Demo about implementing Shor's algorithm with Catalyst (#1010)

**Title:** JIT compiling Shor's algorithm with PennyLane and Catalyst

**Summary:** Over the past year, PennyLane has been integrated with
Catalyst, allowing for quantum just-in-time compilation of classical and
quantum code together. This demo leverages that integration to implement
a version of Shor's factoring algorithm that is just-in-time compiled
from end-to-end, classical control structure and all.

**Relevant references:** See within

**Possible Drawbacks:** As it stands, it's quite long (especially for
the amount of actual output); I am not sure about the ending.

**Related GitHub Issues:** N/A

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If you are writing a demonstration, please answer these questions to
facilitate the marketing process.

* GOALS — Why are we working on this now?

Highlight PennyLane and Catalyst integration, and expose challenges (and
opportunities) in scaling up quantum compilation pipelines.

* AUDIENCE — Who is this for?

Primarily people researchers and developers focused on quantum
compilation.


* KEYWORDS — What words should be included in the marketing post?

quantum compilation; quantum circuits; quantum algorithms; Shor's
algorithm

* Which of the following types of documentation is most similar to your
file?
(more details
[here](https://www.notion.so/xanaduai/Different-kinds-of-documentation-69200645fe59442991c71f9e7d8a77f8))
    
- [ ] Tutorial
- [X] Demo
- [ ] How-to


[sc-75726]

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Co-authored-by: Olivia Di Matteo <dimatteo.olivia@gmail.com>
Co-authored-by: Josh Izaac <josh146@gmail.com>
Co-authored-by: David Ittah <dime10@users.noreply.github.com>
Co-authored-by: justinpickering <justin.r.pickering@gmail.com>
Co-authored-by: Justin Pickering <79890410+justinpickering@users.noreply.github.com>
Co-authored-by: Alvaro Ballon <91897656+alvaro-at-xanadu@users.noreply.github.com>

Author: 6 people