# Control library

In this entry we provide a detailed overview of the specific in-built solutions for various single and multi-qubit operations implemented in Q-CTRL products. Details of the underlying physical controls may be found in the Control Hamiltonian and Control Operations entries.

## Single-qubit driven

The available schemes for implementing single qubit controls are described. In the presentation below $\tau$ denotes the total drive duration, $\Omega_\text{max}$ denotes the maximum single-qubit Rabi and we define $\tau_{\pi} = \pi/\Omega_\text{max}$ as the minimum time ($\pi$-time) for implementing a (primitive) single-qubit $\pi$ rotation.

### Benchmark driven control

#### Primitive

The primitive square control implementing a net $R(\theta,0)$ driven rotation is denoted $\text{PRIMITIVE($\theta$)}$. The control-space array is tabulated

\begin{align} &\text{Cartesian controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \alpha_{x}(t) & \alpha_{y}(t) & \alpha_{z}(t) \\ \hline \theta/\Omega_\text{max} & \Omega_\text{max} & 0 & 0\\ \end{array} \right]\\ \\ &\text{cylindrical controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \Omega(t) & \phi(t) & \Delta(t) \\ \hline \theta/\Omega_\text{max} & \Omega_\text{max} & 0 & 0\\ \end{array} \right] \end{align}

### Control error compensating driven controls

#### BB1

The Wimperis broadband or BB1 sequence (see Wimperis, 1994) implementing a net $R(\theta,0)$ driven rotation is denoted $\text{BB1($\theta$)}$. The control-space array is tabulated

\begin{align} &\text{Cartesian controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \alpha_{x}(t) & \alpha_{y}(t) & \alpha_{z}(t) \\ \hline \theta/\Omega_\text{max}& \Omega_\text{max} & 0 & 0\\ \pi/\Omega_\text{max} & \Omega_\text{max}\cos\left( \phi_{*}\right) & \Omega_\text{max}\sin\left( \phi_{*}\right) &0\\ 2\pi/\Omega_\text{max} & \Omega_\text{max}\cos\left(3 \phi_{*}\right) & \Omega_\text{max}\sin\left(3 \phi_{*}\right) & 0\\ \pi/\Omega_\text{max} & \Omega_\text{max}\cos\left( \phi_{*}\right) & \Omega_\text{max}\sin\left( \phi_{*}\right) & 0\\ \end{array} \right]\\ \\ &\text{cylindrical controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \Omega(t) & \phi(t) & \Delta(t) \\ \hline \theta/\Omega_\text{max} & \Omega_\text{max} & 0 & 0\\ \pi/\Omega_\text{max} & \Omega_\text{max} & \phi_{*} &0\\ 2\pi/\Omega_\text{max} & \Omega_\text{max} & 3 \phi_{*} & 0\\ \pi/\Omega_\text{max} & \Omega_\text{max} & \phi_{*} &0\\ \end{array} \right] \end{align}

where

\begin{align} \phi_{*} = \cos^{-1}\left(-\frac{\theta}{4\pi}\right). \end{align}

#### SCROFULOUS

The sequence called SCROFULOUS (Short Composite ROtation For Undoing Length Over and Under Shoot) (see Cummings, 2003), implementing a net $R(\theta,0)$ driven rotation is denoted $\text{SCROFULOUS($\theta$)}$. The control-space array is tabulated

\begin{align} &\text{Cartesian controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \alpha_{x}(t) & \alpha_{y}(t) & \alpha_{z}(t) \\ \hline \theta_{1}/\Omega_\text{max} & \Omega_\text{max} \cos(\phi_1) & \Omega_\text{max} \sin(\phi_1) & 0\\ \theta_{2}/\Omega_\text{max} & \Omega_\text{max} \cos(\phi_2) & \Omega_\text{max} \sin(\phi_2) &0\\ \theta_{3}/\Omega_\text{max} & \Omega_\text{max} \cos(\phi_3) & \Omega_\text{max} \sin(\phi_3) & 0\\ \end{array} \right]\\ \\ &\text{cylindrical controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \Omega(t) & \phi(t) & \Delta(t) \\ \hline \theta_{1}/\Omega_\text{max} & \Omega_\text{max} & \phi_1 & 0\\ \theta_{2}/\Omega_\text{max} & \Omega_\text{max} & \phi_2 &0\\ \theta_{3}/\Omega_\text{max} & \Omega_\text{max} & \phi_3 & 0\\ \end{array} \right] \end{align}

where

\begin{aligned} \nonumber&\theta_1 = \theta_3 = \text{sinc}^{-1}\left[\frac{2\cos\left(\theta/2\right)}{\pi}\right]\\ \label{eqn:SCROFULOUS}&\theta_2 = \pi\\ \nonumber&\phi_1 = \phi_3 = \cos^{-1}\left[\frac{-\pi\cos\left(\theta_1\right)}{2\theta_1\sin\left(\theta/2\right)}\right]\\ \nonumber&\phi_2 = \phi_1 - \cos^{-1}\left(-\pi/2\theta_1\right) \end{aligned}

