Generic Boundary Matrix Product States Contraction Code

[JoeyT1994]

This PR adds code for doing boundary matrix product state contraction techniques in ITensorNetworks using a 1-site MPS fitting procedure. The result is the ability to do efficient, controlled approximate contractions of 2D and quasi-2D tensor networks.

The BeliefPropagationCache has been separated into an AbstractBeliefPropagationCache and the concrete subtypes BeliefPropagationCache and BoundaryMPSCache in order to achieve this in a streamlined way.

The bmpsc::BoundaryMPSCache is an AbstractBeliefPropagationCache which wraps a bpc::BeliefPropagationCache, an added layer of partitioning around the partitioned_tensornetwork in BeliefPropagationCache and the maximum_virtual_dimension which indicates the maximum permissible size of the virtual links between message tensors.

The underlying partitioned_tensornetwork in the BeliefPropagationCache is assumed to form a grid with only horizontal and vertical edges. Typically vertices are expected to be labeled as integer tuples (column, row) so that edges are of the form (column, row) => (column +- 1, row) or (row, column) => (column, row +-1). Either the rows or the columns can then be partitioned up using the grouping_function kwarg for the constructor (default is the columns). As an 'expert mode', Arbitrary vertex names are actually supported provided the user tells the code how to partition them corresponding planar graph up into a line / ring and then how to sort the vertices within each partition so they go in ascending / descending order.

The message tensors in the BeliefProgpationCache which live between the row/column partitions carry an additional virtual index and thus connect up to form an implicit MPS between partitions.

Functionality for moving the biorthogonal and orthogonal gauge within this implicit MPS is supported and the 1-site boundary MPS routine follows naturally by appropriate ordering of gauge moves and calls to message_update(bp_cache(bmpsc, pe::PartitionEdge)).

Tests are added which show some of the generality of this code (works immediately for complex numbers and real numbers, norm networks and flat tensor networks, on a range of lattices such as hexagonal lattices, square lattices, comb trees etc). The code also immediately works for half-periodic systems where the doubly partitioned network forms a ring and effectively makes the BP approximation on the ring (along with the approximation from the rank of the MPS).

Note this code can be leveraged to do powerful things things like approximately contract "quasi-3D" tensor networks simply by passing a belief propagation cache which partitions up such a network into a flat 2D structure.

Currently the orthogonal gauge is used as default for the MPS fitting as it is stable and robust. The biorthogonal gauge is working but more sensitive to things like the initial message tensors used, the details of the SVD and the ordering of partitionedge updates. Work will need to be done to figure out how to stabilize it.

A number of further improvements are possible.

1. Support for n-site MPS fitting which might help the biorthogonal stability.
2. Support for keeping track of the biorthogonal gauge centre and utilizing it to form channels and efficiently compute more complicated derivatives of the tensor network (currently environment(bmpsc::BoundaryMPSCache, verts::Vector)) is only supported for contiguous vertices which lie in the same column/ row partition.

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