Talk:Quantum logic gate

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The article asserts that $$R(\cos^{-1}\begin{matrix} \frac{3}{5} \end{matrix}))$$ along with a couple of other gates form a "universal" set of gates. What is the significance and/or derivation of 3/5?  Would other angles work as well?  What is the critical property of the particular angle?  --David Battle 01:39, 18 August 2005 (UTC)


 * The fact that it's usable at all is due to the fact that it divided by 2pi is irrational, I believe. For all irrational numbers x all real numbers r in [0,1) and all open intervals that include r, there is an integer n such that nx mod 1 is in the interval. Why this specific angle? Well, $$e^{\theta i} = \cos x + i \sin x$$, so $$e^{\cos^{-1}\begin{matrix} \frac{3}{5} \end{matrix}} = \cos (\cos^{-1}\begin{matrix} \frac{3}{5} \end{matrix}) + i \sin (\cos^{-1}\begin{matrix} \frac{3}{5} \end{matrix})$$, which I believe is $$\frac{3}{5} + i \frac{4}{5}$$, whose absolute value is 1. The associated angle (the argument of this number) is irrational--I think. Actually, using $$e^i$$ for that term of the matrix should work as well, and it's pretty simple, too. Now my question is why an angle is being multiplied by 2pi. --Ihope127 23:10, 19 June 2007 (UTC)


 * You're right, multiplying an angle by 2pi in the phase gate example doesn't make sense. I have removed it. --V79 (talk) 17:15, 1 May 2009 (UTC)

Rotating qubit
Rotating about x axis:
 * $$ R_x(\frac{2\pi}{n}) = \begin{bmatrix} \cos\frac{\pi}{n} & -i\sin\frac{\pi}{n} \\ -i\sin\frac{\pi}{n} & \cos\frac{\pi}{n} \end{bmatrix} $$.

Rotating about y axis:
 * $$ R_y(\frac{2\pi}{n}) = \begin{bmatrix} \cos\frac{\pi}{n} & -\sin\frac{\pi}{n} \\ \sin\frac{\pi}{n} & \cos\frac{\pi}{n} \end{bmatrix} $$.

Rotating about z axis:
 * $$ R_z(\frac{2\pi}{n}) = \begin{bmatrix} \cos\frac{\pi}{n}-i\sin\frac{\pi}{n} & 0 \\ 0 & \cos\frac{\pi}{n}+i\sin\frac{\pi}{n} \end{bmatrix}$$.

http://jquantum.sourceforge.net/jQuantumApplet.html

Quantum gate (PC game) confusion
concerning articles :1:Quantum gate, 2:Quantum Gate and 3:Quantum Gate (PC game) previously, 3 redirected to 2 which held the info about the game, which i thought was very confusing, as it was clear to me that 2 should redirect to 1, with 3 holding the information about the game, so i moved the game information from 2 to 3, and redirected 2 to one, this will be coppied on the other two articles' talk page 193.60.83.241 (talk) 11:47, 14 May 2008 (UTC)

NAND is not unitary
The section in the article which claims to demonstrate how a NAND quantum gate could be physically realised is most strange, because the "NAND gate" produced at the end is not invertible, thus not unitary, and hence not a quantum gate at all! Indeed, if the "gate" produced existed at all, it could be used to create copies of the state of qubits, blatantly violating the no-cloning theorem. However, I'm not sufficiently versed in the electrodynamics to tell whether the whole thing is just nonsense, or perhaps rather a poorly remembered example that was really supposed to produce some other classical gate, such as CNOT. 81.170.129.141 (talk) 23:34, 1 February 2013 (UTC)


 * In cruder terms, I too wish to call bullshit on this section. Reading it seems to suggest you can get NAND by multiplying by a clever choice of phase shift. No citations either. Should be removed. — Preceding unsigned comment added by 71.172.178.157 (talk) 13:48, 12 October 2014 (UTC)


 * I'm new to quantum gates, but I think I have to agree. AFAIK all quantum gates are reversable. No matter how you look at it, AND OR NAND NOR etc. can't be implemented. Only NOT Xor Xnor. (Please correct me if I'm wrong) --User:AltruismAndCake


 * Yeh the section is not Quantum mechanics, the Bloch equations are obtained by taking the saddle points of the full quantum theory resolved explicitly over only product states. This is classical mechanics.

