Corrected speed - Revision history

en>Daniele Pugliesi: removed Category:Turbines; added Category:Turbomachinery using HotCat

2014-04-29T16:05:29Z

removed Category:Turbines; added Category:Turbomachinery using HotCat

← Older revision		Revision as of 18:05, 29 April 2014
Line 1:		Line 1:
			Greetings! I am Marvella and I really feel comfy when individuals use the complete name. He utilized to be unemployed but now he is a meter reader. Body developing is what my family and I appreciate. Puerto Rico is exactly where he's been residing for years and he will by no means move.<br><br>Also visit my web-site :: [http://biztoc.com/mod/groups/topicposts.php?topic=1311479&group_guid=1311468 at home std testing]
	~~[[Image:cnot-compared-to-xor.svg\|thumb\|The classical analog of the CNOT gate is a [[Reversible_computing#Reversible_circuits\|reversible]] [[XOR gate]].]]~~
	~~[[Image:CNOT-QuantumComputation.png\|thumb\|How the CNOT gate can be used (with [[Hadamard gate]]s) in a computation]]~~
	~~[[Image:Cnotf.PNG\|thumb\|Representation of the CNOT gate]]~~
	~~[[Image:functioncnot.PNG\|thumb\|Answer on output depending on input~~ and ~~CNOT function]]~~
	~~[[Image:Cnotsupernot.PNG\|thumb\|The first qubit flips only if the second qubit is 1.]]~~
	~~In [[computing science]], the '''controlled NOT gate''' (also '''C-NOT''' or '''CNOT''') is a [[quantum gate]] that is an essential component in~~ the ~~construction of a [[quantum computer]]~~. ~~It can be used~~ to ~~entangle and disentangle [[Quantum entanglement\|EPR states]]. Specifically, any quantum circuit can~~ be ~~simulated to an arbitrary degree of accuracy using~~ a ~~combination of CNOT gates and single [[qubit]] rotations~~.

	~~==Operation==~~

	~~The CNOT gate flips the second qubit (the target qubit) if and only if the first qubit (the control qubit)~~ is 1.

	~~{\| align="center" class="wikitable"~~
	\|-
	~~! colspan="2" \| Before \|\| colspan="2" \| After~~
	\|-
	~~! Control \|\| Target \|\| Control \|\| Target~~
	~~\|- align=center~~
	~~\| 0 \|\| 0 \|\| 0 \|\| 0~~
	~~\|- align=center~~
	~~\| 0 \|\| 1 \|\| 0 \|\| 1~~
	~~\|- align=center~~
	~~\| 1 \|\| 0 \|\| 1 \|\| 1~~
	~~\|- align=center~~
	~~\| 1 \|\| 1 \|\| 1 \|\| 0~~
	\|}

	~~The resulting value of the second qubit corresponds to the result of a classical [[XOR gate]].~~

	~~The CNOT gate can be represented by the matrix:~~

	~~:<math> \operatorname{CNOT} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}.</math>~~

	~~The first experimental realization of a CNOT gate was accomplished in 1995. Here, a single Beryllium ion in a trap was used. The two qubits were encoded into an optical state~~ and ~~into the vibrational state of the ion within the trap~~. ~~At the time of the experiment, the reliability of the CNOT-operation was measured to be on the order of 90%.~~

	~~In addition to a regular controlled NOT gate, one could construct a function-controlled NOT gate, which accepts an arbitrary number ''n''+1 of qubits as input,~~ where '~~'n''+1 is greater than or equal to 2 (a [[quantum register]]). This gate flips the last qubit of the register if~~ and ~~only if a built-in function, with the first ''n'' qubits as input, returns a 1~~.
	~~The function-controlled NOT gate is an essential element of the [[Deutsch-Jozsa algorithm]].~~
	CNOT is also a kind of universal gate(in the classical sense of the word). It is easy to see that if the CONTROL is set to '1' the TARGET output is always NOT. So, a NOT GATE can be constructed using CNOT.

