TensorKit.artin_braid restricted to unique fusion sector types

[nimaser]

TensorKit.jl/src/fusiontrees/braiding_manipulations.jl

Line 20 in 67e57a0

@assert FusionStyle(I) === UniqueFusion()

This might just be a misunderstanding on my part, but I was hoping to use TensorKit.artin_braid to do passive changes of basis via leaf swapping on a DFib fusion tree (sector labels are of type FibonacciAnyon ⊠ TimeReversed{FibonacciAnyon}, which has a SimpleFusion rule). The doccomment seems to imply an implementation that returns a dict with many keys, but the actual source restricts the input to fusion trees with UniqueFusion sector types, presumably so the output dictionary is guaranteed to be a singleton. Is this the intended final behavior or just a temporary partial implementation?

[lkdvos]

This is somewhat the intended behavior, in the sense that for non UniqueFusion-style sectors we actually take the code path through

TensorKit.jl/src/fusiontrees/braiding_manipulations.jl

Line 81 in b909280

function artin_braid(src::FusionTreeBlock{I, N, 0}, i; inv::Bool = false) where {I, N}

.

Indeed, in that case the output should be a list of all valid trees, and the change of basis is a general unitary matrix instead of a single phase. That docstring does seem to be somewhat outdated, I'll fix that ASAP, thanks for bringing this up!

For reference, here's a small excerpt of how you'd use that:

using TensorKit
using TensorKit: FusionTreeBlock, artin_braid

I = FibonacciAnyon

src = FusionTreeBlock{I}((I.((:τ, :τ, :τ, :τ)), ()), ((false, false, false, false), ()))
dst, U = TensorKit.artin_braid(src, 3)

julia> U
2×2 Matrix{ComplexF64}:
 -0.809017+0.587785im        0.0+0.0im
       0.0+0.0im       -0.309017-0.951057im

Where src and dst tell you which trees go to which.
I hope this helps, let me know if this isn't clear and I'll happily try and explain all of this a bit better.

[nimaser]

I see, thanks! In that case I have a followup question:

• My current setup is that I have a Dict{FusionTree, ComplexF64} storing the amplitudes of each basis fusion tree in my quantum state.
• It's possible that in the future I'll want to store states where the uncoupled charges are fixed, in which case a FusionTreeBlock seems like the ideal datastructure. However, for the moment I would like to deal with fully general states.
  • Would you recommend that I store my state in the dictionary form above, and just convert to/from FusionTreeBlocks right before/after doing each braid move? It seems like this would not be that performant unless the conversion to FusionTreeBlock is nearly allocation-free.
  • Should I instead store an array or dict of FusionTreeBlocks, one per combination of uncoupled charges? My dynamics may mix different blocks, so I will likely have to shift amplitudes from one block to another: I would imagine that a good design would expose a dictionary-like API (like the above fusion tree -> amplitude mapping) and just use FusionTreeBlocks under the hood.

An additional complication is that I'm using this for anyonic fusion basis states, which have a FibonacciAnyon label for the tails of the leaves and the root. So I'd have to have different FusionTreeBlocks with the same uncoupled charges but different tail charges. It doesn't seem like this introduces too much difficulty either way, but I thought I'd mention it in case it's relevant. Thanks for any advice!