Search results for "amplitude amplification"
showing 5 items of 5 documents
Variable time amplitude amplification and quantum algorithms for linear algebra problems
2012
Quantum amplitude amplification is a method of increasing a success probability of an algorithm from a small epsilon>0 to Theta(1) with less repetitions than classically. In this paper, we generalize quantum amplitude amplification to the case when parts of the algorithm that is being amplified stop at different times. We then apply the new variable time amplitude amplification to give two new quantum algorithms for linear algebra problems. Our first algorithm is an improvement of Harrow et al. algorithm for solving systems of linear equations. We improve the running time of the algorithm from O(k^2 log N) to O(k log^3 k log N) where k is the condition number of the system of equations. …
Search by Quantum Walks on Two-Dimensional Grid without Amplitude Amplification
2013
We study search by quantum walk on a finite two dimensional grid. The algorithm of Ambainis, Kempe, Rivosh [AKR05] uses \(O(\sqrt{N \log{N}})\) steps and finds a marked location with probability O(1 / logN) for grid of size \(\sqrt{N} \times \sqrt{N}\). This probability is small, thus [AKR05] needs amplitude amplification to get Θ(1) probability. The amplitude amplification adds an additional \(O(\sqrt{\log{N}})\) factor to the number of steps, making it \(O(\sqrt{N} \log{N})\).
Quantum algorithms for search with wildcards and combinatorial group testing
2012
We consider two combinatorial problems. The first we call "search with wildcards": given an unknown n-bit string x, and the ability to check whether any subset of the bits of x is equal to a provided query string, the goal is to output x. We give a nearly optimal O(sqrt(n) log n) quantum query algorithm for search with wildcards, beating the classical lower bound of Omega(n) queries. Rather than using amplitude amplification or a quantum walk, our algorithm is ultimately based on the solution to a state discrimination problem. The second problem we consider is combinatorial group testing, which is the task of identifying a subset of at most k special items out of a set of n items, given the…
Correcting for Potential Barriers in Quantum Walk Search
2015
A randomly walking quantum particle searches in Grover's $\Theta(\sqrt{N})$ iterations for a marked vertex on the complete graph of $N$ vertices by repeatedly querying an oracle that flips the amplitude at the marked vertex, scattering by a "coin" flip, and hopping. Physically, however, potential energy barriers can hinder the hop and cause the search to fail, even when the amplitude of not hopping decreases with $N$. We correct for these errors by interpreting the quantum walk search as an amplitude amplification algorithm and modifying the phases applied by the coin flip and oracle such that the amplification recovers the $\Theta(\sqrt{N})$ runtime.
Nonadiabatic quantum search algorithms
2007
7 pages, 4 figures.-- PACS nrs.: 03.67.Lx, 05.45.Mt, 72.15.Rn.-- ISI Article Identifier: 000251326400049.-- ArXiv pre-print available at: http://arxiv.org/abs/0706.1139