AbstractQuantumSystem

Abstract type for defining systems.

OpenQuantumSystem(
    H_drift::AbstractMatrix{<:Number},
    H_drives::AbstractVector{<:AbstractMatrix{<:Number}}
    dissipation_operators::AbstractVector{<:AbstractMatrix{<:Number}};
    kwargs...
)
OpenQuantumSystem(
    H_drift::Matrix{<:Number}, H_drives::AbstractVector{Matrix{<:Number}}; 
    dissipation_operators::AbstractVector{<:AbstractMatrix{<:Number}}=Matrix{ComplexF64}[], 
    kwargs...
)
OpenQuantumSystem(H_drift::Matrix{<:Number}; kwargs...)
OpenQuantumSystem(H_drives::Vector{Matrix{<:Number}}; kwargs...)
OpenQuantumSystem(H::Function, n_drives::Int; kwargs...)

Constructs an OpenQuantumSystem object from the drift and drive Hamiltonian terms and dissipation operators.

QuantumSystem(H_drift::Matrix{<:Number}, H_drives::Vector{Matrix{<:Number}}; kwargs...)
QuantumSystem(H_drift::Matrix{<:Number}; kwargs...)
QuantumSystem(H_drives::Vector{Matrix{<:Number}}; kwargs...)
QuantumSystem(H::Function, n_drives::Int; kwargs...)

Constructs a QuantumSystem object from the drift and drive Hamiltonian terms.

get_drift(sys::AbstractQuantumSystem)

Returns the drift Hamiltonian of the system.

get_drives(sys::AbstractQuantumSystem)

Returns the drive Hamiltonians of the system.

CompositeQuantumSystem <: AbstractQuantumSystem

A composite quantum system consisting of subsystems. Couplings between subsystems can be additionally defined. Subsystem drives are always appended to any new coupling drives.

lift(operator::AbstractMatrix{<:Number}, i::Int, subsystem_levels::Vector{Int})
lift(operator::AbstractMatrix{<:Number}, i::Int, n_qubits::Int; kwargs...)
lift(operators::AbstractVector{<:AbstractMatrix{T}}, indices::AbstractVector{Int}, subsystem_levels::Vector{Int})
lift(operators::AbstractVector{<:AbstractMatrix{T}}, indices::AbstractVector{Int}, n_qubits::Int; kwargs...)
lift(operator::AbstractMatrix{T}, indices::AbstractVector{Int}, subsystem_levels::AbstractVector{Int})
lift(operator::AbstractMatrix{T}, indices::AbstractVector{Int}, n_qubits::Int; kwargs...)

Lift an operator acting on the i-th subsystem within subsystem_levels to an operator acting on the entire system spanning subsystem_levels.

A constant dictionary GATES containing common quantum gate matrices as complex-valued matrices. Each gate is represented by its unitary matrix.

• GATES[:I] - Identity: Leaves the state unchanged.
• GATES[:X] - Pauli-X (NOT): Flips the qubit state.
• GATES[:Y] - Pauli-Y: Rotates the qubit state around the Y-axis of the Bloch sphere.
• GATES[:Z] - Pauli-Z: Flips the phase of the qubit state.
• GATES[:H] - Hadamard: Creates superposition by transforming basis states.
• GATES[:CX] - Controlled-X (CNOT): Flips the 2nd qubit (target) if the first qubit (control) is |1⟩.
• GATES[:CZ] - Controlled-Z (CZ): Flips the phase of the 2nd qubit (target) if the 1st qubit (control) is |1⟩.
• GATES[:XI] - Complex: A gate for complex operations.
• GATES[:sqrtiSWAP] - Square root of iSWAP: Partially swaps two qubits with a phase.

The 2×2 Pauli matrics and identity.

AbstractPiccoloOperator

Union type for operators.

EmbeddedOperator

Embedded operator type to represent an operator embedded in a subspace of a larger quantum system.

Fields

• operator::Matrix{ComplexF64}: Embedded operator of size prod(subsystem_levels) x prod(subsystem_levels).
• subspace::Vector{Int}: Indices of the subspace the operator is embedded in.
• subsystem_levels::Vector{Int}: Levels of the subsystems in the composite system.

EmbeddedOperator(
    subspace_operator::AbstractMatrix{<:Number},
    subsystem_indices::AbstractVector{Int},
    subspaces::AbstractVector{<:AbstractVector{Int}},
    subsystem_levels::AbstractVector{Int}
)

Embed the subspace_operator into the provided subspaces of a composite system, where the subsystem_indices list the subspaces at which the operator is defined, and the subsystem_levels list the levels of the subsystems in which the operator is embedded.

EmbeddedOperator(
    subspace_operator::AbstractMatrix{<:Number},
    subsystem_indices::AbstractVector{Int},
    subspaces::AbstractVector{<:AbstractVector{Int}},
    composite_system::CompositeQuantumSystem
)

Embed the subspace_operator into the provided subspaces of a composite system.

EmbeddedOperator(subspace_operator::Matrix{<:Number}, subspace::AbstractVector{Int}, subsystem_levels::AbstractVector{Int})

Create an embedded operator. The operator is embedded at the subspace of the system spanned by the subsystem_levels.

EmbeddedOperator(subspace_operator::AbstractMatrix{<:Number}, system::QuantumSystem; kwargs...)

Embed the subspace_operator into a quantum system.

embed(subspace_operator::AbstractMatrix{<:Number}, embedded_operator::EmbeddedOperator)

Embed the subspace_operator in the subspace of a larger embedded_operator.

embed(operator::AbstractMatrix{<:Number}, subspace::AbstractVector{Int}, levels::Int)

Embed an operator in the subspace of a larger matrix of size levels x levels.

get_enr_subspace_indices(excitation_restriction::Int, subsystem_levels::AbstractVector{Int})

Get the indices for the subspace of the quantum system with an excitation restriction.

get_iso_vec_leakage_indices(subspace::AbstractVector{Int}, levels::Int)
get_iso_vec_leakage_indices(subspaces::AbstractVector{<:AbstractVector{Int}}, subsystem_levels::AbstractVector{Int})
get_iso_vec_leakage_indices(subsystem_levels::AbstractVector{Int}; subspace=1:2)
get_iso_vec_leakage_indices(op::EmbeddedOperator)

Get the indices for the leakage in the isomorphic vector space for operators.

get_iso_vec_subspace_indices(subspace::AbstractVector{Int}, subsystem_levels::AbstractVector{Int})
get_iso_vec_subspace_indices(op::EmbeddedOperator)

Get the indices for the subspace in the isomorphic vector space for operators.

get_leakage_indices(subspace::AbstractVector{Int}, levels::Int)
get_leakage_indices(subspaces::AbstractVector{<:AbstractVector{Int}}, subsystem_levels::AbstractVector{Int})
get_leakage_indices(subsystem_levels::AbstractVector{Int}; subspace=1:2)
get_leakage_indices(op::EmbeddedOperator)

Get the indices for the states that are outside of the provided subspace of the quantum system.

get_subspace_indices(subspace::AbstractVector{Int}, levels::Int)    
get_subspace_indices(subspaces::Vector{<:AbstractVector{Int}}, subsystem_levels::AbstractVector{Int})
get_subspace_indices(subsystem_levels::AbstractVector{Int}; subspace=1:2)
get_subspace_indices(op::EmbeddedOperator)

Get the indices for the provided subspace of the quantum system.

unembed(matrix::AbstractMatrix{<:Number}, subspace::AbstractVector{Int})

Unembed a subspace operator from the matrix. This is equivalent to calling matrix[subspace, subspace].

unembed(op::AbstractMatrix, embedded_op::EmbeddedOperator)

Unembed a sub-matrix from the op at the subspace defined by embedded_op.

unembed(embedded_op::EmbeddedOperator)::Matrix{ComplexF64}

Unembed an embedded operator, returning the original operator.

G(H::AbstractMatrix)::Matrix{Float64}

Returns the isomorphism of $-iH$, i.e. $G(H) = \text{iso}(-iH)$.

See also Isomorphisms.iso, Isomorphisms.H.

H(G::AbstractMatrix{<:Real})

Returns the inverse of $G(H) = iso(-iH)$, i.e. returns H.

