We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N^ 2 3 ) query quantum algorithm. This improves the previous O(N^ 3 4 ) query quantum algorithm of Buhrman this http URL. (quant-ph 0007016) and matches the lower bound by Shi (quant-ph 0112086). The algorithm also solves the generalization of element distinctness in which we have to find k equal items among N items. For this problem, we get an O(N^ k (k+1) ) query quantum algorithm.
Our algorithm is most closely related to the second class. In this direction, @cite_2 have constructed a counterpart of Grover's search @cite_38 based on quantum walk on the hypercube. Childs and Goldstone @cite_1 @cite_17 and @cite_32 have used quantum walk to produce search algorithms on @math -dimensional lattices ( @math ) which is faster than the naive application of Grover's search. This direction is quite closely related to our work. The algorithms by @cite_2 @cite_1 @cite_32 and current paper solve different problems but all have similar structure.
@cite_38 @cite_1 @cite_32 @cite_2 @cite_17
2040927484 1727475799 2126338609 2335111715
Grover's quantum search algorithm provides a way to speed up combinatorial search, but is not directly applicable to searching a physical database. Nevertheless, Aaronson and Ambainis showed that a database of N items laid out in d spatial dimensions can be searched in time of order radical (N) for d>2, and in time of order radical (N) poly(log N) for d=2. We consider an alternative search algorithm based on a continuous-time quantum walk on a graph. The case of the complete graph gives the continuous-time search algorithm of Farhi and Gutmann, and other previously known results can be used to show that radical (N) speedup can also be achieved on the hypercube. We show that full radical (N) speedup can be achieved on a d-dimensional periodic lattice for d>4. In d=4, the quantum walk search algorithm takes time of order radical (N) poly(log N), and in d<4, the algorithm does not provide substantial speedup. In his seminal paper, Grover points out the prospect of faster solutions for an NP-complete problem like SAT. If there are n variables, then an obvious classical deterministic algorithm checks out all 2n truth assignments in about 2n steps, while his quantum search algorithm can find a satisfying truth assignment in about 2n 2 steps. For several NP-complete problems, many sophisticated classical algorithms have been designed. They are still exponential, but much faster than the brute force algorithms. The question arises whether their running time can still be decreased from T(n) to O (√T(n)) by using a quantum computer. Isolated positive examples are known, and some speed-up has been obtained for wider classes. Here, we present a simple method to obtain the full T(n) to O(√T(n)) speed-up for most of the many nontrivial exponential time algorithms for NP-hard problems. The method works whenever the widely used technique of recursive decomposition is employed. This included all currently known algorithms for which such a speedup has not yet been known. I show that for any number of oracle lookups up to about pi 4thinsp radical (N) , Grover close_quote s quantum searching algorithm gives the maximal possible probability of finding the desired element. I explain why this is also true for quantum algorithms which use measurements during the computation. I also show that unfortunately quantum searching cannot be parallelized better than by assigning different parts of the search space to independent quantum computers. copyright ital 1999 ital The American Physical Society We investigate the cost of Grover’s quantum search algorithm when used in the context of pre-image attacks on the SHA-2 and SHA-3 families of hash functions. Our cost model assumes that the attack is run on a surface code based fault-tolerant quantum computer. Our estimates rely on a time-area metric that costs the number of logical qubits times the depth of the circuit in units of surface code cycles. As a surface code cycle involves a significant classical processing stage, our cost estimates allow for crude, but direct, comparisons of classical and quantum algorithms.