The Black Opal package gives you the ability to create noise-suppressing controls which are optimized for the specific noise power spectral densities (in both control noise and ambient dephasing) appearing in your system. After you run our custom optimizer, control solutions will appear in your own Control Library and may be compared against other standard or custom-uploaded controls.

## What the Optimizer Does

At a high-level, a given control is divided into a user-defined number of segments, and the parameters describing the single-qubit control Hamiltonian (in either Cartesian or Cylindrical coordinates) are varied to meet a specific cost function for a target operation enacted in the presence of noise. The output is the control solution showing optimal robustness to this noise.

The graph below shows a control solution implementing a driven "pi" pulse, optimized for simultaneous dephasing and control noise, using only phase modulation on the driving field in the single-qubit control Hamiltonian.

The performance of controls designed to simultaneously suppress both dephasing and control noise is experimentally validated. In the data below, we compare the performance of a Q-CTRL optimized pulse against a conventional control. Here, low values on the Y-axis indicate "good" performance, the X-axis measures the size of an error (as you move away from the center). The purple shading shows the benefit in terms of reduced errors over 10 applications of the control, indicating substantial benefits when using the optimized control. Moreover, this control is about 40% faster than CinBB, the previously best-known control designed to provide similar capabilities.

## How the Optimizer Works

The controls emerging from the Black Opal optimization package are formally "model robust" noise-suppressing controls, such that they continue to perform effectively even as details about the underlying noise processes change. In our case this comes from the specific construction of a cost function ensuring that a **target operation is enacted and that it exhibits error-robustness in the presence of noise**.

This is to be contrasted with "optimal control". What does this difference mean in practice?

Optimal control provides a solution which will work exceptionally well under a very limited set of circumstances (e.g. trying to navigate a multidimensional space without leakage errors). But if, for instance, the details of your Hamiltonian change then the optimal control solution you may have derived will break.

By contrast our solutions use modern algorithmic approaches such as gradient descent, and ensure that even in the presence of small changes in the details, the solutions continue to perform well. This form of robust control allows solutions to simultaneously navigate complex spaces and reduce sensitivity to time-varying noise processes.

We provide two different approaches to control optimization for single-qubit driven controls that we describe below:

- Fixed-frequency control optimization
- Broadband control optimization

Full technical details on the implementation of optimization algorithms appear in our technical documentation.

#### Fixed-frequency control optimization

If this option is selected the cost function for the optimizer will be restricted to the value of the filter function at the defined frequency. This approach allows a high-speed optimization routine at the expense of solution customization for the details of the noise.

The default frequency for optimization is "0" corresponding to the lowest frequency defined in the workspace (generally 1000x smaller than the Rabi rate). In this case controls will generally achieve effective low-frequency noise suppression. This value can be changed to any specific value in the range displayed on the filter function charts (tied to the Rabi rate) and the optimizer will adjust for that frequency.

#### Broadband control optimization

In this case, the optimizer will attempt to minimize the average infidelity over the entire relevant frequency range by calculating the filter function for each trial solution and selecting solutions which minimize the total infidelity. This approach permits a solution customized to your input noise, at the expense of computational runtime for the optimization. You will automatically receive a notification when the optimizer has finished.

## User Inputs

As a user you tell the Black Opal package what to optimize over, what resources you're willing to dedicate to the optimization, and what controls are available to you.

The primary resource of interest in finding *robust* controls is the extended control duration, selectable using the UI. The software will automatically enforce bounds to ensure that solutions outperform the primitive benchmark control.

You can also select the available flexibility in the solution, such as whether or not to include controls enacting a detuning of the drive from resonance, and whether you would like the controls to fix the amplitude (i.e. only implementing "phase modulation") or allowing the amplitude to vary in each segment (i.e. comparable to implementing generic "IQ" modulation)

## Achieving good solutions

In general it can not be guaranteed after running an optimization that the optimal solution has been found due to the presence of local minima in the control landscape. Nevertheless, with the default settings provided (including hard-coded segmentation of the control solution in time), the optimization will typically find a solution that outperforms a primitive pulse.

If the noise you have provided is particularly challenging to optimize against, you may receive a notification that the optimized pulse does not out-perform the equivalent primitive operation, in which case a primitive pulse is returned. In such circumstances there are a few options:

- Increase the duration of the pulse. It is typically easier to find optimized solutions with controls that extend over a longer time. Even in such cases it should take an equivalent amount of computational time to find an new optimized control.
- Run the optimization again with the same settings. Each optimization is run with randomized initial conditions. Hence there is always a chance that a better optimal solution will be found if an optimization is run again.