# Driven controls in the 1-qubit workspace

## Overview

This workspace permits you to create drop-in replacements for single-qubit control operations which are robust to decoherence and control errors. Here you can analyze the performance of non-trivial single qubit operations (driven rotations), as well as error-robust memory (dynamic decoupling) in the presence of time-dependent noise.

We abstract away the physical system and rely on the simplified representation of an ideal single-qubit; control operations are visualized as rotations of the Bloch vector on the Bloch sphere. Here, for instance, is a 90 degree rotation about the x-axis shown in the Bloch sphere framework

This approach works very well for individual superconducting qubits, trapped-ions, semiconductor spin qubits, nitrogen vacancies, and the like (For more complex level structures consider the Qudit workspace).

Key features in this workspace include the ability to:

- Select error-robust controls from a library of solutions or upload your own solution for analysis
- Explore the effect of controls using interactive Bloch spheres and easily accessible charts
- Calculate the noise susceptibility of these controls in the filter function formalism for both control noise and ambient dephasing.
- Define or upload noise power spectra for use in analyses capturing real laboratory environments
- Optimize controls for your noise using a machine-learning toolkit
- Perform error budgeting to identify the best controls and output the control waveforms

## Control and noise assumptions at a glance

### Control Hamiltonian

We consider a qubit with frequency $\omega_{0}$ driven by a near-resonant carrier $B(t) = \Omega(t)\cos\left(\omega_\text{LO}t+\phi(t)\right)$. The control Hamiltonian is

\[\begin{align} &H_{c} = \alpha_{x}(t)\sigma_{x} + \alpha_{y}(t)\sigma_{y} + \alpha_{z}(t)\sigma_{z}\\ \\ &\alpha_{x} = \Omega(t)\cos\phi(t)\\ &\alpha_{y} = \Omega(t)\sin\phi(t)\\ &\alpha_{z} = \Delta(t) = \omega_{0} - \omega_\text{LO} \end{align}\]### Control noise (multiplicative)

\[\begin{align} H_{n,\Omega}(t) = \beta_\Omega(t)\Omega(t)\left(\cos\phi(t)\sigma_x + \sin\phi(t)\sigma_y\right) \end{align}\]This corresponds to a fluctuating drive amplitude proportional to the strength of the driving field. This is captured by a time-dependent random process $S_{\Omega}(\omega) \iff \beta_{\Omega}(t)$

### Ambient dephasing (additive)

\[\begin{align} H_{n,z}(t) = \beta_{z}(t)\sigma_z \end{align}\]This corresponds to an independent additive environment which produces uncontrolled rotations about the quantization axis due to time-dependent process $\beta_{z}(t) \iff S_{z}(\omega)$