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Quantum Walk Search
In the context of quantum computing, the quantum walk search is a quantum algorithm for finding a marked node in a graph. The concept of a quantum walk is inspired by classical random walks, in which a walker moves randomly through a graph or Lattice graph, lattice. In a classical random walk, the position of the walker can be described using a probability distribution over the different nodes of the graph. In a quantum walk, on the other hand, the walker is represented by a quantum state, which can be in a Quantum superposition, superposition of several locations simultaneously. Search algorithms based on quantum walks have the potential to find applications in various fields, including Mathematical optimization, optimization, machine learning, cryptography, and Network analysis (electrical circuits), network analysis. The efficiency and probability of success of a quantum walk search depend heavily on the structure of the Feasible region, search space. In general, quantum walk sea ... [...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] |
Spectral Gap
In mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalues of a matrix or operator; alternately, it is sometimes taken as the smallest non-zero eigenvalue. Various theorems relate this difference to other properties of the system. The spectral gap gets its name from the ''matrix spectrum'', that is, for a matrix, the list of its eigenvalues. It provides insight on diffusion within the graph: corresponding the spectral gap to the smallest non-zero eigenvalue, it is then the mode of the network state that shows the slowest exponential decay over time. See also * Cheeger constant (graph theory) * Cheeger constant (Riemannian geometry) * Eigengap * Spectral gap (physics) * Spectral radius ''Spectral'' is a 2016 Hungarian-American military science fiction action film co-written and directed by Nic Mathieu. Written with Ian Fried (screenwriter), Ian Fried & George Nolfi, the film stars James Badge Dale as DARPA research scientist Ma . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Subspace
In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping''); * linearity of a ''polynomial''. An example of a linear function is the function defined by f(x)=(ax,bx) that maps the real line to a line in the Euclidean plane R2 that passes through the origin. An example of a linear polynomial in the variables X, Y and Z is aX+bY+cZ+d. Linearity of a mapping is closely related to ''Proportionality (mathematics), proportionality''. Examples in physics include the linear relationship of voltage and Electric current, current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships, such as between velocity and kinetic energy, are ''Nonlinear system, nonlinear''. Generalized for functions in more than one dimension (mathematics), dimension, linearity means the property of a function of b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator (physics)
An operator is a function over a space of physical states onto another space of states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Because of this, they are useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory. They play a central role in describing observables (measurable quantities like energy, momentum, etc.). Operators in classical mechanics In classical mechanics, the movement of a particle (or system of particles) is completely determined by the Lagrangian L(q, \dot, t) or equivalently the Hamiltonian H(q, p, t), a function of the generalized coordinates ''q'', generalized velocities \dot = \mathrm q / \mathrm t and its conjugate momenta: :p = \frac If either ''L'' or ''H'' is independent of a generalized coordinate ''q'', meaning the ''L'' and ''H'' do not change when ''q' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Phase Estimation Algorithm
In quantum computing, the quantum phase estimation algorithm is a quantum algorithm to estimate the phase corresponding to an eigenvalue of a given unitary operator. Because the eigenvalues of a unitary operator always have unit modulus, they are characterized by their phase, and therefore the algorithm can be equivalently described as retrieving either the phase or the eigenvalue itself. The algorithm was initially introduced by Alexei Kitaev in 1995. Phase estimation is frequently used as a subroutine in other quantum algorithms, such as Shor's algorithm, the quantum algorithm for linear systems of equations, and the quantum counting algorithm. Overview of the algorithm The algorithm operates on two sets of qubits, referred to in this context as registers. The two registers contain n and m qubits, respectively. Let U be a unitary operator acting on the m- qubit register. The eigenvalues of a unitary operator have unit modulus, and are therefore characterized by thei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Oracle
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. 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, 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 qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amplitude Amplification