	~~==Proof of operation==~~

	~~Let~~ <~~math~~>~~\left\{~~
	~~\|0\rangle =\! \begin{bmatrix} 1 \\ 0 \end{bmatrix},~~
	~~\|1\rangle =\! \begin{bmatrix} 0 \\ 1 \end{bmatrix}~~
	~~\right\}~~<~~/math~~> ~~be the [[orthonormal basis]] (using [[Bra~~-~~ket notation]]).~~

	~~Let <math>\| \psi \rangle = a \| 0 \rangle + b \| 1 \rangle =\! \begin{bmatrix} a \\ b \end{bmatrix}</math>.~~

	~~Let <math>\| \phi \rangle = b \| 0 \rangle + a \| 1 \rangle =\! \begin{bmatrix} b \\ a \end{bmatrix}</math> be the flip qubit of <math> \| \psi \rangle</math>.~~

	~~Recall that <math>\|\alpha\rangle\!\otimes\!\|\beta\rangle~~
	~~= \|\alpha\rangle \|\beta\rangle~~
	~~= \|\alpha,\beta\rangle </math>.~~

	~~===When control qubit is 0===~~

	~~Here is a proof that <math>~~
	~~\operatorname{CNOT}\ \|0,\psi\rangle = \|0,\psi\rangle~~
	~~</math>~~:

	~~Note that here the specific definition of <math> \operatorname{CNOT}\ </math> assumes an eigenbasis of~~

	<math> \|00\rangle = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \|01\rangle = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} , \|10\rangle = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \|11\rangle = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} </math>

	~~It's straightforward to verify that if~~
	~~<math>~~
	~~\|0,\psi\rangle = a\|0\rangle \|0\rangle + b\|0\rangle \|1\rangle = \begin{bmatrix} a \\ b \\ 0 \\ 0 \end{bmatrix}</math>~~

	[[Matrix multiplication\|Then]] <math>\operatorname{CNOT}\ \|0,\psi\rangle = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} a \\ b \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} a \\ b \\ 0 \\ 0 \end{bmatrix} = \|0,\psi\rangle</math>.

	~~Therefore CNOT doesn't change the <math> \|\psi\rangle</math> qubit if the first qubit is 0.~~

	~~===When control qubit is 1===~~

	~~Here is a proof that <math> \operatorname{CNOT}\ \|1,\psi\rangle = \|1,\phi\rangle</math>, which means that the CNOT gate flips the <math>\|\psi\rangle<~~/~~math> qubit.~~

	~~Similarly to the first demonstration, <math> \|1,\psi\rangle = \begin{bmatrix} 0 \\ 0 \\ a \\ b \end{bmatrix}<~~/~~math>~~.

	Then <math> \operatorname{CNOT}\ \|1,\psi\rangle = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ a \\ b \end{bmatrix} = a \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} + b \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}</~~math>~~

	~~It can be seen that~~
	~~<math> \|1,1\rangle = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}<~~/~~math>~~
	~~and~~
	~~<math> \|1,0\rangle = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}<~~/~~math>, using these on the equation above gives~~

	~~<math>~~
	~~\begin{align}~~
	~~\operatorname{CNOT}\ \|1,\psi\rangle~~
	~~& = a \|1,1\rangle + b \|1,0\rangle \\~~
	~~& = \|1\rangle \left( a \|1\rangle + b \|0\rangle \right) \\~~
	~~& = \|1,\phi\rangle \\~~
	~~\end{align}~~
	~~</math>~~

	Therefore the CNOT gate flips the <math>\|\psi\rangle</math> qubit into <math>\|\phi\rangle</math> if the control qubit is set to 1. A simple way to observe this is to multiply the CNOT matrix by a column vector, noticing that the operation on the first bit is identity, and a NOT gate on the second bit.

	=~~=See also==~~
	* [[C-ROT gate]]

	~~==References==~~
	*{{cite book \| author=Nielsen, Michael A. & Chuang, Isaac L.\|title=Quantum Computation and Quantum Information \| publisher=Cambridge University Press \|year=2000 \|isbn=0-521-63235-8}}

	*{{cite journal \| author=Monroe, C. & Meekhof, D. & King, B. & Itano, W. & Wineland, D. \|title=Demonstration of a Fundamental Quantum Logic Gate \| journal=Physical Review Letters \| issue= 25 \| pages=4714–4717 \|year=1995 \|doi=10.1103/PhysRevLett.75.4714 \|volume=75 \| pmid=10059979 \| bibcode=~~1995PhRvL..75.4714M}} [http://tf.nist.gov/general/pdf/140.pdf]~~

