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

Shortest prefix strings containing all subset permutations

AbstractWhat is the length of the shortest string consisting of elements of {1,…n} that contains as subsequences all permutations of any k-element subset? Many authors have considered the special case where k=n. We instead consider an incremental variation on this problem first proposed by Koutas and Hu. For a fixed value of n they ask for a string such that for all values of k⩽n, the prefix containing all permutations of any k-element subset as subsequences is as short as possible. The problem can also be viewed as follows:For k=1 one needs n distinct digits to find each of the n possible permutations. In going from k to k+1, one starts with a string containing all k-element permutations as subsequences, and one adds as few digits as possible to the end of the string so that the new string contains all (k+1)-element permutations.We give a new construction that gives shorter strings than the best previous construction. We then prove a weak form of lower bound for the number of digits added in successive suffixes. The lower bound proof leads to a construction that matches the bound exactly. The length of a shortest prefix string is k(n−2)+[13(k+1)]+3, for k > 2.The lengths for k=1, 2 are n and 2n−1. This proves the natural conjecture that requiring the strings to be prefixes strictly increases the length of the strings required for all but the smallest values of k