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| '''Amplitude amplification''' is a technique in [[quantum computing]] which generalizes the idea behind
| | So, just what this "other side" reminiscent of? Well, it's very nice and relaxing, simply used. Many days I have no struggle with any anxiety at virtually. I feel completely at ease when performing facing groups or interacting with the other random regular people. I enjoy my job and experience, little, if any stress there. I will do everything I belief that I could do generally if i never was affected by anxiety from the start. I have an advanced of self-esteem and self-confidence and rarely, if ever, feel a good "idiot" or "failure," might help my anxiety would have me imagine. I am able to be who it constantly I plan to be in front of other people, that is definitely one that is wonderful feelings in globe.<br><br>What happens though when that "teenage problem", of pimples and zits follows you up? Official statistics say that 25% of adult men and 50% of adult women are influenced by acne. Within the alone nearly 17 million people end up with having acne. Acne accounts for more visits to the doctor than any other skin main issue.<br><br>Rich Presta is their favorite self-help author and self improvement coach. He's a involving best selling ebooks in your area of panic and social anxiety. The Panic Puzzle and Anxiety Free Child Program are 2 of the ebooks in his [https://www.youtube.com/watch?v=kKqUUeVzoLE online therapy for social anxiety disorder] series.<br><br>Loss of control it is all about control with all the anxiety imbalance. Here too, ought to be feel you are probably going to loose control in public, which a person would be catastrophic.<br><br>Start method when you (the parent) have probably the most amount of free time available. When you are an accountant then you'll probably want to hang about until after tax season and a lot more. I would dedicate at least a week if not two.<br><br>Some with the symptoms are extremely broad may can't be around all of these but their family. Other symptoms could be to a novel situation like talking to someone in an excessive position or eating food in front of fantastic of folks. There have been some symptoms go won't be done in chest pains every time they remain people.<br><br>The people who find themselves suffering created by this problem may live associated with their moves. Negative thoughts have led them into believeing they have certain hinders. Where is your mind focus on? This happens, when the when man or woman suffering from this focuses inward on the negative minds. Rather than focusing your attention internally, put your attention out towards the task. Medicine to put your attention outward, also it feel less anxious.<br><br>Meditation, Yoga, and Qigong are all excellent approaches to improve your overall health and start integrating basic breathing and relaxation practices into your life. |
| the [[Grover's algorithm|Grover's search algorithm]], and gives rise to a family of
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| [[quantum algorithm]]s.
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| It was discovered by [[Gilles Brassard]] and
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| [[Peter Høyer]] in 1997,
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| <ref name="brassard1997">
| |
| {{cite journal
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| |author=Gilles Brassard, Peter Høyer
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| |date=June 1997
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| |title=An exact quantum polynomial-time algorithm for Simon's problem
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| |journal=Proceedings of Fifth Israeli Symposium on Theory of Computing and Systems
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| |pages=12–23
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| |publisher=IEEE Computer Society Press
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| |arxiv=quant-ph/9704027
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| |bibcode = 1997quant.ph..4027B }}
| |
| </ref>
| |
| and independently rediscovered by [[Lov K. Grover|Lov Grover]] in 1998. | |
| <ref name="grover1998">
| |
| {{cite journal
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| |author=Grover, Lov K.
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| |date=May 1998
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| |title=Quantum Computers Can Search Rapidly by Using Almost Any Transformation
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| |journal=Phys. Rev. Lett.
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| |volume=80
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| |issue=19
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| |pages=4329–4332
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| |doi=10.1103/PhysRevLett.80.4329
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| |bibcode=1998PhRvL..80.4329G
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| |arxiv = quant-ph/9712011 }}
| |
| </ref>
| |
| | |
| In a quantum computer, amplitude amplification can be used to obtain a
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| quadratic speedup over several classical algorithms.
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| | |
| == Algorithm ==
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| | |
| The derivation presented here roughly follows the one given in
| |
| .<ref name="brassard2000">
| |
| {{cite arxiv
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| |author=Gilles Brassard, Peter Høyer, Michele Mosca, Alain Tapp
| |
| |date=2000-05-15
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| |title=Quantum Amplitude Amplification and Estimation
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| |class=quant-ph
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| |eprint=quant-ph/0005055
| |
| }}
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| </ref>
| |
| Assume we have an N-dimensional [[Hilbert space]] <math>\mathcal{H}</math>
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| representing the [[mathematical formulation of quantum mechanics|state space]] of our quantum system, spanned by the
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| orthonormal computational basis states <math>B := \{ |k\rangle \}_{k=0}^{N-1}</math>.