	~~[[Category:Quantum gates]]~~
	~~[[Category:Quantum information science]~~]

115.113.118.199: /* See also */

2013-03-01T10:51:40Z

See also

← Older revision		Revision as of 12:51, 1 March 2013
Line 1:		Line 1:
	The ~~name~~ of the ~~author~~ is ~~Figures~~. ~~California~~ is where ~~her house~~ is ~~but she requirements~~ to ~~move because~~ of ~~her family members~~. He is ~~really fond~~ of ~~performing ceramics but he~~ is ~~having difficulties~~ to ~~find time for it~~. ~~Hiring has been my occupation for some time but I've currently utilized for another one~~.<br><br>~~Also visit my web site~~ - [~~http:~~//~~www~~.~~blaze16~~.~~com~~/~~blog~~/~~255682~~ http://~~www~~.~~blaze16~~.~~com~~]
			[[Image:cnot-compared-to-xor.svg\|thumb\|The classical analog of the CNOT gate is a [[Reversible_computing#Reversible_circuits\|reversible]] [[XOR gate]].]]
			[[Image:CNOT-QuantumComputation.png\|thumb\|How the CNOT gate can be used (with [[Hadamard gate]]s) in a computation]]
			[[Image:Cnotf.PNG\|thumb\|Representation of the CNOT gate]]
			[[Image:functioncnot.PNG\|thumb\|Answer on output depending on input and CNOT function]]
			[[Image:Cnotsupernot.PNG\|thumb\|The first qubit flips only if the second qubit is 1.]]
			In [[computing science]], the '''controlled NOT gate''' (also '''C-NOT''' or '''CNOT''') is a [[quantum gate]] that is an essential component in the construction of a [[quantum computer]]. It can be used to entangle and disentangle [[Quantum entanglement\|EPR states]]. Specifically, any quantum circuit can be simulated to an arbitrary degree of accuracy using a combination of CNOT gates and single [[qubit]] rotations.

			==Operation==

			The CNOT gate flips the second qubit (the target qubit) if and only if the first qubit (the control qubit) is 1.

			{\| align="center" class="wikitable"
			\|-
			! colspan="2" \| Before \|\| colspan="2" \| After
			\|-
			! Control \|\| Target \|\| Control \|\| Target
			\|- align=center
			\| 0 \|\| 0 \|\| 0 \|\| 0
			\|- align=center
			\| 0 \|\| 1 \|\| 0 \|\| 1
			\|- align=center
			\| 1 \|\| 0 \|\| 1 \|\| 1
			\|- align=center
			\| 1 \|\| 1 \|\| 1 \|\| 0
			\|}

			The resulting value of the second qubit corresponds to the result of a classical [[XOR gate]].

			The CNOT gate can be represented by the matrix:

			:<math> \operatorname{CNOT} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}.</math>

			The first experimental realization of a CNOT gate was accomplished in 1995. Here, a single Beryllium ion in a trap was used. The two qubits were encoded into an optical state and into the vibrational state of the ion within the trap. At the time of the experiment, the reliability of the CNOT-operation was measured to be on the order of 90%.

			In addition to a regular controlled NOT gate, one could construct a function-controlled NOT gate, which accepts an arbitrary number ''n''+1 of qubits as input, where ''n''+1 is greater than or equal to 2 (a [[quantum register]]). This gate flips the last qubit of the register if and only if a built-in function, with the first ''n'' qubits as input, returns a 1.
			The function-controlled NOT gate is an essential element of the [[Deutsch-Jozsa algorithm]].
			CNOT is also a kind of universal gate(in the classical sense of the word). It is easy to see that if the CONTROL is set to '1' the TARGET output is always NOT. So, a NOT GATE can be constructed using CNOT.

			==Proof of operation==

			Let <math>\left\{
			\|0\rangle =\! \begin{bmatrix} 1 \\ 0 \end{bmatrix},
			\|1\rangle =\! \begin{bmatrix} 0 \\ 1 \end{bmatrix}
			\right\}</math> be the [[orthonormal basis]] (using [[Bra-ket notation]]).

