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 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.

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.