• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  What is the use of a Universal-NOT gate?

+ 26 like - 0 dislike

The universal-NOT gate in quantum computing is an operation which maps every point on the Bloch sphere to its antipodal point (see Buzek et al, Phys. Rev. A 60, R2626–R2629). In general, a single qubit quantum state, $|\phi\rangle = \alpha |0\rangle + \beta | 1 \rangle$ will be mapped to $\beta^* |0\rangle - \alpha^*| 1 \rangle$. This operation is not unitary (in fact it is anti-unitary) and so is not something that can be implemented deterministically on a quantum computer.

Optimal approximations to such gates drew quite a lot of interest about 10 years ago (see for example this Nature paper which presents an experimental realization of an optimal approximation).

What has been puzzling me, and what I cannot find in any of the introductions to these papers, is why one would ever want such a gate. Is it actually useful for anything? Moreover, why would one want an approximation, when there are other representations of $SU(2)$ for which there is a unitary operator which anti-commutes with all of the generators?

This question may seem vague, but I believe it has a concrete answer. There presumable is one or more strong reasons why we might want such an operator, and I am simply not seeing them (or finding them). If anyone could enlighten me, it would be much appreciated.

This post has been migrated from (A51.SE)
asked Sep 29, 2011 in Theoretical Physics by Joe Fitzsimons (3,575 points) [ no revision ]
retagged Mar 7, 2014 by dimension10

3 Answers

+ 10 like - 0 dislike

I can imagine a number of reasons why one may want to realize such a gate.

The first is that the universal-NOT exists in classical theory (it is just flipping). This is similar to the case of cloning, that is possible in classical theory but not in quantum theory. So you can look at the study of an approximate universal-NOT as something similar to the study of an approximate cloner (actually, it is easy to argue that if cloning is possible, then universal-NOT is possible: just clone to identify the state, and then rotate it).

The second reason it that the universal-NOT is related to time reversal, and if we want to simulate the latter, we may want to have the former.

The third reason is that the universal-NOT is related to transposition, and as such could be used to test for the presence of entanglement when applied to part of a larger system (partial transposition test).

You can find more recent results and hopefully some more motivation in http://arxiv.org/abs/1104.3039

This post has been migrated from (A51.SE)
answered Sep 29, 2011 by Marco (260 points) [ no revision ]
The first answer you give is exactly the justification they give in the paper, but it's not a reason why you would actually want to be able to do it. As regards the second, you don't actually need a universal NOT for that, you simply need an operator that anti-commutes with the Hamiltonian, and there are potentially far better time reversal techniques (i.e. WaHuHa etc.). The partial transpose test does seem to be one reason to want it though.

This post has been migrated from (A51.SE)
+ 3 like - 0 dislike

[Edited as my original answer misunderstood the question]

The immediate application I can see is in dynamical decoupling. The pulse sequences needed for that are a modified form of the not operation, projecting the state to a point opposite a given symmetry plane on the Bloch sphere. At the moment, the problem there is that the sequences that have been found correct decoherence along one axis on the Bloch sphere. A universal-NOT would be able to generate a universal dynamical decoupler. In essense, for any system and any type of decoherence, we could "run time backwards" and re-extract a coherent system.

(It is maybe interesting to think that, given the links with the no-cloning theorem, there may well be a connection between no cloning and the appearance of a decoherence arrow of time.)

This post has been migrated from (A51.SE)
answered Oct 5, 2011 by Clare (180 points) [ no revision ]
The universal-NOT isn't just a quantum bit flip operator (the Pauli $\sigma_X$ gate fills this role). It maps every state to its antipodal point on the block sphere, and hence anti-commutes with all Pauli operators. I'm not clear on why you bring up feed forward corrections in MBQC, as these all correspond to Pauli operators, *not* universal-NOTs.

This post has been migrated from (A51.SE)
I see. How does it work when a responder has mistaken the question - should I delete my response? Also, I have an answer to the *actual* question, do I put that here or start a new answer?

This post has been migrated from (A51.SE)
You can simply edit this answer if you want.

This post has been migrated from (A51.SE)
Yes, dynamical decoupling is one reason you may want a universal-NOT. However, given that these don't exist, you are likely far better with WaHuHa pulses and refinements there of, than using approximate universal-NOTs. After all, you only need the effective change to add up to zero, and having more than two types of evolution involved doesn't change that end result (see my comment on Marco's answer above). +1 anyway though, as it is indeed a reason for wanting a universal-NOT.

This post has been migrated from (A51.SE)
In terms of implementation, it may indeed be easier as a first step to tailor the decoupler to the specific system. The idea though would be to use a U-NOT as a *universal* decoupler, not dependent on any of the specifics of the state in question or the algorithm (or the error model). It would be a very powerful piece of kit. Practically, though, I wouldn't spend time trying to develop one just yet.

This post has been migrated from (A51.SE)
I think universal decoupling pulses may be possible, but we know for a fact that the universal-NOT isn't.

This post has been migrated from (A51.SE)
+ 2 like - 0 dislike

I suppose, the reason for U-Not gate is more clear in wider framework of research of universal quantum machines, conducted by V. Buzek et. al. So U-Not is coming in good company with question about universal quantum cloning (it is also impossible to do precisely, so it is question about most perfect approximation) and other elementary operations. An introduction to U-Not may be found here http://arxiv.org/abs/quant-ph/9901053 (seems it is just online version of second reference in Nature paper cited above).

This post has been migrated from (A51.SE)
answered Oct 7, 2011 by Alex V (300 points) [ no revision ]
The article you cite is already mentioned in the question. It's the preprint of the Phys Rev A article.

This post has been migrated from (A51.SE)
Likely yes. I see. It is "the second reference in Nature" I mentioned instead.

This post has been migrated from (A51.SE)

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights