qiskit_dynamics.models.rotating_wave_approximation#

rotating_wave_approximation(model, cutoff_freq, return_signal_map=False)[source]#

Construct a new model by performing the rotating wave approximation with a given cutoff frequency. The outputs of this function can be used in JAX-transformable functions, however this function itself cannot (see below).

Performs elementwise rotating wave approximation (RWA) with cutoff frequency cutoff_freq on each operator in a model, returning a new model. The new model contains a modified list of signal coefficients, and setting the optional argument return_signal_map=True results in the additional return of the function f which maps the signals of the input model to those of the output RWA model, such that the code blocks:

model.signals = new_signals
rwa_model = rotating_wave_approximation(model, cutoff_freq)

and

rwa_model, f = rotating_wave_approximation(model, cutoff_freq, return_signal_map=True)
rwa_model.signals = f(new_signals)

result in an rwa_model with the same updated signals.

Note

The rotating_wave_approximation function itself cannot be included in a function to-be JAX-transformed, however the resulting model and signal_map can. For example, the following function is not JAX-transformable:

def function_with_rwa(t):
    operators = ...
    signals = ...
    model = GeneratorModel(operators=operators, signals=signals)
    rwa_model = rotating_wave_approximation(model, cutoff_freq)
    return rwa_model(t)

Whereas, defining:

rwa_model, signal_map = rotating_wave_approximation(
    model,
    cutoff_freq,
    return_signal_map=True
)

The following function is JAX-transformable:

def jax_transformable_func(t):
    rwa_model.signals = signal_map(new_signals)
    return rwa_model(t)

In this way, the outputs of rotating_wave_approximation can be used in JAX-transformable functions, however rotating_wave_approximation itself cannot.

We now describe the formalism. When considering $s_{i} (t) e^{- t F} G_{i} e^{t F}$ , in the basis in which $F$ is diagonal, the $(j, k)$ element of $G_{i}$ has effective frequency ${\tilde{ν}}_{i j k}^{\pm} = \pm ν_{i} + I m [- d_{j} + d_{k}] / 2 π$ , where the $\pm ν_{i}$ comes from expressing $s_{i} (t) = R e [a_{i} (t) e^{2 π i ν_{i} t}] = a_{i} (t) e^{i (2 π ν_{i} t + ϕ_{i})} / 2 + c . c .$ and the other term comes from the rotating frame. Define $G_{i}^{\pm}$ the matrix whose entries $(G_{i}^{\pm})_{j k}$ are the entries of $G_{i}$ s.t. $| ν_{i j k}^{\pm} | < ν_{*}$ for some cutoff frequency $ν_{*}$ . Then, after the RWA, we may write

s_{i} (t) G_{i} \to G_{i}^{+} a_{i} e^{i (2 π ν_{i} t + ϕ_{i})} / 2 + G_{i}^{-} \overset{―}{a_{i}} e^{- i (2 π ν_{i} t + ϕ_{i})} / 2.

When we regroup these to use only the real components of the signal, we find that

s_{i} (t) G_{i} \to s_{i} (t) (G_{i}^{+} + G_{i}^{-}) / 2 + s_{i}^{'} (t) (i G_{i}^{+} - i G_{i}^{-})

where $s_{i}^{'} (t)$ is a signal with the same frequency and amplitude as $s_{i}$ , but with a phase shift of $ϕ_{i} - π / 2$ .

Parameters:

model (BaseGeneratorModel) – The model to approximate.
cutoff_freq (float) – The cutoff frequency for the approximation.
return_signal_map (Optional[bool]) – Whether to also return the function for mapping pre-RWA signals to post-RWA signals.

Return type:

BaseGeneratorModel

Returns:

GeneratorModel with twice as many terms, and, if return_signal_map, also the function f.

Raises:

ValueError – If the model has no signals.