Spin shuttling model defined by a stochastic field, the realization of the stochastic field is specified by the paths of the shuttled spins.

Arguments

• n::Int: Number of spins
• Ψ::Vector{<:Number}: Initial state of the spin system, the length of the vector must be `2^n
• T::Real: Maximum time
• N::Int: Time discretization
• B::GaussianRandomField: Noise field
• X::Tuple{Vararg{Function,N}}: Shuttling paths, the length of the vector must be `n
• initialize::Bool: Initialize the random function

Fields

• R::GaussianRandomFunction: Random function defined by the noise field and the paths

Example

model = ShuttlingModel(1, [1+0im, 0+0im], 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]), [t->t])

General one spin shuttling model initialized at initial state |Ψ₀⟩, with arbitrary shuttling path x(t).

Arguments

• Ψ::Vector{<:Number}: Initial state of the spin system, the length of the vector must be `2^n
• T::Real: Maximum time
• N::Int: Time discretization
• B::GaussianRandomField: Noise field
• x::Function: Shuttling path
• initialize::Bool: Initialize the random function

Example

model = OneSpinModel(1 / √2 * [1+0im, 1+0im], 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]), t->t)

One spin shuttling model initialzied at |Ψ₀⟩=|+⟩. The qubit is shuttled at constant velocity along the path x(t)=L/T*t, with total time T in μs and length L in μm.

Arguments

• T::Real: Maximum time
• L::Real: Length of the path
• N::Int: Time discretization
• B::GaussianRandomField: Noise field

Example

model = OneSpinModel(1.0, 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]))

One spin shuttling model initialzied at |Ψ₀⟩=|+⟩. The qubit is shuttled at constant velocity along a forth-back path x(t, T, L) = t<T/2 ? 2L/T*t : 2L/T*(T-t), with total time T in μs and length L in μm.

Arguments

• T::Real: Maximum time
• L::Real: Length of the path
• N::Int: Time discretization
• B::GaussianRandomField: Noise field

Example

model = OneSpinForthBackModel(1.0, 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]))

General two spin shuttling model initialized at initial state |Ψ₀⟩, with arbitrary shuttling paths x₁(t), x₂(t).

Arguments

• Ψ::Vector{<:Number}: Initial state of the spin system, the length of the vector must be `2^n
• T::Real: Maximum time
• N::Int: Time discretization
• B::GaussianRandomField: Noise field
• x₁::Function: Shuttling path for the first spin
• x₂::Function: Shuttling path for the second spin

Example

model = TwoSpinModel(1 / √2 * [1+0im, 1+0im, 1+0im, 1+0im], 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]), t->t, t->t+1)

Two spin shuttling model initialized at the singlet state |Ψ₀⟩=1/√2(|↑↓⟩-|↓↑⟩). The qubits are shuttled at constant velocity along the path x₁(t)=L/T₁*t and x₂(t)=L/T₁*(t-T₀). The delay between the them is T₀ and the total shuttling time is T₁+T₀. It should be noticed that due to the exclusion of fermions, x₁(t) and x₂(t) cannot overlap.

Arguments

• Ψ::Vector{<:Number}: Initial state of the spin system, the length of the vector must be 4, default is 1/√2(|↑↓⟩-|↓↑⟩)
• T₀::Real: Delay time
• T₁::Real: Shuttling time
• L::Real: Length of the path
• N::Int: Time discretization
• B::GaussianRandomField: Noise field

Example

model = TwoSpinSequentialModel(1.0, 1.0, 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]))

Two spin shuttling model initialized at the singlet state |Ψ₀⟩=1/√2(|↑↓⟩-|↓↑⟩). The qubits are shuttled at constant velocity along the 2D path x₁(t)=L/T*t, y₁(t)=0 and x₂(t)=L/T*t, y₂(t)=D. The total shuttling time is T and the length of the path is L in μm.