Superpositions within Matrices
Do the logic gates allow superpositional values to act as operators or instructions? If they do, that's not really coming through in the article. If they do not, then is this a fundamental limitation of the quantum phenomena, or is it just much simpler to build a QC with "ordinary" logic gates? While the answers to these questions might turn out to be rather awkward, I think they'd go a long way to making the idea of quantum computing more accessible to a larger number of people. Thank you! -- 17:38, 22 April 2014 (UTC) — Preceding unsigned comment added by TheLastWordSword (talk • contribs)


 * Superpositions are those values that are not aligned perfectly along the base vectors, which represents the observable states of the variable. It is a medium-large project to 1) figure out *which* pieces of knowledge are the basics of quantum computing, and 2) to wrap the (mine/your) head around them -- so expecting it to just jump out at you and explain itself is a bit.. much to ask. Of course it would be awesome if every article was awesome that way, but we are just humans. Also, ***you*** can enhance it. 2001:2002:51E3:8007:B66D:83FF:FE0E:C298 (talk) 20:36, 7 September 2017 (UTC)

Universality via Deutsch gates
Assume Deutsch gates are permitted to take an arbitrary rotation as parameter, e.g. set via a potentiometer-like control attached to each Deutsch gate that can be rotated smoothly about 360 degrees when manually constructing the circuit. Presumably this would permit realizing any unitary operation on n qubits with finitely many quantum gates.

As a function of n, what is the minimum number of Deutsch gates needed to be sure of being able to implement any unitary operation?

It seems likely this has already been determined. If so this section of the article should say something about it. Vaughan Pratt (talk) 18:09, 1 December 2015 (UTC)

Physical Manifestation?
Where can I learn about physically realizations of these various gates?

Meekohi (talk) 21:01, 2 December 2019 (UTC)


 * Check out the "Physical Implementations" section of the infobox at the bottom of the page. The two most popular models of quantum computing are the superconducting qubit (particularly transmon) and trapped ion computers. Those pages have some information on how gates are implemented, but it will vary by architecture (and potentially even by device). --Rxtreme (talk) 05:41, 10 December 2019 (UTC)

Pauli Y operator is Hermitian
The section on Unitary inversion of gates claims that the Pauli Y gate is not Hermitian. The Pauli Y gate is Hermitian because the Pauli y matrix/operator is hermitian (as are all the Pauli matrices). I have edited the page to give the Ising (XX) gate instead as an example. 65.130.201.235 (talk) 19:33, 9 February 2020 (UTC)
 * that's correct, of course, thanks. Maybe a more standard gate like $$\pi/8 (T)$$ would be a better example? --Qcomp (talk) 22:20, 9 February 2020 (UTC)
 * Thank you for this correction! I am adding phase shift gates as examples to the article. Omnissiahs hierophant (talk) 20:05, 28 February 2020 (UTC)

