Related papers: Quantum complexities of ordered searching, sorting…
We consider quantum search algorithms that have access to a noisy oracle that, for every oracle call, with probability $p>0$ completely depolarizes the query registers, while otherwise working properly. Previous results had not ruled out…
This paper studies the average complexity on the number of comparisons for sorting algorithms. Its information-theoretic lower bound is $n \lg n - 1.4427n + O(\log n)$. For many efficient algorithms, the first $n\lg n$ term is easy to…
Quantum query complexity is typically characterized in terms of XOR queries |x,y> to |x,y+f(x)> or phase queries, which ensure that even queries to non-invertible functions are unitary. When querying a permutation, another natural model is…
We consider the problem of finding the minimum element in a list of length $N$ using a noisy comparator. The noise is modelled as follows: given two elements to compare, if the values of the elements differ by at least $\alpha$ by some…
A new quantum algorithm for a search problem and its computational complexity are discussed. It is shown in the search problem containing 2^n objects that our algorithm runs in polynomial time.
One of the most important quantum algorithms ever discovered is Grover's algorithm for searching an unordered set. We give a new lower bound in the query model which proves that Grover's algorithm is exactly optimal. Similar to existing…
A recent breakthrough by Ambainis, Balodis, Iraids, Kokainis, Pr\=usis and Vihrovs (SODA'19) showed how to construct faster quantum algorithms for the Traveling Salesman Problem and a few other NP-hard problems by combining in a novel way…
This paper presents an enhancement to Grover's search algorithm for instances where the number of items (or the size of the search problem) $N$ is not a power of 2. By employing an efficient algorithm for the preparation of uniform quantum…
The quantum search algorithm of Chen and Diao, which finds with certainty a single target item in an unsorted database, is modified so as to be capable of searching for an arbitrary specified number of target items. If the number of…
We derive new time-space tradeoff lower bounds and algorithms for exactly computing statistics of input data, including frequency moments, element distinctness, and order statistics, that are simple to calculate for sorted data. We develop…
This paper describes a quantum algorithm for finding the maximum among N items. The classical method for the same problem takes O(N) steps because we need to compare two numbers in one step. This algorithm takes O(sqrt(N)) steps by…
We provide a tight analysis of Grover's recent algorithm for quantum database searching. We give a simple closed-form formula for the probability of success after any given number of iterations of the algorithm. This allows us to determine…
Sorting is a fundamental problem in computer science. In the classical setting, it is well-known that $(1\pm o(1)) n\log_2 n$ comparisons are both necessary and sufficient to sort a list of $n$ elements. In this paper, we study the Noisy…
The set equality problem is to decide whether two sets $A$ and $B$ are equal or disjoint, under the promise that one of these is the case. Some other problems, like the Graph Isomorphism problem, is solvable by reduction to the set quality…
The set equality problem is to tell whether two sets $A$ and $B$ are equal or disjoint under the promise that one of these is the case. This problem is related to the Graph Isomorphism problem. It was an open problem to find any $\omega(1)$…
We consider the problem of search of an unstructured list for a marked element, when one is given advice as to where this element might be located, in the form of a probability distribution. The goal is to minimise the expected number of…
It has recently been shown that starting with a classical query algorithm (decision tree) and a guessing algorithm that tries to predict the query answers, we can design a quantum algorithm with query complexity $O(\sqrt{GT})$ where $T$ is…
The method for analyzing algorithmic runtime complexity using decision trees is discussed using the sorting algorithm. This method is then extended to optimal algorithms which may find all cliques of size q in network N, or simply the first…
The edge list model is arguably the simplest input model for graphs, where the graph is specified by a list of its edges. In this model, we study the quantum query complexity of three variants of the triangle finding problem. The first asks…
Quantum computation has attracted much attention since it was shown by Shor and Grover the possibility to implement quantum algorithms able to realize, respectively, factoring and searching in a faster way than any other known classical…