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...algorithm 3)). Furthermore, Szegedy [31] have shown that, for a class of Markov chains, quantum walk algorithms are quadratically faster than the corresponding classical algorithm. Buhrman and Spalek =-=[12]-=- have used Szegedy’s result to construct an O(n 5/3 ) quantum algorithm for verifying if a product of two n × n matrices A and B is equal to a third matrix C. 2 Preliminaries Let [N] denote {1,... ,N}...

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...lly smaller than the classical average hitting time. Since then, in the framework of Ambainis or Szegedy, numerous new algorithms emerged with substantially better complexity in a variety of contexts =-=[5, 19, 12, 18]-=-. This work develops a new schema for quantum search algorithms, based on any ergodic Markov chain. We extend the Szegedy quantization of classical Markov chains to possibly non-symmetric Markov chain...

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...erification of matrix products. For triangle finding, the best upper bound is O ( n 1.3) [20] and the best lower bound is Ω(n). For verification of matrix products, the best upper bound is O ( n 5/3) =-=[10]-=- and the best lower bound is Ω ( n 3/2) . Again, the adversary method cannot give better bounds, but the polynomial method might. The semidefinite programming method [7] gives an exact characterizatio...

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...query problem, for any polynomial amount of processing on the graph. We also obtain a new quantum algorithm for BMM in the query complexity setting, improving the previous best by Buhrman and ˇSpalek =-=[9]-=-: Theorem 1.5 There is an Õ(min{n1.3L17/30, n2 + L47/60n13/15})-query quantum algorithm for computing the product of two n× n Boolean matrices, where L is the number of ones in the output matrix. The...

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... grids was taken by Ambainis, Kempe and Rivosh [9]. The frameworks of Ambainis and Szegedy were used in various contexts to find algorithms with substantial complexity gains over simple Grover search =-=[26, 12, 23, 15]-=-. In a recent work, Magniez, Nayak, Roland and Santha [25] proposed a new quantum walk based search method that expanded the scope of the previous approaches. The MNRS search algorithm is also concept...

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...e problems of GROUP COMMUTATIVITY and TRIANGLE FINDING thus give strong evidence that the walks due to Ambainis are not comparable with the ones due to Szegedy. A recent result of Buhrman and ˇSpalek =-=[Bv06]-=- on matrix product verification also relies on the Szegedy quantization for its worst case time complexity. However, for the worst case instances, when there is at most one erroneous entry, the approa...

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...rkov chain adds (if |S| = M) or removes (if |S| = M + 1) a random element from the set. 4.3 Applications of element distinctness There are several results that build on element distinctness algorithm =-=[27, 18, 16]-=-. Triangle finding. A graph G with N vertices is specified by ( N) 2 variables xij with xij = 1 if there is an edge between vertices i and j. The access to xij is by queries to a black box. The task i...