StarkRamseyXYAmpScanAnalysis¶

class StarkRamseyXYAmpScanAnalysis[source]¶

Ramsey XY analysis for the Stark shifted phase sweep.

Overview

This analysis is a variant of RamseyXYAnalysis. In both cases, the X and Y data are treated as the real and imaginary parts of a complex oscillating signal. In RamseyXYAnalysis, the data are fit assuming a phase varying linearly with the x-data corresponding to a constant frequency and assuming an exponentially decaying amplitude. By contrast, in this model, the phase is assumed to be a third order polynomial \(\theta(x)\) of the x-data. Additionally, the amplitude is not assumed to follow a specific form. Techniques to compute a good initial guess for the polynomial coefficients inside a trigonometric function like this are not trivial. Instead, this analysis extracts the raw phase and runs fits on the extracted data to a polynomial \(\theta(x)\) directly.

The measured P1 values for a Ramsey X and Y experiment can be written in the form of a trignometric function taking the phase polynomial \(\theta(x)\):

\[\begin{split}P_X = \text{amp}(x) \cdot \cos \theta(x) + \text{offset},\\ P_Y = \text{amp}(x) \cdot \sin \theta(x) + \text{offset}.\end{split}\]

Hence the phase polynomial can be extracted as follows

\[\theta(x) = \tan^{-1} \frac{P_Y}{P_X}.\]

Because the arctangent is implemented by the atan2 function defined in \([-\pi, \pi]\), the computed \(\theta(x)\) is unwrapped to ensure continuous phase evolution.

We call attention to the fact that \(\text{amp}(x)\) is also Stark tone amplitude dependent because of the qubit frequency dependence of the dephasing rate. In general \(\text{amp}(x)\) is unpredictable due to dephasing from two-level systems distributed randomly in frequency or potentially due to qubit heating. This prevents us from precisely fitting the raw \(P_X\), \(P_Y\) data. Fitting only the phase data makes the analysis robust to amplitude dependent dephasing.

In this analysis, the phase polynomial is defined as

\[\theta(x) = 2 \pi t_S f_S(x)\]

where

\[f_S(x) = c_1 x + c_2 x^2 + c_3 x^3 + f_{\rm err},\]

denotes the Stark shift. For the lowest order perturbative expansion of a single driven qubit, the Stark shift is a quadratic function of \(x\), but linear and cubic terms and a constant offset are also considered to account for other effects, e.g. strong drive, collisions, TLS, and so forth, and frequency mis-calibration, respectively.

Fit model

This is the curve fitting analysis. The following equation(s) are used to represent curve(s).

\[\theta^\nu(x) = c_1^\nu x + c_2^\nu x^2 + c_3^\nu x^3 + f_{\rm err},\]

where \(\nu \in \{+, -\}\). The Stark shift is asymmetric with respect to \(x=0\), because of the anti-crossings of higher energy levels. In a typical transmon qubit, these levels appear only in \(f_S < 0\) because of the negative anharmonicity. To precisely fit the results, this analysis uses different model parameters for positive (\(x > 0\)) and negative (\(x < 0\)) shift domains.

Fit parameters

The following fit parameters are estimated during the analysis.

Descriptions

\(c_1^+\): The linear term coefficient of the positive Stark shift (fit parameter: stark_pos_coef_o1).
\(c_2^+\): The quadratic term coefficient of the positive Stark shift. This parameter must be positive because Stark amplitude is chosen to induce blue shift when its sign is positive. Note that the quadratic term is the primary term (fit parameter: stark_pos_coef_o2).
\(c_3^+\): The cubic term coefficient of the positive Stark shift (fit parameter: stark_pos_coef_o3).
\(c_1^-\): The linear term coefficient of the negative Stark shift. (fit parameter: stark_neg_coef_o1).
\(c_2^-\): The quadratic term coefficient of the negative Stark shift. This parameter must be negative because Stark amplitude is chosen to induce red shift when its sign is negative. Note that the quadratic term is the primary term (fit parameter: stark_neg_coef_o2).
\(c_3^-\): The cubic term coefficient of the negative Stark shift (fit parameter: stark_neg_coef_o3).
\(f_{\rm err}\): Constant phase accumulation which is independent of the Stark tone amplitude. (fit parameter: stark_ferr).

