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Controlled NOT gate

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In <a href="Computer_science" title="Computer science">computer science</a>, the controlled NOT gate (also C-NOT or CNOT), controlled-X gate, controlled-bit-flip gate, Feynman gate or controlled Pauli-X is a <a href="Quantum_logic_gate" title="Quantum logic gate">quantum logic gate</a> that is an essential component in the construction of a <a href="Quantum_circuit" title="Quantum circuit">gate-based</a> <a href="Quantum_computer" class="mw-redirect" title="Quantum computer">quantum computer</a>. It can be used to <a href="Quantum_entanglement" title="Quantum entanglement">entangle</a> and disentangle <a href="Bell_state" title="Bell state">Bell states</a>. Any quantum circuit can be simulated to an arbitrary degree of accuracy using a combination of CNOT gates and single <a href="Qubit" title="Qubit">qubit</a> rotations.<a href="#cite_note-Barenco-1">[1]</a><a href="#cite_note-Nielsen-Chuang-2">[2]</a> The gate is sometimes named after <a href="Richard_Feynman" title="Richard Feynman">Richard Feynman</a> who developed an early notation for quantum gate diagrams in 1986.<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a>

The CNOT can be expressed in the <a href="Pauli_matrices" title="Pauli matrices">Pauli basis</a> as:

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<annotation encoding="application/x-tex">{\displaystyle {\mbox{CNOT}}=e^{i{\frac {\pi }{4}}(I_{1}-Z_{1})(I_{2}-X_{2})}=e^{-i{\frac {\pi }{4}}(I_{1}-Z_{1})(I_{2}-X_{2})}.}</annotation>

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</math><img src="./4c295d9b1a4f0bcdd2097d4a6422de16114f5513.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:46.742ex; height:3.176ex;" alt="{\displaystyle {\mbox{CNOT}}=e^{i{\frac {\pi }{4}}(I_{1}-Z_{1})(I_{2}-X_{2})}=e^{-i{\frac {\pi }{4}}(I_{1}-Z_{1})(I_{2}-X_{2})}.}" loading="lazy"></dd></dl>

Being both <a href="Unitary_matrix" title="Unitary matrix">unitary</a> and <a href="Hermitian_matrix" title="Hermitian matrix">Hermitian</a>, CNOT <a href="Sylvester's_formula" title="Sylvester's formula">has the property</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\theta U}=(\cos \theta )I+(i\sin \theta )U}">

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<annotation encoding="application/x-tex">{\displaystyle e^{i\theta U}=(\cos \theta )I+(i\sin \theta )U}</annotation>

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</math><img src="./5385692ef2d165b264048e15ab96eaef161a24c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.538ex; height:3.176ex;" alt="{\displaystyle e^{i\theta U}=(\cos \theta )I+(i\sin \theta )U}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=e^{i{\frac {\pi }{2}}(I-U)}=e^{-i{\frac {\pi }{2}}(I-U)}}">

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<annotation encoding="application/x-tex">{\displaystyle U=e^{i{\frac {\pi }{2}}(I-U)}=e^{-i{\frac {\pi }{2}}(I-U)}}</annotation>

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</math><img src="./d97db0380952a5f664e1982ea353b8ced9f8218d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:25.52ex; height:3.176ex;" alt="{\displaystyle U=e^{i{\frac {\pi }{2}}(I-U)}=e^{-i{\frac {\pi }{2}}(I-U)}}" loading="lazy">, and is <a href="Involutory_matrix" title="Involutory matrix">involutory</a>.