Arguments

• Ψ::Vector{<:Number}: Initial state of the spin system, the length of the vector must be 4, default is 1/√2(|↑↓⟩-|↓↑⟩)
• T::Real: Total time
• D::Real: Distance between the two qubits
• L::Real: Length of the path
• N::Int: Time discretization
• B::GaussianRandomField: Noise field

Example

model = TwoSpinParallelModel(1.0, 1.0, 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]))

Calculate the dephasing matrix of a given spin shuttling model.

Arguments

• model::ShuttlingModel: The spin shuttling model
• method::Symbol: The method to calculate the dephasing factor, :trapezoid, :simpson, :adaptive

Returns

• W::Matrix{<:Real}: The dephasing matrix

Example

model = OneSpinModel(1.0, 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]), t->t)
W = dephasingmatrix(model)

Sample the dephasing matrix array for a given normal random vector.

Arguments

• model::ShuttlingModel: The spin shuttling model
• randseq::Vector{<:Real}: The random sequence
• isarray::Bool: Return the dephasing matrix array for each time step

Calculate the dephasingfactor according to a special combinator of the noise sequence.

Arguments

• model::ShuttlingModel: The spin shuttling model
• c::Vector{Int}: The combinator of the noise sequence, which should have the same length as the number of spins.
• method::Symbol: The method to calculate the characteristic value, :trapezoid, :simpson, :adaptive

Returns

• Real: The dephasing factor

Example

model = OneSpinModel(1.0, 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]), t->t)
c = [1, 1, 1]
dephasingfactor(model, c)

Sample the dephasing factor for a given normal random vector and a specific combinator.

Arguments

• model::ShuttlingModel: The spin shuttling model
• randseq::Vector{<:Real}: The random sequence
• c::Vector{Int}: The combinator of the noise sequence, which should have the same length as the number of spins.
• isarray::Bool: Return the dephasing factor array for each time step

Return the dephasing factor of a single spin on a series of discrete time points.

Arguments

• n::Int: The number of discrete time points at which to calculate the dephasing factor.
• N::Int: Intervals between each time points.
• model::ShuttlingModel: The shuttling model, which must represent a single spin (model.n == 1).

Returns

• Vector{<:Real}: A vector containing the dephasing factor at each of the n discrete time points.

Sampling an observable that defines on a specific spin shuttling model

Arguments

• model::ShuttlingModel: The spin shuttling model
• objective::Function: The objective function objective(mode::ShuttlingModel; randseq)`
• M::Int: Monte-Carlo sampling size

MixingUnitaryChannel <: QuantumProcess A quantum process that applies a set of unitary operations with given probabilities.

Fields

• Us::Vector{Matrix{Complex{Float64}}}: Vector of unitary matrices, each of size 2^n × 2^n for n qubits.
• Ps::Vector{Float64}: Vector of mixing probabilities, which sum to 1

processtomography(krausops::Vector{<:Matrix{<:Complex}}, basis::Vector{<:Matrix{<:Complex}}; normalized::Bool=false) Compute the transfer matrix for a quantum channel defined by a set of Kraus operators.

Arguments

• krausops::Vector{<:Matrix{<:Complex}}: Vector of Kraus operators, each of size 2^n × 2^n for n qubits.
• basis::Vector{<:Matrix{<:Complex}}: Vector of basis matrices, each of size 2^n × 2^n for n qubits.
• normalized::Bool=false: If true, the transfer matrix is normalized by the number of Kraus operators.

Returns

• Matrix{Float64}: The process tomography transfer matrix of size `N × N

paulitransfermatrix(krausops::Vector{<:Matrix{<:Complex}}; normalized::Bool=false) Compute the Pauli transfer matrix for a set of Kraus operators.

Arguments

• krausops::Vector{<:Matrix{<:Complex}}: Vector of Kraus operators, each of size 2^n × 2^n for n qubits.
• normalized::Bool=false: If true, the transfer matrix is normalized by the number of Kraus operators.

Returns

• Matrix{Float64}: The Pauli transfer matrix of size 4^n × 4^n.

krausops(channel::MixingUnitaryChannel)::Vector{Matrix{Complex{Float64}}} Convert a MixingUnitaryChannel to its Kraus operators.

