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::Vector{Function}: Shuttling paths, the length of the vector must be `n
• R::RandomFunction: Random function of the sampled noises on paths

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

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.

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
• v::Real: Velocity of the shuttling

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

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.

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.

Calculate the dephasing matrix of a given spin shuttling model.

Sample a phase integral of the process. The integrate of a random function should be obtained from directly summation without using high-order interpolation (Simpson or trapezoid).

Monte-Carlo sampling of any objective function. The function must return Tuple{Real,Real} or Tuple{Vector{<:Real},Vector{<:Real}}

Arguments

• samplingfunction::Function: The function to be sampled
• M::Int: Monte-Carlo sampling size

Returns

• Tuple{Real,Real}: The mean and variance of the sampled function
• Tuple{Vector{<:Real},Vector{<:Real}}: The mean and variance of the sampled function

Example

f(x) = x^2
sampling(f, 1000)

Reference

https://en.wikipedia.org/wiki/Standarddeviation#Rapidcalculation_methods

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

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

Arguments

• model::ShuttlingModel: The spin shuttling model

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-Brownian
• 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 * exp(-|t₁ - t₂|/θ_t) * exp(-|x₁-x₂|/θ_x) where t is time and x is position.

Pink-Brownian 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|/θ) where expinti is the exponential integral function.

Similar type of RandomFunction in Mathematica. Can be used to generate a time series on a given time array subject to a Gaussian random process traced from a Gaussian random field.

Arguments

• μ::Vector{<:Real}: mean of the process
• P::Vector{<:Point}: time-position array
• Σ::Symmetric{<:Real}: covariance matrices
• C::Cholesky: Cholesky decomposition of the covariance matrices

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::RandomFunction: a direct sum of random processes R₁⊕ R₂⊕ ... ⊕ Rₙ
• c::Vector{Int}: a vector of coefficients

Returns

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

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
• process::GaussianRandomField: a Gaussian random field

Auto-Covariance matrix of a Gaussian random process.

Arguments

• P::Vector{<:Point}: time-position array
• process::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
• process<:GaussianRandomField: a Gaussian random field, e.g. OrnsteinUhlenbeckField or PinkBrownianField