References missing
The top of the article has a Template:More citations needed but the article has already many references. It would be better if we could have in-line templates or specific templates per section.--ReyHahn (talk) 18:31, 10 May 2021 (UTC)
 * Thank you so much for your contributions with rotation operator gates/SU(2)! :D It is joyful to read your texts. -- The references can be made better, though. I have been thinking a lot about the references, and basically the entire content of the article is just mathematical consequences derived from 1) Born rule, 2) how serial gates works, 3) how parallel gates works. Given just these three anchors, it's possible although time-consuming to figure the rest out by oneself. ← This was pretty much what I did, and I sometimes flip out and go over the math again, and again, and again, as maybe I got something wrong regarding Measurement and the two sections about entanglement. It HOPE that is possible to read the article from top to bottom without encountering anything upsetting. (Except maybe for "note3" in the section about the Hadamard Transform. I have not found a source for that somewhat wild statement, although honestly I think it's correct.) Also that someone has identified the gates listed in the article, that they are real and not something we made up (although, it does not matter which gates one use, but anyways). · · · Omnissiahs hierophant (talk) 05:06, 11 May 2021 (UTC)
 * I agree as many of these facts are almost mathematical not everything has to be referenced (like the gate part) but we have to be sure we are using the right conventions (note 3 seems relevant, let me see what can I find). For the basic gates (Clifford gates, T, rotation gates) a simple ref to Nielsen & Chuang would suffice. I have just read up to the universal gates part, I will check the rest later but I see that there are some ref already. For quick verification for quantum gates we can also use Qiskit textbook/documentation, Quantum Inspire knowledge base] or N&C.--ReyHahn (talk) 08:58, 11 May 2021 (UTC)

Some random ideas about this article, and how to make it more complete

 * $$CZ = SWAP \times CZ$$, which is the reason why CZ has such a weird circuit representation (just a line)
 * Extend the section of "Measurement on pairwise entangled qubits" to also talk about quantum parallelism in terms that the rest of the world also uses. Or maybe add a new section. That is, $$|x \otimes y\rangle \mapsto |x \otimes (F(x) \oplus y)\rangle$$. ← This is also what the circuits for Shors algorithm, phase estimation, et.c. looks like. The CU-gates they talk about is the "$$F(x) \oplus$$"-part..
 * Maybe a section on physical realizations of gates, how to go from hamiltonians and the schrödinger equation to gates.
 * This article, and others as well, would probably benefit from having something written about measurement basis. Pauli-Z, X, Y, computational basis (Z), Hadamard basis, Bell basis. Basis shift can be defined as application of gates.

The end of the article kindof ends with something that looks more like a tutorial than anything else. This article feels like it's about 10x easier to read than most books on the subject. I really like that :D Not sure that it is "the wikipedia way" though. · · · Omnissiahs hierophant (talk) 23:51, 13 May 2021 (UTC)
 * Some quick answers:
 * Sure about CZ, I was thinking we could remove the image with CX, CY, CZ and put the usual images for CNOT, CY, and the symmetrical CZ.
 * I do not know if "coupling gates" is a term on its own. Usually, in the physical system you have some couplings that you can exploit to create multi-qubit gates like the Ising gates, I guess that's it.
 * I think the page is the right size but going further will make it too large.
 * --ReyHahn (talk) 09:34, 14 May 2021 (UTC)

Removal of the Ising gates
I was the one that expanded on the Ising gates and the related complex SWAPs. But maybe it makes the section of the gates longer and does not really add to it. Also we do not have a circuit representation for those gates. Should we remove them from the article for better fluidity?--ReyHahn (talk) 20:09, 6 August 2021 (UTC)
 * No please! They are interesting! It is our only anchor to iSWAP and trapped ion machines, and maybe gates that are somewhat close to hardware gives some depth :) · · · Omnissiahs hierophant (talk) 23:30, 6 August 2021 (UTC)

Mølmer–Sørensen Gate
Is there any reason the Mølmer–Sørensen gate is excluded in this article? There being no objections I could write up a section on MS gates. — Preceding unsigned comment added by Andrewvh4 (talk • contribs) 19:15, 11 October 2021 (UTC)
 * Go ahead, with the proper references.--ReyHahn (talk) 19:18, 11 October 2021 (UTC)
 * Yes please write about it :) It seems to me from just reading the mapping on the basis states (g = 0, e = 1) that this gate is maybe something like a phase-shifting Hadamard-CNOT? Tried a little to find the matrix that does exactly that mapping, they don't give it as a single matrix in the sources.. · · · Omnissiahs hierophant (talk) 11:01, 15 November 2021 (UTC)

Controlled gates
I deleted this section:

A controlled function or gate behaves conceptually differently depending on whether one approaches it from before measurement, or after measurement:


