Add state overlap optimisation to measure expressivity of circuits

[AdamSmith-physics]

First Pass

• [x] Test Dense vs Sparse eigensolvers
• [x] Use Jamie's sparse Hamiltonian creation
• [x] Get ground state as vector
• [x] Implement basic circuit to state evaluation
• [x] Implement Polar Decomposition based overlap optimisation
• [x] Tidy up and add progress bar

Second Pass

There is a big issue with compilation times when splitting a state vector into a tensor and using this for contraction. Try using MPS instead where all of the tensor will be rank 2, 3 or 4 at most.

• [x] Fix some aspects of MatrixProductStates and merge to main
• [x] Implement state vector to MPS function (MatrixProductState)
• [x] Implement MPS version of circuit to state evaluation
• [x] Implement MPS version of Polar Decomposition based overlap optimisation
• [x] Try using SparseArrayKit to produce sparse Hamiltonian from MPO construction.

[AdamSmith-physics]

Compilation times are a killer using TensorOperations. As you go to larger system sizes, you have to convert the state to a high-rank tensor and the higher the rank the longer the compilation times for some reason!

It took 1163 seconds to compile for N=16, and only 69 seconds to perform 500 iterations with M=4! This is ridiculous!

Might have to move to an exact MPS representation in order to reach larger system sizes. Then the tensors will all be rank-3.

[JamieMair]

Thanks Adam for this. Is this ready for a PR?

[AdamSmith-physics]

Yes, I think so actually. I will raise the request now.