Arguments

• channel::MixingUnitaryChannel: The mixing unitary channel.

Returns

• Vector{Matrix{Complex{Float64}}}: Vector of Kraus operators,

Calculate the state fidelity of a spin shuttling model using numerical integration of the covariance matrix.

Arguments

• model::ShuttlingModel: The spin shuttling model
• method::Symbol: The method to calculate the dephasing factor, :trapezoid, :simpson, :adaptive

Returns

• Real: The state fidelity

Sample the state fidelity of a spin shuttling model using Monte-Carlo sampling.

Arguments

• model::ShuttlingModel: The spin shuttling model
• randseq::Vector{<:Real}: The random sequence
• isarray::Bool: Return the dephasing matrix array for each time step

concurrence(ρ) -> Float64

Compute the Wootters concurrence of a two‑qubit (4×4) density matrix ρ.

Arguments

• ρ::AbstractMatrix{<:Number}: Hermitian, positive‑semidefinite matrix with size == (4,4) and tr(ρ) ≈ 1.

Returns

• Float64 in the closed interval [0,1].

Algorithm

1. Build the spin‑flip operator Y ⊗ Y (Kronecker product of Pauli‑Y).
2. Form spin-flip matrix ρ̃.
3. Compute the eigenvalues λ_i of ρ * ρ̃, take their square‑roots, sort descending.
4. Return max(0, λ₁ − λ₂ − λ₃ − λ₄).

Reference

W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).

vonneumannentropy(ρ::AbstractMatrix{<:Number})::Float64

Compute the von Neumann entropy of a quantum state represented by a density matrix ρ.

Arguments

• ρ::AbstractMatrix{<:Number}: Hermitian, positive-semidefinite matrix representing the quantum state.

Returns

• Float64: The von Neumann entropy, a non-negative real number.

returns the process fidelity of a quantum channel defined by a transfer matrix Λ and a target process S.

Arguments

• Λ::Matrix{<:Number}: Transfer matrix of the quantum channel.
• S::Matrix{<:Number}: Target process matrix.
• d::Int=0: Dimension of the Hilbert space. If d

is 0, it is inferred from the size of Λ.

Returns

• Float64: The process fidelity, a real number in the closed interval **[0

Arguments

• krausops::Vector{Matrix{<:Number}}: Vector of Kraus operators, each of size 2^n × 2^n for n qubits.
• U::Matrix{<:Number}: The target unitary matrix.
• d::Int=0: Dimension of the Hilbert space. If d

is 0, it is inferred from the size of U.

Returns

• Float64: The process fidelity, a real number in the closed interval **[0

processfidelity(E::MixingUnitaryChannel, S::Matrix{<:Number}) Compute the process fidelity of a quantum channel defined by a MixingUnitaryChannel and a target process S.

Arguments

• E::MixingUnitaryChannel: The mixing unitary channel.
• U::Matrix{<:Number}: The target unitary matrix.
• d::Int=0: Dimension of the Hilbert space. If d is 0, it is inferred from the size of E.

Returns

• Float64: The process fidelity, a real number in the closed interval **[0

[1] L. H. Pedersen, K. Molmer, and N. M. Moller, Fidelity of quantum operations, Physics Letters A 367, 47 (2007).

Calculate the average gate fidelity between two unitary operations.

Arguments

• U0::Matrix{<:Number}: The first unitary operation, represented as a matrix of size 2^n × 2^n for n qubits.
• U1::Matrix{<:Number}: The second unitary operation, represented as a matrix of size 2^n × 2^n for n qubits.

Returns

• Real: The average gate fidelity between the two unitary operations.

Calculate the average gate fidelity between a quantum channel defined by Kraus operators and a unitary operation.

Arguments

• krausops::Vector{<:Matrix{<:Number}}: Vector of Kraus operators, each of size 2^n × 2^n for n qubits.
• U::Matrix{<:Number}: The unitary operation, represented as a matrix of size 2^n × 2^n for n qubits.

