qtcad.qubit.dynamics module

Class related to simulating dynamics (EDSR)

class qtcad.qubit.dynamics.Dynamics

Bases: qubit.core.dynamics.Dynamics

Useful tools for simulating dynamics of two-level systems.

H_RF(H0, delta_V, omega)

Write the full Hamiltonian \(H(t) = H_0 + \delta V \cos(\omega t)\) in a frame rotating at frequency \(\omega\) around the \(z\) axis.

Parameters

  • H0 (2D array) – Diagonalized system Hamiltonian.

  • delta_V (2D array) – Drive that can initiate transitions between eigenstates of H0.

  • omega (float) – Frequency of the drive and of the rotating frame.

Returns

qutip Qobj – Hamiltonian matrix in the rotating frame.

Note

The operators H0 and delta_V should be input in units where \(\hbar\) = 1.

fidelity(U, res, H0=None)

Computes the fidelity of a gate.

At time \(t\), the fidelity \(F(t)\) of the quantum-logic gate that is represented by the unitary operator \(U\) in an ideal setting is given by:

\[F(t) = |\langle\psi(t)|U|\psi(0)\rangle|^2,\]

where \(|\psi(0)\rangle\) is the initial state of the system (before application of the gate), and \(|\psi(t)\rangle\) is the time-dependent state of the system under a (potentially) non-ideal implementation of the quantum-logic gate, e.g., realistic driving in the presence of a stochastic time-dependent field.

The above equation gives the fidelity in the laboratory frame. In a frame that rotates with a free Hamiltonian \(H_0\), the fidelity output by this function is, instead, given by [WM08]

\[F(t) = |\langle\psi(t)|\mathrm e^{iH_0t}U|\psi(0)\rangle|^2.\]
Parameters

  • U (2D array/Qobj) – The gate that is ideally implemented.

  • res (QuTiP Result object) – A Result object that contains information about the dynamics of the system, as output by the mesolve function of QuTiP.

  • H0 (2D array/Qobj) – If None, the output fidelity will be given in the lab frame. If not None, the output fidelity will be given in the rotating frame with respect to H0.

Returns

1D array – Fidelity of the gate U for the times specified by res.

Note

For the fidelity, we use the definition of Eq. (16.8) in Ref. [WM08] by Walls and Milburn. We remark that this fidelity is the square of the (square-root) fidelity defined by Nielssen and Chuang [NC10].

plot_2_level(result)

Produces a plot of the probability to be in the up and down states as a function of time from the simulation of a two-level system dynamics.

Parameters

result (qutip result object) – The result object obtained from simulating the dynamics of a two-level system.

Note

This function can work in conjunction with ‘transition_2_levels’ to visualize the results of a simple EDSR simulation.

projectors(n_states)

Generate operators (qutip Qobj) that project a ket onto a specific basis state.

Parameters

n_states (integer) – Dimensionality of the subspace under consideration, i.e. number of basis states that span the Hilbert space.

Returns

list of Qobj objects – Proj, a list containing the projectors for each basis state.

transition_2_levels(i, j, H0, delta_V, omega=None, T=None, npts=1000, plot=False)

Project system and drive Hamiltonians onto a relevant 2 level subspace.

Parameters

  • i (integer) – Index of the up state. The convention chosen for the Pauli z operator for this qubit is the same one as qutip: sigma_z = |up><up| - |down><down|

  • j (integer) – Index of down state. This state is taken to be the initial state in the simulation.

  • H0 (2D array) – System Hamiltonian.

  • delta_V (2D array) – EDSR perturbation.

  • omega (float) – Driving frequency of the perturbation.

  • T (float) – Time over which the dynamics will be evaluated.

  • npts (integer) – Number of points in the simulation.

  • plot (boolean) – Whether or not a plot should be produced at the end of the calculation of the probability to be in the initial and final states as a function of time.

Returns

tuple – a tuple (h0, u, omega0, omega_rabi, result) where h0 (qutip Qobj) is the projected system Hamiltonian, u (qutip Qobj) is the projected perturbation, omega0 (float) is the resonance frequency of the transition, omega_rabi (float) is the Rabi frequency of the transition, and result (qutip mesolve) is the result of the simulation of the dynamics of the two level system by qutip.

Note

This function can be used to get a rough understanding of the relevant frequencies of a transition that is being investigated. The calculation performed by this function is a simplified version of a more accurate calculation may be required to adequately investigate EDSR for a given system.

Note

This function may only be useful when the perturbing potential, delta_V, is a linear function of the the voltage applied to the gate generating the perturbation.

Note

In the situation where a higher energy state is the down state and a lower energy state is the up state, the output resonance frequency will be negative.