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| A '''Continuous-time quantum walk (CTQW)''' is a walk on a given [[Connectivity (graph theory)|connected]] [[Graph (mathematics)|graph]] that is dictated by a time-varying unitary matrix that relies on the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the quantum system and the [[adjacency matrix]]. CTQW belongs to what is known as [[Quantum random walk|''Quantum walks'']], which also consists of the Discrete-time quantum walk. The concept of Continuous-time quantum walk is believed to have been first considered for quantum computation by Edward Farhi and Sam Gutmann.<ref name="Farhi:1998">E. Farhi and S. Gutmann, ''Quantum computation and decision trees'', Phys. Rev. A '''58''', 915 (1998)</ref> Though, it may be possible to consider CTQW for directed graphs, we focus on this area as it applies to undirected graphs unless stated otherwise.
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| ==Introduction==
| | Real estate planning needs a very special kind of lawyer, one that understands the financial markets and tendencies, in addition to how they apply to your specific condition. However it could be luring to draw increase your personal strategies and goals, you would be smart to have a legal expert to help you in the process. Go through on this page about things to search for in the great property lawyer or attorney.<br><br>You will have a straight to anticipate crystal clear, regular connection along with your lawyer. He or she should be able to offer you a standard summery of whatever you can assume when it comes to your situation, as well as what possible problems may possibly come up and what could be carried out to treatment these problems.<br><br>Examine on the internet to learn what sort of service fees are typical for the legal issues. This can help you steer clear of being swindled down the road. Select a attorney who may have an effective background and a great deal of practical experience to make certain that they don't overcharge you with regard to their very own time-spending mistakes.<br><br>Hang on to signal a contract with the legal professional till you experience the condition is correct. Request a bid from your attorney so you know approximately the number of hrs the individual intends on adding in to the case. In the event the legal professional refuses, proceed. A collection is appropriate, however it is not honest if you have no idea what you are entering into.<br><br>Always be mindful of how well you will be communicated with by the legal professional. Whether it will take time or perhaps weeks to get replies to cell phone calls or emails, then two options are present. Either your attorney does not consider a priority, or they are overstressed. Neither of them scenario is perfect for you.<br><br>Make sure you recognize how much you are going to spend before you decide to employ a lawyer. Lawyers can be extremely pricey. Their on an hourly basis charges can split your finances just before they even start to assist you. Be sure you know the expenses advance and if you cannot manage a specific legal professional, search for one that you can afford to pay for.<br><br>When getting together with using a would-be lawyer, question her or him the person you will mainly be talking to about your situation. In certain situations, lawyers give part of their caseload to your junior associate. If you believe you undoubtedly get in touch with a specific attorney, all of a sudden learning you may be utilizing another individual can be quite disconcerting. These emotions could possibly be exacerbated should you don't get along with the other individual, also.<br><br>Discover how possible attorneys respond to your own needs. The important thing to a excellent lawyer is now being happy to accomodate the needs and desires with their customers. If you come across a legal professional which is not thinking about fulfilling this, then you happen to be presented adequate alert you should hire a company else to fully handle your case.<br><br>Whenever you think that you could have got a suit, it really is appealing to phone one of those lawyers from the advertisements on tv. Despite the fact that these legal professionals are skilled, they will not be capable to practice legislation in your state, depending on exactly where they required the Club exam. It is always good to check out this and look at the fine print.<br><br>If you find a breakup legal professional and they also seem to be very interested in generating your boyfriend or girlfriend "pay out" for every thing they already have carried out to you, seek out someone else. While you may have revenge within your coronary heart, your legal professional should certainly believe directly at all times. [http://www.youtube.com/watch?v=iUzzhAbdWLM David Gruber] Becoming biased is not really an effective work ethic, even should it be within your prefer.<br><br>As you now recognize more about how estate preparation operates and exactly how legal counsel might help, it is possible to move forward with a effectively-informed plan. Search for recommendations, do your homework and have your brand new legal professional on board the instant you can.It can be never past too far to organize your future. |
| Ever since the advent of quantum mechanics and the discovery by Shor of how to achieve factorization of large semi-primes efficiently (polynomial time) using quantum computation, scientists have been intrigued by the realm of possibilities. In many cases where the quantum algorithms are derived for problems, the complexity of the algorithms are faster than their classical counterpart. Some of these include [[Shor's algorithm|Shor's factorization algorithm]], which is exponentially faster than known classical factoring algorithms, and [[Grover's algorithm|Grover's searching algorithms]], which is polynomially faster than any possible black-box classical algorithm. Many of these algorithms use (or can be modified to use) the concept of [[Quantum Fourier transform|''quantum fourier transform'']].