 * From the perspective of before measurement, the controlled function simultaneously executes on all those basis states, of a superposition (i.e. linear combination) of possible outcomes, that match the controls. Such an operation is a unitary transformation on both the control qubit and the target qubit, and is described by a controlled-U gate.
 * If one instead approaches it from the perspective of after measurement, the controlled function has either executed, or not, depending on which basis state that was measured. Such an operation is a measurement on the control qubit, with its result controlling the operations on the target qubit (i.e. a classically controlled function). For example, this is used in the typical quantum teleportation protocol where Alice measures her system and then transmits the result through classical channel to Bob, so that Bob can perform the corrsponding operation to retrieve the unknown quantum state.

End of quote.

I began writing this section, didn't add any real sources for the actual if-statement, but only for quantum parallelism. Then it morphed away as stuff sometimes do on wikipedia, and now I am no longer sure how true the 2nd half of the 2nd bullet is.

The sources (esp. Ömer) has good examples for what "quantum-if" are, and how they differ from normal if's.

Splitting the article
Should we make a separate article i.e. List of quantum logic gates? that way we could focus on a few main ones here and add even more in the new article. Note that the article is reaching 100 kB of WP:readable prose so it is suggested to be split.--ReyHahn (talk) 15:30, 5 September 2022 (UTC)
 * No hurry here but my proposal will be the following leave Clifford gates, Toffoli and maybe T gate in the main article and the rest goes into more detail in List of quantum logic gates.--ReyHahn (talk) 16:31, 7 September 2022 (UTC)
 * any objections? --ReyHahn (talk) 14:01, 10 September 2022 (UTC)
 * I am okay with that, it will be lots of work updating the links in other articles though. we could add grovers diffusion operator as a gate in that list, also, maybe. and so on, a lot of oft-used "sub-routines" are actually n-qubit gates :) · · · Omnissiahs hierophant (talk) 14:31, 10 September 2022 (UTC)
 * Let me work on a draft, and we can discuss it.--ReyHahn (talk) 14:55, 10 September 2022 (UTC)

