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Quantum Algorithm For Linear Systems Of Equations
The quantum algorithm for linear systems of equations, also called HHL algorithm, designed by Aram Harrow, Avinatan Hassidim, and Seth Lloyd, is a quantum algorithm published in 2008 for solving linear systems. The algorithm estimates the result of a scalar measurement on the solution vector to a given linear system of equations. The algorithm is one of the main fundamental algorithms expected to provide a speedup over their classical counterparts, along with Shor's factoring algorithm, Grover's search algorithm, and the quantum fourier transform. Provided the linear system is sparse and has a low condition number \kappa, and that the user is interested in the result of a scalar measurement on the solution vector, instead of the values of the solution vector itself, then the algorithm has a runtime of O(\log(N)\kappa^2), where N is the number of variables in the linear system. This offers an exponential speedup over the fastest classical algorithm, which runs in O(N\kappa) (or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aram Harrow
Aram Wettroth Harrow (born 1980) is a professor of physics in the Massachusetts Institute of Technology's Center for Theoretical Physics. Harrow works in quantum information science and quantum computing. Together with Avinatan Hassidim and Seth Lloyd, he designed a quantum algorithm for linear systems of equations, which in some cases exhibits an exponential advantage over the best classical algorithms. The algorithm has wide application in quantum machine learning. He is a steering committee member of Quantum Information Processing (QIP), the largest annual conference in the field of quantum computing. Harrow is a co-administrator of SciRate, a free and open access scientific collaboration network. He also co-runs a blog, ''The Quantum Pontiff''. His collaborators include Peter Shor Peter Williston Shor (born August 14, 1959) is an American professor of applied mathematics at MIT. He is known for his work on quantum computation, in particular for devising Shor's algori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amplitude Amplification
Amplitude amplification is a technique in quantum computing which generalizes the idea behind the 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 in. Assume we have an N-dimensional Hilbert space \mathcal representing the state space of our 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 used to partition \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Machine Learning
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data, known as training data, in order to make predictions or decisions without being explicitly programmed to do so. Machine learning algorithms are used in a wide variety of applications, such as in medicine, email filtering, speech recognition, agriculture, and computer vision, where it is difficult or unfeasible to develop conventional algorithms to perform the needed tasks.Hu, J.; Niu, H.; Carrasco, J.; Lennox, B.; Arvin, F.,Voronoi-Based Multi-Robot Autonomous Exploration in Unknown Environments via Deep Reinforcement Learning IEEE Transactions on Vehicular Technology, 2020. A subset of machine learning is closely related to computational statistics, which focuses on making predicti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least-squares Fit
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals (a residual being the difference between an observed value and the fitted value provided by a model) made in the results of each individual equation. The most important application is in data fitting. When the problem has substantial uncertainties in the independent variable (the ''x'' variable), then simple regression and least-squares methods have problems; in such cases, the methodology required for fitting errors-in-variables models may be considered instead of that for least squares. Least squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical regress ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Finance
Computational finance is a branch of applied computer science that deals with problems of practical interest in finance.Rüdiger U. Seydel, '' tp://nozdr.ru/biblio/kolxo3/F/FN/Seydel%20R.U.%20Tools%20for%20Computational%20Finance%20(4ed.,%20Springer,%202009)(ISBN%203540929282)(O)(348s)_FN_.pdf Tools for Computational Finance', Springer; 3rd edition (May 11, 2006) 978-3540279235 Some slightly different definitions are the study of data and algorithms currently used in finance and the mathematics of computer programs that realize financial models or systems.Cornelis A. Los, ''Computational Finance'' World Scientific Pub Co Inc (December 2000) Computational finance emphasizes practical numerical methods rather than mathematical proofs and focuses on techniques that apply directly to economic analyses.Mario J. Miranda and Paul L. Fackler, ''Applied Computational Economics and Finance'', The MIT Press (September 16, 2002) It is an interdisciplinary field between mathematical fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Element Method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain. The simpl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radar Cross-section
Radar cross-section (RCS), also called radar signature, is a measure of how detectable an object is by radar. A larger RCS indicates that an object is more easily detected. An object reflects a limited amount of radar energy back to the source. The factors that influence this include: *the material with which the target is made; *the size of the target relative to the wavelength of the illuminating radar signal; *the absolute size of the target; *the incident angle (angle at which the radar beam hits a particular portion of the target, which depends upon the shape of the target and its orientation to the radar source); *the reflected angle (angle at which the reflected beam leaves the part of the target hit; it depends upon incident angle); *the polarization of the transmitted and the received radiation with respect to the orientation of the target. While important in detecting targets, strength of emitter and distance are not factors that affect the calculation of an RCS becaus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preconditioner
In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for Numerical mathematics, numerical solving methods. Preconditioning is typically related to reducing a condition number of the problem. The preconditioned problem is then usually solved by an iterative method. Preconditioning for linear systems In linear algebra and numerical analysis, a preconditioner P of a matrix A is a matrix such that P^A has a smaller condition number than A. It is also common to call T=P^ the preconditioner, rather than P, since P itself is rarely explicitly available. In modern preconditioning, the application of T=P^, i.e., multiplication of a column vector, or a block of column vectors, by T=P^, is commonly performed in a Matrix-free methods, matrix-free fashion, i.e., where neither P, nor T=P^ (and often not even A) are explicitly available in a matrix form. Preconditioners are use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stefanie Barz
Stefanie Barz is a German physicist and Professor of Quantum Information and Technology at the University of Stuttgart. She studies quantum physics and quantum information in photonics. Early life and education Barz studied mathematics, physics and computer sciences at the Johannes Gutenberg University Mainz. During her undergraduate studies she was an Erasmus Programme student at the KTH Royal Institute of Technology. She earned her PhD in Vienna before moving to the University of Oxford, where she worked in quantum photonics. She was awarded the University of Vienna LaudiMaxima Prize for her dissertation. Her research created the means to demonstrate blind computing using entangled photons. The photons were generated using a nonlinear crystal, and the entangled photons represent qubits of information. Whilst the sender knows the initial state of entanglement, companies in control of data processing will be unaware, making it impossible to decode the information without destro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PSPACE
In computational complexity theory, PSPACE is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space. Formal definition If we denote by SPACE(''t''(''n'')), the set of all problems that can be solved by Turing machines using ''O''(''t''(''n'')) space for some function ''t'' of the input size ''n'', then we can define PSPACE formally asArora & Barak (2009) p.81 :\mathsf = \bigcup_ \mathsf(n^k). PSPACE is a strict superset of the set of context-sensitive languages. It turns out that allowing the Turing machine to be nondeterministic does not add any extra power. Because of Savitch's theorem,Arora & Barak (2009) p.85 NPSPACE is equivalent to PSPACE, essentially because a deterministic Turing machine can simulate a non-deterministic Turing machine without needing much more space (even though it may use much more time).Arora & Barak (2009) p.86 Also, the complements of all problems in PSPACE are also in PSPACE, meaning t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Values
In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self-adjoint operator T^*T (where T^* denotes the adjoint of T). The singular values are non-negative real numbers, usually listed in decreasing order (''σ''1(''T''), ''σ''2(''T''), …). The largest singular value ''σ''1(''T'') is equal to the operator norm of ''T'' (see Min-max theorem). If ''T'' acts on Euclidean space \Reals ^n, there is a simple geometric interpretation for the singular values: Consider the image by T of the unit sphere; this is an ellipsoid, and the lengths of its semi-axes are the singular values of T (the figure provides an example in \Reals^2). The singular values are the absolute values of the eigenvalues of a normal matrix ''A'', because the spectral theorem can be applied to obtain unitary diagonalization of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
