. We review the syntax of several existing quantum programming languages and discuss their features and limitations.

• Project funded by National Science Centre
• Number: N N516 481840
• Contract number: 4818/B/T02/2011/40
• Dates: 20.04.2011-19.04.2014