The CNOT gate can be further decomposed as products of <a href="List_of_quantum_logic_gates#Rotation_operator_gates" title="List of quantum logic gates">rotation operator gates</a> and exactly one <a href="List_of_quantum_logic_gates#Two_qubit_interaction_gates" title="List of quantum logic gates">two qubit interaction gate</a>, for example

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<annotation encoding="application/x-tex">{\displaystyle {\mbox{CNOT}}=e^{-i{\frac {\pi }{4}}}R_{y_{1}}(-\pi /2)R_{x_{1}}(-\pi /2)R_{x_{2}}(-\pi /2)R_{xx}(\pi /2)R_{y_{1}}(\pi /2).}</annotation>

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</math><img src="./7887bdf51bfe9e5dc77b7b613f9098f0fd3aae1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:66.85ex; height:3.843ex;" alt="{\displaystyle {\mbox{CNOT}}=e^{-i{\frac {\pi }{4}}}R_{y_{1}}(-\pi /2)R_{x_{1}}(-\pi /2)R_{x_{2}}(-\pi /2)R_{xx}(\pi /2)R_{y_{1}}(\pi /2).}" loading="lazy"></dd></dl>

In general, any single qubit <a href="Unitary_matrix#Properties" title="Unitary matrix">unitary gate</a> can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=e^{iH}}">

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<annotation encoding="application/x-tex">{\displaystyle U=e^{iH}}</annotation>

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</math><img src="./53b453df132d014e83dd49eb29775c93d92c5cc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.223ex; height:2.676ex;" alt="{\displaystyle U=e^{iH}}" loading="lazy">, where H is a <a href="Hermitian_matrix" title="Hermitian matrix">Hermitian matrix</a>, and then the controlled U is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle CU=e^{i{\frac {1}{2}}(I_{1}-Z_{1})H_{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle CU=e^{i{\frac {1}{2}}(I_{1}-Z_{1})H_{2}}}</annotation>

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The CNOT gate is also used in classical <a href="Reversible_computing" title="Reversible computing">reversible computing</a>.

<div class="mw-heading mw-heading2"><h2 id="Operation">Operation</h2></div>

The CNOT gate operates on a <a href="Quantum_register" title="Quantum register">quantum register</a> consisting of 2 qubits. The CNOT gate flips the second qubit (the target qubit) if and only if the first qubit (the control qubit) is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle }">

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If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|0\rangle ,|1\rangle \}}">

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More generally, the inputs are allowed to be a linear superposition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|0\rangle ,|1\rangle \}}">

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<annotation encoding="application/x-tex">{\displaystyle a|00\rangle +b|01\rangle +c|10\rangle +d|11\rangle }</annotation>

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into:

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The action of the CNOT gate can be represented by the matrix (<a href="Permutation_matrix" title="Permutation matrix">permutation matrix</a> form):

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {CNOT} ={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}}.}</annotation>

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</math><img src="./7902c186299d0d276596546bb94df25cf3f35c63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:26.122ex; height:12.509ex;" alt="{\displaystyle \operatorname {CNOT} ={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}}.}" loading="lazy"></dd></dl>

The first experimental realization of a CNOT gate was accomplished in 1995. Here, a single <a href="Beryllium" title="Beryllium">Beryllium</a> ion in a <a href="Ion_trap#Trapped_ion_quantum_computer" title="Ion trap">trap</a> 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%.<a href="#cite_note-6">[6]</a>

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 <a href="Quantum_register" title="Quantum register">quantum register</a>). 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 <a href="Deutsch%E2%80%93Jozsa_algorithm" title="Deutsch–Jozsa algorithm">Deutsch–Jozsa algorithm</a>.

<div class="mw-heading mw-heading2"><h2 id="Behaviour_in_the_Hadamard_transformed_basis">Behaviour in the Hadamard transformed basis</h2></div>

When viewed only in the computational basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|0\rangle ,|1\rangle \}}">

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Insight can be gained by expressing the CNOT gate with respect to a Hadamard transformed basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|+\rangle ,|-\rangle \}}">

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<annotation encoding="application/x-tex">{\displaystyle \{|+\rangle ,|-\rangle \}}</annotation>

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</math><img src="./7b2e661261ce77235609b1b9e6c51b552dab1905.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.078ex; height:2.843ex;" alt="{\displaystyle \{|+\rangle ,|-\rangle \}}" loading="lazy">. The Hadamard transformed basis<a href="#cite_note-7">[a]</a> of a one-qubit <a href="Quantum_register" title="Quantum register">register</a> is given by

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<mfrac>

<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle |+\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle ),\qquad |-\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle -|1\rangle ),}</annotation>