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| In recent times, some scientists have decided to propose a new form of looking at quantum computation algorithms. The reason is that many classical algorithms are based on (classical) random walks. This led to the intuitive question of whether a "quantum random walk" could be proposed. Though named "quantum random walk" by many research articles, the more appropriate name will be "quantum walk" given the sense of the randomness used. One such model of a quantum walk is the Continuous-time quantum walk (CTQW) proposed by E. Farhi et al.<ref name="Farhi:1998"/> There are a number of variations to representing a CTQW, but we follow the definition below.
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| ===Mathematical Definition===
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| A continuous-time quantum walk (CTQW) on a graph ''G'' = (''V,E''), where ''V'' is the set of vertices (nodes) and ''E'' is the set of edges connecting the nodes, is defined as follows:<br> Let ''A'' be the ''|V| <math>\times</math> |V|'' [[adjacency matrix]] of ''G'' with elements
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| :<math>
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| A_{u,v} = \left\{
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| \begin{matrix}
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| 1 & \text{if } \textrm{\{u,v\}} \text{ } \epsilon\text{ } E\\
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| 0 & \text{otherwise}
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| \end{matrix}\right.
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| </math>
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| and ''D'' be the ''|V| <math>\times</math> |V|'' [[degree matrix]] of ''G'' (for which the diagonal entry corresponding to vertex ''v'' is degree(''v'')), and let ''L'' = ''D'' - ''A'', be the corresponding [[Laplacian matrix]] which is positive semidefinite. The continuous-time quantum walk on the graph ''G'' is then defined by the unitary matrix
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| :<math>\,U(t) = e^{-itL}</math>
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| <br>where <math>i</math> is the [[imaginary unit]] and <math>t\,\, \epsilon \text{ }\mathbb{R}</math>. The probability of a walk starting at vertex <math>u</math> ending up at vertex <math>v</math> at time <math>t</math> is given by <math>\left| \langle v| U(t) | u \rangle \right|^2</math>. | |
| Consequently starting from the quantum state <math>|\psi_0\rangle</math> and performing a quantum walk for time <math>t</math> will result in the new state
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| <math>|\psi_t\rangle = U(t)|\psi_0\rangle</math> and measuring will thus localize the walk in vertex <math>v</math> with the probability
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| <math>\left| \langle v| U(t)|\psi_0\rangle \right|^2</math>.
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| <ref name="Gerhardt:2003">H. Gerhardt and J. Watrous, ''Continuous-time quantum walks on the symmetric group'', quant-ph/0305182</ref>
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| ==Quantum walk==
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| To discuss the quantum walk, it is useful to define the more familiar topic of (classical) random walk (CRW). The random walk is well-defined in random processes with the [http://mathworld.wolfram.com/MarkovProcess.html Markov process] model.
