The Development of Transport in the Czech Republic

Before 1989, transport in the former Czechoslovakia met its tasks based on the controlling principles of planned economy, focused eastwards and oriented on cooperation between the Eastern Bloc countries within COMECOM. Due to the preference for raw material extraction and heavy industry, the transport sector dealt mainly with transporting commodities of these branches with high demands in volume. The planned economic principles were also reflected by the consistent division of transport work with a preference for stack substrate transport by rail. The change of the political and economic circumstances in November 1989 influenced the life and needs of society substantially. A market economy has come, focused on the market of developed European countries and having an impact on the transport sector as such, individual transport systems, transport preferences and transported commodities [2]. As at 1 January 1993, Czechoslovakia has been divided into two independent countries, i.e. the Czech Republic and Slovakia. Therefore the following data from the Transport Statistics of the Czech Republic [1] are comparable starting from 1994. The authors of the article had data available until 2006

Quantum Circuits for the Unitary Permutation Problem

We consider the Unitary Permutation problem which consists, given $n$ unitary gates $U_1, \ldots, U_n$ and a permutation $\sigma$ of $\{1,\ldots, n\}$, in applying the unitary gates in the order specified by $\sigma$, i.e. in performing $U_{\sigma(n)}\ldots U_{\sigma(1)}$. This problem has been introduced and investigated by Colnaghi et al. where two models of computations are considered. This first is the (standard) model of query complexity: the complexity measure is the number of calls to any of the unitary gates $U_i$ in a quantum circuit which solves the problem. The second model provides quantum switches and treats unitary transformations as inputs of second order. In that case the complexity measure is the number of quantum switches. In their paper, Colnaghi et al. have shown that the problem can be solved within $n^2$ calls in the query model and $\frac{n(n-1)}2$ quantum switches in the new model. We refine these results by proving that $n\log_2(n) +\Theta(n)$ quantum switches are necessary and sufficient to solve this problem, whereas $n^2-2n+4$ calls are sufficient to solve this problem in the standard quantum circuit model. We prove, with an additional assumption on the family of gates used in the circuits, that $n^2-o(n^{7/4+\epsilon})$ queries are required, for any $\epsilon >0$. The upper and lower bounds for the standard quantum circuit model are established by pointing out connections with the permutation as substring problem introduced by Karp.Comment: 8 pages, 5 figure