Gabs.jl

Gabs.jl is a numerical tooling package for simulating Gaussian quantum information.

Gaussian states and operators have the convenient property that they can be characterized by low-dimensional matrices in the phase space representation. Thus, a large class of continuous variable quantum information can be efficiently simulated on a classical computer. Gabs.jl provides a high-level Julia interface for straightforward and high performance implementations of Gaussian quantum systems.

See the detailed suggested readings & references page for a background on quantum information with Gaussian states.

Installation

To install Gabs.jl, start Julia and run the following command:

using Pkg
Pkg.add("Gabs")

To use the package, run the command

using Gabs

Now, the entire library is loaded into the current workspace, with access to its high-level interface and predefined objects.

Examples

Gaussian States

A wide variety of predefined methods to create a specific instance of the GaussianState type are available. For a full description of the API for Gaussian states, see the State Zoo section of the documentation. Let's examine a few well-known examples with the Julia REPL:

julia> using Gabs

julia> sympbasis = QuadPairBasis(1)
QuadPairBasis(1)

julia> s1 = vacuumstate(sympbasis)
GaussianState for 1 mode.
  symplectic basis: QuadPairBasis
mean: 2-element Vector{Float64}:
 0.0
 0.0
covariance: 2×2 Matrix{Float64}:
 1.0  0.0
 0.0  1.0

julia> s2 = coherentstate(sympbasis, rand(ComplexF64))
GaussianState for 1 mode.
  symplectic basis: QuadPairBasis
mean: 2-element Vector{Float64}:
 1.1000447533324929
 0.38900397266196973
covariance: 2×2 Matrix{Float64}:
 1.0  0.0
 0.0  1.0

julia> s3 = squeezedstate(sympbasis, rand(Float64), rand(Float64))
GaussianState for 1 mode.
  symplectic basis: QuadPairBasis
mean: 2-element Vector{Float64}:
 0.0
 0.0
covariance: 2×2 Matrix{Float64}:
 0.414711   0.0537585
 0.0537585  0.609798

Tensor products of Gaussian states are called with ⊗ or tensor:

julia> tp = s1 ⊗ s2 ⊗ s3
GaussianState for 3 modes.
  symplectic basis: QuadPairBasis
mean: 6-element Vector{Float64}:
 0.0
 0.0
 1.1000447533324929
 0.38900397266196973
 0.0
 0.0
covariance: 6×6 Matrix{Float64}:
 1.0  0.0  0.0  0.0  0.0        0.0
 0.0  1.0  0.0  0.0  0.0        0.0
 0.0  0.0  1.0  0.0  0.0        0.0
 0.0  0.0  0.0  1.0  0.0        0.0
 0.0  0.0  0.0  0.0  0.414711   0.0537585
 0.0  0.0  0.0  0.0  0.0537585  0.609798

Partial traces of Gaussian states are called with ptrace:

julia> ptrace(tp, [1, 3])
GaussianState for 2 modes.
  symplectic basis: QuadPairBasis
mean: 4-element Vector{Float64}:
 0.0
 0.0
 0.0
 0.0
covariance: 4×4 Matrix{Float64}:
 1.0  0.0  0.0        0.0
 0.0  1.0  0.0        0.0
 0.0  0.0  0.414711   0.0537585
 0.0  0.0  0.0537585  0.609798

Gaussian Operators

Gabs.jl contains many predefined methods to create instances of GaussianUnitary and GaussianChannel types. For a full description of the API for Gaussian operators, see the Operator Zoo section of the documentation. Let's see a few well-known examples:

julia> using Gabs

julia> sympbasis = QuadBlockBasis(1)
QuadBlockBasis(1)

julia> un = displace(sympbasis, rand(ComplexF64))
GaussianUnitary for 1 mode.
  symplectic basis: QuadBlockBasis
displacement: 2-element Vector{Float64}:
 1.0534226538602067
 0.16968892481522801
symplectic: 2×2 Matrix{Float64}:
 1.0  0.0
 0.0  1.0

julia> ch = amplifier(sympbasis, rand(Float64), rand(Int64))
GaussianChannel for 1 mode.
  symplectic basis: QuadBlockBasis
displacement: 2-element Vector{Float64}:
 0.0
 0.0
transform: 2×2 Matrix{Float64}:
 1.05403  0.0
 0.0      1.05403
noise: 2×2 Matrix{Float64}:
 5.93927e17  0.0
 0.0         5.93927e17

Applications of these operators on states can be called in-place with apply! and out-of-place with *:

julia> s = vacuumstate(sympbasis);

julia> apply!(s, un); s
GaussianState for 1 mode.
  symplectic basis: QuadBlockBasis
mean: 2-element Vector{Float64}:
 1.0534226538602067
 0.16968892481522801
covariance: 2×2 Matrix{Float64}:
 1.0  0.0
 0.0  1.0

julia> tp = (ch ⊗ ch ⊗ ch) * (s ⊗ s ⊗ s)
GaussianState for 3 modes.
  symplectic basis: QuadBlockBasis
mean: 6-element Vector{Float64}:
 1.1103377371945864
 1.1103377371945864
 1.1103377371945864
 0.17885700114374567
 0.17885700114374567
 0.17885700114374567
covariance: 6×6 Matrix{Float64}:
 5.93927e17  0.0         0.0         0.0         0.0         0.0
 0.0         5.93927e17  0.0         0.0         0.0         0.0
 0.0         0.0         5.93927e17  0.0         0.0         0.0
 0.0         0.0         0.0         5.93927e17  0.0         0.0
 0.0         0.0         0.0         0.0         5.93927e17  0.0
 0.0         0.0         0.0         0.0         0.0         5.93927e17