Initial Guess

\(c_1^+\): 0.
\(c_2^+\): 1e6.
\(c_3^+\): 0.
\(c_1^-\): 0.
\(c_2^-\): -1e6.
\(c_3^-\): 0.
\(f_{\rm err}\): 0

Boundaries

\(c_1^+\): None
\(c_2^+\): [0, inf]
\(c_3^+\): None
\(c_1^-\): None
\(c_2^-\): [-inf, 0]
\(c_3^-\): None
\(f_{\rm err}\): None

Analysis options

These are the keyword arguments of the run() method.

Options

Defined in the class StarkRamseyXYAmpScanAnalysis:
- pulse_len (float)
  
  Default value: None
  
  Duration of effective Stark pulse in units of sec.
Defined in the class BaseCurveAnalysis:
- plotter (BasePlotter)
  
  Default value: Instance of CurvePlotter
  
  A curve plotter instance to visualize the analysis result.
- plot_raw_data (bool)
  
  Default value: False
  
  Set True to draw processed data points, dataset without formatting, on canvas. This is False by default.
- plot_residuals (bool)
  
  Default value: False
  
  Set True to draw the residuals data for the fitting model. This is False by default.
- return_fit_parameters (bool)
  
  Default value: True
  
  (Deprecated) Set True to return all fit model parameters with details of the fit outcome. Default to False.
- data_processor (Callable)
  
  Default value: None
  
  A callback function to format experiment data. This can be a DataProcessor instance that defines the self.__call__ method.
- normalization (bool)
  
  Default value: False
  
  Set True to normalize y values within range [-1, 1]. Default to False.
- average_method (Literal[“sample”, “iwv”, “shots_weighted”])
  
  Default value: "shots_weighted"
  
  Method to average the y values when the same x values appear multiple times. One of “sample”, “iwv” (i.e. inverse weighted variance), “shots_weighted”. See mean_xy_data() for details. Default to “shots_weighted”.
- p0 (Dict[str, float])
  
  Default value: {}
  
  Initial guesses for the fit parameters. The dictionary is keyed on the fit parameter names.
- bounds (Dict[str, Tuple[float, float]])
  
  Default value: {}
  
  Boundary of fit parameters. The dictionary is keyed on the fit parameter names and values are the tuples of (min, max) of each parameter.
- fit_method (str)
  
  Default value: "least_squares"
  
  Fit method that LMFIT minimizer uses. Default to least_squares method which implements the Trust Region Reflective algorithm to solve the minimization problem. See LMFIT documentation for available options.
- lmfit_options (Dict[str, Any])
  
  Default value: {}
  
  Options that are passed to the LMFIT minimizer. Acceptable options depend on fit_method.
- x_key (str)
  
  Default value: "xval"
  
  Circuit metadata key representing a scanned value.
- fit_category (str)
  
  Default value: "freq"
  
  Name of dataset in the scatter table to fit.
- result_parameters (List[Union[str, ParameterRepr])
  
  Default value: []
  
  Parameters reported in the database as a dedicated entry. This is a list of parameter representation which is either string or ParameterRepr object. If you provide more information other than name, you can specify [ParameterRepr("alpha", "α", "a.u.")] for example. The parameter name should be defined in the series definition. Representation should be printable in standard output, i.e. no latex syntax.
- extra (Dict[str, Any])
  
  Default value: {}
  
  A dictionary that is appended to all database entries as extra information.
- fixed_parameters (Dict[str, Any])
  
  Default value: {}
  
  Fitting model parameters that are fixed during the curve fitting. This should be provided with default value keyed on one of the parameter names in the series definition.
- filter_data (Dict[str, Any])
  
  Default value: {}
  
  Dictionary of experiment data metadata to filter. Experiment outcomes with metadata that matches with this dictionary are used in the analysis. If not specified, all experiment data are input to the curve fitter. By default, no filtering condition is set.
- data_subfit_map (Dict[str, Dict[str, Any]])
  
  Default value: {"Xpos": ("Xpos", {"series": ("series", "X"), "direction": ("direction", "pos")}), "Ypos": ("Ypos", {"series": ("series", "Y"), "direction": ("direction", "pos")}), "Xneg": ("Xneg", {"series": ("series", "X"), "direction": ("direction", "neg")}), "Yneg": ("Yneg", {"series": ("series", "Y"), "direction": ("direction", "neg")})}
  
  The mapping of experiment result data to sub-fit models. This dictionary is keyed on the LMFIT model name, and the value is a sorting key-value pair that filters the experiment results, and the filtering is done based on the circuit metadata.
Defined in the class BaseAnalysis:
- figure_names (str or List[str])
  
  Default value: None
  
  Identifier of figures that appear in the experiment data to sort figures by name.