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The question whether surreal or hyperreal numbers (that both contain the reals, even if they have the same cardinality) could be useful to provide a more satisfactory theory of QM is maybe more interesting. -yuggib

Background

I indirectly ended up on this stack, much to my surprise. Little by little I have been working to piece the puzzle together.

I have been pondering this question for a while now, but was not able to formulate it so succinctly until I saw the above quote.

I still don't feel able to properly convey my intuition around why this is a good approach..

Abstract

Star

is not zero, but neither positive nor negative, and is therefore said to be fuzzy and confused with (a fourth alternative that means neither "less than", "equal to", nor "greater than") 0

Fuzzy logic

is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1

Qubit

—usually taken to have the value “0” and “1”, like a bit. However, whereas the state of a bit can only be either 0 or 1, the general state of a qubit according to quantum mechanics can be a superposition of both.

Motivation

"It is likely to be a fresh research question, and you the person one most interested in the entire world in finding an answer --- in which case it is most probably up to you (and your opportunity!) to obtain the answer first." -Niel

While the motivation to wanting to make this work may not be apparent at first, a big piece of what I'm wanting to accomplish is creating a quantum algorithm based on surreal constructions for a quantum intelligence that can use game theory for computation.

I have placed a couple different bounties in an attempt to push along this research.

Question

How can surreal maths be used in quantum computing?

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    $\begingroup$ I'm voting to close this question as off-topic because this appears to ask about speculative models of physics. (This is not to say it is not an interesting question in principle, but it is not appropriate for this site at this point --- at least not without more development to present how it could be used. Before jumping all the way to "is this useful for QM", it is worth considering whether it makes any sense in relation to the framework of quantum computation; and before you do that, you should consider what a "surreal-valued-amplitude" model of computation might look like.) $\endgroup$
    – Niel de Beaudrap
    Commented Jul 13, 2018 at 9:53
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    $\begingroup$ @meowzz: If you are interested in exploring the idea --- and there is at least some potential in it --- I recommend that you read / research a little about counting complexity, and see how surreal numbers might be incorporated into that line of thought. This won't necessarily end up coinciding with quantum computation, but it is the right way (in my opinion) to explore how such ideas might relate to quantum computation. $\endgroup$
    – Niel de Beaudrap
    Commented Jul 13, 2018 at 14:34
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    $\begingroup$ The reason I did not vote to close is because it is about quantum theory, and while this SE is called "quantum computing" (so perhaps this is more appropriate for the physics SE), I don't have a problem if it gets asked here. "Quantum information", "quantum communication", "quantum metrology" and "quantum foundations" are also fine in my opinion. The IQC in Waterloo has quantum foundations people that don't do "quantum computing" per say, but it's still called "IQC". This is a "quantum foundations" question as far as I see. $\endgroup$
    – user1271772 No more free time
    Commented Jul 13, 2018 at 14:44
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    $\begingroup$ I have cleaned up the comments on this question. Please remember comments are for clarification of the question. If you want to discuss this question, please also take it to chat. See also this meta post about this question. $\endgroup$
    – auden
    Commented Jul 13, 2018 at 15:29
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    $\begingroup$ @meowzz Categorical quantum mechanics is not a surreal approach to QM. In case you're interested about CQM in particular, ask about it in a new question. Making that change to your question now, would render the existing answer invalid, which is unfair. $\endgroup$
    – Sanchayan Dutta
    Commented Jul 13, 2018 at 16:00

1 Answer 1

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In quantum field theory there are Feynman path integrals that diverge and for this there is a concept of "renormalization". At least one approach to this uses surreal numbers but it is not very mainstream.

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  • $\begingroup$ I currently have a bounty open on that question! $\endgroup$
    – user820789
    Commented Jul 13, 2018 at 14:48
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    $\begingroup$ I hope you get a satisfactory answer! $\endgroup$
    – user1271772 No more free time
    Commented Jul 13, 2018 at 14:50

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