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| ===Classical Random Walk (CRW)===
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| Here, a graph is traversed in an order that predicts the probability of being at a node <math>v</math> in time <math>t</math> given that the walk starts from a node <math>u</math> in the graph. To elaborate this concept, consider the case of a classical walk on a line. Assume a person, John, stands at the origin of a straight line and he has a fair coin. John wants to exhibit a random walk in which the probability of moving a step to the left or right is equal (=1/2). John flips his coin and the outcome dictates where he goes next. If the outcome is head (H), he takes one step to the right and if it is tail (T), he takes one step to the left. Following this rule, then the probability of John being at position <math>d</math> after <math>N</math> steps can be shown to be
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| :<math>P_N(d) = \frac{1}{2^N}\begin{pmatrix}N\\ \frac{d+N}{2}\end{pmatrix}</math>
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| Since John has a fair coin, the average of his distribution (where he's expected to be on the average) is the origin (position 0), and the standard deviation can be shown to be <math>\sqrt{N}</math>.<ref name="Travaglione:2001">B.C. Travaglione and G.J. Milburn, ''Implementing the quantum random walk'', quant-ph/0109076</ref> A table showing the probability of this distribution for up to the fourth step is shown below:
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| <!-- Unsourced image removed: [[Image:prob_dist_classic.png||center|{{deletable image-caption|1=Monday, 31 December 2007}}|{{deletable image-caption|1=Monday, 31 December 2007}}]] -->
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| ===Continuous-time quantum walk (CTQW)===
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| Now, the fair coin in John's possession is replaced with a qubit. This qubit is defined in terms of states <math>|\downarrow\rangle</math> and <math>|\uparrow\rangle</math> different from the <math>|0\rangle</math>, <math>|1\rangle</math> basis. Next, a Hadamard operation is applied to the initial state. A Hadamard on the up state will give a superposition of up and down as follows
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| :<math>H|\uparrow\rangle = \frac{1}{\sqrt{2}}|\uparrow\rangle + \frac{1}{\sqrt{2}}|\downarrow\rangle</math>
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| and
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| :<math>H=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix}</math>
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| After applying the Hadamard operator, the spin-1/2 rotation in the z-direction is applied, thereby stepping left or right based on the resulting state. Overall, the applied operator becomes
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| :<math>U=e^{i\hat{p}\sigma_z}H</math>
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| where <math>\sigma_z</math> is the Pauli-Z operator and <math>\hat{p}</math> is the momentum operator of the qubit particle. This evolving operator in the CTQW is applied N times to the resulting qubit states. When the initial state representing the origin of the line is chosen to be <math>\frac{1}{\sqrt{2}}|\uparrow\rangle + \frac{1}{\sqrt{2}}|\downarrow\rangle</math>, the standard deviation is shown to be approximately <math>\frac{3}{5}N</math> <ref name=Travaglione:2001/> which is in quadratic form of the standard deviation from classical algorithm. In this case the probability of being at position ''d'' after ''N'' steps is different from the classical case as depicted below:
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| <!-- Image with unknown copyright status removed: [[Image:prob_dist_quantum.png||center]] -->
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| Note that the description employs a discrete-time quantum walk, but it provides an intuitive use on the graph problem for the continuous-time quantum walk where the [[Hamiltonian (quantum mechanics)|Hamiltonian]] evolves the state of the system at any given time.
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| ==Difference between CTQW and CRW<ref name=Farhi:2001>A.M. Childs, E. Farhi, and S. Gutmann, ''An example of the difference between quantum and classical random walks'', quant-ph/0103020</ref>==
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| Farhi et al. showed<ref name=Farhi:2001/> that on a symmetric graph known as <math>G_{4}</math>, which consists of two binary trees of depth four(4) merged at their children nodes, it is possible for the quantum random walk to achieve exponential speed-up over its classical counterpart. [[Image:G4 graph.png||thumb||right||G4 graph]]
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| It could be deduced that the probability of reaching the EXIT node from the ENTRANCE node of a <math>G_{n}</math> is less than <math>2^{-n}</math>, but in the quantum case, the limiting distribution probability is at least <math>{(2n+1)}^{-1}</math>. This is a tremendous increase, but it only states that the time to reach the EXIT is faster ''as contrasted from'' the algorithmic speedup. However, it was later shown by the same author <ref name="Farhi:2003">A.M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, and D.A. Spielman, ''Exponential Algorithmic Speedup by a Quantum walk'' quant-ph/0209131</ref> that indeed an exponential algorithmic speedup can be achieved by the quantum walk.