#### SK1

The first-order Solovay-Kitaev sequence (see Brown, 2004 and Brown, 2005) implementing a net $R(\theta,0)$ driven rotation is denoted $\text{SK1($\theta$)}$. The control-space array is tabulated

\begin{align} &\text{Cartesian controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \alpha_{x}(t) & \alpha_{y}(t) & \alpha_{z}(t) \\ \hline \theta/\Omega_\text{max} & \Omega_\text{max} & 0 & 0\\ 2\pi/\Omega_\text{max} & \Omega_\text{max}\cos\left(+ \phi_{*}\right) & \Omega_\text{max}\sin\left(+ \phi_{*}\right) &0\\ 2\pi/\Omega_\text{max} & \Omega_\text{max}\cos\left(- \phi_{*}\right) & \Omega_\text{max}\sin\left(- \phi_{*}\right) & 0\\ \end{array} \right]\\ \\ &\text{cylindrical controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \Omega(t) & \phi(t) & \Delta(t) \\ \hline \theta/\Omega_\text{max} & \Omega_\text{max} & 0 & 0\\ 2\pi/\Omega_\text{max} & \Omega_\text{max} & + \phi_{*} &0\\ 2\pi/\Omega_\text{max} & \Omega_\text{max} & - \phi_{*} & 0\\ \end{array} \right] \end{align}

where

\begin{align} \phi_{*} = \cos^{-1}\left(-\frac{\theta}{4\pi}\right). \end{align}

### Dephasing error compensating driven controls

#### CORPSE

The sequence called CORPSE (compensating for off-resonance with a pulse sequence) (see Brando, 2003), implementing a net $R(\theta,0)$ driven rotation is denoted $\text{CORPSE($\theta$)}$. the control-space array is tabulated

\begin{align} &\text{Cartesian controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \alpha_{x}(t) & \alpha_{y}(t) & \alpha_{z}(t) \\ \hline \theta_{1}/\Omega_\text{max} & \Omega_\text{max} & 0 & 0\\ \theta_{2}/\Omega_\text{max} & -\Omega_\text{max} & 0 &0\\ \theta_{3}/\Omega_\text{max} & \Omega_\text{max} & 0 & 0\\ \end{array} \right]\\ \\ &\text{cylindrical controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \Omega(t) & \phi(t) & \Delta(t) \\ \hline \theta_{1}/\Omega_\text{max} & \Omega_\text{max} & 0 & 0\\ \theta_{2}/\Omega_\text{max} & \Omega_\text{max} & \pi &0\\ \theta_{3}/\Omega_\text{max} & \Omega_\text{max} & 0 & 0\\ \end{array} \right] \end{align}

where

\begin{aligned} \nonumber&\theta_1 = 2\pi + \frac{\theta}{2} - \sin^{-1}\left[\frac{\sin\left(\theta/2\right)}{2}\right]\\ \label{eqn:CORPSE}&\theta_2 = 2\pi - 2\sin^{-1}\left[\frac{\sin\left(\theta/2\right)}{2}\right]\\ \nonumber&\theta_3 = \frac{\theta}{2} - \sin^{-1}\left[\frac{\sin\left(\theta/2\right)}{2}\right]\\ \end{aligned}

#### WAMF1

The sequence called WAMF1 (first-order Walsh amplitude-modulated filter) (see Ball, 2015), implementing a net $R(\theta,0)$ driven rotation is denoted $\text{WAMF1($\theta$)}$. The control-space array is tabulated

\begin{align} &\text{Cartesian controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \alpha_{x}(t) & \alpha_{y}(t) & \alpha_{z}(t) \\ \hline \tau/4 & \Omega_\text{max} & 0 & 0\\ \tau/4 & 2\theta/\tau - \Omega_\text{max} & 0 &0\\ \tau/4 & 2\theta/\tau - \Omega_\text{max} & 0 & 0\\ \tau/4 & \Omega_\text{max} & 0 & 0\\ \end{array} \right]\\ \\ &\text{cylindrical controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \Omega(t) & \phi(t) & \Delta(t) \\ \hline \tau/4 & \Omega_\text{max} & 0 & 0\\ \tau/4 & 2\theta/\tau - \Omega_\text{max} & 0 &0\\ \tau/4 & 2\theta/\tau - \Omega_\text{max} & 0 & 0\\ \tau/4 & \Omega_\text{max} & 0 & 0\\ \end{array} \right] \end{align}

This construction may be optimized by finding the minimum value of $\tau$ is found such to produce maximal suppression against dephasing noise.