</semantics>

</math><img src="./e5d82130b9afaf042d85679c57dfff678f37a29f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:47.266ex; height:6.176ex;" alt="{\displaystyle |+\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle ),\qquad |-\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle -|1\rangle ),}" loading="lazy"></dd></dl>

and the corresponding basis of a 2-qubit register is

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<annotation encoding="application/x-tex">{\displaystyle |++\rangle =|+\rangle \otimes |+\rangle ={\frac {1}{2}}(|0\rangle +|1\rangle )\otimes (|0\rangle +|1\rangle )={\frac {1}{2}}(|00\rangle +|01\rangle +|10\rangle +|11\rangle )}</annotation>

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</math><img src="./23c5b10a5b648075c8e0d97f6cbbe765385897f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:77.884ex; height:5.176ex;" alt="{\displaystyle |++\rangle =|+\rangle \otimes |+\rangle ={\frac {1}{2}}(|0\rangle +|1\rangle )\otimes (|0\rangle +|1\rangle )={\frac {1}{2}}(|00\rangle +|01\rangle +|10\rangle +|11\rangle )}" loading="lazy">,</dd></dl>

etc. Viewing CNOT in this basis, the state of the second qubit remains unchanged, and the state of the first qubit is flipped, according to the state of the second bit. (For details see below.) "Thus, in this basis the sense of which bit is the control bit and which the target bit has reversed. But we have not changed the transformation at all, only the way we are thinking about it."<a href="#cite_note-RieffelPolak2011-8">[7]</a>

The "computational" basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|0\rangle ,|1\rangle \}}">

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<annotation encoding="application/x-tex">{\displaystyle \{|0\rangle ,|1\rangle \}}</annotation>

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</math><img src="./7a72099dd8347d05e92c98f19db855b1651f8402.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.787ex; height:2.843ex;" alt="{\displaystyle \{|0\rangle ,|1\rangle \}}" loading="lazy"> is the eigenbasis for the spin in the Z-direction, whereas the Hadamard basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|+\rangle ,|-\rangle \}}">

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<annotation encoding="application/x-tex">{\displaystyle \{|+\rangle ,|-\rangle \}}</annotation>

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</math><img src="./7b2e661261ce77235609b1b9e6c51b552dab1905.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.078ex; height:2.843ex;" alt="{\displaystyle \{|+\rangle ,|-\rangle \}}" loading="lazy"> is the eigenbasis for spin in the X-direction. Switching X and Z and qubits 1 and 2, then, recovers the original transformation."<a href="#cite_note-HeisenbergRepresentationGottesman-9">[8]</a> This expresses a fundamental symmetry of the CNOT gate.

The observation that both qubits are (equally) affected in a CNOT interaction is of importance when considering information flow in entangled quantum systems.<a href="#cite_note-InformationFlowDeutschHayden-10">[9]</a>

<div class="mw-heading mw-heading3"><h3 id="Details_of_the_computation">Details of the computation</h3></div>

We now proceed to give the details of the computation. Working through each of the Hadamard basis states, the results on the right column show that the first qubit flips between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |+\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |+\rangle }</annotation>

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</math><img src="./6f39109e5dd0d63a152bd539e71a233b78e23eb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |+\rangle }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |-\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |-\rangle }</annotation>

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</math><img src="./7da62e696eae51c2d89944b647e24b41cafb2f1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |-\rangle }" loading="lazy"> when the second qubit is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |-\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |-\rangle }</annotation>

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</math><img src="./7da62e696eae51c2d89944b647e24b41cafb2f1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |-\rangle }" loading="lazy">:

<th>Initial state in Hadamard basis</th>

<th>Equivalent state in computational basis</th>

<th>Apply operator</th>

<th>State in computational basis after CNOT</th>

<th>Equivalent state in Hadamard basis

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<annotation encoding="application/x-tex">{\displaystyle |++\rangle }</annotation>

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</math><img src="./023494f194d3b3cdab7a48097c71553fc30ce59d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.2ex; height:2.843ex;" alt="{\displaystyle |++\rangle }" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}(|00\rangle +|01\rangle +|10\rangle +|11\rangle )}</annotation>

