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BHT Algorithm
In quantum computing, the Brassard-Høyer-Tapp algorithm or BHT algorithm is a quantum algorithm that solves the collision problem. In this problem, one is given ''n'' and an ''r''-to-1 function f:\,\\rightarrow\ and needs to find two inputs that ''f'' maps to the same output. The BHT algorithm only makes O(n^) queries to ''f'', which matches the lower bound of \Omega(n^) in the black box model. The algorithm was discovered by Gilles Brassard, Peter Høyer, and Alain Tapp in 1997. It uses Grover's algorithm, which was discovered the year before. Algorithm Intuitively, the algorithm combines the square root speedup from the birthday paradox In probability theory, the birthday problem asks for the probability that, in a set of randomly chosen people, at least two will share a birthday. The birthday paradox is that, counterintuitively, the probability of a shared birthday exceeds 5 ... using (classical) randomness with the square root speedup from Grover's (quantum) algori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Computing
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though current quantum computers may be too small to outperform usual (classical) computers for practical applications, larger realizations are believed to be capable of solving certain computational problems, such as integer factorization (which underlies RSA encryption), substantially faster than classical computers. The study of quantum computing is a subfield of quantum information science. There are several models of quantum computation with the most widely used being quantum circuits. Other models include the quantum Turing machine, quantum annealing, and adiabatic quantum computation. Most models are based on the quantum bit, or " qubit", which is somewhat analogous to the bit in classical computation. A qubit can be in a 1 or 0 quan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Algorithm
In quantum computing, a quantum algorithm is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical (or non-quantum) algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a problem, where each step or instruction can be performed on a classical computer. Similarly, a quantum algorithm is a step-by-step procedure, where each of the steps can be performed on a quantum computer. Although all classical algorithms can also be performed on a quantum computer, the term quantum algorithm is usually used for those algorithms which seem inherently quantum, or use some essential feature of quantum computation such as quantum superposition or quantum entanglement. Problems which are undecidable using classical computers remain undecidable using quantum computers. What makes quantum algorithms interesting is that they might be able to solve some problem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Collision Problem
The r-to-1 collision problem is an important theoretical problem in complexity theory, quantum computing, and computational mathematics. The collision problem most often refers to the 2-to-1 version: given n even and a function f:\,\\rightarrow\, we are promised that f is either 1-to-1 or 2-to-1. We are only allowed to make queries about the value of f(i) for any i\in\. The problem then asks how many such queries we need to make to determine with certainty whether f is 1-to-1 or 2-to-1. Classical solutions Deterministic Solving the 2-to-1 version deterministically requires \frac+1 queries, and in general distinguishing r-to-1 functions from 1-to-1 functions requires \frac + 1 queries. This is a straightforward application of the pigeonhole principle: if a function is r-to-1, then after \frac + 1 queries we are guaranteed to have found a collision. If a function is 1-to-1, then no collision exists. Thus, \frac + 1 queries suffice. If we are unlucky, then the first n/r quer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Black Box
In science, computing, and engineering, a black box is a system which can be viewed in terms of its inputs and outputs (or transfer characteristics), without any knowledge of its internal workings. Its implementation is "opaque" (black). The term can be used to refer to many inner workings, such as those of a transistor, an engine, an algorithm, the human brain, or an institution or government. To analyse an open system with a typical "black box approach", only the behavior of the stimulus/response will be accounted for, to infer the (unknown) ''box''. The usual representation of this ''black box system'' is a data flow diagram centered in the box. The opposite of a black box is a system where the inner components or logic are available for inspection, which is most commonly referred to as a white box (sometimes also known as a "clear box" or a "glass box"). History The modern meaning of the term "black box" seems to have entered the English language around 1945. In electr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory Of Computing
''Theory of Computing'' is a peer-reviewed open access scientific journal covering theoretical computer science. The journal was established in 2005 and is published by the Department of Computer Science of the University of Chicago. The editor-in-chief is László Babai László "Laci" Babai (born July 20, 1950, in Budapest) a fellow of the American Academy of Arts and Sciences, and won the Knuth Prize. Babai was an invited speaker at the International Congresses of Mathematicians in Kyoto (1990), Zürich (1994, ... (University of Chicago). External links * Publications established in 2005 Creative Commons Attribution-licensed journals Computer science journals University of Chicago {{compu-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gilles Brassard
Gilles Brassard, is a faculty member of the Université de Montréal, where he has been a Full Professor since 1988 and Canada Research Chair since 2001. Education and early life Brassard received a Ph.D. in Computer Science from Cornell University in 1979, working in the field of cryptography with John Hopcroft as his advisor. Research Brassard is best known for his fundamental work in quantum cryptography, quantum teleportation, quantum entanglement distillation, quantum pseudo-telepathy, and the classical simulation of quantum entanglement.Herzberg runner-up: Gilles Brassard Natural Sc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grover's Algorithm
In quantum computing, Grover's algorithm, also known as the quantum search algorithm, refers to a quantum algorithm for unstructured search that finds with high probability the unique input to a black box function that produces a particular output value, using just O(\sqrt) evaluations of the function, where N is the size of the function's domain. It was devised by Lov Grover in 1996. The analogous problem in classical computation cannot be solved in fewer than O(N) evaluations (because, on average, one has to check half of the domain to get a 50% chance of finding the right input). Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani proved that any quantum solution to the problem needs to evaluate the function \Omega(\sqrt) times, so Grover's algorithm is asymptotically optimal. Since classical algorithms for NP-complete problems require exponentially many steps, and Grover's algorithm provides at most a quadratic speedup over the classical solution for un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birthday Attack
A birthday attack is a type of cryptographic attack that exploits the mathematics behind the birthday problem in probability theory. This attack can be used to abuse communication between two or more parties. The attack depends on the higher likelihood of collisions found between random attack attempts and a fixed degree of permutations (pigeonholes). With a birthday attack, it is possible to find a collision of a hash function in \sqrt = 2^, with 2^n being the classical preimage resistance security. There is a general (though disputed) result that quantum computers can perform birthday attacks, thus breaking collision resistance, in \sqrt = 2^. Although there are some digital signature vulnerabilities associated with the birthday attack, it cannot be used to break an encryption scheme any faster than a brute-force attack. Understanding the problem As an example, consider the scenario in which a teacher with a class of 30 students (n = 30) asks for everybody's birthday (for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Element Distinctness Problem
In computational complexity theory, the element distinctness problem or element uniqueness problem is the problem of determining whether all the elements of a list are distinct. It is a well studied problem in many different models of computation. The problem may be solved by sorting the list and then checking if there are any consecutive equal elements; it may also be solved in linear expected time by a randomized algorithm that inserts each item into a hash table and compares only those elements that are placed in the same hash table cell. Several lower bounds in computational complexity are proved by reducing the element distinctness problem to the problem in question, i.e., by demonstrating that the solution of the element uniqueness problem may be quickly found after solving the problem in question. Decision tree complexity The number of comparisons needed to solve the problem of size n, in a comparison-based model of computation such as a decision tree or algebraic decisio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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