### Dephasing and control error compensating driven controls

#### CORPSEinBB1

The sequence called CORPSEinBB1 (CORPSE concatenated within BB1) (see Kabytayev, 2014), implementing a net $R(\theta,0)$ driven rotation is denoted $\text{CORPSEinBB1($\theta$)}$. The control-space array is tabulated

\begin{align} \\ &\text{cylindrical controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} &\Omega(t) &\phi(t) & \Delta(t) \\ \hline (2\pi + \frac{\theta}{2}-k)/\Omega_\text{max} & \Omega_\text{max} & 0 & 0\\ (2\pi-2k)/\Omega_\text{max} & \Omega_\text{max} & \pi &0\\ (\frac{\theta}{2} - k)/\Omega_\text{max} & \Omega_\text{max} & 0 & 0\\ \pi/\Omega_\text{max} & \Omega_\text{max} & \phi_{*} &0\\ 2\pi/\Omega_\text{max} & \Omega_\text{max} & 3\phi_{*} & 0\\ \pi/\Omega_\text{max} & \Omega_\text{max} & \phi_{*} &0\\ \end{array} \right] \end{align}

where

\begin{align} k &= \sin^{-1}\left[\frac{\sin(\theta/2)}{2}\right]\\ \phi_{*} &= \cos^{-1}\left[-\frac{\theta}{4\pi}\right]. \end{align}

#### CORPSEinSCROFULOUS

The sequence called CORPSEinSCROFULOUS (CORPSE concatenated within a SCROFULOUS, i.e. each element of the SCROFULOUS is replaced by a CORPSE, see Brando, 2003 and Cummings, 2003), implementing a net $R(\theta,0)$ driven rotation is denoted $\text{CORPSEinSCROFULOUS($\theta$)}$. The control-space array is tabulated

\begin{align} &\text{Cartesian controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \alpha_{x}(t) & \alpha_{y}(t) & \alpha_{z}(t) \\ \hline \Gamma_1^{\theta_{1}}/\Omega_\text{max} & \Omega_\text{max} \cos(\phi_1) & \Omega_\text{max} \sin(\phi_1) & 0\\ \Gamma_2^{\theta_{1}}/\Omega_\text{max} & \Omega_\text{max} \cos(\phi_1) & - \Omega_\text{max} \sin(\phi_1) & 0\\ \Gamma_3^{\theta_{1}}/\Omega_\text{max} & \Omega_\text{max} \cos(\phi_1) & \Omega_\text{max} \sin(\phi_1) & 0\\ % ------------------------------------------------------------------------------ % ------------------------------------------------------------------------------ \Gamma_1^{\theta_{2}}/\Omega_\text{max} & \Omega_\text{max} \cos(\phi_2) & \Omega_\text{max} \sin(\phi_2) & 0\\ \Gamma_2^{\theta_{2}}/\Omega_\text{max} & \Omega_\text{max} \cos(\phi_2) & - \Omega_\text{max} \sin(\phi_2) & 0\\ \Gamma_3^{\theta_{2}}/\Omega_\text{max} & \Omega_\text{max} \cos(\phi_2) & \Omega_\text{max} \sin(\phi_2) & 0\\ % ------------------------------------------------------------------------------ % ------------------------------------------------------------------------------ \Gamma_1^{\theta_{3}}/\Omega_\text{max} & \Omega_\text{max} \cos(\phi_3) & \Omega_\text{max} \sin(\phi_3) & 0\\ \Gamma_2^{\theta_{3}}/\Omega_\text{max} & \Omega_\text{max} \cos(\phi_3) & - \Omega_\text{max} \sin(\phi_3) & 0\\ \Gamma_3^{\theta_{3}}/\Omega_\text{max} & \Omega_\text{max} \cos(\phi_3) & \Omega_\text{max} \sin(\phi_3) & 0\\ % ------------------------------------------------------------------------------ % ------------------------------------------------------------------------------ \end{array} \right]\\ \\ &\text{cylindrical controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \Omega(t) & \phi(t) & \Delta(t) \\ \hline \Gamma_1^{\theta_{1}}/\Omega_\text{max} & \Omega_\text{max} & \phi_1 & 0\\ \Gamma_2^{\theta_{1}}/\Omega_\text{max} & \Omega_\text{max} & \phi_1 + \pi& 0\\ \Gamma_3^{\theta_{1}}/\Omega_\text{max} & \Omega_\text{max} & \phi_1 & 0\\ % ------------------------------------------------------------------------------ % ------------------------------------------------------------------------------ \Gamma_1^{\theta_{2}}/\Omega_\text{max} & \Omega_\text{max} & \phi_2 & 0\\ \Gamma_2^{\theta_{2}}/\Omega_\text{max} & \Omega_\text{max} & \phi_2 + \pi& 0\\ \Gamma_3^{\theta_{2}}/\Omega_\text{max} & \Omega_\text{max} & \phi_2 & 0\\ % ------------------------------------------------------------------------------ % ------------------------------------------------------------------------------ \Gamma_1^{\theta_{3}}/\Omega_\text{max} & \Omega_\text{max} & \phi_3 & 0\\ \Gamma_2^{\theta_{3}}/\Omega_\text{max} & \Omega_\text{max} & \phi_3 + \pi& 0\\ \Gamma_3^{\theta_{3}}/\Omega_\text{max} & \Omega_\text{max} & \phi_3 & 0\\ \end{array} \right] \end{align}

