Related papers: Speed from Repetition
Understanding quantum speed-up over classical computing is fundamental for the development of efficient quantum algorithms. In this paper, we study such problem within the framework of the Quantum Query Model, which represents the…
Suppose we have k matrices of size n by n. We are given an oracle that knows all the entries of k matrices, that is, we can query the oracle an (i,j) entry of the l-th matrix. The goal is to test if each pair of k matrices commute with each…
The maximal clique problem, to find the maximally sized clique in a given graph, is classically an NP-complete computational problem, which has potential applications ranging from electrical engineering, computational chemistry,…
A central problem in scheduling is to schedule $n$ unit size jobs with precedence constraints on $m$ identical machines so as to minimize the makespan. For $m=3$, it is not even known if the problem is NP-hard and this is one of the last…
The oracle identification problem (OIP) is, given a set $S$ of $M$ Boolean oracles out of $2^{N}$ ones, to determine which oracle in $S$ is the current black-box oracle. We can exploit the information that candidates of the current oracle…
Sorting is one of the fundamental problems in computer science. Playing a role in many processes, it has a lower complexity bound imposed by $\mathcal{O}(n\log{n})$ when executing on a sequential machine. This limit can be brought down to…
Perceptrons, which perform binary classification, are the fundamental building blocks of neural networks. Given a data set of size~$N$ and margin~$\gamma$ (how well the given data are separated), the query complexity of the best-known…
Query complexity is a model of computation in which we have to compute a function $f(x_1, \ldots, x_N)$ of variables $x_i$ which can be accessed via queries. The complexity of an algorithm is measured by the number of queries that it makes.…
Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this…
Quantum computers are expected to revolutionize our ability to process information. The advancement from classical to quantum computing is a product of our advancement from classical to quantum physics -- the more our understanding of the…
We study the fundamental problem of guessing cryptographic keys, drawn from some non-uniform probability distribution $D$, as e.g. in LPN, LWE or for passwords. The optimal classical algorithm enumerates keys in decreasing order of…
Quantum computers leverage the principles of quantum mechanics to do computation with a potential advantage over classical computers. While a single classical computer transforms one particular binary input into an output after applying one…
Recently a great deal of attention has focused on quantum computation following a sequence of results suggesting that quantum computers are more powerful than classical probabilistic computers. Following Shor's result that factoring and the…
Quantum walks, being the quantum analogue of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the `glued trees' algorithm, which provides…
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.…
We explain the mechanism of the quantum speed-up - quantum algorithms requiring fewer computation steps than their classical equivalent - for a family of algorithms. Bob chooses a function and gives to Alice the black box that computes it.…
Recent research has demonstrated that quantum computers can solve certain types of problems substantially faster than the known classical algorithms. These problems include factoring integers and certain physics simulations. Practical…
Quantum computing is a popular topic in computer science, which has recently attracted many studies in various areas such as machine learning and network. However, the topic of quantum data structures seems neglected. There is an open…
We study the probability of making an error if, by querying an oracle a fixed number of times, we declare constant a randomly chosen n-bit Boolean function. We compare the classical and the quantum case, and we determine for how many…
Learning a hidden parity function from noisy data, known as learning parity with noise (LPN), is an example of intelligent behavior that aims to generalize a concept based on noisy examples. The solution to LPN immediately leads to decoding…