MPS emulator for LUCJ circuits

[bartandrews]

The aims for this pull request:

• Add a method to convert a molecular Hamiltonian to an MPO.
• Add a method that can construct an LUCJ circuit as an MPS.
• Create a "how-to guide" jupyter notebook.

[bartandrews]

Note that the mkl and mkl-devel packages (and cython) needs to be installed in the environment for TeNPy to be compiled against MKL for faster performance, see https://tenpy.readthedocs.io/en/latest/install/extra.html I'm not sure of the best way to incorporate this. It works fine without it, but power users need to make sure that TeNPy is compiled optimally.

Note that by default, i.e. after running pip install physics-tenpy, the cython code will be compiled, but the numpy/scipy versions will be linked against OpenBLAS, and not MKL. It seems debatable whether MKL is better than OpenBLAS. If we did want MKL, it is better to use conda rather than pip, since conda ships numpy/scipy packages linked against MKL (and these packages are maintained). pip does have packages like intel-numpy and intel-scipy but these packages are out-of-date (i.e. do not support the latest python version).

I think that this issue transcends tenpy and becomes a more general question of whether or not to link numpy/scipy against MKL, which is a matter of personal preference. Therefore, I suggest that we leave this as is.

[kevinsung]

Thank you Bart! I'll begin with some high-level comments that will involve some code restructuring.

[bartandrews]

Thank you Bart! I'll begin with some high-level comments that will involve some code restructuring.

Thank you for reviewing this! 😃 I will implement these changes asap

[kevinsung]

I see that in the gate tests, you do the following:

1. Apply the gate to a product state as a state vector
2. Apply the gate to a product state as an MPS
3. Check that the expectation value on a molecular Hamiltonian matches in both cases.

Is it possible to retrieve the state vector amplitudes from the MPS? If so, then we don't need to introduce a Hamiltonian. We can just directly check that amplitudes stored by the MPS match the exact ones.

Another question is whether we can apply the evolution not to a product state, but to a random state instead. The basic gates often have trivial action on a product state. For example, applying a Givens rotation between two occupied orbitals doesn't do anything.

[bartandrews]

It is possible to convert an MPS to statevector using the mps_to_full function. Yes, I agree that we should remove the MolecularHamiltonian from these gate tests, and keep the tests for the molecular Hamiltonian separate.

Generating a completely random MPS is non-trivial, since a generic random state from Hilbert space will have volume law entanglement, whereas MPS are good at representing states with area law entanglement. I presume that you mean a completely random state, and not just a random product state? One solution is to start with a random product state and then perform a TEBD RandomUnitaryEvolution to introduce some short-range entanglement. This will not result in a completely random state but it will be entangled to as much as the computational cost / bond dimension will allow.

[kevinsung]

A uniformly random state would be ideal, but perhaps unnecessary. We should be able to represent such a state for small test sizes though, right? You can generate a uniformly random state vector using ffsim.random.random_state_vector. Is it possible to convert such a state vector into an MPS using Tenpy? Otherwise, I think your idea of generating a random evolution should be fine.

[bartandrews]

Yes, for small system sizes you may be able to convert a completely random statevector into an MPS using the full_to_mps function. However, the bond dimension will blow up quite quickly. I will try this in the first instance and if it's not tractable, I will try the random unitary evolution approach.

[bartandrews]

Another thought on this, currently we have a bitstring_to_mps utility function that can convert a product state into an MPS. If we have an arbitrary state in ffsim, which is some superposition of bitstrings, perhaps we can construct the corresponding MPS by reading off the amplitudes and then using the MPS.add function? I am imagining something like a state_to_mps function, which is higher level than bitstring_to_mps. This may be easier than trying to decipher TeNPy's internal functions.

[kevinsung]

Sure, that sounds fine with me. Is the full_to_mps function not suitable for this task?

[bartandrews]

It may be... I will check what works best