where

\begin{aligned} \nonumber&\Gamma_1^{\theta'} = 2\pi + \frac{\theta'}{2} - \sin^{-1}\left[\frac{\sin\left(\theta'/2\right)}{2}\right]\\ \label{eqn:CORPSE_Angles}&\Gamma_2^{\theta'} = 2\pi - 2\sin^{-1}\left[\frac{\sin\left(\theta'/2\right)}{2}\right]\\ \nonumber&\Gamma_3^{\theta'} = \frac{\theta'}{2} - \sin^{-1}\left[\frac{\sin\left(\theta'/2\right)}{2}\right]\\ \end{aligned}

and $\theta’ \in {\theta_1, \theta_2, \theta_3}$ are the SCROFULOUS angles:

\begin{aligned} \nonumber&\theta_1 = \theta_3 = \text{sinc}^{-1}\left[\frac{2\cos\left(\theta/2\right)}{\pi}\right]\\ \label{eqn:SCROFULOUS_Angles}&\theta_2 = \pi\\ \nonumber&\phi_1 = \phi_3 = \cos^{-1}\left[\frac{-\pi\cos\left(\theta_1\right)}{2\theta_1\sin\left(\theta/2\right)}\right]\\ \nonumber&\phi_2 = \phi_1 - \cos^{-1}\left(-\pi/2\theta_1\right) \end{aligned}

#### CORSPEinSK1

The sequence called CORPSEinSK1 (CORPSE concatenated within SK1) (see Kabytayev, 2014), implementing a net $R(\theta,0)$ driven rotation is denoted $\text{CORPSEinSK1($\theta$)}$. The control-space array is tabulated

\begin{align} \\ &\text{cylindrical controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \Omega(t) & \phi(t) & \Delta(t) \\ \hline (2\pi + \frac{\theta}{2}-k)/\Omega_\text{max} & \Omega_\text{max} & 0 & 0\\ (2\pi-2k)/\Omega_\text{max} & \Omega_\text{max} & \pi &0\\ (\theta/2 - k)/\Omega_\text{max} & \Omega_\text{max} & 0 & 0\\ 2\pi/\Omega_\text{max} & \Omega_\text{max} & -\phi_{*} &0\\ 2\pi/\Omega_\text{max} & \Omega_\text{max} & +\phi_{*} & 0\\ \end{array} \right] \end{align}

where

\begin{align} k &= \sin^{-1}\left[\frac{\sin(\theta/2)}{2}\right]\\ \phi_{*} &= \cos^{-1}\left[-\frac{\theta}{4\pi}\right]. \end{align}

## Dynamical decoupling sequences

A Dynamic Decoupling Sequence (DDS) is canonically defined as a series of $n$-instantaneous unitary operations, often $\pi$-pulses, executed at time offsets

$\{t_j\}_{j=1}^{n}$

over the time interval with a total duration $\tau$. We define a set of error-robust common DDS below. In each case we use the notation $J_\theta$ to indicate an instantaneous unitary which is a rotation of $\theta$ about the $J = X, Y$ or $Z$ axis. This is related to DDS control format variables rabi rotation $\omega$, azimuthal angle $\phi$ and detuning rotation $\delta$ as follows:

$\begin{array}{c|c|c|c} \; & \omega & \phi & \delta \\ \hline X_\theta & \theta & 0 & 0 \\ Y_\theta & \theta & \pi/2 & 0 \\ Z_\theta & 0 & 0 & \theta \end{array}$

In practice all DDS typically have a $X_{\pi/2}$ operation at the start and finish. This is because it is assumed that the qubit is initially in the $|0\rangle$ state and a superposition needs to be created and removed to make the qubit sensitive to dephasing. For brevity, these operations are not included in any of the definitions below.

### Benchmark dynamical decoupling sequences

#### Ramsey

The Ramsey DDS is simply free evolution between the start and finish of the DDS. Technically, the Ramsey DDS does not dynamically decouple from the environment, nevertheless, it is a useful sequence for characterization and testing protocols and hence it is included. Ramsey DDS is parameterized by just the duration $\tau$ and contains no offsets in between the start and finish of the sequence.

### Dephasing error compensating dynamical decoupling sequences

#### Spin echo

The spin echo (SE) DDS is the simplest DDS. It is parameterized by duration $\tau$. There is a single $X_\pi$ unitary operation at $t_1=\frac{\tau}{2}$.

#### Carr-Purcell

The Carr-Purcell (CP) DDS is parameterized by the number of offsets $n$ and duration $\tau$. The DDS is made up of a set of $X_\pi$ operations applied at

$t_i = \frac{\tau}{n} \left( \frac{1}{2} + i-1 \right),$

where $i=1 \cdots n$.

#### Carr-Purcell-Meiboom-Gill

The Carr-Purcell-Meiboom-Gill (CPMG) DDS has the same timing and number of offsets as a CP DDS,

$t_i = \frac{\tau}{n} \left( \frac{1}{2} + i-1 \right),$

where $i=1 \cdots n$. However, due to a $\pi/2$ phase shift between the initial and final phases, the operations are $Y_\pi$.