| |
| Furthermore assume we have a [[Hermitian]] [[projection operator]] <math>P: \mathcal{H} \to \mathcal{H}</math>.
| |
| Alternatively, <math>P</math> may be given in terms of a
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| Boolean [[Oracle_machine#Definitions|oracle]] function
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| <math>\chi:\mathbb{Z} \to \{0,1\}</math>
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| and an orthonormal operational basis
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| <math>B_{\text{op}} := \{|\omega_k\rangle \}_{k=0}^{N-1}</math>,
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| in which case | |
| :<math>P := \sum_{\chi(k)=1} |\omega_k \rangle \langle \omega_k|</math>.
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| <math>P</math> can be used to partition
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| <math>\mathcal{H}</math> into a direct sum of two mutually orthogonal subspaces,
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| the ''good'' subspace <math>\mathcal{H}_1</math> and
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| the ''bad'' subspace <math>\mathcal{H}_0</math>:
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| :<math>
| |
| \begin{align}
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| \mathcal{H}_1 &:= \text{Image}\; P &= \operatorname{span}\{|\omega_k\rangle \in B_{\text{op}} \;|\; \chi(x) = 1\},\\
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| \mathcal{H}_0 &:= \text{Ker} \; P &= \operatorname{span}\{|\omega_k\rangle \in B_{\text{op}} \;|\; \chi(x) = 0\}.
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| \end{align}
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| </math> | |
| | |
| Given a normalized state vector <math>|\psi\rangle \in \mathcal{H}</math> which has nonzero overlap with both subspaces, we can
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| uniquely decompose it as
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| :<math>|\psi\rangle = \sin(\theta) |\psi_1\rangle +\cos(\theta) |\psi_0\rangle</math>,
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| where <math>\theta = \arcsin\left( \left| P |\psi\rangle \right| \right) \in [0, \pi/2]</math>,
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| and | |
| <math>|\psi_1\rangle</math> and <math>|\psi_0\rangle</math> are the
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| normalized projections of <math>|\psi\rangle</math> into the
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| subspaces <math>\mathcal{H}_1</math> and <math>\mathcal{H}_0</math>,
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| respectively.
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| This decomposition defines a two-dimensional subspace
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| <math>\mathcal{H}_\psi</math>, spanned by the vectors
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| <math>|\psi_0\rangle</math> and <math>|\psi_1\rangle</math>.
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| <!-- (The degenerate case, where<math>|\psi\rangle</math> lies entirely in one of the subspaces ...) -->
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| The probability of finding the system in a ''good'' state when measured
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| is <math>\sin^2(\theta)</math>.
| |
| | |
| Define a unitary operator
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| <math>Q(\psi, P) := -S_\psi S_P \,\!</math>, | |
| where
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| :<math>
| |
| \begin{align}
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| S_\psi &= I -2|\psi\rangle \langle\psi|\quad \text{and}\\
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| S_P &= I -2 P.
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| \end{align}
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| </math>
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| <math>S_P</math> flips the phase of the states in the ''good'' subspace, whereas
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| <math>S_\psi</math> flips the phase of the initial state <math>|\psi\rangle</math>.
| |
| | |
| The action of this operator on <math>\mathcal{H}_\psi</math> is given by | |
| :<math>Q |\psi_0\rangle = -S_\psi |\psi_0\rangle = (2\cos^2(\theta)-1)|\psi_0\rangle +2 \sin(\theta) \cos(\theta) |\psi_1\rangle</math> and
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| :<math>Q |\psi_1\rangle = S_\psi |\psi_1\rangle = -2 \sin(\theta) \cos(\theta) |\psi_0\rangle +(1-2\sin^2(\theta))|\psi_1\rangle</math>.
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| Thus in the <math>\mathcal{H}_\psi</math> subspace <math>Q</math>
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| corresponds to a rotation by the angle <math>2\theta\,\!</math>:
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| :<math>Q = \begin{pmatrix}
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| \cos(2\theta) & \sin(2\theta)\\
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| -\sin(2\theta) & \cos(2\theta)
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| \end{pmatrix}</math>.
| |
| | |
| <!-- Outside <math>\mathcal{H}_\psi</math> the operator <math>Q</math> acts simply as <math>-S_\chi</math>...-->
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| | |
| Applying <math>Q</math> <math>n</math> times on the state
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| <math>|\psi\rangle</math>
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| gives
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| :<math>Q^n |\psi\rangle = \cos((2n+1) \theta) |\psi_0\rangle +\sin((2n+1)\theta) |\psi_1\rangle</math>,
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| rotating the state between the ''good'' and ''bad'' subspaces.