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</math><img src="./a6c01b0d3641ab6427dd44eb043f98bb89b9be9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.835ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}(|00\rangle +|01\rangle +|10\rangle +|11\rangle )}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}(|00\rangle +|01\rangle +|11\rangle +|10\rangle )}</annotation>

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</math><img src="./db202e126aee2ebb9321b17d80ea7240012487f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.835ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}(|00\rangle +|01\rangle +|11\rangle +|10\rangle )}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle |++\rangle }</annotation>

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</math><img src="./023494f194d3b3cdab7a48097c71553fc30ce59d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.2ex; height:2.843ex;" alt="{\displaystyle |++\rangle }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle |+-\rangle }</annotation>

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</math><img src="./3b8a1269f7d9ccf800848ccd57d3195dcbea21b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.2ex; height:2.843ex;" alt="{\displaystyle |+-\rangle }" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}(|00\rangle -|01\rangle +|10\rangle -|11\rangle )}</annotation>

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</math><img src="./a08326265486a70b0d621c0b1667d5eba6c7d97f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.835ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}(|00\rangle -|01\rangle +|10\rangle -|11\rangle )}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}(|00\rangle -|01\rangle +|11\rangle -|10\rangle )}</annotation>

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</math><img src="./ed41aafe7c2975cc35bf7430828b3109389aba21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.835ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}(|00\rangle -|01\rangle +|11\rangle -|10\rangle )}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle |--\rangle }</annotation>

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</math><img src="./c8127907de1e2fce01485e18bca1a108302090fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.2ex; height:2.843ex;" alt="{\displaystyle |--\rangle }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle |-+\rangle }</annotation>

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</math><img src="./0a9a5fe36775a2c9b49d615a61c82ba9772cdcbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.2ex; height:2.843ex;" alt="{\displaystyle |-+\rangle }" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}(|00\rangle +|01\rangle -|10\rangle -|11\rangle )}</annotation>

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</math><img src="./fd8210ba7d9a79468128dccb67db041087238c99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.835ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}(|00\rangle +|01\rangle -|10\rangle -|11\rangle )}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}(|00\rangle +|01\rangle -|11\rangle -|10\rangle )}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle |-+\rangle }</annotation>

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</math><img src="./0a9a5fe36775a2c9b49d615a61c82ba9772cdcbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.2ex; height:2.843ex;" alt="{\displaystyle |-+\rangle }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle |--\rangle }</annotation>

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</math><img src="./c8127907de1e2fce01485e18bca1a108302090fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.2ex; height:2.843ex;" alt="{\displaystyle |--\rangle }" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}(|00\rangle -|01\rangle -|10\rangle +|11\rangle )}</annotation>

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</math><img src="./f79c94377bd07eed393dc306e7f26771394ef5a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.835ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}(|00\rangle -|01\rangle -|10\rangle +|11\rangle )}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}(|00\rangle -|01\rangle -|11\rangle +|10\rangle )}</annotation>

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</math><img src="./03dd9d91a7a6e90e39669cb705432588ce121bde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.835ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}(|00\rangle -|01\rangle -|11\rangle +|10\rangle )}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle |+-\rangle }</annotation>

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</math><img src="./3b8a1269f7d9ccf800848ccd57d3195dcbea21b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.2ex; height:2.843ex;" alt="{\displaystyle |+-\rangle }" loading="lazy">

</td></tr></tbody></table>

A quantum circuit that performs a Hadamard transform followed by CNOT then another Hadamard transform, can be described as performing the CNOT gate in the Hadamard basis (i.e. a <a href="Change_of_basis#Endomorphisms" title="Change of basis">change of basis</a>):

(H1 ⊗ H1)−1 . CNOT . (H1 ⊗ H1)

The single-qubit Hadamard transform, H1, is <a href="Hermitian_matrix" title="Hermitian matrix">Hermitian</a> and therefore its own inverse. The tensor product of two Hadamard transforms operating (independently) on two qubits is labelled <a href="Hadamard_transform#Definition" title="Hadamard transform">H2</a>. We can therefore write the matrices as:

H2 . CNOT . H2

When multiplied out, this yields a matrix that swaps the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |01\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |01\rangle }</annotation>