#### Uhrig

The Uhrig DDS is parameterized by duration $\tau$ and number of offsets $n$. The Uhrig DDS is the application of $Y_\pi$ operations at offsets given by

$t_i = \tau \sin^2 \left( \frac{\pi i}{2(n+1)} \right),$

where $i=1 \cdots n$.

#### Periodic

The periodic DDS is parameterized by duration $\tau$ and number of offsets $n$. The periodic DDS is made of $X_\pi$ operations at offsets

$t_i = \frac{\tau}{n+1},$

where $i=1 \cdots n$.

#### Walsh

The Walsh DDS is parameterized by the duration $\tau$ and paley order $k$. The Walsh DDS is made of $X_\pi$ operations at offsets defined by a $k$th order Walsh Function given by

$PAL_k(x)=\prod_{j=1}^m R_j(x)^{b_j}, x\in [0, 1],$

The $j$th Rademacher function $R_j(.)$ is defined as

$R_j(x) := \rm{sgn}(\sin(2^j\pi x)),$

where $(b_m, b_{m-1}, \ldots, b_{0})$ is the binary representation of $k$ and $m$ is the index of most significant bit (hence $b_m=1$ by definition).

The Quadratic DDS is parameterized by duration $\tau$, number of inner offsets $n_1$ and number of outer offsets $n_2$. The outer sequence consists of $n_2$ pulses of type $X_\pi$, which partition the time-domain into $n_2+1$ sub-intervals on which inner sequences consisting of $n_1$ pulses of type $Z_\pi$ are nested. The total number of offsets $n$ is $n = n_1+n_2(n_1+1)$.

The pulse times for outer sequence ($X_\pi^1,\ldots, X_\pi^{n_2}$) are defined according to the Uhrig DDS on the time-domain $t\in{0,\tau}$. The $j$th $X_\pi$ pulse, $X_\pi^{j}$, therefore has timing offset defined by

$t_x^i = \tau \sin^2[\frac{\pi i}{2(n_2+1)}],$

where $j=1 \cdots n_2$. On each sub-interval defined by the outer sequence, an inner sequence ($Z_\pi^1,\ldots, Z_\pi^{n_1}$) is implemented. The pulse times for the inner sequences are also defined according to Uhrig DDS. The $k$th $Z_\pi$ pulse of the $j$th inner sequence has timing offset defined by

$t_z^{(k,j)} = (t_x^j-t_x^{j-1}) \sin^2[\frac{\pi k}{2(n_1+1)}] + t_{x}^{j-1},$

where $k=1 \cdots n_1$ and $j=1 \cdots n_2+1$.

#### X and XY concatenated

The $X$ or $XY$ concatenated DDS are constructed by recursively concatenating DDS structures. They are parameterized by the concatenation order $l$ and the duration $\tau$ of the total sequence. Let the $l$th order of concatenation be denoted as $C_l(\tau)$. In this scheme, zeroth order concatenation of duration $\tau$ is defined as free evolution over a period of $\tau$. We use a notation of $1 (\tau)$ to mean free evolution over duration $\tau$. We define the base DSS to be:

$C_{0}(\tau)=\mathbb{1}(\tau).$

The $l$th order X-CDD can be recursively defined as

$C_l(\tau)=C_{l-1}(\frac{\tau}{2})X_\pi C_{l-1}(\frac{\tau}{2})X_\pi.$

XY-CDD is similarly defined by

$C_l(\tau)=C_{l-1}(\frac{\tau}{4})X_\pi C_{l-1}(\frac{\tau}{4})Y_\pi C_{l-1}(\frac{\tau}{4})X_\pi C_{l-1}(\frac{\tau}{4})Y_\pi.$

## Two-qubit parametric drive (iSWAP subspace)

The available control solutions are described for implementing parametrically-driven two-qubit iSWAP gates. For notational simplicity the standard notation for control-coordinates is streamlined by replacing the subscripted symbols with the un-subscripted symbols:

\begin{align} &\text{parametric coupling rate:} \hspace{1cm} &&\Omega_\text{iSWAP}(t)&&\longrightarrow&&\Lambda(t)\\ &\text{parametric drive phase:} \hspace{1cm} &&\phi_\text{iSWAP}(t)&&\longrightarrow&&\xi(t)\\ &\text{qubit 1 Rabi rate:} \hspace{1cm} &&\Omega_\text{qubit1}(t)&&\longrightarrow&&\Omega(t)\\ &\text{qubit 1 drive phase:} \hspace{1cm} &&\phi_\text{qubit1}(t)&&\longrightarrow&&\phi(t) \end{align}

