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My apologies if this question is not proper for this site, but I could not figure out the following. Can anyone provide insight? It is almost certain that stochastic differential equations (SDEs) can be used to model noise in quantum computation, and that it might be possible to maintain coherence by controlling the environment through proper estimation of the SDE parameters. In this context, many methods must have been used to estimate these parameters. However, since rough path theory provides path wise convergence under many circumstances, methods based on rough path theory should theoretically be better than other approaches for parameter estimation. In other words the signatures of rough path can be helpful to model the noise and the parameter to maintain coherence. Why has no one worked on this? Why are there no references addressing this issue?

Or I am not searching the web properly. Any comment?

EDIT: The following link shows and SDE applicable. https://arxiv.org/pdf/2405.14283

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  • $\begingroup$ can you include some references for the type of SDE systems that you have in mind? $\endgroup$
    – Thomas Kojar
    Commented Oct 7 at 2:46
  • $\begingroup$ @ThomasKojar I have added a link for a reference. Can we use tough path theory for the SDE mentioned? $\endgroup$
    – Creator
    Commented Oct 7 at 3:33
  • $\begingroup$ In the book "a course on rough paths" Chapter 9 Stochastic differential equations Theorem 9.1, they go over the pathwise lift for SDE solutions. The regularity there for the volatility is C3_b, so I am not sure if that is useful in your context. $\endgroup$
    – Thomas Kojar
    Commented Oct 11 at 1:37
  • $\begingroup$ @ThomasKojar Thank you for your comment. I understand volatility can be a limitation. Considering allowable volatility, I was expecting to see work related to extending application of rough path to Non-Markovian settings. $\endgroup$
    – Creator
    Commented Oct 11 at 2:48
  • $\begingroup$ yes for example the fractional Brownian motion is not Markov and one can use Rough paths for it e.g. see here mathoverflow.net/questions/351549/…. The Rough paths are really a deterministic construction i.e. it doesn't use martingale/Markov structure. In that book they also do the construction for very general Gaussian processes. Also check out the Friz-Victoir book page.math.tu-berlin.de/~friz/master4_May6th.pdf Multidimensional Stochastic Processes as Rough Paths: Theory and Applications $\endgroup$
    – Thomas Kojar
    Commented Oct 11 at 3:24

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A nice very similar context of studying reverse-time SDEs using rough paths is in the work "Generative Fractional Diffusion Models" https://arxiv.org/pdf/2310.17638v2. Here they have to deal with a non-Markovian setting too as in your equation (18).

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  • $\begingroup$ Thank you for the reference. I will have a look. Have you found anything related to quantum computation? $\endgroup$
    – Creator
    Commented Oct 16 at 0:52
  • $\begingroup$ No I didn't search. I was mostly searching for the particular SDE you are interested in. $\endgroup$
    – Thomas Kojar
    Commented Oct 16 at 0:59
  • $\begingroup$ No worries. I searched a lot, but never found one. I was wondering if it is possible to write a paper in that area. May I ask your view? $\endgroup$
    – Creator
    Commented Oct 16 at 1:59
  • $\begingroup$ sounds very interesting. If you encounter any specific technical difficulties, feel free to post them on Mathoverflow. $\endgroup$
    – Thomas Kojar
    Commented Oct 16 at 3:20

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