			Let <math>\| \psi \rangle = a \| 0 \rangle + b \| 1 \rangle =\! \begin{bmatrix} a \\ b \end{bmatrix}</math>.

			Let <math>\| \phi \rangle = b \| 0 \rangle + a \| 1 \rangle =\! \begin{bmatrix} b \\ a \end{bmatrix}</math> be the flip qubit of <math> \| \psi \rangle</math>.

			Recall that <math>\|\alpha\rangle\!\otimes\!\|\beta\rangle
			= \|\alpha\rangle \|\beta\rangle
			= \|\alpha,\beta\rangle </math>.

			===When control qubit is 0===

			Here is a proof that <math>
			\operatorname{CNOT}\ \|0,\psi\rangle = \|0,\psi\rangle
			</math>:

			Note that here the specific definition of <math> \operatorname{CNOT}\ </math> assumes an eigenbasis of

			<math> \|00\rangle = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \|01\rangle = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} , \|10\rangle = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \|11\rangle = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} </math>

			It's straightforward to verify that if
			<math>
			\|0,\psi\rangle = a\|0\rangle \|0\rangle + b\|0\rangle \|1\rangle = \begin{bmatrix} a \\ b \\ 0 \\ 0 \end{bmatrix}</math>

			[[Matrix multiplication\|Then]] <math>\operatorname{CNOT}\ \|0,\psi\rangle = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} a \\ b \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} a \\ b \\ 0 \\ 0 \end{bmatrix} = \|0,\psi\rangle</math>.

			Therefore CNOT doesn't change the <math> \|\psi\rangle</math> qubit if the first qubit is 0.

			===When control qubit is 1===

			Here is a proof that <math> \operatorname{CNOT}\ \|1,\psi\rangle = \|1,\phi\rangle</math>, which means that the CNOT gate flips the <math>\|\psi\rangle</math> qubit.

			Similarly to the first demonstration, <math> \|1,\psi\rangle = \begin{bmatrix} 0 \\ 0 \\ a \\ b \end{bmatrix}</math>.

			Then <math> \operatorname{CNOT}\ \|1,\psi\rangle = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ a \\ b \end{bmatrix} = a \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} + b \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}</math>

			It can be seen that
			<math> \|1,1\rangle = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}</math>
			and
			<math> \|1,0\rangle = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}</math>, using these on the equation above gives

			<math>
			\begin{align}
			\operatorname{CNOT}\ \|1,\psi\rangle
			& = a \|1,1\rangle + b \|1,0\rangle \\
			& = \|1\rangle \left( a \|1\rangle + b \|0\rangle \right) \\
			& = \|1,\phi\rangle \\
			\end{align}
			</math>

			Therefore the CNOT gate flips the <math>\|\psi\rangle</math> qubit into <math>\|\phi\rangle</math> if the control qubit is set to 1. A simple way to observe this is to multiply the CNOT matrix by a column vector, noticing that the operation on the first bit is identity, and a NOT gate on the second bit.

			==See also==
			* [[C-ROT gate]]

			==References==
			*{{cite book \| author=Nielsen, Michael A. & Chuang, Isaac L.\|title=Quantum Computation and Quantum Information \| publisher=Cambridge University Press \|year=2000 \|isbn=0-521-63235-8}}

			*{{cite journal \| author=Monroe, C. & Meekhof, D. & King, B. & Itano, W. & Wineland, D. \|title=Demonstration of a Fundamental Quantum Logic Gate \| journal=Physical Review Letters \| issue= 25 \| pages=4714–4717 \|year=1995 \|doi=10.1103/PhysRevLett.75.4714 \|volume=75 \| pmid=10059979 \| bibcode=1995PhRvL..75.4714M}} [http://tf.nist.gov/general/pdf/140.pdf]

			[[Category:Quantum gates]]
			[[Category:Quantum information science]]

86.185.64.17: Remove redundant brackets; correct typo

2012-03-23T17:40:00Z

Remove redundant brackets; correct typo

New page

The name of the author is Figures. California is where her house is but she requirements to move because of her family members. He is really fond of performing ceramics but he is having difficulties to find time for it. Hiring has been my occupation for some time but I've currently utilized for another one.<br><br>Also visit my web site - [http://www.blaze16.com/blog/255682 http://www.blaze16.com]