In the presentation below, $\Lambda_\text{max}$ and $\Omega_\text{max}$ denote the maximum parametric coupling rate and maximum single-qubit Rabi respectively, and we define $\tau_\text{iSWAP} = \pi/\Lambda_\text{max}$ and $\tau_\text{qubit} = \pi/\Omega_\text{max}$ as the minimum time ($\pi$-time) for implementing an iSWAP $\pi$ rotation and single-qubit $\pi$ rotation respectively. The control solutions presented below implement the net iSWAP operation

\begin{align} U_{\text{iSWAP}}(\pi,0) = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 0 & -i & 0 \\ 0 & -i & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}. \end{align}

### Benchmark control

#### Primitive

The primitive iSWAP drive is similar to the unmodulated single-qubit primitive control, but implementing the iSWAP operation, $U_{\text{iSWAP}}(\pi,0)$. The control-space array is tabulated

\begin{align} &\text{polar controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \Lambda(t) & \xi(t) & \Omega(t) & \phi(t) \\ \hline \tau_\text{iSWAP} & \Lambda_\text{max} & 0 & 0 & 0 \\ \end{array} \right] \end{align}

### Dephasing error compensating controls

#### CPMG in WAMF

The iSWAP control phasor is modulated with a WAMF envelope, the segments of which are interleaved with single-qubit $pi$ rotations, forming a CPMG (Carr-Purcell-Meiboom-Gill) timing pattern. The single-qubit controls in each segment (here implemented as primitive rotations) may be replaced with any predefined single-qubit control, as long as it implements a net $\pi$ rotation. The control-space array is tabulated

\begin{align} &\text{polar controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \Lambda(t) & \xi(t) & \Omega(t) & \phi(t) \\ \hline \tau_\text{iSWAP} & \Lambda_\text{max} & 0 & 0 & 0 \\ \tau_\text{qubit} & 0 & 0 & \Omega_\text{max} & 0 \\ \tau_\text{iSWAP} & \Lambda_\text{max} & 0 & 0 & 0 \\ \tau_\text{qubit} & 0 & 0 & \Omega_\text{max} & 0 \\ \tau_\text{iSWAP} & \Lambda_\text{max}/2 & 0 & 0 & 0 \\ \tau_\text{iSWAP} & \Lambda_\text{max}/2 & 0 & 0 & 0 \\ \tau_\text{qubit} & 0 & 0 & \Omega_\text{max} & 0 \\ \tau_\text{iSWAP} & \Lambda_\text{max} & 0 & 0 & 0 \\ \tau_\text{qubit} & 0 & 0 & \Omega_\text{max} & 0 \\ \tau_\text{iSWAP} & \Lambda_\text{max} & 0 & 0 & 0 \\ \end{array} \right] \end{align}

#### Interleaved

For the interleaved control solution iSWAP control segments are temporally interleaved with single-qubit control segments. The iSWAP and single-qubit phasors (where nonzero) are constrained to fixed-amplitude drives: $\Lambda(t) = \Lambda_\text{max}$ and $\Omega(t) = \Omega_\text{max}$. This control solution relies strictly on phase modulation in order to achieve error robustness. Segment durations are allowed to vary such that individual segments implement arbitrary unitary rotations on the relevant control subspace. The specific phase-modulation pattern has been determined using the control optimizer, for a specific ratio $\Omega_\text{max}/\Lambda_\text{max}$. The control-space array is tabulated

\begin{align} &\text{polar controls}&& && \begin{array}{c} \\ 1\\ 2\\ 3 \\ 4 \\ \vdots \\ m-1\\ m \\ \end{array} \left[ \begin{array}{c|c|c|c} \tau_{i} & \Lambda(t) & \xi(t) & \Omega(t) & \phi(t) \\ \hline \tau_{1} & 0 &0 & \Omega_\text{max} & \phi_{1} \\ \tau_{2} & \Lambda_\text{max} & \xi_{2} & 0 &0\\ \tau_{3} & 0 &0 & \Omega_\text{max} & \phi_{3} \\ \tau_{4} & \Lambda_\text{max} & \xi_{4} & 0 &0\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ \tau_{m-1} & 0 &0 & \Omega_\text{max} & \phi_{m-1} \\ \tau_{m} & \Lambda_\text{max} & \xi_{m} & 0 &0\\ \end{array} \right] \end{align}

#### Concurrent

For the concurrent control solution iSWAP and single-qubit phasors are allowed to be simultaneously active in every segment. The iSWAP and single-qubit phasors are constrained to fixed-amplitude drives: $\Lambda(t) = \Lambda_\text{max}$ and $\Omega(t) = \Omega_\text{max}$. This control solution relies on simultaneous phase modulation on both iSWAP and single-qubit drives in order to achieve error robustness. For an $m$-segment control solution, segment durations are fixed as equal divisions $\tau/m$ of the total drive duration $\tau$. The specific phase-modulation pattern has been determined using the control optimizer, for a specific ratio $\Omega_\text{max}/\Lambda_\text{max}$. The control-space array is tabulated

\begin{align} &\text{polar controls}&& &&\left[ \begin{array}{c|c|c|c} \tau_{i} & \Lambda(t) & \xi(t) & \Omega(t) & \phi(t) \\ \hline \tau/m & \Lambda_\text{max} & \xi_{1} & \Omega_\text{max} & \phi_{1} \\ \tau/m & \Lambda_\text{max} & \xi_{2} & \Omega_\text{max} & \phi_{2} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ \tau/m & \Lambda_\text{max} & \xi_{m} & \Omega_\text{max} & \phi_{m} \\ \end{array} \right] \end{align}