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</math><img src="./20574b999c62f66d7995a0c2e662af619f83c08a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.876ex; height:2.843ex;" alt="{\displaystyle |01\rangle }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |11\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |11\rangle }</annotation>

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</math><img src="./a30390d3d468e3c5993820c84f8c5c5407d14e96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.876ex; height:2.843ex;" alt="{\displaystyle |11\rangle }" loading="lazy"> terms over, while leaving the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |00\rangle }">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |00\rangle }</annotation>

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</math><img src="./79016644d7bb5d4282f69bf8af1befa3131445fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.876ex; height:2.843ex;" alt="{\displaystyle |00\rangle }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |10\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |10\rangle }</annotation>

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</math><img src="./7ea8f8133d5db75de4b048da145e24ff71c02c7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.876ex; height:2.843ex;" alt="{\displaystyle |10\rangle }" loading="lazy"> terms alone. This is equivalent to a CNOT gate where qubit 2 is the control qubit and qubit 1 is the target qubit:<a href="#cite_note-11">[b]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}{\begin{bmatrix}{\begin{array}{rrrr}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{array}}\end{bmatrix}}.{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}}.{\frac {1}{2}}{\begin{bmatrix}{\begin{array}{rrrr}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{array}}\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&0&0&1\\0&0&1&0\\0&1&0&0\end{bmatrix}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Constructing_a_Bell_state">Constructing a Bell state</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

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</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Bell_state" title="Bell state">Bell state</a></div>

A common application of the CNOT gate is to maximally entangle two qubits into the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Phi ^{+}\rangle }">

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle |\Phi ^{+}\rangle }</annotation>

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</math><img src="./4570afa1cbf91ef240b83f9ad1476558042f4361.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.74ex; height:3.009ex;" alt="{\displaystyle |\Phi ^{+}\rangle }" loading="lazy"> <a href="Bell_state" title="Bell state">Bell state</a>; this forms part of the setup of the <a href="Superdense_coding" title="Superdense coding">superdense coding</a>, <a href="Quantum_teleportation" title="Quantum teleportation">quantum teleportation</a>, and entangled <a href="Quantum_cryptography" title="Quantum cryptography">quantum cryptography</a> algorithms.

To construct <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Phi ^{+}\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |\Phi ^{+}\rangle }</annotation>

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</math><img src="./4570afa1cbf91ef240b83f9ad1476558042f4361.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.74ex; height:3.009ex;" alt="{\displaystyle |\Phi ^{+}\rangle }" loading="lazy">, the inputs A (control) and B (target) to the CNOT gate are

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )_{A}}</annotation>

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</math><img src="./0c926daa8373be83de8df243a2688ca22d99f14d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.477ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )_{A}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle _{B}}">

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<annotation encoding="application/x-tex">{\displaystyle |0\rangle _{B}}</annotation>

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</math><img src="./c388407409059d0762a5b62c60b0104e28f24a9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.194ex; height:2.843ex;" alt="{\displaystyle |0\rangle _{B}}" loading="lazy">.</dd></dl>

After applying CNOT, the resulting Bell state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )}">

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<annotation encoding="application/x-tex">{\textstyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )}</annotation>

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</math><img src="./3950ec7dfc42d19aa9bcdece7b9ea11c423b69f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.429ex; height:4.176ex;" alt="{\textstyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )}" loading="lazy"> has the property that the individual qubits can be measured using any basis and will always present a 50/50 chance of resolving to each state. In effect, the individual qubits are in an undefined state. The correlation between the two qubits is the complete description of the state of the two qubits; if we both choose the same basis to measure both qubits and compare notes, the measurements will perfectly correlate.

When viewed in the computational basis, it appears that qubit A is affecting qubit B. Changing our viewpoint to the Hadamard basis demonstrates that, in a symmetrical way, qubit B is affecting qubit A.

