# Control Operations

Overview of the key physical operations implemented by the controls described in our products

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 $U_{\text{iSWAP}} = \exp[-iH_{c,\text{iSWAP}}\tau]$ 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 $(4\times 4)$ 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 ${\hat{S}}^{(\mu)}_{\varphi}\rightarrow{\hat{S}}^{(\mu)}_{x}$. 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, $\mathcal{H}_{1}(t) = \int_{0}^{t} H_{c}(t')dt'$ 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.

The ideal Mølmer-Sørensen gate implements the unitary operation \begin{aligned} U_\text{MS} = \exp \Bigg( \pm i \frac{\pi}{2} \sum_{\mu,\nu=1}^{N} \hat{S}^{(\mu)}_{x} \hat{S}^{(\nu)}_{x} \otimes \mathbb{I} \Bigg). \end{aligned} To achieve unitary evolution $U_{c}(\tau) = U_\text{MS}$ at the end ($t=\tau$) of the interaction, the ideal control drive must satisfy the following conditions \begin{aligned} &\text{target entangling phase:} && && \left\lvert \varphi_{\mu\nu}(\tau) \right\rvert &&=&& \frac{\pi}{2} \\ &\text{spin-mode decoupling:} && && U_\text{spin-mode}(\tau) &&=&& \mathbb{I} \end{aligned} implying the ideal decoupling condition $\alpha_{k}^{(\mu)}(\tau)\equiv0$.
Another way to assess the decoupling condition is to examine the probabilities of measuring the ions in the various possible states. For a Mølmer-Sørensen interaction involving 2 addressed ions, let $P_{n}(t)$ denote the probability of measuring $n$ ions in the $|1\rangle$ state at time $t\in\{0,\tau\}$, for $n\in\{0,1,2\}$. These populations are given by the following equations \begin{aligned} P_{0}(t) &= \frac{1}{8} \left[ 2+ \exp\left({-\sum_{k=1}^{M}\left| \alpha^{(1)}_k + \alpha^{(2)}_k \right|^2 (\bar n_k +1/2)}\right) +\exp\left({-\sum_{k=1}^{M}\left| \alpha^{(1)}_k - \alpha^{(2)}_k \right|^2 (\bar n_k +1/2)}\right) \right. \\ &\left. +4\cos\left(\varphi_{1,2}(t)\right) \exp\left({-\sum_{k=1}^{M}\left|\alpha_{k}^{(1)}(t)\right|^2(\bar{n}_{k}+1/2)}\right) \right]\\ \\ P_{1}(t) &= \frac{1}{4} \left[ 2- \exp\left({-\sum_{k=1}^{M}\left| \alpha^{(1)}_k + \alpha^{(2)}_k \right|^2 (\bar n_k +1/2)}\right) -\exp\left({-\sum_{k=1}^{M}\left| \alpha^{(1)}_k - \alpha^{(2)}_k \right|^2 (\bar n_k +1/2)}\right) \right]\\ \\ P_{2}(t) &= \frac{1}{8} \left[ 2+ \exp\left({-\sum_{k=1}^{M}\left| \alpha^{(1)}_k + \alpha^{(2)}_k \right|^2 (\bar n_k +1/2)}\right) +\exp\left({-\sum_{k=1}^{M}\left| \alpha^{(1)}_k - \alpha^{(2)}_k \right|^2 (\bar n_k +1/2)}\right) \right. \\ &\left. -4\cos\left[\varphi_{1,2}(t)\right] \exp\left({-\sum_{k=1}^{M}\left|\alpha_{k}^{(1)}(t)\right|^2(\bar{n}_{k}+1/2)}\right) \right] \end{aligned} where $\bar{n}_{k}$ is the average phonon occupancy of the $k$th mode, and $\varphi_{1,2}(t)$ is the entangling phase between spin 1 and 2. By inspection, we see the decoupling condition corresponds to $P_{1}(\tau) = 0$. These expressions are valid for the case of strictly 2 ions in the ion trap. However similar expressions may be obtained for the general case of $N$ ions. These expressions are built into the Q-CTRL package.