Returns

• Real: The average gate fidelity between the quantum channel and the unitary operation.

Analytical dephasing factor of a one-spin shuttling model.

Arguments

• T::Real: Total time
• L::Real: Length of the path
• B<:GaussianRandomField: Noise field, Ornstein-Uhlenbeck or Pink-Lorentzian
• path::Symbol: Path of the shuttling model, :straight or :forthback

Analytical dephasing factor of a sequenced two-spin EPR pair shuttling model.

Ornstein-Uhlenbeck field, the correlation function of which is σ^2/(4*κₜ *κₓ) * exp(-|t₁ - t₂|*κₜ) * exp(-|x₁-x₂|*κₓ) where t is time and x is position.

Pink-Lorentzian Field, the correlation function of which is σ^2 * (expinti(-γ[2]abs(t₁ - t₂)) - expinti(-γ[1]abs(t₁ - t₂)))/log(γ[2]/γ[1]) * exp(-κ|x₁-x₂|) where expinti is the exponential integral function.

Pink-Gaussian Field, the correlation function of which is σ^2 * (expinti(-γ[2]abs(t₁ - t₂)) - expinti(-γ[1]abs(t₁ - t₂)))/log(γ[2]/γ[1]) * exp(-|x₁-x₂|^2/(2κ^2)) where expinti is the exponential integral function.

Pink-HeavisidePi Field, the correlation function of which is σ^2 * (expinti(-γ[2]abs(t₁ - t₂)) - expinti(-γ[1]abs(t₁ - t₂)))/log(γ[2]/γ[1]) * sinc(-κ|x₁-x₂|) where expinti is the exponential integral function.

Pink-Delta Field, the correlation function of which is σ^2 * (expinti(-γ[2]abs(t₁ - t₂)) - expinti(-γ[1]abs(t₁ - t₂)))/log(γ[2]/γ[1]) * δ(x₁-x₂) where expinti is the exponential integral function. δ is the Dirac delta function.

A Gaussian random function defined as a finite sample generated from a valid covariance matrix using Cholesky decomposition.

Create a new random function composed by a linear combination of random processes. The input random function represents the direct sum of these processes. The output random function is a tensor contraction from the input.

Arguments

• R::GaussianRandomFunction: a direct sum of random processes R₁⊕ R₂⊕ ... ⊕ Rₙ
• c::Vector{Int}: a vector of coefficients
• initialize::Bool=true: whether to initialize the Cholesky decomposition of the covariance matrix

Returns

• GaussianRandomFunction: a new random function composed by a linear combination of random processes

Restrict the random field along a parameteized curve.

Arguments

• X::Tuple: a vector of functions [x₁(t), x₂(t), ...]
• t::AbstractArray: time array
• B::GaussianRandomField: a Gaussian random field
• initialize::Bool: initialize the random function

Returns

• GaussianRandomFunction: a new random function restricted along the curve

Initialize the Cholesky decomposition of the covariance matrix of a random function.

Compute the characteristic functional of the process from the numerical quadrature of the covariance matrix. Using Simpson's rule by default.

Compute the final phase of the characteristic functional of the process from the numerical quadrature of the covariance matrix. Using Simpson's rule by default.

Covariance matrix of a Gaussian random field. When P₁=P₂, it is the auto-covariance matrix of a Gaussian random process. When P₁!=P₂, it is the cross-covariance matrix between two Gaussian random processes.

Arguments

• P₁::Vector{<:Point}: time-position array
• P₂::Vector{<:Point}: time-position array
• GRF::GaussianRandomField: a Gaussian random field

Auto-Covariance matrix of a Gaussian random field.

Arguments

• P::Vector{<:Point}: time-position array
• GRF::GaussianRandomField: a Gaussian random field

Returns

• Symmetric{Real}: auto-covariance matrix

Covariance function of Gaussian random field.

Arguments

• p₁::Point: time-position array
• p₂::Point: time-position array
• GRF<:GaussianRandomField: a Gaussian random field, e.g. OrnsteinUhlenbeckField or PinkLorentzianField