Quick Start
Generating a noise series from a stochastic process
Construct time-position array and define a Gaussian random field
P=collect(zip(t, v.*t))
-B=OrnsteinUhlenbeckField(0,[κₜ,κₓ],σ) Define a Gaussian random process (random function) by projecting the Gaussian random field along the time-space array P. Then we can use R() to invoke the process and generating a random time series.
R=RandomFunction(P, B)
-plot(R()) Shuttling of a single spin
Define premeters
σ = sqrt(2) / 20; # variance of the process
+Quick Start · SpinShuttling.jl Quick Start
Basic model
$
H{noise}(t)=g \muB \sumj \left[B0(x^cj)+\tilde{B}(x^c{j},t)\right] Sj^z $ We assume the electrons are adiabatically transported in moving-wave potential with their wave-functions well localized at xj^c$. Then the effective magnetic noise $\tilde{B}(x_j^c, t)$ can be modeled by a Gaussian random field.
In the case of pure dephasing, the system dynamics cna be explicitly writen out. $ U(t)=\exp(-\frac{i}{\hbar} \int0^t H{noise}(\tau)\mathrm{d} \tau) $
If we label a realization of the random process by $k$, then the pure dephasing channel can be expressed as a mixing unitary process. $ \mathcal{E}(\rho)=\frac{1}{M} \sum{k=1}^M Uk \rho Uk^\dagger =\sumk Ek \rho Ek^\dagger, \quad Ek= Uk /\sqrt{M} $ The pure dephasing of such a system can be analytically solved and efficiently obtained via a matrix of dephasing factors. While more general system dynamics involving other interactions can be numerically solved by Monte-Carlo sampling.
$
\mathcal{H}(t)=H{noise}(t)+H{int}(t) $
Generating a noise series from a stochastic field
Import the package.
using SpinShuttling
+using Plots
We first define an 2D Ornstein-Uhlenbeck field, specified by three parameters.
κₜ=1/20; # inverse correlation time
+κₓ=1/0.1; # inverse correlation length
+σ = 1; # noise strength
+B=OrnsteinUhlenbeckField(0,[κₜ,κₓ],σ); # mean is zero
Specify a trajectory (t,x(t)) on the 2D plane, in this example case it's just a line.
t=range(1,20,200); # time step
+v=2; #velocity
+P=collect(zip(t, v.*t));
A Gaussian random process (random function) can be obtained by projecting the Gaussian random field along the time-space array P. Then we can use R() to invoke the process and generating a random time series.
R=RandomFunction(P, B)
+plot(t, R(), xlabel="t", ylabel="B(t)", size=(400,300))
Shuttling of a single spin
We can follow the above approach to define a single spin shuttling model.
σ = sqrt(2) / 20; # variance of the process
κₜ=1/20; # temporal correlation
-κₓ=1/0.1; # spatial correlation
Create an Ornstein-Uhlenbeck Process
B=OrnsteinUhlenbeckField(0,[κₜ,κₓ],σ)
Consider the shuttling of a single spin at constant velocity v. We need to specify the initial state, travelling time T and length L=v*T, and the stochastic noise expreienced by the spin qubit.
T=400; # total time
+κₓ=1/0.1; # spatial correlation
+B=OrnsteinUhlenbeckField(0,[κₜ,κₓ],σ)
Consider the shuttling of a single spin at constant velocity v. We need to specify the initial state, travelling time T and length L=v*T, and the stochastic noise expreienced by the spin qubit.
T=400; # total time
L=10; # shuttling length
v=L/T;
The package provided a simple encapsulation for the single spin shuttling, namely by OneSpinModel. We need to specify the discretization size and monte-carlo size to create a model.
M = 10000; # monte carlo sampling size
N=301; # discretization size
-model=OneSpinModel(T,L,N,B)
-println(model)
The fidelity of the spin state after shuttling can be calculated using numerical integration of the covariance matrix.
f1=averagefidelity(model)
The fidelity can also be obtained from Monte-Carlo sampling.
f2, f2_err=sampling(model, fidelity, M)
For the single spin shuttling at constant velocity, analytical solution is also available.
f3=1/2*(1+W(T,L,B))
We can compare the results form the three methods.
@assert isapprox(f1, f3,rtol=1e-2)
+model=OneSpinModel(T,L,N,B)
The output will be <pre> Model for spin shuttling Spin Number: n=1 Initial State: |Ψ₀⟩=[0.707, 0.707] Noise Channel: OrnsteinUhlenbeckField(0, [0.05, 10.0], 0.07071067811865475) Time Discretization: N=301 Process Time: T=400 Shuttling Paths: ┌──────────────────────────────┐ 10 │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋│ x1(t) │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀│ │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀│ │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ │⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ │⠀⠀⠀⣀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 0 │⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ └──────────────────────────────┘ ⠀0⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀400⠀ </pre> The fidelity of the spin state after shuttling can be calculated using numerical integration of the covariance matrix.