Here is draft of the article in question User:ReyHahn/List_of_quantum_logic_gates. Comments are welcome. Note that I removed most of the comments on the Clifford gates as those, Toffoli and controlled gates should remain in the article.--ReyHahn (talk) 17:31, 10 September 2022 (UTC) And for the Grover diffusor we could include a section of algorithm-based gates with names but not necessarily with matrix representation.--ReyHahn (talk) 19:33, 10 September 2022 (UTC)
 * Lets move it :) Its especially nice that the properties of the gates are being listed in that table, determinant, and so on. That page looks really nice :D · · · Omnissiahs hierophant (talk) 10:21, 11 September 2022 (UTC)
 * Thanks! I have created List of quantum logic gates. Please help me verify that everything is in there before erasing anything. We can also discuss somethings like should controlled phase be listed? Or should Toffoli and Deutsch go into the "other" list (at least Toffoli has its own article so it would not hurt).--ReyHahn (talk) 12:07, 11 September 2022 (UTC)
 * Also we have to be careful with the linking.--ReyHahn (talk) 12:10, 11 September 2022 (UTC)
 * Quick update: I have decided to leave the following in the main article (I,X,Y,Z,H,S,T,CNOT,SWAP, controlled gates, Toffoli and Deutsch), Deutsch is discussed in some other parts in the main text and Toffoli has its own article so it does not hurt if we put them all in the same table in the list article.--ReyHahn (talk) 12:23, 11 September 2022 (UTC)
 * It reduced 23% of the prose now the article is safe from additional splitting from the time being.--ReyHahn (talk) 09:39, 12 September 2022 (UTC)
 * Ok. Thanks :)! When reading the article now, I realize that maybe I need to sprinkle some citations/references for various statements. The "this is real"-feeling only comes with references, and most of the refs come only after the Notable gates-section. But easily done :) ... or maybe not? It's just mathematical statements, and 7 sources/refs were given in the section header Notable gates (I think the gates listed is in all those sources, so super-easy to verify?) · · · Omnissiahs hierophant (talk) 15:45, 13 September 2022 (UTC)
 * The more the references the better. There is a gate that is bothering me though. Is that phase shift gate $P$. I cannot find a good book that includes it. It certainly available in most SDK (Qiskit,Quantum Inspire and so on) but it is hard to find a basic textbook that includes it.--ReyHahn (talk) 06:59, 14 September 2022 (UTC)
 * Yeah, ok. In this paper the P gate appears (with the name E) on page 11: - the gate comes from the mathematical properties of the phase gates in general, basically. Idk, maybe it needs this explanation: If we have one phase gate, we can make any phase gate. Ph(y)Rz(2y)=P(y). (if Ph exist, then also P, and then also T and S, and their adjoints). And Ph exist because Ph*Rz*Ry*Rx is a factorization of any gate in U(2), and Ph is included in many instruction sets. Imho we should not remove P plz!
 * --- page 163 mentions the "phase shift gate" with the name R, with the P-gates matrix.
 * --- page 180-181 lists the phase shift gate, but does not give it an operator name, just shows its matrix. It does explain it somewhat the same way our wikitext does, however. — Preceding unsigned comment added by Omnissiahs hierophant (talk • contribs)
 * I think you are assuming that one always has the rotation gates but some QPU only have phase gate P to start with. Not a problem with keeping it but on finding a resources that is old. I just found one! I have added it in List of quantum logic gates. I did not know about this book but it is recommended by some on stackexchange.--ReyHahn (talk) 14:55, 14 September 2022 (UTC)
 * It is not super-awesome (Quantum computing for computer scientists), poor layout, But it teaches the basics of quantum circuit design, and how it relates to normal computing, and probabilistic computing. Its aimed at computer engineers, kindof 2nd year BSc-student level math :) · · · Omnissiahs hierophant (talk) 15:47, 14 September 2022 (UTC)
 * Yanufki and Manucci works for me too.--ReyHahn (talk) 14:57, 14 September 2022 (UTC)
 * I am approaching quantum logic gates from the perspective that the user has a quantum processor with a universal instruction set. and therefore all gates in U(2^n) are available, at least in principle. And especially true for gates on one or two qubits. I think this also is the case with almost all quantum languages around. But you are thinking about if from an engineers perspective? :) (bottom up/top down perspective) · · · Omnissiahs hierophant (talk) 15:17, 14 September 2022 (UTC)

Maybe it is that. I just like minimal set gates with no continuous parameters (also gates with names). BTW Mølmer-Sørensen gate is defined as a non-parametrized gate in the wiki article, but in some other articles it has an angle. What's up with that? Any good ref?--ReyHahn (talk) 15:29, 14 September 2022 (UTC) I am under the impression that there is a difference between the MG gate and the general MG gates.--ReyHahn (talk) 15:41, 14 September 2022 (UTC)
 * In the papers i read they were parameterized. That was some time ago and that article (Mølmer-Sørensen gate) has evolved since then. i think the authors wrote it can implement the CNOT gate (like, in the abstract), and.. that's it all i know. · · · Omnissiahs hierophant (talk) 15:55, 14 September 2022 (UTC)
 * Qiskit defines the MG gate as equivalent to R_xx https://quantum-computing.ibm.com/composer/docs/iqx/operations_glossary and defines also a general one for three qubits https://qiskit.org/documentation/stubs/qiskit.circuit.library.GMS.html. I think there is conflicting information and thus conflicting information here in Wikipedia.--ReyHahn (talk) 15:59, 14 September 2022 (UTC)
 * Oh noes! Maybe Mølmer-Sørensen gate and Ising gates needs some fact-checking? ofc everything always needs fact-checking, not trust wikipedia blindly (do the math, check sources)..... and update. · · · Omnissiahs hierophant (talk) 16:19, 14 September 2022 (UTC)
 * Mama mia! Cirq has also two different MS gates https://quantumai.google/reference/python/cirq_ionq/ionq_native_gates/MSGate . Yes, we need to thread these gates carefully. And possibly update the MS article.--ReyHahn (talk) 16:25, 14 September 2022 (UTC)
 * :D · · · Omnissiahs hierophant (talk) 16:28, 14 September 2022 (UTC)

