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Recently, it has been shown that one-dimensional quantum walks can mix more quickly than classical random walks, suggesting that quantum Monte Carlo algorithms can outperform their classical counterparts. We study two quantum walks on the n-dimensional hypercube, one in discrete time and one in continuous time. In both cases we show that the quantum walk(More)
A graph G is said to be d-distinguishable if there is a d-coloring of G which no non-trivial automorphism preserves. It was conjectured that if jGj > jAut(G)j and the Aut(G) action on G has no singleton orbits, then G is 2-distinguishable. We give an example where this fails. We partially repair the conjecture by showing that when " enough motion occurs, "(More)
The hidden subgroup problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonabelian case, however, remains open; an efficient solution would give rise to an efficient quantum algorithm for graph(More)
We give a simple proof of the Alon–Roichman theorem, which asserts that the Cayley graph obtained by selecting c ε log |G| elements, independently and uniformly at random, from a finite group G has expected second eigenvalue no more than ε; here c ε is a constant that depends only on ε. In particular, such a graph is an expander with constant probability.(More)
The Hidden Subgroup Problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over Abelian groups and this was used in Simon's algorithm and Shor's Factoring and Discrete Log algorithms. The non-Abelian case is open; an efficient solution would give rise to an efficient quantum algorithm for Graph Iso-morphism. We(More)
We study the computational complexity of solving systems of equations over a finite group. An equation over a group is an expression of the form Û½ ¡ Û¾ ¡ ¡ ¡ ¡ ¡ Û id where each Û is either a variable, an inverted variable, or group constant and id is the identity element of. A solution to such an equation is an assignment of the variables (to values in)(More)
The symmetric encryption problem which manifests itself when two parties must securely transmit a message m with a short shared secret key is considered in conjunction with a computationally unbounded adversary. As the adversary is unbounded, any encryption scheme must leak information about m; in particular, the mutual information between m and its(More)
In the leader election problem, n players wish to elect a random leader. The difficulty is that some coalition of players may conspire to elect one of its own members. We adopt the perfect information model: all communication is by broadcast, and the bad players have unlimited computational power. Protocols proceed in rounds: though players are synchronized(More)
In this paper we investigate the problem of computing the maximum number of entries which can be added to a partially filled latin square. The decision version of this question is known to be NP-complete. We present two approximation algorithms for the optimization version of this question. We first prove that the greedy algorithm achieves a factor of 1/3.(More)
We study the hidden subgroup problem (HSP) over groups of the form G n where G is a group of constant size. While these groups are structurally simpler than the symmetric groups S n , for which solving the HSP would yield a quantum algorithm for Graph Isomorphism, they share an important property with S n : almost all of their irreducible representations(More)