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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 Amplitude amplification ： ウィキペディア英語版 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 $\backslash mathcal$ representing the state space of our quantum system, spanned by the orthonormal computational basis states $B\; :=\; \backslash \_^$. Furthermore assume we have a Hermitian projection operator $P:\; \backslash mathcal\; \backslash to\; \backslash mathcal$. Alternatively, $P$ may be given in terms of a Boolean oracle function $\backslash chi:\backslash mathbb\; \backslash to\; \backslash$ and an orthonormal operational basis $B\_\_^$, in which case :$P\; :=\; \backslash sum\_\; |\backslash omega\_k\; \backslash rangle\; \backslash langle\; \backslash omega\_k|$. $P$ can be used to partition $\backslash mathcal$ into a direct sum of two mutually orthogonal subspaces, the ''good'' subspace $\backslash mathcal\_1$ and the ''bad'' subspace $\backslash mathcal\_0$: :$$ \begin \mathcal_1 &:= \text\; P &= \operatorname\,\\ \mathcal_0 &:= \text \; P &= \operatorname\. \end Given a normalized state vector $|\backslash psi\backslash rangle\; \backslash in\; \backslash mathcal$ which has nonzero overlap with both subspaces, we can uniquely decompose it as :$|\backslash psi\backslash rangle\; =\; \backslash sin(\backslash theta)\; |\backslash psi\_1\backslash rangle\; +\backslash cos(\backslash theta)\; |\backslash psi\_0\backslash rangle$, where $\backslash theta\; =\; \backslash arcsin\backslash left(\; \backslash left|\; P\; |\backslash psi\backslash rangle\; \backslash right|\; \backslash right)\; \backslash in\; (\backslash pi/2)$, and $|\backslash psi\_1\backslash rangle$ and $|\backslash psi\_0\backslash rangle$ are the normalized projections of $|\backslash psi\backslash rangle$ into the subspaces $\backslash mathcal\_1$ and $\backslash mathcal\_0$, respectively. This decomposition defines a two-dimensional subspace $\backslash mathcal\_\backslash psi$, spanned by the vectors $|\backslash psi\_0\backslash rangle$ and $|\backslash psi\_1\backslash rangle$. The probability of finding the system in a ''good'' state when measured is $\backslash sin^2(\backslash theta)$. Define a unitary operator $Q(\backslash psi,\; P)\; :=\; -S\_\backslash psi\; S\_P\; \backslash ,\backslash !$, where :$$ \begin S_\psi &= I -2|\psi\rangle \langle\psi|\quad \text\\ S_P &= I -2 P. \end $S\_P$ flips the phase of the states in the ''good'' subspace, whereas $S\_\backslash psi$ flips the phase of the initial state $|\backslash psi\backslash rangle$. The action of this operator on $\backslash mathcal\_\backslash psi$ is given by :$Q\; |\backslash psi\_0\backslash rangle\; =\; -S\_\backslash psi\; |\backslash psi\_0\backslash rangle\; =\; (2\backslash cos^2(\backslash theta)-1)|\backslash psi\_0\backslash rangle\; +2\; \backslash sin(\backslash theta)\; \backslash cos(\backslash theta)\; |\backslash psi\_1\backslash rangle$ and :$Q\; |\backslash psi\_1\backslash rangle\; =\; S\_\backslash psi\; |\backslash psi\_1\backslash rangle\; =\; -2\; \backslash sin(\backslash theta)\; \backslash cos(\backslash theta)\; |\backslash psi\_0\backslash rangle\; +(1-2\backslash sin^2(\backslash theta))|\backslash psi\_1\backslash rangle$. Thus in the $\backslash mathcal\_\backslash psi$ subspace $Q$ corresponds to a rotation by the angle $2\backslash theta\backslash ,\backslash !$: :$Q\; =\; \backslash begin$ \cos(2\theta) & \sin(2\theta)\\ -\sin(2\theta) & \cos(2\theta) \end. Applying $Q$ $n$ times on the state $|\backslash psi\backslash rangle$ gives :$Q^n\; |\backslash psi\backslash rangle\; =\; \backslash cos((2n+1)\; \backslash theta)\; |\backslash psi\_0\backslash rangle\; +\backslash sin((2n+1)\backslash theta)\; |\backslash psi\_1\backslash rangle$, rotating the state between the ''good'' and ''bad'' subspaces. After $n$ iterations the probability of finding the system in a ''good'' state is $\backslash sin^2((2n+1)\backslash theta)\backslash ,\backslash !$. The probability is maximized if we choose :$n\; =\; \backslash left\backslash lfloor\backslash frac\backslash right\backslash rfloor$. Up until this point each iteration increases the amplitude of the ''good'' states, hence the name of the technique. 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「Amplitude amplification」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.