This provides us an overview of the model. It's a single spin shuttling problem with initial state Ψ₀ and an Ornstein-Uhlenbeck noise. The total time of simulation is T, which is discretized into N steps.
The state fidelity after such a quantum process can be obtained by different numerical methods.
f1=averagefidelity(model); # direct integration
+
+f2, f2_err=sampling(model, fidelity, M); # Monte-Carlo sampling
For the single spin shuttling at constant velocity, analytical solution is also available.
f3=1/2*(1+W(T,L,B));
We can compare the results form the three methods and check their consistency.
@assert isapprox(f1, f3,rtol=1e-2)
@assert isapprox(f2, f3, rtol=1e-2)
println("NI:", f1)
println("MC:", f2)
-println("TH:", f3)
Shuttling of entangled spin pairs.
Settings
This document was generated with Documenter.jl version 1.3.0 on Monday 27 May 2024. Using Julia version 1.10.3.
NI:0.5172897445804854
+MC:0.5210640948921192
+TH:0.5183394145238882The pure dephasing channel is computationaly simple, and can be represented by a dephasing matrix $w$, such that the final density state after the channel is given by $\mathcal{E}(\rho)=w \odot\rho$. Here $\odot$ is a element-wise Hadmard product.
Ψ= model.Ψ
+ρ=Ψ*Ψ'
+w=dephasingmatrix(model)
+w.*ρThe final density matrix is shown in output.
2×2 Matrix{Float64}:
+ 0.5 0.0172897
+ 0.0172897 0.5We can check that the fidelity between the initial and final state is consistent with results above.
f=(Ψ'*ρt*Ψ)0.5172897445804852Dephasing of entangled spin pairs during shuttling.
Following the approach above, we can further explore the multi-spin system. The general abstraction on such a problem is given by the data type ShuttlingModel.
ShuttlingModel(n, Ψ, T, N, B, X, R)User can freely define a n-qubit system with arbitrary initial state. Here, X=[x1,x2...] is an array of function, containing spin trajectories $x_i(t)$. R is a random function constructed from the specific noise process.
One more example is the shuttling of two spin pairs. We can define such a two spin system.
L=10; σ =sqrt(2)/20; M=5000; N=501; T1=100; T0=25; κₜ=1/20; κₓ=1/0.1;
+B=OrnsteinUhlenbeckField(0,[κₜ,κₓ],σ)
+model=TwoSpinModel(T0, T1, L, N, B)<pre> Model for spin shuttling Spin Number: n=2 Initial State: |Ψ₀⟩=[0.0, 0.707, -0.707, 0.0] Noise Channel: OrnsteinUhlenbeckField(0, [0.05, 10.0], 0.07071067811865475) Time Discretization: N=501 Process Time: T=125 Shuttling Paths: ┌──────────────────────────────┐ 10 │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⡹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ x1(t) │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ x2(t) │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ │⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ │⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ │⠀⠀⠀⠀⠀⡴⠋⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ │⠀⠀⠀⣠⠞⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ │⠀⢀⡜⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 0 │⣴⣋⣀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ └──────────────────────────────┘ ⠀0⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀200⠀ </pre> The system is initialized in the Bell state $\ket{\Psi^-}$. The model encapsulated a model of two spin shuttled in a sequential manner, as we can see from the two trajectories x1(t) and x2(t). One spin goes first and then follows another, with waiting time T0. This is modeled by the piece-wise linear trajectories. We can see some quite interesting covariance from such a system.
heatmap(collect(model.R.Σ), title="cross covariance matrix, two spin EPR")
We can check that the dephasing of the system and calculate its fidelity as before.
f1=averagefidelity(model)
+f2, f2_err=sampling(model, fidelity, M)
+f3=1/2*(1+W(T0, T1, L,B))
+
+println("NI:", f1)
+println("MC:", f2)
+println("TH:", f3)NI:0.970066663028708
+MC:0.9699908129499957
+TH:0.9700818676163352The density matrix after the channel can be given by the dephasing matrix.
Ψ= model.Ψ
+ρ=Ψ*Ψ'
+w=dephasingmatrix(model)
+
+ρt=w.*ρ4×4 Matrix{Float64}:
+ 0.0 0.0 -0.0 0.0
+ 0.0 0.5 -0.470067 0.0
+ -0.0 -0.470067 0.5 -0.0
+ 0.0 0.0 -0.0 0.0