## Mølmer-Sørensen drive

The available schemes for implementing Mølmer-Sørensen controls are described. In the presentation below $\tau$ denotes the total drive duration, $\Omega_\text{max}$ denotes the maximum Rabi. The control solutions presented below implement the net Mølmer-Sørensen unitary

\begin{aligned} U_\text{MS} = \exp \Bigg( i\frac{\pi}{2}\sum_{\mu,\nu=1}^{N} \hat{S}^{(\mu)}_{x} \hat{S}^{(\nu)}_{x} \otimes \mathbb{I} \Bigg). \end{aligned}

### Benchmark control

#### Primitive

Assume a fixed Rabi rate $\Omega(t) = \Omega_\text{MS}\le\Omega_\text{max}$ and drive phase $\phi(t)=0$. The primitive implementation of $U_\text{MS}$ is tabulated

\begin{align} &\text{polar controls}&& &&\left[ \begin{array}{c|c|c} \tau_{i} & \Omega(t) & \phi(t) \\ \hline \tau_\text{MS} & \Omega_\text{MS} & 0 \\ \end{array} \right] \end{align}

where $\tau_\text{MS}$ is the drive duration such that

\begin{aligned} \left\lvert \varphi_{\mu\nu}(\tau_\text{MS}) \right\rvert = \frac{\pi}{2} \end{aligned}

where the entangling phase in this case is evaluated

\begin{aligned} \varphi_{\mu\nu}(\tau) = \Omega_\text{MS}^2 \text{Im}\left[ \sum_{k=1}^M \eta_{k}^{(\mu)} \eta_{k}^{(\nu)} \int_{0}^{\tau}dt_1 \int_{0}^{t_1}dt_2 e^{-i\delta_{k}(t_1-t_2)} \right] = \Omega_\text{MS}^2 \sum_{k=1}^M \eta_{k}^{(\mu)} \eta_{k}^{(\nu)} \frac{\sin(\delta_{k}\tau)-\delta_{k}\tau}{\delta_{k}^2}. \end{aligned}

### Control and detuning error compensating controls

#### Amplitude modulation

The amplitude modulation scheme (see Choi, 2014) is designed to decouple a set of motional modes in a Mølmer-Sørensen entangling gate by applying a time-dependent drive on the system. Similar to the phase-modulation scheme described above, this is achieved by ensuring that the phase-space trajectories of the targeted modes return to their starting point by the end of the gate. For a number $J$ of target modes, this leads to a set of $2J$ conditions for the closure of phase-space loops. This, together with the requirement of the correct entangling phase for the gate, defines the total $2J+1$ conditions for the correct gate implementation.

Consider a system with $M$ total oscillator modes and define a subset with $J<=M$ elements containing the modes to be decoupled. For the amplitude modulation solution, we split the total gate time ($\tau$) into $2J+1$ segments and assign to each segment $j$ a Rabi frequency $\Omega_j = \vert \Omega_j \vert e^{i \phi_j}$, where the phases $$\phi_j \in \{0,\pi \}$$ simply determine the sign of the drive amplitude. This allows us to write the total decoupling condition $$( \alpha_{k}^{(\mu)}(\tau)\equiv0 )$$ explicitly for each of the $J$ target modes by integrating the phase-space trajectories:

\begin{aligned} \alpha_{k}^{(\mu)}(\tau) = \Omega_j \eta_k^{(\mu)} e^{-i \delta_k \, j \, \tau_j } \left(e^{i \delta_k\tau_j} - 1\right) \equiv 0 \end{aligned}

By solving these equations, the control-space array is tabulated

\begin{align} &\text{polar controls}&& && \begin{array}{c} \\ 1\\ 2\\ \vdots \\ m\\ \vdots \\ 2 J+1 \\ \end{array} \left[ \begin{array}{c|c} \tau_{i} & \vert \Omega(t) \vert & \phi(t)\\ \hline \tau_{s} & \vert \Omega_\text{1} \vert & \phi_{1} \\ \tau_{s} & \vert \Omega_\text{2} \vert & \phi_{2} \\ \vdots & \vdots & \vdots \\ \tau_{s} & \vert \Omega_{m} \vert & \phi_{m} \\ \vdots & \vdots & \vdots \\ \tau_{s} & \vert \Omega_{2 J +1} \vert & \phi_{2 J +1} \\ \end{array} \right], && && \tau_{s} = \frac{\tau}{2 J + 1}, \end{align}

where the drive amplitude has been scaled such that the entangling phase satisfies

\begin{aligned} \vert \varphi_{\mu\nu}(\tau) \vert = \frac{\pi}{2}, && && \vert \Omega_\text{m} \vert \le\Omega_\text{max}, \hspace{0.5 cm} \forall m \in \{1, 2, \dots, 2J+1 \}. \end{aligned}

