# Control parameters

## Single-qubit Rabi rate

The Rabi rate, denoted $\Omega(t)$, and modified Rabi rate, denoted $\Omega’(t)$ capture the instantaneous rate of rotation (in angular frequency) for resonantly-driven $(\Delta(t) = 0)$ and off-resonantly-driven $(\Delta(t) \ne 0)$ single-qubit driven operations respectively. This maps to the rate at which the Bloch vector rotates about the relevant control axis on the Bloch sphere. To see this, recall the single-qubit control Hamiltonian may be written

\begin{align} H_{c} = \frac{1}{2}\boldsymbol{\alpha}(t)\boldsymbol{\sigma}, \hspace{1.5cm} \boldsymbol{\alpha}(t) = \begin{bmatrix} \alpha_{x}(t), & \alpha_{y}(t), &\alpha_{z}(t) \end{bmatrix}, \hspace{1.5cm} \boldsymbol{\sigma} = \begin{bmatrix} \sigma_{x} \\ \sigma_{y} \\ \sigma_{z} \end{bmatrix} \end{align}

where

\begin{align} \alpha_{x}(t) &&=&&I(t) &&=&&\Omega(t) \cos(\phi(t)), \\ \alpha_{y}(t) &&=&&Q(t) &&=&&\Omega(t) \sin(\phi(t)), \\ \alpha_{z}(t) &&=&& \Delta(t) &&=&& \omega_{0}-\omega_\text{LO}(t). \end{align}

Or equivalently,

\begin{align} H_{c} = \frac{1}{2}\Omega'(t)\hat{\boldsymbol{n}}(t)\boldsymbol{\sigma}, \hspace{1.5cm} \hat{\boldsymbol{n}}(t) = \frac{\boldsymbol{\alpha}(t)}{|\boldsymbol{\alpha}(t)|}, \hspace{1.5cm} \Omega'(t) = {|{\boldsymbol{\alpha}(t)|}}. \end{align}

In this form, the instantaneous rotation generator $\hat{\boldsymbol{n}}(t)\boldsymbol{\sigma} \equiv n_x(t)\sigma_x+n_y(t)\sigma_y+n_z(t)\sigma_z$ drives the infinitesimal rotation $d\theta = \Omega’(t) dt$ about the rotation axis $\hat{\boldsymbol{n}}(t)\in\mathbb{R}^3$ at time $t$, with instantaneous rotational frequency

\begin{align} \frac{d\theta}{dt} = \Omega'(t). \hspace{1.5cm} \end{align}

For the the general case of an off-resonant drive ($\Delta(t) \ne 0$) we obtain

\begin{align} &\text{modified Rabi rate:} \hspace{1.5cm} \Omega'(t) =\sqrt{\alpha_{x}(t)^2+\alpha_{y}(t)^2 + \alpha_{z}(t)^2} =\sqrt{I(t)^2+Q(t)^2 + \Delta(t)^2} =\sqrt{\Omega(t)^2+\Delta(t)^2}. \end{align}

In the limit of zero detuning ($\Delta(t) = 0$) the modified Rabi rate reduces to $\Omega’(t) \rightarrow \Omega(t)$, where the on-resonance Rabi rate satisfies

\begin{align} &\text{Rabi rate (on resonance):}&& \Omega(t) =\sqrt{\alpha_{x}(t)^2+\alpha_{y}(t)^2} =\sqrt{I(t)^2+Q(t)^2} \end{align}

corresponding to the rotation rate about the axis

\begin{align} \hat{\boldsymbol{n}}(t) = \begin{bmatrix} \cos\phi(t), & \sin\phi(t), & 0 \end{bmatrix}, \end{align}

in the $xy$-plane (or equatorial plane) of the Bloch sphere, with orientation completely determined by the drive phase $\phi(t)$. This is consistent with the physical interpretation that the Rabi rate is proportional the amplitude of the resonant driving field. We note that in this definition the Rabi rate is strictly positive, with rotational direction determined by the orientation of $\hat{\boldsymbol{n}}(t)$.

The maximum possible amplitude of the driving field in a real hardware system establishes a maximum possible value of the on-resonance Rabi rate. We denote this upper bound as $\Omega_\text{max}$, and require

\begin{align} &\text{maximum Rabi rate:}&& \Omega(t)\le\Omega_\text{max}, \hspace{1cm} t\in[0,\tau]. \end{align}

That is, the instantaneous Rabi rate over the duration of the control, $t\in[0,\tau]$, cannot exceed this upper bound. In particular, we require

\begin{align} \Omega_\text{peak}\le \Omega_\text{max} \end{align}

where

\begin{align} &\text{peak Rabi rate:}&& \Omega_\text{peak} = \text{max}_{t}\big[\Omega(t)\big]_{0}^{\tau} \end{align}

defines the peak Rabi rate over the duration $t\in[0,\tau]$ of the control drive. $$\Omega_\text{max}$$ therefore bounds the speed at which quantum operations may be driven due to the physical limitations of the control system. Control solutions which exceed these bounds cannot be physically implemented.

## Single-qubit rotation angle

In the case of a resonantly-driven rotation enacted over time $\tau$, with a constant Rabi rate $\Omega$ and drive phase $\phi$, we write the rotation operator

\begin{align} R(\theta,\phi) \equiv \exp\left[-i\frac{\theta}{2}\hat{\boldsymbol{n}}(\phi)\boldsymbol{\sigma}\right], \hspace{1.5cm} \theta = \Omega\tau, \hspace{1.5cm} \hat{\boldsymbol{n}}(\phi) = \begin{bmatrix} \cos\phi, & \sin\phi, & 0 \end{bmatrix} \end{align}

and refer to $\theta$ as the net rotation angle.