The input state can alternately be viewed as

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<annotation encoding="application/x-tex">{\displaystyle |+\rangle _{A}}</annotation>

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</math><img src="./d8ea436e713a9884eb47c011ab7c38a514ec5d58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.824ex; height:2.843ex;" alt="{\displaystyle |+\rangle _{A}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2}}}(|+\rangle +|-\rangle )_{B}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2}}}(|+\rangle +|-\rangle )_{B}}</annotation>

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</math><img src="./8e76b7f00a3e658cf567b30ba90c472ee4a6c836.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:16.783ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2}}}(|+\rangle +|-\rangle )_{B}}" loading="lazy">.</dd></dl>

In the Hadamard view, the control and target qubits have conceptually swapped and qubit A is inverted when qubit B is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |-\rangle _{B}}">

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<annotation encoding="application/x-tex">{\displaystyle |-\rangle _{B}}</annotation>

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</math><img src="./87864377424e8842934f2ada2a98323b53f6fb4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.839ex; height:2.843ex;" alt="{\displaystyle |-\rangle _{B}}" loading="lazy">. The output state after applying the CNOT gate is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\sqrt {2}}}(|++\rangle +|--\rangle ),}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {2}}}(|++\rangle +|--\rangle ),}</annotation>

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</math><img src="./68f69f5e12d7f5065d1fa5c35152ae6f5bf3878e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.724ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{\sqrt {2}}}(|++\rangle +|--\rangle ),}" loading="lazy"> which can be shown as follows:

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<annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{\sqrt {2}}}(|+\rangle _{A}|+\rangle _{B}+|-\rangle _{A}|-\rangle _{B})}</annotation>

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</math><img src="./ed83991e561c626cadacb2b7bcd8e0af114c2d2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:30.365ex; height:6.176ex;" alt="{\displaystyle ={\frac {1}{\sqrt {2}}}(|+\rangle _{A}|+\rangle _{B}+|-\rangle _{A}|-\rangle _{B})}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{2{\sqrt {2}}}}((|0\rangle _{A}+|1\rangle _{A})(|0\rangle _{B}+|1\rangle _{B})+(|0\rangle _{A}-|1\rangle _{A})(|0\rangle _{B}-|1\rangle _{B}))}</annotation>

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</math><img src="./c4b01d715fac5451d91a2a1bebb00b0f2a314e46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:64.288ex; height:6.176ex;" alt="{\displaystyle ={\frac {1}{2{\sqrt {2}}}}((|0\rangle _{A}+|1\rangle _{A})(|0\rangle _{B}+|1\rangle _{B})+(|0\rangle _{A}-|1\rangle _{A})(|0\rangle _{B}-|1\rangle _{B}))}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{2{\sqrt {2}}}}((|00\rangle +|01\rangle +|10\rangle +|11\rangle )+(|00\rangle -|01\rangle -|10\rangle +|11\rangle ))}</annotation>

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</math><img src="./84cfe88286825db3315122ba985befae8ef62243.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:63.872ex; height:6.176ex;" alt="{\displaystyle ={\frac {1}{2{\sqrt {2}}}}((|00\rangle +|01\rangle +|10\rangle +|11\rangle )+(|00\rangle -|01\rangle -|10\rangle +|11\rangle ))}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle ).}</annotation>

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</math><img src="./cef75769ecdc73bfd5d75be7727f56710139dc7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.437ex; height:6.176ex;" alt="{\displaystyle ={\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle ).}" loading="lazy"></dd></dl>

The C-ROT gate (controlled <a href="Rabi_cycle" title="Rabi cycle">Rabi rotation</a>) is equivalent to a C-NOT gate except for a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /2}">

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<annotation encoding="application/x-tex">{\displaystyle \pi /2}</annotation>

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</math><img src="./2b44e3d874a0b229fded7ffce67a0677dd5b8b67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /2}" loading="lazy"> rotation of the nuclear spin around the z axis.<a href="#cite_note-12">[10]</a><a href="#cite_note-13">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="Implementations">Implementations</h2></div>

<a href="Trapped_ion_quantum_computer" class="mw-redirect" title="Trapped ion quantum computer">Trapped ion quantum computers</a>:

<ul><li><a href="Cirac%E2%80%93Zoller_controlled-NOT_gate" title="Cirac–Zoller controlled-NOT gate">Cirac–Zoller controlled-NOT gate</a></li>

