A driven operation on a generic quantum system refers to the dynamic transformation of the system under the control Hamiltonian $H_{c}(t)$ associated with an applied control drive. This corresponds to the unitary evolution operator, $U_{c}(\tau)$, governing the system evolution over the duration $t\in[0,\tau]$ of the interaction. Below we discuss this for a variety of quantum systems treated at Q-CTRL.

## Single-Qubit Driven

Driven operations on a single qubit may be visualized in terms of rotations on the Bloch sphere , parameterized by the time dependent control phasors and amplitudes. For simplicity consider the single-qubit control Hamiltonian where the drive phase and control amplitudes are constant in time. In this case the control Hamiltonian is written

where

where

so that $\Omega’ = \sqrt{\Omega^2+\Delta^2}$ is the *modified Rabi rate* expressed in terms of
the *on-resonance Rabi rate*, $\Omega$, and the drive detuning, $\Delta$. Solving the Schrodinger equation, the unitary evolution operator therefore takes the form

After an interaction of duration $\tau$, the rotation generator $\hat{\boldsymbol{n}}\boldsymbol{\sigma} \equiv n_x\sigma_x+n_y\sigma_y+n_z\sigma_z$ drives a rotation though an angle $\theta = \Omega’\tau$ about the axis defined by the unit vector $\hat{\boldsymbol{n}}\in\mathbb{R}^3$, reflecting the homeomorphism between SU(2) and SO(3). The result of this rotation is denoted

In the case of a resonant drive ($\Delta = \alpha_{z} = 0$) the modified Rabi rate reduces to the on-resonance Rabi rate, $\Omega’ = \Omega$, and the axis of rotation $\hat{\boldsymbol{n}}$ lies in the $xy$-plane (or *equatorial plane*) of the Bloch sphere, with orientation completely determined by the drive phase $\phi$. Namely,

and the target evolution operator reduces to

In this case the driven rotation is written

## Single-Qubit Dynamic Decoupling

Dynamic decoupling combines a sequence of single-qubit driven operations
$
R(\theta,\hat{\boldsymbol{n}})
=
\exp\left[-i\frac{\theta}{2}\hat{\boldsymbol{n}}\boldsymbol{\sigma}\right],
$
interleaved with identity operators, $\mathbb{I}$. This typically implement a *net* identity operator for the qubit. The timing and rotations of the individual operations depend on the specific sequence being deployed. A variety of well-known dynamic decoupling sequences are incorporated in the Q-CTRL control library.

## Two-Qubit Parametric Drive

Parametrically-driven two-qubit gates may be implemented between two capacitively-coupled transmon qubits consisting of one fixed- and one tunable-frequency transmon. A control flux drive $\Phi(t)$ is applied to the tunable-frequency transmon. This modulates the transition frequency $\omega_{T}(t)$ of the tuneable qubit and, via the capacitive coupling, generates a modulated effective two-qubit coupling. Target two-qubit gates are then driven by tuning the flux modulation resonantly with the desired transition, captured by the Hamiltonian

A detailed description of the underlying physical system and the derivation of the associated Hamiltonians can be found in Didier, 2017 , Caldwell, 2018 , and Reagor, 2018 . See here for a description of the physical parameters appearing in this Hamiltonian. Implementation of target two-qubit operations are discussed bellow for various subspaces of interest.

### iSWAP Subspace

After tuning the flux drive to resonantly activate the iSWAP interaction, the control Hamiltonian takes the form

where the control parameters $\Lambda(t)$ and $\xi(t)$ relate to the frequency and phase of the flux drive, calibrated for a particular system such that the effective control operators may be implemented, and $|10\rangle\hspace{-0.07cm}\langle 01|$ is the non-Hermitian component of the operator defining the iSWAP control axis. For constant coupling rate, $\Lambda$, and drive phase, $\xi$, the control Hamiltonian is constant in time. The primitive iSWAP operation is then implemented via the evolution operator at time $\tau$, taking the form

The primitive iSWAP gate with zero phase (analogous to the $\sigma_{x}$ axis) is then implemented by setting $\theta=\pi$ and $\xi=0$, yielding

where we have restricted attention to the relevant iSWAP subspace, spanned by the eigenstates

## Mølmer-Sørensen Drive

For a system with $N$ ions and $M$ motional modes the Mølmer-Sørensen control Hamiltonian ($\hbar=1$) takes the form

where the control phasor $\gamma_\text{MS}(t)$ and non-Hermitian operator $C_\text{MS}$ are defined

If the optical phases for the red and blue sidebands are tuned such that $\varphi_{r} = -\varphi_{b}$, the phase parameter $\varphi=0$ and the spin operator for the $\mu$th ion (embedded in the $N$-ion Hilbert space) takes the form . In this configuration the spin-mode interaction generates a state-dependent force resulting in an effective spin-spin coupling governed by the unitary evolution operator

This expression is obtained using a Magnus expansion on $H_{c}(t)$, where $\varphi_{\mu\nu}(t)$ is the entangling phase between the $\mu$th and $\nu$th spins, is the first order Magnus term, and all higher-order terms are identically zero due to commutation properties (Milne, 2017 ). These objects take the form

where we define

The *geometric phasor* $\kappa_{k}^{(\mu)}(t)$ captures the dynamic evolution in phase-space of the $\mu$th spin coupled to the $k$th mode under the applied drive, and the function $\alpha_{k}^{(\mu)}(t)$ maps the corresponding trajectory over the duration of the interaction. Evolution contributed by the first-order Magnus term $\exp
\left(
-i
\mathcal{H}_{1}(t)
\right)$ captures spin-mode coupling over the duration of the interaction, equivalent to the unitary operation

This resembles a displacement operator on the total spin-mode system, with $\alpha^{(\mu)}_{k}(t)$ therefore interpreted as the amount of displacement in phase space associated with the $\mu$th spin coupled to the $k$th mode.