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Zeno machines (abbreviated ZM, and also called accelerated Turing machine, ATM) are a hypothetical computational model related to Turing machines that allows a countably infinite number of algorithmic steps to be performed in finite time. -Wikipedia

The quantum Zeno effect (also known as the Turing paradox) is a feature of quantum-mechanical systems allowing a particle's time evolution to be arrested by measuring it frequently enough with respect to some chosen measurement setting. -Wikipedia

This seems fitting for a continuous-variable environment (eg. photonics). Has any research been done on quantum Zeno machines? (Google: No results found for "quantum zeno machine".)

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    $\begingroup$ Surely the anti-Zeno effect, where things get sped up, is more interesting? $\endgroup$
    – DaftWullie
    Commented Nov 29, 2018 at 6:22
  • $\begingroup$ Could both be leveraged simultaneously? $\endgroup$
    – user820789
    Commented Nov 29, 2018 at 6:50
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    $\begingroup$ I suppose that depends what you want to achieve. By the way, just because two things (Zeno machine and Zeno effect) have "Zeno" in the name doesn't mean that they are related. I think a zeno machine (quantum or not) would require infinite energy to implement. $\endgroup$
    – DaftWullie
    Commented Nov 29, 2018 at 8:22
  • $\begingroup$ @DaftWullie If the Zeno effect was applied to the thing being measured (eg. an arbitrary quantum system) & the anti-Zeno effect was applied to the measuring apparatus, perhaps a higher fidelity of measurement could be achieved. Regarding infinite energy: Potential? Actual? $\omega$? $\aleph_0$? $\endgroup$
    – user820789
    Commented Nov 30, 2018 at 22:24

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There are two notions of Zeno topics related to quantum computation. The first, which is controversial is usually called hypercomputation, which deals with the possibility of surpassing the limitations of the Church-Turing thesis by means of quantum computation. It is related to the Zeno effect through the fact that if it could be realized, it may solve the halting problem. Contogo refers to this option as a "Quantum Zeno machine". In this context, please see also:Nielsen exploring this possibility.

The second topic which goes by the name Quantum Zeno effect (as referred to in the Wikipedia page in the question) is well established and experimentally verified, (please see Kwiat, White, Mitchell, Nairz, Weihs, Weinfurter, and Zeilinger ).

Kwiat et al. were motivated by one of the most striking examples of this effect: the Elitzur-Vaidman bomb testing problem, (please see the following review by Vaidman).

This problem deals with bombs which can only interact with the outside world by means of their trigger, classically, testing the bomb would cause every good bomb to explode, but quantum mechanically, one can reach, using the Zeno effect, almost 100% detection probability without exploding the bomb. This is an example of a non-demolition measurement which is not accompanied by state reduction.

Translated to quantum computation terminology, this effect is called by Jozsa counterfactual quantum computation, according to which a quantum computer, programmed to solve a problem, can give the result even without running.

A detailed account of the Elitzur-Vaidman bomb testing problem is given by Penrose in his popular book: Shadows of the mind.

One of the most important application of the quantum Zeno effect is its exploitation to keep a system inside a decoherence free subspace (by performing repeated measurements) as proposed by: Beige, Braun, Tregenna, and Knigh. Very recently, this proposal was adapted to holonomic quantum computation by Mousolou and Sjöqvist .

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    $\begingroup$ I find this answer very insightful wrt infinite-dimensional Hilbert space. $\endgroup$
    – user820789
    Commented Nov 29, 2018 at 16:27
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    $\begingroup$ Re: controversial: arxiv.org/abs/1309.0144 ? $\endgroup$
    – user820789
    Commented Nov 29, 2018 at 16:34
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    $\begingroup$ I am not familiar with supertasks, I'll try to read the reference. Roger Penrose has a chapter (no. 16) about infinities in physics in his book The road to reality, in which he expresses the opinion that issue of higher infinities and computability may become important to physics in the future. In contrast, other authors don't see any need beyond potential infinities; infinite dimensional Hilbert spaces used in physics can be understood within the traditional mathematics, without involving higher infinities. $\endgroup$
    – David Bar Moshe
    Commented Nov 29, 2018 at 17:02
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    $\begingroup$ @meowzz: To put physicist-intuitions into some more turn-of-20th-century foundations-of-maths terminology, Hilbert-space being infinite dimensional is akin to having a potential infinity: there is no known a priori limit to the Hilbert space and the state that it can have, but this is not the same as making use of those dimensions to perform supertasks. (By analogy: if you own the Hilbert Hotel, it does not automatically mean that you have infinitely many people staying at your hotel at once: you just don't have any finite limit on the number of customers you can accommodate.) $\endgroup$
    – Niel de Beaudrap
    Commented Nov 30, 2018 at 13:13
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    $\begingroup$ @meowzz: As for infinitesimals, the question many people would ask is what to use them for. The whole notion of 'infinitesimal' is essentially 'undetectably small' --- which poses a problem for obtaining a detectable advantage from any system which somehow makes use of them. This is not to say that infinitesimals are necessarily useless to physics and technology, but any way to 'use' them in an essential way to achieve an interesting outcome would basically require new models of physics (which are beyond the scope of this particular StackExchange). $\endgroup$
    – Niel de Beaudrap
    Commented Nov 30, 2018 at 13:17
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Suppose we are given the ($n\times n$ adjacency matrix $M_0$ of graph $G_0$ and $M_1$ of graph $G_1$, and we wish to know whether $G_0\simeq G_1$. It is a folklore result that if we can prepare states:

$$\vert\alpha_G\rangle=\sum\limits_{\sigma\in S_n}\vert \sigma (G)\rangle,$$

with $S_n$ being the symmetric group on $n$ elements, we can prepare such a state for $G=G_0$ and another for $G=G_1$, and run the $\mathsf{SWAP}$ test between them, to answer the graph isomorphism problem.

Following the ideas of Aharonov and Ta-Shma we can propose a "quantum Zeno effect" preparation for such a state $\vert \alpha_G\rangle$.

That is, Aharanov and Ta-Shma speak of adiabatic state preparation in terms of this "quantum Zeno effect."

Of course, although the quantum Zeno effect was discussed/envisioned by Turing, there is no other relation between the quantum Zeno effect and "Zeno" hypercomputation, as mentioned above.

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