Dynamics Simulation Results

The solver.Result Class

Before embarking on simulating the dynamics of quantum systems, we will first look at the data structure used for returning the simulation results. This object is a Result class that stores all the crucial data needed for analyzing and plotting the results of a simulation. A generic Result object result contains the following properties for storing simulation data:

Property

Description

result.solver

String indicating which solver was used to generate the data.

result.times

List/array of times at which simulation data is calculated.

result.expect

List/array of expectation values, if requested.

result.e_data

Dictionary of expectation values, if requested.

result.states

List/array of state vectors/density matrices calculated at times, if requested.

result.final_state

State vector or density matrix at the last time of the evolution.

result.stats

Various statistics about the evolution.

Accessing Result Data

To understand how to access the data in a Result object we will use an example as a guide, although we do not worry about the simulation details at this stage. Like all solvers, the Master Equation solver used in this example returns an Result object, here called simply result. To see what is contained inside result we can use the print function:

>>> print(result)
<Result
  Solver: mesolve
  Solver stats:
    method: 'scipy zvode adams'
    init time: 0.0001876354217529297
    preparation time: 0.007544517517089844
    run time: 0.001268625259399414
    solver: 'Master Equation Evolution'
    num_collapse: 1
  Time interval: [0, 1.0] (2 steps)
  Number of e_ops: 1
  State not saved.
>

The first line tells us that this data object was generated from the Master Equation solver mesolve. Next we have the statistics including the ODE solver used, setup time, number of collpases. Then the integration interval is described, followed with the number of expectation value computed. Finally, it says whether the states are stored.

Now we have all the information needed to analyze the simulation results. To access the data for the two expectation values one can do:

expt0 = result.expect[0]
expt1 = result.expect[1]

Recall that Python uses C-style indexing that begins with zero (i.e., [0] => 1st collapse operator data). Alternatively, expectation values can be obtained as a dictionary:

e_ops = {"sx": sigmax(), "sy": sigmay(), "sz": sigmaz()}
...
expt_sx = result.e_data["sx"]

When e_ops is a list, e_data ca be used with the list index. Together with the array of times at which these expectation values are calculated:

times = result.times

we can plot the resulting expectation values:

plot(times, expt0)
plot(times, expt1)
show()

State vectors, or density matrices, are accessed in a similar manner, although typically one does not need an index (i.e [0]) since there is only one list for each of these components. Some other solver can have other output, heomsolve’s results can have ado_states output if the options store_ados is set, similarly, fmmesolve can return floquet_states.

Multiple Trajectories Solver Results

Solver which compute multiple trajectories such as the Monte Carlo Equations Solvers or the Stochastics Solvers result will differ depending on whether the trajectories are flags to be saved. For example:

>>> mcsolve(H, psi, np.linspace(0, 1, 11), c_ops, e_ops=[num(N)], ntraj=25, options={"keep_runs_results": False})
>>> np.shape(result.expect)
(1, 11)

>>> mcsolve(H, psi, np.linspace(0, 1, 11), c_ops, e_ops=[num(N)], ntraj=25, options={"keep_runs_results": True})
>>> np.shape(result.expect)
(1, 25, 11)

When the runs are not saved, the expectation values and states are averaged over all trajectories, while a list over the runs are given when they are stored. For a fix output format, average_expect return the average, while runs_states return the list over trajectories. The runs_ output will return None when the trajectories are not saved. Standard derivation of the expectation values is also available:

Reduced result

Trajectories results

Description

average_states

runs_states

State vectors or density matrices calculated at each times of tlist

average_final_state

runs_final_state

State vectors or density matrices calculated at the last time of tlist

average_expect

runs_expect

List/array of expectation values, if requested.

std_expect

List/array of standard derivation of the expectation values.

average_e_data

runs_e_data

Dictionary of expectation values, if requested.

std_e_data

Dictionary of standard derivation of the expectation values.

Multiple trajectories results also keep the trajectories seeds to allows recomputing the results.

seeds = result.seeds

One last feature specific to multi-trajectories results is the addition operation that can be used to merge sets of trajectories.

>>> run1 = smesolve(H, psi, np.linspace(0, 1, 11), c_ops, e_ops=[num(N)], ntraj=25)
>>> print(run1.num_trajectories)
25
>>> run2 = smesolve(H, psi, np.linspace(0, 1, 11), c_ops, e_ops=[num(N)], ntraj=25)
>>> print(run2.num_trajectories)
25
>>> merged = run1 + run2
>>> print(merged.num_trajectories)
50

This allows one to improve statistics while keeping previous computations.