See also Isomorphisms.iso, Isomorphisms.G.

ad_vec(H::AbstractMatrix{ℂ}; anti::Bool=false) where ℂ <: Number

Returns the vectorized adjoint action of a matrix H:

\[\text{ad_vec}(H) = \mqty(1 & 0 \\ 0 & 1) \otimes H - (-1)^{\text{anti}} \mqty(0 & 1 \\ 1 & 0) \otimes H^*\]

density_to_iso_vec(ρ::AbstractMatrix{<:Number})

Returns the isomorphism ρ⃗̃ = ket_to_iso(vec(ρ)) of a density matrix ρ

iso(H::AbstractMatrix{<:Number})

Returns the isomorphism of $H$:

\[iso(H) = \widetilde{H} = \mqty(1 & 0 \\ 0 & 1) \otimes \Re(H) + \mqty(0 & -1 \\ 1 & 0) \otimes \Im(H)\]

where $\Im(H)$ and $\Re(H)$ are the imaginary and real parts of $H$ and the tilde indicates the standard isomorphism of a complex valued matrix:

\[\widetilde{H} = \mqty(1 & 0 \\ 0 & 1) \otimes \Re(H) + \mqty(0 & -1 \\ 1 & 0) \otimes \Im(H)\]

See also Isomorphisms.G, Isomorphisms.H.

iso_D(L::AbstractMatrix{ℂ}) where ℂ <: Number

Returns the isomorphic representation of the Lindblad dissipator L.

iso_operator_to_iso_vec(Ũ::AbstractMatrix{ℝ}) where ℝ <: Real

Convert a real matrix Ũ representing an isomorphism operator into a real vector.

iso_operator_to_operator(Ũ)

iso_to_ket(ψ̃::AbstractVector{<:Real})

Convert a real isomorphism vector ψ̃ into a ket vector.

iso_vec_to_density(ρ⃗̃::AbstractVector{<:Real})

Returns the density matrix ρ from its isomorphism ρ⃗̃

iso_vec_to_iso_operator(Ũ⃗::AbstractVector{ℝ}) where ℝ <: Real

Convert a real vector Ũ⃗ into a real matrix representing an isomorphism operator.

iso_vec_to_operator(Ũ⃗::AbstractVector{ℝ}) where ℝ <: Real

Convert a real vector Ũ⃗ into a complex matrix representing an operator.

ket_to_iso(ψ::AbstractVector{<:Number})

Convert a ket vector ψ into a complex vector with real and imaginary parts.

mat(x::AbstractVector)

Convert a vector x into a square matrix. The length of x must be a perfect square.

operator_to_iso_operator(U)

operator_to_iso_vec(U::AbstractMatrix{ℂ}) where ℂ <: Number

Convert a complex matrix U representing an operator into a real vector.

annihilate(levels::Int)

Get the annihilation operator for a system with levels.

create(levels::Int)

Get the creation operator for a system with levels.

haar_identity(n::Int, radius::Number)

Generate a random unitary matrix close to the identity matrix using the Haar measure for an n-dimensional system with a given radius. The smaller the radius, the closer the matrix will be to the identity.

haar_random(n::Int)

Generate a random unitary matrix using the Haar measure for an n-dimensional system.

ket_from_bitstring(ket::String)

Get the state vector for a qubit system given a ket string ket of 0s and 1s.

ket_from_string(
    ket::String,
    levels::Vector{Int};
    level_dict=Dict(:g => 0, :e => 1, :f => 2, :h => 2),
    return_states=false
)

Construct a quantum state from a string ket representation.

operator_from_string(operator::String; lookup=PAULIS)

Reduce the string (each character is one key) via operators from a dictionary.

is_reachable(gate::AbstractMatrix{<:Number}, system::AbstractQuantumSystem; kwargs...)

Check if the gate is reachable using the given system.

Keyword Arguments

• use_drift::Bool=true: include drift Hamiltonian in the generators
• kwargs...: keyword arguments for is_reachable

is_reachable(gate, hamiltonians; kwargs...)

Check if the gate is reachable using the given hamiltonians.

Arguments

• gate::AbstractMatrix: target gate
• hamiltonians::AbstractVector{<:AbstractMatrix}: generators of the Lie algebra

Keyword Arguments

• subspace::AbstractVector{<:Int}=1:size(gate, 1): subspace indices
• compute_basis::Bool=true: compute the basis or use the Hamiltonians directly
• remove_trace::Bool=true: remove trace from generators
• verbose::Bool=true: print information about the operator algebra
• atol::Float32=eps(Float32): absolute tolerance

See also QuantumSystemUtils.operator_algebra.

operator_algebra(generators; kwargs...)

Compute the Lie algebra basis for the given generators.

Arguments

• generators::Vector{<:AbstractMatrix}: generators of the Lie algebra

Keyword Arguments

• return_layers::Bool=false: return the Lie tree layers
• normalize::Bool=false: normalize the basis
• verbose::Bool=false: print information
• remove_trace::Bool=true: remove trace from generators