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| After <math>n</math> iterations the probability of finding the
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| system in a ''good'' state is <math>\sin^2((2n+1)\theta)\,\!</math>.<br/>
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| The probability is maximized if we choose
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| :<math>n = \left\lfloor\frac{\pi}{4\theta}\right\rfloor</math>.
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| Up until this point each iteration increases the amplitude of the ''good'' states, hence
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| the name of the technique. | |
| | |
| == Applications ==
| |
| | |
| Assume we have an unsorted database with N elements, and an oracle function
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| <math>\chi</math> which can recognize the ''good'' entries we are searching for, and <math>B_{\text{op}} = B</math> for simplicity.
| |
| | |
| If there are G such entries in the database in total, then we can find them by
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| initializing the quantum computer into a uniform superposition
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| :<math>|\psi\rangle = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} |k\rangle</math>
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| of all the database elements,
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| and running the above algorithm. In this case the overlap of the initial state with the ''good'' subspace is equal to the
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| square root of the frequency of the ''good'' entries in the database, <math>\sin(\theta) = |P |\psi\rangle|= \sqrt{G/N}</math>.
| |
| If <math>\sin(\theta) \ll 1</math>, we can approximate the number of required iterations as
| |
| :<math>n = \left\lfloor\frac{\pi}{4\theta}\right\rfloor
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| \approx \left\lfloor\frac{\pi}{4 \sin(\theta)}\right\rfloor
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| = \left\lfloor\frac{\pi}{4} \sqrt{\frac{N}{G}}\right\rfloor = O(\sqrt{N}).
| |
| </math>
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| Measuring the state will now give one of the ''good'' entries with a high probability. Since each application of <math>S_P</math> requires a single oracle query (assuming that the oracle is implemented as a quantum gate),
| |
| we can find a ''good'' entry with just <math>O(\sqrt{N})</math> oracle queries, thus obtaining a quadratic speedup over the best possible classical algorithm.
| |
| | |
| If we set G to one, the above scenario essentially reduces to the original Grover search.
| |
| | |
| == References ==
| |
| {{reflist}}
| |
| | |
| {{Quantum computing}}
| |
| | |
| {{DEFAULTSORT:Amplitude Amplification}}
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| [[Category:Quantum algorithms]]
| |
| [[Category:Search algorithms]]
| |
So, just what this "other side" reminiscent of? Well, it's very nice and relaxing, simply used. Many days I have no struggle with any anxiety at virtually. I feel completely at ease when performing facing groups or interacting with the other random regular people. I enjoy my job and experience, little, if any stress there. I will do everything I belief that I could do generally if i never was affected by anxiety from the start. I have an advanced of self-esteem and self-confidence and rarely, if ever, feel a good "idiot" or "failure," might help my anxiety would have me imagine. I am able to be who it constantly I plan to be in front of other people, that is definitely one that is wonderful feelings in globe.
What happens though when that "teenage problem", of pimples and zits follows you up? Official statistics say that 25% of adult men and 50% of adult women are influenced by acne. Within the alone nearly 17 million people end up with having acne. Acne accounts for more visits to the doctor than any other skin main issue.
Rich Presta is their favorite self-help author and self improvement coach. He's a involving best selling ebooks in your area of panic and social anxiety. The Panic Puzzle and Anxiety Free Child Program are 2 of the ebooks in his online therapy for social anxiety disorder series.
Loss of control it is all about control with all the anxiety imbalance. Here too, ought to be feel you are probably going to loose control in public, which a person would be catastrophic.
Start method when you (the parent) have probably the most amount of free time available. When you are an accountant then you'll probably want to hang about until after tax season and a lot more. I would dedicate at least a week if not two.
Some with the symptoms are extremely broad may can't be around all of these but their family. Other symptoms could be to a novel situation like talking to someone in an excessive position or eating food in front of fantastic of folks. There have been some symptoms go won't be done in chest pains every time they remain people.
The people who find themselves suffering created by this problem may live associated with their moves. Negative thoughts have led them into believeing they have certain hinders. Where is your mind focus on? This happens, when the when man or woman suffering from this focuses inward on the negative minds. Rather than focusing your attention internally, put your attention out towards the task. Medicine to put your attention outward, also it feel less anxious.
Meditation, Yoga, and Qigong are all excellent approaches to improve your overall health and start integrating basic breathing and relaxation practices into your life.