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Coefficients#

The resulting plots of our experiments as PDF can be found in plotting/rplots/img-gen, while a PNG version can be found on this website.

Summary#

Our Figure 16 in the paper summarised the coefficient mean \(\mu_c(\boldsymbol{\omega})\) for 1D (\(R_X\) and \(R_Y\) encoding), and 2D (\(R_X R_Y\) encoding) inputs and all qubits, only depicting the zero and maximum frequency-coefficient, shown here: Coefficients Mean Summary Coefficients Mean Summary

Absolute coefficient mean \(\mu_c(\boldsymbol{\omega})\) for the lowest frequency \(\boldsymbol{\omega}=\boldsymbol{0}\) and highest frequency \(\boldsymbol{\omega} = \boldsymbol{\omega}_\text{max}\) in the respective spectrum under the influence of varying noise levels. We considered one-dimensional encodings \(R_X\) and \(R_Y\) (\(D\) = 1), and a two-dimensional encoding \(R_X R_Y\) (\(D\) = 2). The y-axis for each facet row are equal throughout the respective \(\boldsymbol{\omega}\), but differs in between.

The corresponding relative standard deviation is (Our Figure 17 in the paper) Coefficients Mean Summary Coefficients Mean Summary

Relative standard deviation \(\sigma_c(\boldsymbol{\omega})\) for the absolute coefficient mean values from Fig. 16.

The detailed mean and standard deviations for varying qubit counts and encodings can be found in the corresponding pages on coefficients RX, coefficients RY and 2D-coefficients.

Influence of Input Encoding#

We measure the coefficients for \([3\dots 6]\) qubits and \(R_{\{X, Y, Z\}}\) encodings over frequencies in a noiseless setting.

Our Figure 2a from the paper: Coefficients Encoding Mean Coefficients Encoding Mean

Absolute coefficient mean \(\mu_c(\boldsymbol{\omega})\).

And the corresponding relative standard deviation (Our Figure 2b in the paper): Coefficients Encoding Standard Deviation Coefficients Encoding Standard Deviation

Relative standard deviation \(\sigma_c(\boldsymbol{\omega})\) for the absolute coefficient mean values from Fig. 2a.

Our Figure 3 from the paper: Coefficients Encoding Real/Imag Coefficients Encoding Real/Imag

Coefficients, separated into real and imaginary parts for a circuit with six qubits and different single qubit Pauli-encodings. The individual frequency components are colour-coded.

Effect of Coherent Noise#

Our Figure 7 in the paper: Coefficients Num Freq Coefficients Num Freq

Absolute coefficient mean \(\mu_c(\boldsymbol{\omega})\) for 6 qubits and \(R_Y\) encodings over frequencies for different noise levels \([0,\dots,3]\%\) and different ansätze. Noise is applied either on the encoding gates (\(\boldsymbol{\epsilon}_{\boldsymbol{x}}\)), the trainable gates (\(\boldsymbol{\epsilon}_{\boldsymbol{\theta}}\)) or on the full VQC (\(\boldsymbol{\epsilon}_{\boldsymbol{x}}\) and \(\boldsymbol{\epsilon}_{\boldsymbol{\theta}}\)).

Our Figure 8 in the paper: Coefficients Num Freq Coefficients Num Freq

Number of frequencies in the spectrum with and without applying a coherent gate error for \(D\)-dimensional inputs and only those circuits where the maximum possible number of frequencies is not achieved in the noiseless case.