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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Quantum algorithm for linear systems of equations ： ウィキペディア英語版 Quantum algorithm for linear systems of equations The quantum algorithm for linear systems of equations, designed by Aram Harrow, Avinatan Hassidim, and Seth Lloyd, is a quantum algorithm for solving linear systems formulated in 2009. The algorithm estimates the result of a scalar measurement on the solution vector to a given linear system of equations.〔(Quantum algorithm for solving linear systems of equations, by Harrow et al. ).〕 The algorithm is one of the main fundamental algorithms expected to provide an exponential speedup over their classical counterparts, along with Shor's factoring algorithm, Grover's search algorithm and quantum simulation. Provided the linear system is a sparse and has a low condition number $\backslash kappa$, 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 $O(\backslash log(N)\backslash kappa^2)$, where $N$ is the number of variables in the linear system.. This offers an exponential speedup over the fastest classical algorithm, which runs in $O(N\backslash kappa)$ (or $O(N\backslash sqrt)$ for positive semidefinite matrices). An implementation of the quantum algorithm for linear systems of equations was first demonstrated in 2013 by Cai et al., Barz et al.and Pan et al. in parallel. The demonstrations consisted of simple linear equations on specially designed quantum devices.〔(Experimental quantum computing to solve systems of linear equations by Cai et al. ).〕 〔(Experimental realization of quantum algorithm for solving linear systems of equations, by Pan et al. ).〕 Due to the prevalence of linear systems in virtually all areas of science and engineering, the quantum algorithm for linear systems of equations has the potential for widespread applicability.〔(Quantum Computer Runs The Most Practically Useful Quantum Algorithm, by Lu and Pan ).〕 == Procedure == The problem we are trying to solve is: given a Hermitian $N\backslash times\; N$ matrix $A$ and a unit vector $\backslash overrightarrow$, find the solution vector $\backslash overrightarrow$ satisfying $A\backslash overrightarrow=\backslash overrightarrow$. This algorithm assumes that the user is not interested in the values of $\backslash overrightarrow$ itself, but rather the result of applying some operator $M$ onto x, $\backslash langle\; x|M|x\backslash rangle$. First, the algorithm represents the vector $\backslash overrightarrow$ as a quantum state of the form: : $|b\backslash rangle=\backslash sum\_^N\; b\_i|i\backslash rangle.$ Next, Hamiltonian simulation techniques are used to apply the unitary operator $e^$ to $|b\backslash rangle$ for a superposition of different times $t$. The ability to decompose $|b\backslash rangle$ into the eigenbasis of $A$ and to find the corresponding eigenvalues $\backslash lambda\_j$ is facilitated by the use of quantum phase estimation. The state of the system after this decomposition is approximately: :$\backslash sum\_^N\; \backslash beta\_j|u\_j\backslash rangle|\backslash lambda\_j\backslash rangle,$ where $u\_j$ is the eigenvector basis of $A$, and $|b\backslash rangle=\backslash sum\_^N\; \backslash beta\_j|u\_j\backslash rangle$. We would then like to perform the linear map taking $|\backslash lambda\_j\backslash rangle$ to $C\backslash lambda^\_j|\backslash lambda\_j\backslash rangle$, where $C$ is a normalizing constant. The linear mapping operation is not unitary and thus will require a number of repetitions as it has some probability of failing. After it succeeds, we uncompute the $|\backslash lambda\_j\backslash rangle$ register and are left with a state proportional to: : $\backslash sum\_^N\; \backslash beta\_i\backslash lambda^\_j|u\_j\backslash rangle=A^|b\backslash rangle=|x\backslash rangle,$ Where $|x\backslash rangle$ is a quantum-mechanical representation of the desired solution vector ''x''. To read out all components of ''x'' would require the procedure be repeated at least ''N'' times. However, it is often the case that one is not interested in $x$ itself, but rather some expectation value of a linear operator ''M'' acting on ''x''. By mapping ''M'' to a quantum-mechanical operator and performing the quantum measurement corresponding to ''M'', we obtain an estimate of the expectation value $\backslash langle\; x|M|x\backslash rangle$. This allows for a wide variety of features of the vector ''x'' to be extracted including normalization, weights in different parts of the state space, and moments without actually computing all the values of the solution vector ''x''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Quantum algorithm for linear systems of equations」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.