Editing Toffoli gate (section)

== Related logic gates ==
{{more|List of quantum logic gates}}
[[File:Qcircuit Fredkin.svg|thumb|126x126px|The Fredkin gate]]
[[Image:Qcircuit ToffolifromCNOT.svg|thumb|405x405px|The Toffoli gate can be constructed from single qubit [[Quantum logic gate#Phase shift gates|T]]- and [[Hadamard transform#Quantum computing applications|Hadamard]]-gates, and a minimum of six [[Controlled NOT gate|CNOT]]s.]]
* The [[Fredkin gate]] is a universal reversible 3-bit gate that swaps the last two bits if the first bit is 1; a controlled-swap operation.
* The ''n''-bit Toffoli gate is a generalization of the Toffoli gate. It takes ''n'' bits ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub> as inputs and outputs ''n'' bits. The first ''n''&nbsp;−&nbsp;1 output bits are just ''x''<sub>1</sub>, ..., ''x''<sub>''n''−1</sub>. The last output bit is (''x''<sub>1</sub> AND ... AND ''x''<sub>''n''&minus;1</sub>) XOR ''x''<sub>''n''</sub>.
* The Toffoli gate can be realized by five two-[[qubit]] [[quantum gate]]s,<ref>
{{cite journal
| last1 = Barenco | first1 = Adriano
| last2 = Bennett | first2 = Charles H.
| last3 = Cleve | first3 = Richard
| last4 = DiVincenzo | first4 = David P.
| last5 = Margolus | first5 = Norman
| last6 = Shor | first6 = Peter | authorlink6 = Peter Shor
| last7 = Sleator| first7 = Tycho
| last8 = Smolin| first8 = John A.
| last9 = Weinfurter| first9 = Harald
| date = Nov 1995
| title = Elementary gates for quantum computation
| journal = Physical Review A
| volume = 52
| issue = 5
| pages = 3457–3467
| doi = 10.1103/PhysRevA.52.3457
| arxiv = quant-ph/9503016
| pmid=9912645|bibcode = 1995PhRvA..52.3457B| s2cid = 8764584
}}</ref> but it can be shown that it is not possible using fewer than five.<ref>{{Cite journal |last1=Yu |first1=Nengkun |last2=Duan |first2=Runyao |last3=Ying |first3=Mingsheng |date=2013-07-30 |title=Five two-qubit gates are necessary for implementing the Toffoli gate |journal=Physical Review A |volume=88 |issue=1 |pages=010304 |doi=10.1103/physreva.88.010304 |arxiv=1301.3372 |bibcode=2013PhRvA..88a0304Y |s2cid=55486826 |issn=1050-2947}}</ref>
* Another universal gate, the [[Quantum logic gates#Deutsch gate|Deutsch gate]], can be realized by five optical pulses with neutral atoms.<ref>
{{cite journal
| last1 = Shi | first1 = Xiao-Feng
| date = May 2018
| title = Deutsch, Toffoli, and CNOT Gates via Rydberg Blockade of Neutral Atoms
| journal = Physical Review Applied
| volume = 9
| issue = 5
| pages = 051001
| doi = 10.1103/PhysRevApplied.9.051001
| arxiv = 1710.01859 | bibcode= 2018PhRvP...9e1001S
| s2cid = 118909059
}}</ref> The Deutsch gate is a universal gate for quantum computing.<ref>{{Cite journal |last=Deutsch |first=D. |date=1989 |title=Quantum Computational Networks |journal=Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences |volume=425 |issue=1868 |pages=73–90 |doi=10.1098/rspa.1989.0099 |jstor=2398494 |bibcode=1989RSPSA.425...73D |s2cid=123073680 |issn=0080-4630}}</ref>
* The Margolus gate (named after [[Norman Margolus]]), also called simplified Toffoli, is very similar to a Toffoli gate but with a −1 in the diagonal: RCCX = diag(1, 1, 1, 1, 1, −1, ''X''). The Margolus gate is also universal for reversible circuits and acts very similar to a Toffoli gate, with the advantage that it can be constructed with about half of the CNOTs compared to the Toffoli gate.<ref>{{Cite journal |last=Maslov |first=Dmitri |date=2016-02-10 |title=Advantages of using relative-phase Toffoli gates with an application to multiple control Toffoli optimization |journal=Physical Review A |language=en |volume=93 |issue=2 |pages=022311 |doi=10.1103/PhysRevA.93.022311 |bibcode=2016PhRvA..93b2311M |issn=2469-9926 |doi-access=free |arxiv=1508.03273 }}</ref>
* The iToffoli gate was implemented in superconducting qubits with pair-wise coupling by simultaneously applying noncommuting operations. <ref>{{cite journal |last1=Kim |first1=Y. |last2=Morvan |first2=A. |last3=Nguyen |first3=L.B. |last4=Naik |first4=R.K.|last5=Jünger |first5=C.|last6=Chen |first6=L.|last7=Kreikebaum |first7=J.M.|last8=Santiago |first8=D.I.|last9=Siddiqi |first9=I.|title=High-fidelity three-qubit iToffoli gate for fixed-frequency superconducting qubits |journal=Nature Physics |date=2 May 2022 |volume=18 |issue=5 |pages=783–788  |doi=10.1038/s41567-022-01590-3 |bibcode=2022NatPh..18..783K |doi-access=free |arxiv=2108.10288}}</ref>