Editing Quantum logic gate (section)

=== Controlled gates ===
{{Further|Controlled NOT gate}}
[[Image:Controlled gate.svg|upright=0.6|thumb|Circuit representation of controlled-''U'' gate]]
Controlled gates act on 2 or more qubits, where one or more qubits act as a control for some operation.<ref name="Barenco" /> For example, the [[controlled NOT gate]] (or CNOT or CX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is {{nowrap|<math>|1\rangle</math>,}} and otherwise leaves it unchanged. With respect to the basis {{nowrap|<math>|00\rangle</math>,}} {{nowrap|<math>|01\rangle</math>,}} {{nowrap|<math>|10\rangle</math>,}} {{nowrap|<math>|11\rangle</math>,}} it is represented by the [[Hermitian matrix|Hermitian]] [[Unitary matrix|unitary]] matrix:

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

The CNOT (or controlled Pauli-''X'') gate can be described as the gate that maps the basis states <math>|a,b\rangle \mapsto |a,a \oplus b\rangle</math>, where <math>\oplus</math> is [[Exclusive or|XOR]].

The CNOT can be expressed in the [[Pauli matrices|Pauli basis]] as:

:<math> \mbox{CNOT} = e^{i\frac{\pi}{4}(I-Z_1)(I-X_2)}=e^{-i\frac{\pi}{4}(I-Z_1)(I-X_2)}. </math>

Being a Hermitian unitary operator, CNOT [[Sylvester's formula|has the property]] that <math> e^{i\theta U}=(\cos \theta)I+(i\sin \theta) U</math> and <math> U =e^{i\frac{\pi}{2}(I-U)}=e^{-i\frac{\pi}{2}(I-U)}</math>, and is [[Involutory matrix|involutory]].

More generally if ''U'' is a gate that operates on a single qubit with matrix representation
:<math> U =  \begin{bmatrix} u_{00} & u_{01} \\ u_{10} & u_{11} \end{bmatrix} , </math>
then the ''controlled-U gate'' is a gate that operates on two qubits in such a way that the first qubit serves as a control. It maps the basis states as follows.{{Multiple image
| image1            = Qcircuit_CNOT.svg
| image2            = Qcircuit_CY.svg
| image3            = Qcircuit CC.svg
| total_width       = 400
| footer            = Circuit diagrams of controlled Pauli gates (from left to right): CNOT (or controlled-X), controlled-Y and controlled-Z.
}}
:<math> | 0 0 \rangle \mapsto | 0 0 \rangle </math>
:<math> | 0 1 \rangle \mapsto | 0 1 \rangle </math>
:<math> | 1 0 \rangle \mapsto | 1 \rangle \otimes U |0 \rangle = | 1 \rangle \otimes (u_{00} |0 \rangle + u_{10} |1 \rangle) </math>
:<math> | 1 1 \rangle \mapsto | 1 \rangle \otimes U |1 \rangle = | 1 \rangle \otimes (u_{01} |0 \rangle + u_{11} |1 \rangle) </math>

The matrix representing the controlled ''U'' is

:<math> \mbox{C}U = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & u_{00} & u_{01} \\  0 & 0 & u_{10} & u_{11} \end{bmatrix}.</math>
When ''U'' is one of the Pauli operators, ''X'',''Y'', ''Z'', the respective terms "controlled-''X''", "controlled-''Y''", or "controlled-''Z''" are sometimes used.{{r|Nielsen-Chuang|pages=177–185}} Sometimes this is shortened to just C''X'', C''Y'' and C''Z''.

In general, any single qubit [[Unitary matrix#Properties|unitary gate]] can be expressed as <math> U =  e^{iH} </math>, where ''H'' is a [[Hermitian matrix]], and then the controlled ''U'' is <math> CU =  e^{i\frac{1}{2}(I-Z_1)H_2}.</math>

Control can be extended to gates with arbitrary number of qubits<ref name="Barenco" /> and functions in programming languages.<ref name="adjoint-controlled-qsharp"/> Functions can be conditioned on superposition states.<ref name="Oemer-structured-programming">{{cite web|title=Structured Quantum Programming|last=Ömer|first=Bernhard|url=http://tph.tuwien.ac.at/~oemer/doc/structquprog.pdf|pages=72, 92–107|date=2 September 2009|publisher=Institute for Theoretical Physics, Vienna University of Technology|archive-url=https://web.archive.org/web/20220327170025/tph.tuwien.ac.at/~oemer/doc/structquprog.pdf|archive-date=March 27, 2022}}</ref><ref name="Oemer2"/>

==== Classical control ====
{{Further|Deferred measurement principle}}
[[File:Example for classic controlled quantum gate.png|thumb|upright=0.8|'''Example:''' The qubit <math>\phi</math> is [[#Measurement|measured]], and the result of this measurement is a [[Boolean data type|Boolean]] value, which is consumed by the classical computer. If <math>\phi</math> measures to 1, then the classical computer tells the quantum computer to apply the U gate on <math>\psi</math>.<br>In circuit diagrams, single lines are [[qubit]]s, and doubled lines are [[bit]]s.]] Gates can also be controlled by classical logic. A quantum computer is controlled by a [[classical computing|classical computer]], and behaves like a [[coprocessor]] that receives instructions from the classical computer about what gates to execute on which qubits.{{r|Oemer|pages=42–43}}<ref name="cryo-controller"/> Classical control is simply the inclusion, or omission, of gates in the instruction sequence for the quantum computer.{{r|Nielsen-Chuang|pages=26–28}}{{r|Williams|pages=87–88}}

{{anchor|Phase shift|Phase shift gates|Phase shift gate|Phase gate}}