LocalOperators.jl

LocalOperators.jl is a simple package that exports a single struct type, LocalOperator representing an operator acting on a local Hilbert space. When adding, subtracting or multiplying two LocalOperator types together,the appropriate number of identity matrices are padded to each on the left and right such that they are compatible.

Usage

LocalOperators.jl exports only a single struct type:

LocalOperators.LocalOperator — Type

LocalOperator{T<:Number} <: AbstractMatrix{T}

Concrete type corresponding to a matrix operator acting on a tensor product of local vector spaces. The operator acts as the matrix stored in the field data on the local vector spaces defined by the field support, and as the identity elsewhere.

Fields

data::Matrix{T}: stores the matrix represention of the operator
support::UnitRange{Int}: stores the sites that the operator has support on

source

A LocalOperator can be constructed like so:

julia> Z = Matrix{ComplexF64}([1 0; 0 -1]); # Pauli Z matrix

julia> r = 0:0;                             # Support on site 0 only

julia> a = LocalOperator(Z, r)
2×2 1-local LocalOperator{ComplexF64} on sites 0 to 0:
 1.0+0.0im   0.0+0.0im
 0.0+0.0im  -1.0+0.0im

We can also supply an integer instead of a range to easily construct a $1$-local operator,

julia> b = LocalOperator(Z, 2)
2×2 1-local LocalOperator{ComplexF64} on sites 2 to 2:
 1.0+0.0im   0.0+0.0im
 0.0+0.0im  -1.0+0.0im

or omit this entirely to use the default index $0$. We can now multiply a and b together to perform $(a \otimes \mathbb{I}) \cdot (\mathbb{I} \otimes b)$:

julia> a * b 
4×4 2-local LocalOperator{ComplexF64} on sites 0 to 1:
 1.0+0.0im   0.0+0.0im   0.0+0.0im  0.0+0.0im
 0.0+0.0im  -1.0+0.0im   0.0+0.0im  0.0+0.0im
 0.0+0.0im   0.0+0.0im  -1.0+0.0im  0.0+0.0im
 0.0+0.0im   0.0+0.0im   0.0+0.0im  1.0+0.0im

We can also do addition and subtraction

julia> a + b 
4×4 2-local LocalOperator{ComplexF64} on sites 0 to 1:
 2.0+0.0im  0.0+0.0im  0.0+0.0im   0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im   0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im   0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im  -2.0+0.0im

julia> b - a 
4×4 2-local LocalOperator{ComplexF64} on sites 0 to 1:
 0.0+0.0im   0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  -2.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im   0.0+0.0im  2.0+0.0im  0.0+0.0im
 0.0+0.0im   0.0+0.0im  0.0+0.0im  0.0+0.0im

If two LocalOperator types have support on the identical sites, then we fall back to the regular operations:

julia> a - a
2×2 1-local LocalOperator{ComplexF64} on sites 0 to 0:
 0.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im

Operations between LocalOperator types with differing local dimension are not supported:

julia> a = LocalOperator([1 0; 0 -1], 0:0, 2);

julia> b = LocalOperator([1 0 0; 0 0 0 ; 0 0 -1], 1:1, 3);

julia> a * b
ERROR: LocalDimensionMismatch: local dimensions must match: a has local dim 2, b has local dim 3
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