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| It is also important to note that the quantum walk 'collapses' to a classical random walk with the degree of decoherence.<ref name="Kempe:2003">J. Kempe, ''Quantum random walks - an introductory overview'', quant-ph/0303081</ref> It is possible to obtain the classical if one measures the positions in the quantum work at every step. In other words, the inability to have superposition restricts the classical domain of walks in graph theory. | |
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| ==Potential Applications==
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| The area of CTQW as well as discrete-time quantum walk has given additional insight on how to achieve quantum computation. It is an area of research that is still captivating interest and a number of algorithm using CTQW has been proposed. The three main applications to date are
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| #Quantum search algorithm<ref name="Shenvi:2002">N. Shenvi, J. Kempe, and K.B. Whaley, ''A quantum random walk search algorithm'', quant-ph/0210064</ref>
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| #Graph Traversal problem<ref name="Farhi:2003"/>
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| #[http://info.phys.unm.edu/~alandahl/iqiwiki/index.php/Quantum_Scattering_NAND-Tree_Algorithm Quantum scattering NAND-tree]
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| In each case, the algorithmic speedup is not different from what was expected in existing quantum algorithms like the Grover search algorithm that takes <math>O(\sqrt{N})</math>.
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| ==References==
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| <references/>
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| ==External links==
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| *[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=151590 Quantum Walk on arxiv.org]
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| *[http://demonstrations.wolfram.com/ContinuousTimeQuantumWalk/ CTQW on Wolfram Demonstrations]
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| *[http://machida.stat.t.u-tokyo.ac.jp/pukiwiki/index.php?Quantum%20walk Quantum walk]
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| {{DEFAULTSORT:Continuous-Time Quantum Walk}}
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| [[Category:Quantum mechanics]]
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Real estate planning needs a very special kind of lawyer, one that understands the financial markets and tendencies, in addition to how they apply to your specific condition. However it could be luring to draw increase your personal strategies and goals, you would be smart to have a legal expert to help you in the process. Go through on this page about things to search for in the great property lawyer or attorney.
You will have a straight to anticipate crystal clear, regular connection along with your lawyer. He or she should be able to offer you a standard summery of whatever you can assume when it comes to your situation, as well as what possible problems may possibly come up and what could be carried out to treatment these problems.
Examine on the internet to learn what sort of service fees are typical for the legal issues. This can help you steer clear of being swindled down the road. Select a attorney who may have an effective background and a great deal of practical experience to make certain that they don't overcharge you with regard to their very own time-spending mistakes.
Hang on to signal a contract with the legal professional till you experience the condition is correct. Request a bid from your attorney so you know approximately the number of hrs the individual intends on adding in to the case. In the event the legal professional refuses, proceed. A collection is appropriate, however it is not honest if you have no idea what you are entering into.
Always be mindful of how well you will be communicated with by the legal professional. Whether it will take time or perhaps weeks to get replies to cell phone calls or emails, then two options are present. Either your attorney does not consider a priority, or they are overstressed. Neither of them scenario is perfect for you.
Make sure you recognize how much you are going to spend before you decide to employ a lawyer. Lawyers can be extremely pricey. Their on an hourly basis charges can split your finances just before they even start to assist you. Be sure you know the expenses advance and if you cannot manage a specific legal professional, search for one that you can afford to pay for.
When getting together with using a would-be lawyer, question her or him the person you will mainly be talking to about your situation. In certain situations, lawyers give part of their caseload to your junior associate. If you believe you undoubtedly get in touch with a specific attorney, all of a sudden learning you may be utilizing another individual can be quite disconcerting. These emotions could possibly be exacerbated should you don't get along with the other individual, also.
Discover how possible attorneys respond to your own needs. The important thing to a excellent lawyer is now being happy to accomodate the needs and desires with their customers. If you come across a legal professional which is not thinking about fulfilling this, then you happen to be presented adequate alert you should hire a company else to fully handle your case.
Whenever you think that you could have got a suit, it really is appealing to phone one of those lawyers from the advertisements on tv. Despite the fact that these legal professionals are skilled, they will not be capable to practice legislation in your state, depending on exactly where they required the Club exam. It is always good to check out this and look at the fine print.
If you find a breakup legal professional and they also seem to be very interested in generating your boyfriend or girlfriend "pay out" for every thing they already have carried out to you, seek out someone else. While you may have revenge within your coronary heart, your legal professional should certainly believe directly at all times. David Gruber Becoming biased is not really an effective work ethic, even should it be within your prefer.
As you now recognize more about how estate preparation operates and exactly how legal counsel might help, it is possible to move forward with a effectively-informed plan. Search for recommendations, do your homework and have your brand new legal professional on board the instant you can.It can be never past too far to organize your future.