<li><a href="M%C3%B8lmer%E2%80%93S%C3%B8rensen_gate" title="Mølmer–Sørensen gate">Mølmer–Sørensen gate</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Regulation">Regulation</h2></div>

In May, 2024, Canada implemented <a href="Export_restriction" title="Export restriction">export restrictions</a> on the sale of quantum computers containing more than 34 <a href="Qubits" class="mw-redirect" title="Qubits">qubits</a> and error rates below a certain CNOT <a href="Threshold_theorem" title="Threshold theorem">error threshold</a>, along with restrictions for quantum computers with more qubits and higher error rates.<a href="#cite_note-14">[12]</a> The same restrictions quickly popped up in the UK, France, Spain and the Netherlands. They offered few explanations for this action, but all of them are <a href="Wassenaar_Arrangement" title="Wassenaar Arrangement">Wassenaar Arrangement</a> states, and the restrictions seem related to <a href="National_security" title="National security">national security</a> concerns potentially including <a href="Quantum_cryptography" title="Quantum cryptography">quantum cryptography</a> or <a href="Protectionism" title="Protectionism">protection from competition</a>.

<ul><li><a href="Toffoli_gate" title="Toffoli gate">Toffoli gate</a> (controlled-controlled-NOT gate)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle |+\rangle }</annotation>

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</math><img src="./6f39109e5dd0d63a152bd539e71a233b78e23eb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |+\rangle }" loading="lazy"> can be constructed by applying a <a href="Hadamard_gate" class="mw-redirect" title="Hadamard gate">Hadamard gate</a> to a qubit set to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation>

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</math><img src="./ed066a3ad158da0ad6d6a421a606b1c8a35eb95b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }" loading="lazy">, and similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |-\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |-\rangle }</annotation>

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</math><img src="./7da62e696eae51c2d89944b647e24b41cafb2f1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |-\rangle }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle H_{2}\cdot \operatorname {CNOT} \cdot H_{2}=\operatorname {SWAP} \cdot \operatorname {CNOT} \cdot \operatorname {SWAP} }</annotation>

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</math><img src="./82819c47c40acacbff052f28a768e285f711ce47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:42.454ex; height:2.509ex;" alt="{\displaystyle H_{2}\cdot \operatorname {CNOT} \cdot H_{2}=\operatorname {SWAP} \cdot \operatorname {CNOT} \cdot \operatorname {SWAP} }" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SWAP} }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {SWAP} }</annotation>

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</math><img src="./c7f2b81d8d7d51be935f9c9bb5d7979cb7d98d45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.007ex; height:2.176ex;" alt="{\displaystyle \operatorname {SWAP} }" loading="lazy"> is the <a href="Quantum_logic_gate#Swap_gate" title="Quantum logic gate">SWAP gate</a>.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBarencoBennettCleveDiVincenzo1995" class="citation journal cs1">Barenco, Adriano; Bennett, Charles H.; Cleve, Richard; DiVincenzo, David P.; Margolus, Norman; Shor, Peter; Sleator, Tycho; Smolin, John A.; Weinfurter, Harald (1995-11-01). "Elementary gates for quantum computation". Physical Review A. 52 (5). American Physical Society (APS): 3457–3467. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/9503016">quant-ph/9503016</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995PhRvA..52.3457B">1995PhRvA..52.3457B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysreva.52.3457">10.1103/physreva.52.3457</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1050-2947">1050-2947</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9912645">9912645</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8764584">8764584</a>.</cite>

</li>

<li id="cite_note-Nielsen-Chuang-2"><a href="#cite_ref-Nielsen-Chuang_2-0">^</a> <cite id="CITEREFNielsenChuang2000" class="citation book cs1"><a href="Michael_Nielsen" title="Michael Nielsen">Nielsen, Michael A.</a>; <a href="Isaac_Chuang" title="Isaac Chuang">Chuang, Isaac</a> (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0521632358</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/43641333">43641333</a>.</cite>

</li>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=lsmIO1vdlL4&t=15m16s">Michael Westmoreland: "Isolation and information flow in quantum dynamics" - discussion around the Cnot gate</a></li></ul></div><div><div class="zim-footer">

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