Borrow some interface ideas from BlockArrays and other packages

[mtfishman]

It could be useful to adopt some interface ideas from other Julia packages that implement blocked arrays/matrices, for example:

• BlockArrays.jl
• BlockSparseMatrices.jl
• BlockBandedMatrices.jl
• BlockDiagonals.jl

For example, in BlockArrays.jl, the "block index" (the label of the block) has a special type Block. This allows notation like:

julia> using BlockArrays

julia> A = PseudoBlockArray(ones(3,3), [1,2], [2,1]) # A dense array split into 4 blocks
2×2-blocked 3×3 PseudoBlockArray{Float64,2}:
 1.0  1.0  │  1.0
 ──────────┼─────
 1.0  1.0  │  1.0
 1.0  1.0  │  1.0

julia> @view A[Block(2,1)]  # Similar to our `blockview` function
2×2 view(::PseudoBlockArray{Float64,2,Array{Float64,2},Tuple{BlockedUnitRange{Array{Int64,1}},BlockedUnitRange{Array{Int64,1}}}}, BlockSlice(Block(2),2:3), BlockSlice(Block(1),1:2)) with eltype Float64:
 1.0  1.0
 1.0  1.0

julia> A[BlockIndex((2,1), (2,2))] = 2  # Set the (2,2) index of the (2,1) block
2

julia> A
2×2-blocked 3×3 PseudoBlockArray{Float64,2}:
 1.0  1.0  │  1.0
 ──────────┼─────
 1.0  1.0  │  1.0
 1.0  2.0  │  1.0

It would be nice to use similar concepts in NDTensors for the BlockSparseTensor and DiagBlockSparseTensor.

[mtfishman]

Note that this may be a breaking change for ITensors when it is introduced, since it would make sense for functions like nzblocks to return a vector of Block objects.

[emstoudenmire]

Maybe it’s kind of sneaky, but I think it’s ok to make occasional changes which are technically breaking if it’s to a part of a code that no one is (likely) relying on at the moment. Especially for the 0.x series of releases. I believe Julia itself did this a bit throughout the 0.x series and it wasn’t too bad.

[mtfishman]

Yeah, it could be a pretty silent change, if we make sure all of the functions that were using Block-like objects now accept Block, and make Block act like an NTuple in every relevant way.

I think we can have the strategy that breaking changes can happen from 0.x.y to 0.x+1.0, if we can't deprecate something (if we can deprecate, we will have warnings with the jump to 0.x+1.0, then errors in the jump to 0.x+2.0).

[mtfishman]

Some BlockArrays.jl functionality to add:

• blocksize(::Tensor) - get the number of blocks in each dimension as a tuple (like our current nblocks function).
• blocklength(::Tensor) = prod(blocksize(T)) - get the total number of blocks (zero and nonzero).

Also these functions for ITensor.