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We are finalizing the following note, in case anyone is interested in having a look:
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David Jaz Myers, $\;$ Hisham Sati, $\;$ U.S.:
Abstract. Despite the evident necessity of topological protection for realizing scalable quantum computers, the conceptual underpinnings of topological quantum logic gates had arguably remained shaky, both regarding their (elusive) physical realization as well as their quantum information-theoretic nature. Building on recent results on defect branes in string/M-theory [SS23a] and on their holographically dual anyonic defects in condensed matter theory [SS23b], here we explain (as announced in [SS22]) how the specification of realistic topological quantum gates, operating by anyon defect braiding in topologically ordered quantum materials, has a surprisingly slick formulation in parameterized point-set topology, which is so fundamental that it lends itself to certification in modern homotopically typed programming languages, such as cubical
Agda
.We propose that this remarkable confluence of concepts may jointly kickstart the development of topological quantum programming proper as well as the real-world application of homotopy type theory, both of which have arguably been falling behind their high expectations; in any case it provides a powerful paradigm for simulating and verifying topological quantum computing architectures with high-level certification languages aware of the actual physical principles of realistic topological quantum hardware.
In a companion article (announced earlier) we explain how further passage to “dependent linear” homotopy types naturally extends this scheme to a full-blown quantum programmin/certification language in which our topological quantum gates may be compiled into verified quantum circuits with quantum measurement gates and classical control.
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Comments are welcome. If you do want to take a look, please grab our latest version pdf from behind the above link.
The very first reference in the abstract is broken.
By the way, when such references are posted to the nForum, they get mysteriously damaged, here is the first hyperlink as it appears in the comment above:
This appears to be a software bug.
Thanks for alerting me.
I had tried to use relative URLs (following what the technical team has been urging me to do) but clearly either me or else the software is not handling them properly.
This pointer should work now: Despite the evident necessity…
Looks fascinating. A couple of typos from p. 2:
from cursory peruse
perusal
paramterized
And the “curiously” of footnote 1, p. 4 suggests to me that the reason is as yet unknown, when in fact you go on to suggest why. Maybe “intriguingly”.
Hopefully some more interesting reflections later.
Thanks! All fixed now (here).
There’s a box between (66) and (67) which has $\Delta_{\gamma}: \Delta_{\gamma}$ twice rather than $\delta: \Delta_{\gamma}$
implyin (p.38); path linfting (p. 39); eqivalently (p. 45); togeneralized (p. 48)
heading runs into box in (74)
Thanks! All fixed now (here).
Last week at QFT and Cobordism @ CQTS we had a talk (here) on
This made me think that a fun way to advertize synthetic homotopy theory in HoTT would be as
I feel that this fun pun captures a whole lot of the difference in approaches one sees these days, related to whether or not to go for identification of fundamental laws or instead fall back on blind statistics. E.g. it’s not a coincidence that Machine Learning of (algebraic) topology has been prominently applied to the “landscape-scanning” approach on string theory (e.g. He 2020), while the Machine Knowing of algebraic topology connects to identifying the missing fundamental laws, such as in Hypothesis H.
I made a feeble attempt to try this out on the remaining audience in an advertisement of TQG in HoTT during the very last 20 minutes of the meeting; but I’ll want to try again when I have more (or any) leisure to prepare (that conference week was immensely hectic, as in parallel to all the hosting activity we also finalized a contribution for QPL2023, with deadline within hours of the end of our conference).
while the Machine Knowing of algebraic topology connects to identifying the missing fundamental laws, such as in Hypothesis H.
In the sense that it’s not induction (or abduction) that’s needed, but something deductive, where the synthetic principles are built into the machine?
I am not sure what you may be asking beyond what you certainly know? Otherwise please clarify what you are getting at.
Incidentally, now I am reminded of a discussion I had with IK, my favorite intellectual antagonist on the ontology of fundamental physics, years ago (2015?) on a walk through late night Prague.
As you may recall from discussion we had elsewhere on the nForum at some point (now I can’t find it) he favors a positivistic approach to fundamental physics where only the available data is taken as real and hypotheses about deeper principles (such as GUTs or Strings) are maybe not outright rejected but not regarded worth the intellectural effort. In pressing to get to the bottom of this sentiment I eventually wondered whether he would be content with giving up all BSM model building and instead just hook up a deep neural network to the LHC, assuming it would eventually be strong enough to start making correct predictions. I remember his answer was Yes!, which, while I don’t share the sentiment, I find an interesting position to take. This is the extreme Learning-position on ontology. On the other side is the Knowing position, where we press for identification of the fundamental laws (at higher risk of failure, of course, but maybe with a higher goal in mind, too).
To be clearer, I was contrasting the ability to extrapolate from data that goes on in machine learning, which we might call induction, with the computerised deduction of consequences from first principles, which is presumably what you meant by machine knowing. It seems farfetched to hope that the machine can come up with laws such as Hypothesis H by itself.
I’m still fond of my association of the triad deduction, induction, abduction with category theory’s ways of completing a triangle of morphisms: composition, extension, lift, and so an understanding of Hypothetico-deductivism as lift followed by composition.
It’s not for the machine to come up with something like Hypothesis H. Instead, the point is that the ingredients of Hypothesis H are so elementarily algebro-topological (cohomology of Cohomotopy cocycle spaces) that a HoTT machine already knows about it.
A special case of this is the punchline of the article headlining this thread: That topological quantum gates are secretly so elementarily algebro-topological that a HoTT machine alread knows all about them: We don’t have to statistically machine-learn how conformal block monodromies are associated to braids, instead the HoTT machine secretly already knows all about how this works in full detail.
Right. So all very Michael Friedman-like, we bake the fundamental constitutive principles into the system.
Re #12
The issue in singling out this fact the machine already knows from all the other homotopical facts it knows… “Coming up with” vs isolating a fact seems to me a rather fine distinction.
I don’t get the impression we are communicating. But that’s okay, I should better stick to posting edit logs.
@Urs which comment are you replying to? I’m happy to have you explain more of your thought as to why my puzzlement is misplaced.
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