Normal Order for bosonic operators and related simplification tools

`QuantumSymbolics.jl` can represent bosonic creation and annihilation operators, their products and tensor products: ``` julia> using SymbolicUtils, QuantumSymbolics julia> (Create*Create*Destroy)⊗(Destroy*Create) a†a†a⊗aa† julia> (Create*Create*Destroy)⊗(Destroy*Create) |> typeof STensorOperator (alias for QuantumSymbolics.STensor{QuantumInterface.AbstractOperator}) julia> (Create*Create*Destroy)⊗(Destroy*Create) |> arguments 2-element Vector{Any}: a†a†a aa† julia> (Create*Create*Destroy)⊗(Destroy*Create) |> arguments .|> typeof 2-element Vector{DataType}: SMulOperator SMulOperator julia> (Create*Create*Destroy)⊗(Destroy*Create) |> arguments .|> arguments 2-element Vector{Vector{Any}}: [a†, a†, a] [a, a†] ``` It can also run simplification/canonicalization rules: ``` julia> qsimplify_fock(Destroy*CoherentState(0.5)) 0.5|0.5⟩ ``` ### This bounty is about implementing a "Normal Form" canonicalization for products of bosonic creation and annihilation operators The [Normal Form](https://en.wikipedia.org/wiki/Normal_order) is an important canonical way to write products of creation and annihilation operators. To implement it in QuantumSymbolics, one would need to have: - an internal function that with high performance gives the normalized representation, potentially using more efficient internal representation; that internal function can assume it is always given a product of creation and annihilation operators - a public function that does the chores related to detecting products of creation and annihilation operators in arbitrary expressions and then calling the internal function only on the appropriate subexpression In the end, the following should work: `normal_form( 3.5*(some_operator*a*a'*a*a')⊗(a*a') + a_constant*other_operator⊗another_operator )` giving a new expression in which all subexpressions which were products of creation and annihilation operators are now in normal form. The most useful references would be: - [the overall documentation of QuantumSymbolics](https://doc.quantumsavory.org/QuantumSymbolics.jl/dev/) - [normal form on wikipedia](https://en.wikipedia.org/wiki/Normal_order) - [a julia library specifically on normal forms of operator products -- QuantumAlgebra.jl](https://github.com/jfeist/QuantumAlgebra.jl) For this bounty to be completed, one would need: - implementation of the aforementioned functions - documentation - tests, potentially against other existing libraries (e.g. QuantumAlgebra.jl) If you are new to julia, make sure to: - install julia using `juliaup` for version management - look through the [julia manual](https://docs.julialang.org/en/v1/) and [modern workflows](https://modernjuliaworkflows.org/) - try out VSCode with the [Julia plugin](https://www.julia-vscode.org/) for good IDE support