## Two-qubit parametric coupling rate

For parametrically-driven two-qubit controls the parametric coupling rate, denoted $\Lambda(t)$, captures the rate of rotation (in angular frequency) for driving rotations associated with the two-qubit parametric operations. This is analogous to the single-qubit Rabi rate for single-qubit controls. We illustrate this below for iSWAP operations.

### iSWAP subspace

We consider control solutions implementing the control Hamiltonian, $H_{c}(t) = H_{c,\text{iSWAP}}(t) + H_{c,\text{qubit1}}(t)$, incorporating both parametric iSWAP controls, and single-qubit controls on qubit 1, where

\begin{align} &H_{c,\text{iSWAP}}(t) = \gamma_{\text{iSWAP}}(t) C_{\text{iSWAP}} + \text{H.C.}, \hspace{0cm} && && \gamma_{\text{iSWAP}}(t) = \Lambda(t)e^{+i \xi(t)}, \hspace{0cm} && && C_{\text{iSWAP}} = \frac{1}{2}|10\rangle\hspace{-0.07cm}\langle 01|, \\ \\ &H_{c,\text{qubit1}}(t) = \gamma_{\text{qubit1}}(t) C_{\text{qubit1}} + \text{H.C.}, \hspace{0cm} && && \gamma_{\text{qubit1}}(t) = \Omega(t)e^{+i \phi(t)} \hspace{0cm} && && C_{\text{qubit1}} = \frac{1}{2}|0\rangle\hspace{-0.07cm}\langle 1|\otimes\mathbb{I}. \end{align}

The non-Hermitian operators $$|10\rangle\hspace{-0.07cm}\langle 01|$$ and $$|0\rangle\hspace{-0.07cm}\langle 1|$$ drive the iSWAP and single-qubit interactions respectively, and $\mathbb{I}$ is the identity associated with the 2-dimensional Hilbert space of qubit 2. The control parameters $\Lambda(t)$ and $\xi(t)$ for the parametric drive relate to the frequency and phase of the flux drive $\Phi(t)$, and are analogous to the single qubit Rabi rate and drive phase. The maximum value of the parametric coupling rate is constrained by limitations imposed by the hardware system. As with the maximum Rabi rate for single-qubit controls, we denote this upper bound as $\Lambda_\text{max}$, and require

\begin{align} &\text{maximum parametric coupling rate:}&& \Lambda(t)\le\Lambda_\text{max}, \hspace{1cm} t\in[0,\tau]. \end{align}

That is, the instantaneous parametric coupling rate over the duration of the control, $t\in[0,\tau]$, cannot exceed this upper bound. In particular, we require

\begin{align} \Lambda_\text{peak}\le \Lambda_\text{max} \end{align}

where

\begin{align} &\text{peak parametric coupling rate:}&& \Lambda_\text{peak} = \text{max}_{t}\big[\Lambda(t)\big]_{0}^{\tau} \end{align}

defines the peak parametric coupling rate over the duration $t\in[0,\tau]$ of the control drive. $$\Lambda_\text{max}$$ therefore bounds the speed at which quantum operations may be driven due to the physical limitations of the control system. Control solutions which exceed these bounds cannot be physically implemented.

## Mølmer-Sørensen Rabi rate

The Mølmer-Sørensen control Hamiltonian cast in the standard form used at Q-CTRL takes the form

\begin{aligned} H_{c}(t) &= \gamma_\text{MS}(t) C_\text{MS} +\text{H.C.}, \hspace{2cm} \gamma_\text{MS}(t) = \Omega(t)e^{+i\phi(t)}, \hspace{2cm} C_\text{MS} = \sum_{\mu=1}^{N} \sum_{k=1}^{M} \eta_{k}^{(\mu)}e^{+i\delta_{k} t} \hat{S}_{\varphi}^{(\mu)} \otimes \hat{a}_{k} \end{aligned}

where $\delta_{k}$ is the detuning of the $k$th mode from the laser; $\eta_{k}^{(\mu)}$ is Lamb-Dicke parameter capturing the mode-laser coupling; and $\Omega(t)$ and $\phi(t)$ are the Rabi rate and drive phase for resonantly-driven qubit transitions. The maximum possible amplitude of the driving field in a real hardware system establishes a maximum possible value of the on-resonance Rabi rate. We denote this upper bound as $\Omega_\text{max}$, and require

\begin{align} &\text{maximum Rabi rate:}&& \Omega(t)\le\Omega_\text{max}, \hspace{1cm} t\in[0,\tau]. \end{align}

That is, the instantaneous Rabi rate over the duration of the control, $t\in[0,\tau]$, cannot exceed this upper bound. In particular, we require

\begin{align} \Omega_\text{peak}\le \Omega_\text{max} \end{align}

where

\begin{align} &\text{peak Rabi rate:}&& \Omega_\text{peak} = \text{max}_{t}\big[\Omega(t)\big]_{0}^{\tau} \end{align}

defines the peak Rabi rate over the duration $t\in[0,\tau]$ of the control drive. $$\Omega_\text{max}$$ therefore bounds the speed at which quantum operations may be driven due to the physical limitations of the control system. Control solutions which exceed these bounds cannot be physically implemented.