Amplitude amplification is a technique in quantum computing that generalizes the idea behind Grover's search algorithm, and gives rise to a family of quantum algorithms. It was discovered by Gilles Brassard and Peter Høyer in 1997, and independently rediscovered by Lov Grover in 1998. In a quantum computer, amplitude amplification can be used to obtain a quadratic speedup over several classical algorithms. Algorithm The derivation presented here roughly follows the one given by Brassard et al. in 2000. Assume we have an N-dimensional Hilbert space \mathcal representing the state space of a quantum system, spanned by the orthonormal computational basis states B := \_^. Furthermore assume we have a Hermitian projection operator P\colon \mathcal \to \mathcal. Alternatively, P may be given in terms of a Boolean oracle function \chi\colon\mathbb \to \ and an orthonormal operational basis B_ := \_^, in which case :P := \sum_ , \omega_k \rangle \langle \omega_k, . P can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schematic View Of Quantum Phase Estimation On Walk Operator
A schematic, or schematic diagram, is a designed representation of the elements of a system using abstract, graphic symbols rather than realistic pictures. A schematic usually omits all details that are not relevant to the key information the schematic is intended to convey, and may include oversimplified elements in order to make this essential meaning easier to grasp, as well as additional organization of the information. For example, a subway map intended for passengers may represent a subway station with a dot. The dot is not intended to resemble the actual station at all but aims to give the viewer information without unnecessary visual clutter. A schematic diagram of a chemical process uses symbols in place of detailed representations of the vessels, piping, valves, pumps, and other equipment that compose the system, thus emphasizing the functions of the individual elements and the interconnections among them and suppresses their physical details. In an electronic circuit d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Register
In quantum computing, a quantum register is a system comprising multiple qubits. It is the quantum analogue of the classical processor register. Quantum computers perform calculations by manipulating qubits within a quantum register. Definition It is usually assumed that the register consists of qubits. It is also generally assumed that registers are not density matrices, but that they are pure, although the definition of "register" can be extended to density matrices. An n size quantum register is a quantum system comprising n pure qubits. The Hilbert space, \mathcal, in which the data is stored in a quantum register is given by \mathcal = \mathcal\otimes\mathcal\otimes\ldots\otimes\mathcal where \otimes is the tensor product. The number of dimensions of the Hilbert spaces depends on what kind of quantum systems the register is composed of. Qubits are 2-dimensional complex spaces (\mathbb^2), while qutrits are 3-dimensional complex spaces (\mathbb^3), etc. For a regist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mario Szegedy
Mario Szegedy (born October 23, 1960) is a Hungarian-American computer scientist, professor of computer science at Rutgers University. He received his Ph.D. in computer science in 1989 from the University of Chicago after completing his dissertation titled ''Algebraic Methods in Lower Bounds for Computational Models''. He held a Lady Davis Postdoctoral Fellowship at the Hebrew University of Jerusalem (1989–90), a postdoc at the University of Chicago, 1991–92, and a postdoc at Bell Laboratories (1992). Szegedy's research areas include computational complexity theory, quantum computing, computational geometry, and computational theory. He was awarded the Gödel Prize twice, in 2001 and 2005, for his work on probabilistically checkable proofs and on the space complexity of approximating the frequency moments in streamed data. His work on streaming algorithms and the resulting data analysis was also recognized by the 2019 Paris Kanellakis Theory and Practice Award. With comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chain Mixing Time
In probability theory, the mixing time of a Markov chain is the time until the Markov chain is "close" to its steady state distribution. More precisely, a fundamental result about Markov chains is that a finite state irreducible aperiodic chain has a unique stationary distribution and, regardless of the initial state, the time-''t'' distribution of the chain converges to as ''t'' tends to infinity. Mixing time refers to any of several variant formalizations of the idea: how large must ''t'' be until the time-''t'' distribution is approximately ? One variant, ''total variation distance mixing time'', is defined as the smallest ''t'' such that the total variation distance of probability measures is small: :t_(\varepsilon) = \min \left\. Choosing a different \varepsilon, as long as \varepsilon < 1/2, can only change the mixing time up to a constant factor (depending on ) and so one often fixes and simply writes [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greedy Algorithm
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time. For example, a greedy strategy for the travelling salesman problem (which is of high computational complexity) is the following heuristic: "At each step of the journey, visit the nearest unvisited city." This heuristic does not intend to find the best solution, but it terminates in a reasonable number of steps; finding an optimal solution to such a complex problem typically requires unreasonably many steps. In mathematical optimization, greedy algorithms optimally solve combinatorial problems having the properties of matroids and give constant-factor approximations to optimization problems with the submodular structure. Specifics Greedy algori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
