# Performance metrics for single- and multi-qubit operations

We use the filter function framework to help you perform error budgeting for the controls you select, giving you actionable information on whether a selected control is right for your application.

We present this data in a simple-to-read visual format so you can judge which protocol is best at a glance. In the BLACK OPAL interface these are represented for both control noise and ambient dephasing in the case of single qubits, or more specific types of noise in various multiqubit operations.

In this framework, the average infidelity of an operation is approximately given by the overlap integral of the noise power spectral density and the filter function describing the control. We use an exponential expression for infidelity which approximates the effect of adding higher-order terms to the calculation.

\[\begin{align} &\mathcal{I}_\text{av} \approx \frac{1}{2}\left(1-e^{-2\sum_{k=1}^{p}\mathcal{O}_k}\right)\\ &\mathcal{O}_{k} = \frac{1}{2\pi}\int_{-\infty}^{\infty}S_{k}(\omega)F_{k}(\omega) d\omega \end{align}\]Moreover, this expression is employed in such a way as to combine terms from both noise processes. For convenience we also calculate and display a worst-case error bound.

In addition, for certain gates like the Mølmer-Sørensen gate where the dominant error process comes from the noise-free operation (resulting in imperfect qubit-motional decoupling), we also provide a noise-free metric describing how the gate performs in the absence of any power spectral density. This is described in detail in our documentation.

The accuracy of the error budgeting process depends on the strength of the noise to which your qubit is subjected and the control selected. For instance, the filter function formalism is most accurate in the small error regime, where most experimental teams in quantum information are aiming to operate. In practice we find that the filter function provides good predictive power up to error rates of about 10%.

Please note that, any changes to the control inputs or the details of the noise power spectral densities in use will see these performance metrics recalculated.

Further details on the relevant single-qubit control Hamiltonian and noise Hamiltonian can be found in our detailed technical documentation.

## Further reading

For a detailed discussion of the formal bounds on the validity of the filter function formalism see Section 6 and the Appendix of Arbitrary quantum control of qubits in the presence of universal noise.

For a detailed discussion of the exponential fidelity metric in use please see the Supplementary Information for Experimental noise filtering by quantum control.