← Older revision		Revision as of 18:05, 29 April 2014
Line 1:		Line 1:
			Greetings! I am Marvella and I really feel comfy when individuals use the complete name. He utilized to be unemployed but now he is a meter reader. Body developing is what my family and I appreciate. Puerto Rico is exactly where he's been residing for years and he will by no means move.<br><br>Also visit my web-site :: [http://biztoc.com/mod/groups/topicposts.php?topic=1311479&group_guid=1311468 at home std testing]
	~~[[Image:cnot-compared-to-xor.svg\|thumb\|The classical analog of the CNOT gate is a [[Reversible_computing#Reversible_circuits\|reversible]] [[XOR gate]].]]~~
	~~[[Image:CNOT-QuantumComputation.png\|thumb\|How the CNOT gate can be used (with [[Hadamard gate]]s) in a computation]]~~
	~~[[Image:Cnotf.PNG\|thumb\|Representation of the CNOT gate]]~~
	~~[[Image:functioncnot.PNG\|thumb\|Answer on output depending on input~~ and ~~CNOT function]]~~
	~~[[Image:Cnotsupernot.PNG\|thumb\|The first qubit flips only if the second qubit is 1.]]~~
	~~In [[computing science]], the '''controlled NOT gate''' (also '''C-NOT''' or '''CNOT''') is a [[quantum gate]] that is an essential component in~~ the ~~construction of a [[quantum computer]]~~. ~~It can be used~~ to ~~entangle and disentangle [[Quantum entanglement\|EPR states]]. Specifically, any quantum circuit can~~ be ~~simulated to an arbitrary degree of accuracy using~~ a ~~combination of CNOT gates and single [[qubit]] rotations~~.

	~~==Operation==~~

	~~The CNOT gate flips the second qubit (the target qubit) if and only if the first qubit (the control qubit)~~ is 1.

	~~{\| align="center" class="wikitable"~~
	\|-
	~~! colspan="2" \| Before \|\| colspan="2" \| After~~
	\|-
	~~! Control \|\| Target \|\| Control \|\| Target~~
	~~\|- align=center~~
	~~\| 0 \|\| 0 \|\| 0 \|\| 0~~
	~~\|- align=center~~
	~~\| 0 \|\| 1 \|\| 0 \|\| 1~~
	~~\|- align=center~~
	~~\| 1 \|\| 0 \|\| 1 \|\| 1~~
	~~\|- align=center~~
	~~\| 1 \|\| 1 \|\| 1 \|\| 0~~
	\|}

	~~The resulting value of the second qubit corresponds to the result of a classical [[XOR gate]].~~

	~~The CNOT gate can be represented by the matrix:~~

	~~:<math> \operatorname{CNOT} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}.</math>~~

	~~The first experimental realization of a CNOT gate was accomplished in 1995. Here, a single Beryllium ion in a trap was used. The two qubits were encoded into an optical state~~ and ~~into the vibrational state of the ion within the trap~~. ~~At the time of the experiment, the reliability of the CNOT-operation was measured to be on the order of 90%.~~

	~~In addition to a regular controlled NOT gate, one could construct a function-controlled NOT gate, which accepts an arbitrary number ''n''+1 of qubits as input,~~ where '~~'n''+1 is greater than or equal to 2 (a [[quantum register]]). This gate flips the last qubit of the register if~~ and ~~only if a built-in function, with the first ''n'' qubits as input, returns a 1~~.
	~~The function-controlled NOT gate is an essential element of the [[Deutsch-Jozsa algorithm]].~~
	CNOT is also a kind of universal gate(in the classical sense of the word). It is easy to see that if the CONTROL is set to '1' the TARGET output is always NOT. So, a NOT GATE can be constructed using CNOT.

