Full tomography, Flux Optimizers, Observers.jl

[GTorlai]

This PR applies the following changes:

• Conversion from dense vector/matrix to an ITensor.
• Definition of informationally-complete set of bases and input states.
• Function to sample many shots in the same basis, which only requires a basis rotation once (both QST/QPT)
• Refined interface of getsamples
• Various function introduced to handle ITensors (instead of TN), e.g. choi matrix, sampling, distances etc.
• Introduce Optimisers.jl. Since a dense 1d flattening is required, extra care is needed to make sure that the array function in ITensors parses the dimension of two similar TNs (e.g. an MPS and its gradients) in a consistent way. We do so by fixing the order of the indices just before the TN is flattened.
• Introduces Observers.jl to replace the old one. For tomography, this can contain a function called "earlystop", where if a certain condition dependent on the train/test loss is met, the training is halted.
• Full quantum state and process tomography with the following methods: i) linear inversion (with a correction to satisfy rho>0), ii) least-squares and iii) maximum likelihood, which both relies on convex optimization.
• Merge the two tomography function (in the qst and qpt files) into a single one, now in quantumtomography.jl. This file contains also the interface for full tomography, which the other 3 files now contains lower level functions of these methods.
• Relax the input requirements of tomography from Matrix to include vectors of vectors.
• Added a standard early stopping criteria that can be acivate using earlystop=true in the tomography function. This avoid to having to redefine the early stop function in the observer every time. However, if such function is defined it supersedes the default one.
• Increase testing to saturate codecov, according to the latest report.

[codecov-commenter]

Codecov Report

Merging #231 (51acb6d) into master (11c250b) will increase coverage by 5.40%. The diff coverage is 85.50%.

@@            Coverage Diff             @@
##           master     #231      +/-   ##
==========================================
+ Coverage   80.30%   85.71%   +5.40%     
==========================================
  Files          21       22       +1     
  Lines        2326     2401      +75     
==========================================
+ Hits         1868     2058     +190     
+ Misses        458      343     -115     

Impacted Files	Coverage Δ
src/PastaQ.jl	100.00% <ø> (ø)
src/deprecated.jl	75.00% <ø> (+75.00%)	:arrow_up:
src/circuits/getsamples.jl	64.04% <67.93%> (-24.49%)	:arrow_down:
src/circuits/productstates.jl	86.66% <73.33%> (+30.41%)	:arrow_up:
src/io.jl	71.95% <74.28%> (ø)
src/tomography/quantumtomography.jl	85.71% <85.71%> (ø)
src/utils.jl	74.77% <96.55%> (+6.17%)	:arrow_up:
src/distances.jl	95.00% <96.72%> (+1.66%)	:arrow_up:
src/optimizers.jl	96.96% <96.96%> (+47.60%)	:arrow_up:
src/tomography/fulltomography.jl	97.43% <97.43%> (ø)
... and 19 more

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[GTorlai]

Ready to merge once Observers.jl is registered (late Tuesday evening)

[mtfishman]

It looks like there has been some work splitting off the Flux Optimisers to a separate package (https://github.com/FluxML/Optimisers.jl), should we try using that so we don't have the full Flux dependency?

[GTorlai]

Right now we only use the optimizer, but once we merge the variationalcircuit branch we will also use AD. Maybe there is a way to also only import chunks of it.

[mtfishman]

You don't need all of Flux for AD, just Zygote. Flux basically just adds a neural network framework on top of Zygote.