About global phase
Global phase is weird. I am starting to think that maybe I have been wrong about it even existing, doubting myself. But, in the books they kindof say that it does exist. For example, by saying that gates belong to U(2), and not SU(2). They also reference it by including it in factorizations.. But, however, IBM says that the Hamiltonian for it is useless (link).

It has to exist, otherwise single-qubit gates would not belong to U(2), but SU(2). And intuitively (although I have not tested this), if one inverts the global phase early in the Grover's algorithm, then that would cause the algorithm amplify (with Grover's diffusion operator) all the wrong values, instead of the correct value(s) - essentially inverting the search query.

Why are people suggesting global phase does not exist?? Is it because it does not matter at all in normal quantum physics? · · · Omnissiahs hierophant (talk) 16:10, 20 January 2023 (UTC)
 * Global phases do nothing in quantum circuits, as long as it is truly global. Note that you could write all quantum circuits with SU(2) gates and it will work the same. We just use U(2) gates because it is easier to write some gates in this form. For example, we could use iX instead of X as our bitflip gate, but then controlled-iX is not equivalent to CNOT (which is one of the most helpful ones), so we prefer use U(2). I do not see why a global phase would make the Grover's algorithm not work, could you provide more details? --ReyHahn (talk) 16:44, 20 January 2023 (UTC)


 * IDK, Maybe I should not have written that, without proof or anything. And maybe this topic is moot since I think there is no disagreement in that global phase is real and exist. The reason I picked this up and made it a topic is that I see various people saying online that SU(n) is the symmetry group, and not U(n). Because phase (or even relative phase) is not visible anyways. But it is! Global phase can be converted to relative phase (via P(x)Rx(2x)=Ph(x)) and relative phase is visible through usage of alternative basis (slot in the X or Y matrix instead of Z as the projection operator). Maybe these misunderstandings go away if more details are written about measurement. This article always uses computational basis (ie Z), and it is maybe not clear that other basis are equally valid idk.. "The expectation value of an observable A for a system in a state $$\psi$$ is given by the inner product $$\langle \psi, A \psi \rangle$$." // Dirac–von Neumann axioms. IDK maybe I should just be silent, but this has been bothering me for so long, and derp. I've already written this blob of text already now. This whole thing can be explained better in the articles about quantum this-and-that maybe. Also maybe I want to know if my world-view is wrong, if it actually is SU(n) and what the reason is for people saying this. We dont need it, but it is there. · · · Omnissiahs hierophant (talk) 14:42, 7 October 2023 (UTC)

Reads like an advanced quantum mechanics text book.
I am pretty lay when it comes to the math behind quantum mechanics. I think a lot of people may have an interest in stuff like this. However, in just a few paragraphs the article jumps into things that only physics grads are going to understand. It would be nice if there was some more general and less intimidating information about how this works and its applications. The kinds of info a lay person could understand. As I understand it this page should really just describe what a Q logic gate is and how they can be applied. There should be another page with types of gates and the more complex descriptions of the mathematical formulas. I imagine that your average person reads of few lines on this page and gives up right away. Since this is an encyclopedia, it should contain the simplest way to understand how things work alongside the complex information. Thanks 162.246.112.154 (talk) 19:51, 21 April 2024 (UTC)
 * This seems easy compared to actual quantum mechanics though. For example the article is written with a simpler (but sufficient) version of "measurement" than what is usually given. Anyways, maybe you can contribute with more layman text in the beginning or give more elaborate suggestions. Wikipedia is a community effort after all :) · · · Omnissiahs hierophant (talk) 20:59, 24 June 2024 (UTC)