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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 using (classical) randomness with the square root speedup from Grover's (quantum) algorithm. First, ''n''1/3 inputs to ''f'' are selected at random and ''f'' is queried at all of them. If there is a collision among these inputs, then we return the colliding pair of inputs. Otherwise, all these inputs map to distinct values by ''f''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Computing
A quantum computer is a computer that exploits quantum mechanical phenomena. On small scales, physical matter exhibits properties of wave-particle duality, both particles and waves, and quantum computing takes advantage of this behavior using specialized hardware. Classical physics cannot explain the operation of these quantum devices, and a scalable quantum computer could perform some calculations Exponential growth, exponentially faster than any modern "classical" computer. Theoretically a large-scale quantum computer could post-quantum cryptography, break some widely used encryption schemes and aid physicists in performing quantum simulator, physical simulations; however, the current state of the art is largely experimental and impractical, with several obstacles to useful applications. The basic unit of information in quantum computing, the qubit (or "quantum bit"), serves the same function as the bit in classical computing. However, unlike a classical bit, which can be in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Algorithm
In quantum computing, a quantum algorithm is an algorithm that 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 generally reserved for algorithms that seem inherently quantum, or use some essential feature of quantum computation such as quantum superposition or quantum entanglement. Problems that are undecidable using classical computers remain undecidable using quantum computers. What makes quantum algorithms interesting is that they might be able to solve some problems fa ... [...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 queri ... [...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 analyze 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 electroni ... [...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 (199 ... (University of Chicago). External links * Academic journals 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 Sciences and Engineering Research Council of Canada, retrieved January 24, 2010. Some of these concepts have been implemented in the laboratory. In 1984, together with Charles H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grover's Algorithm
In quantum computing, Grover's algorithm, also known as the quantum search algorithm, is 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 of a function, domain. It was devised by Lov Grover in 1996. The analogous problem in classical computation would have a query complexity O(N) (i.e., the function would have to be evaluated O(N) times: there is no better approach than trying out all input values one after the other, which, on average, takes N/2 steps). Charles H. Bennett (physicist), 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 algorithm, asymptotically optimal. Since classical algorithms for NP-completenes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birthday Attack
A birthday attack is a bruteforce collision 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). Let H be the number of possible values of a hash function, with H=2^l. With a birthday attack, it is possible to find a collision of a hash function with 50% chance in \sqrt = 2^, where l is the bit length of the hash output, and with 2^ being the classical preimage resistance security with the same probability. 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-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 decision t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