#### Custom control

In the customized amplitude modulation control, the condition on the number of segments being $2J+1$ is relaxed and the complexity of the control solution can be set by selecting the total number of segments, $N_s$, at will. In this case, an optimization problem is solved to determine the drive amplitude for each segment $j$:

\begin{align} &\text{polar controls}&& && \begin{array}{c} \\ 1\\ 2\\ \vdots \\ m\\ \vdots \\ N_s \\ \end{array} \left[ \begin{array}{c|c} \tau_{i} & \vert \Omega(t) \vert & \phi(t)\\ \hline \tau_{s} & \vert \Omega_\text{1} \vert & \phi_{1} \\ \tau_{s} & \vert \Omega_\text{2} \vert & \phi_{2} \\ \vdots & \vdots & \vdots \\ \tau_{s} & \vert \Omega_{m} \vert & \phi_{m} \\ \vdots & \vdots & \vdots \\ \tau_{s} & \vert \Omega_{N_s} \vert & \phi_{N_s} \\ \end{array} \right], && && \tau_{s} = \frac{\tau}{N_s}. \end{align}

#### Phase modulation

The phase-modulation scheme (see Green, 2015) is designed to suppress the dominant source of infidelity in implementing Mølmer-Sørensen entangling gates. This scheme uses a fixed-amplitude drive, with $\Omega(t) = \Omega_\text{MS}\le\Omega_\text{max}$, and relies exclusively on discrete shifts in the drive phase to ensure multiple oscillator modes are decoupled, and to suppress the effects of fluctuations in the driving field. The phase-modulation pattern is determined analytically to satisfy the decoupling condition for $J$ targeted modes, such that their phase-space trajectories close (return to the origin) at the end of the interaction. Consider a system with $M$ oscillator modes in total and relative detunings denoted $\delta_{k}$ for $$k\in\{1,...,M\}.$$ Define the mode-closure sequence

\begin{aligned} &\text{mode closure sequence:} && && \begin{pmatrix} \delta_{\rho_{1}}, & \delta_{\rho_{2}}, & \dots & \delta_{\rho_{J}} \end{pmatrix}, && && \rho_{j}\in\{1,...,M\}, && && \hspace{0.5cm} \forall j\in\{1,...,J\}. \end{aligned}

Namely, an ordered sequence selected from the set of $M$ modes, where the sequence length $J$ may be greater than, equal to, or less than $M$. This therefore includes any subset, permutation and combination the the $M$ modes, including repetitions, but such that every $\delta_{\rho_{j}}$ corresponds to one of the $M$ oscillator modes. To decouple the $J$ targeted modes, the drive duration is partitioned into $m=2^J$ segments of length $\tau_{s} = \tau/2^J$. The shift in the drive phase at the $\ell$th segment may be expressed

\begin{aligned} \phi_{\ell} &= -\sum_{j=0}^{q(\ell)} b_{j}(\ell) 2^{j} \delta_{\rho_{j+1}} \tau_{s} - h(\ell) \pi, && && \hspace{1cm} \ell\in\{1,...,m\} \end{aligned}

where $q(\ell)$ is the most significant binary digit in the binary representation of $\ell$, and $h(\ell)$ is the Hamming weight:

\begin{aligned} &\ell &&\underset{\text{dec.}}{=}&& \sum_{j=0}^{q(\ell)}b_{j}(\ell)2^{j} &&\underset{\text{bin.}}{=}&& b_{q}(\ell),b_{q-1}(\ell),...,b_{1}(\ell)b_{0}(\ell)\\ &h(\ell) &&=&& \sum_{j=0}^{q(\ell)}b_{j}(\ell) \end{aligned}

With these definitions, the control-space array is tabulated

\begin{align} &\text{polar controls}&& && \begin{array}{c} \\ 1\\ 2\\ \vdots \\ \ell\\ \vdots \\ m \\ \end{array} \left[ \begin{array}{c|c} \tau_{i} & \Omega(t) & \phi(t)\\ \hline \tau_{s} & \Omega_\text{MS} & \phi_{1} \\ \tau_{s} & \Omega_\text{MS} & \phi_{2} \\ \vdots & \vdots & \vdots \\ \tau_{s} & \Omega_\text{MS} & \phi_{\ell} \\ \vdots & \vdots & \vdots \\ \tau_{s} & \Omega_\text{MS} & \phi_{m} \\ \end{array} \right], && && \tau_{s} = \frac{\tau}{m}, && \hspace{1cm} && m = 2^{J}, \end{align}

where the drive amplitude has been scaled such that the entangling phase satisfies

\begin{aligned} \left \lvert \varphi_{\mu\nu}(\tau) \right\rvert = \frac{\pi}{2}, && && \Omega_\text{MS}\le\Omega_\text{max}. \end{aligned}