	~~==Proof of operation==~~

	~~Let~~ <~~math~~>~~\left\{~~
	~~\|0\rangle =\! \begin{bmatrix} 1 \\ 0 \end{bmatrix},~~
	~~\|1\rangle =\! \begin{bmatrix} 0 \\ 1 \end{bmatrix}~~
	~~\right\}~~<~~/math~~> ~~be the [[orthonormal basis]] (using [[Bra~~-~~ket notation]]).~~

	~~Let <math>\| \psi \rangle = a \| 0 \rangle + b \| 1 \rangle =\! \begin{bmatrix} a \\ b \end{bmatrix}</math>.~~

	~~Let <math>\| \phi \rangle = b \| 0 \rangle + a \| 1 \rangle =\! \begin{bmatrix} b \\ a \end{bmatrix}</math> be the flip qubit of <math> \| \psi \rangle</math>.~~

	~~Recall that <math>\|\alpha\rangle\!\otimes\!\|\beta\rangle~~
	~~= \|\alpha\rangle \|\beta\rangle~~
	~~= \|\alpha,\beta\rangle </math>.~~

	~~===When control qubit is 0===~~

	~~Here is a proof that <math>~~
	~~\operatorname{CNOT}\ \|0,\psi\rangle = \|0,\psi\rangle~~
	~~</math>~~:

	~~Note that here the specific definition of <math> \operatorname{CNOT}\ </math> assumes an eigenbasis of~~

	<math> \|00\rangle = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \|01\rangle = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} , \|10\rangle = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \|11\rangle = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} </math>

	~~It's straightforward to verify that if~~
	~~<math>~~
	~~\|0,\psi\rangle = a\|0\rangle \|0\rangle + b\|0\rangle \|1\rangle = \begin{bmatrix} a \\ b \\ 0 \\ 0 \end{bmatrix}</math>~~

	[[Matrix multiplication\|Then]] <math>\operatorname{CNOT}\ \|0,\psi\rangle = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} a \\ b \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} a \\ b \\ 0 \\ 0 \end{bmatrix} = \|0,\psi\rangle</math>.

	~~Therefore CNOT doesn't change the <math> \|\psi\rangle</math> qubit if the first qubit is 0.~~

	~~===When control qubit is 1===~~

	~~Here is a proof that <math> \operatorname{CNOT}\ \|1,\psi\rangle = \|1,\phi\rangle</math>, which means that the CNOT gate flips the <math>\|\psi\rangle<~~/~~math> qubit.~~

	~~Similarly to the first demonstration, <math> \|1,\psi\rangle = \begin{bmatrix} 0 \\ 0 \\ a \\ b \end{bmatrix}<~~/~~math>~~.

	Then <math> \operatorname{CNOT}\ \|1,\psi\rangle = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ a \\ b \end{bmatrix} = a \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} + b \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}</~~math>~~

	~~It can be seen that~~
	~~<math> \|1,1\rangle = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}<~~/~~math>~~
	~~and~~
	~~<math> \|1,0\rangle = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}<~~/~~math>, using these on the equation above gives~~

	~~<math>~~
	~~\begin{align}~~
	~~\operatorname{CNOT}\ \|1,\psi\rangle~~
	~~& = a \|1,1\rangle + b \|1,0\rangle \\~~
	~~& = \|1\rangle \left( a \|1\rangle + b \|0\rangle \right) \\~~
	~~& = \|1,\phi\rangle \\~~
	~~\end{align}~~
	~~</math>~~

	Therefore the CNOT gate flips the <math>\|\psi\rangle</math> qubit into <math>\|\phi\rangle</math> if the control qubit is set to 1. A simple way to observe this is to multiply the CNOT matrix by a column vector, noticing that the operation on the first bit is identity, and a NOT gate on the second bit.

	=~~=See also==~~
	* [[C-ROT gate]]

	~~==References==~~
	*{{cite book \| author=Nielsen, Michael A. & Chuang, Isaac L.\|title=Quantum Computation and Quantum Information \| publisher=Cambridge University Press \|year=2000 \|isbn=0-521-63235-8}}

	*{{cite journal \| author=Monroe, C. & Meekhof, D. & King, B. & Itano, W. & Wineland, D. \|title=Demonstration of a Fundamental Quantum Logic Gate \| journal=Physical Review Letters \| issue= 25 \| pages=4714–4717 \|year=1995 \|doi=10.1103/PhysRevLett.75.4714 \|volume=75 \| pmid=10059979 \| bibcode=~~1995PhRvL..75.4714M}} [http://tf.nist.gov/general/pdf/140.pdf]~~

	~~[[Category:Quantum gates]]~~
	~~[[Category:Quantum information science]~~]