Control operations

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

\[\begin{align} H_{c} = \frac{1}{2}\boldsymbol{\alpha}\boldsymbol{\sigma} = \frac{1}{2}\Omega'\hat{\boldsymbol{n}}\boldsymbol{\sigma}, \end{align}\]

where

\[\begin{align} \boldsymbol{\alpha} = \begin{bmatrix} \alpha_{x}, & \alpha_{y}, &\alpha_{z} \end{bmatrix}, \hspace{1.5cm} \boldsymbol{\sigma} = \begin{bmatrix} \sigma_{x} \\ \sigma_{y} \\ \sigma_{z} \end{bmatrix}, \hspace{1.5cm} \hat{\boldsymbol{n}} = \frac{\boldsymbol{\alpha}}{|\boldsymbol{\alpha}|}, \hspace{1.5cm} \Omega' = {|{\boldsymbol{\alpha}|}}=\sqrt{\alpha_{x}^2+\alpha_{y}^2+\alpha_{z}^2}, \end{align}\]

where

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

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 Schrödinger equation, the unitary evolution operator therefore takes the form

\[\begin{align} U_{c}(t; \Omega', \hat{\boldsymbol{n}})\equiv \exp\left[-i\frac{\Omega't}{2}\hat{\boldsymbol{n}}\boldsymbol{\sigma}\right]. \end{align}\]

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

\[\begin{align} R(\theta,\hat{\boldsymbol{n}}) \equiv U(\tau; \Omega', \hat{\boldsymbol{n}}) = \exp\left[-i\frac{\theta}{2}\hat{\boldsymbol{n}}\boldsymbol{\sigma}\right], \hspace{1cm} \theta \equiv \Omega'\tau. \end{align}\]

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,

\[\begin{align} \hat{\boldsymbol{n}}(\phi) = \frac{\boldsymbol{\alpha}}{|{\boldsymbol{\alpha}|}} = \begin{bmatrix} \cos\phi, & \sin\phi, & 0 \end{bmatrix}, \end{align}\]

and the target evolution operator reduces to

\[\begin{align} U(t; \Omega, \hat{\boldsymbol{n}}(\phi))\equiv \exp\left[-i\frac{\Omega t}{2}\hat{\boldsymbol{n}}(\phi)\boldsymbol{\sigma}\right], \hspace{1cm} \hat{\boldsymbol{n}}(\phi)\boldsymbol{\sigma} = \cos\phi\sigma_{x}+\sin\phi\sigma_{y}. \end{align}\]

In this case the driven rotation is written

\[\begin{align} R(\theta,\phi) \equiv \exp\left[-i\frac{\theta}{2}\hat{\boldsymbol{n}}(\phi)\boldsymbol{\sigma}\right], \hspace{1cm} \theta \equiv \Omega\tau. \end{align}\]

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

\[\begin{equation} \begin{aligned} H_{int}(t) &=g(t) \sum_{n=-\infty}^{\infty}J_{n}\left(\frac{\tilde{\omega}_{T}}{2\omega_{p}}\right)e^{+i(2\omega_{p}t+2\theta_{p})n}\\ &\times\Big\{ e^{-it\Delta}\left|10\rangle\hspace{-0.07cm}\langle 01\right| && \hspace{1cm} && \text{(iSWAP)}\\ &+\sqrt{2}e^{-i(\Delta +|\eta_{F}|)t}\left|20\rangle\hspace{-0.07cm}\langle 11\right| && \hspace{1cm} && (\text{CZ}_{20})\\ &+\sqrt{2}e^{-i(\Delta -|\eta_{T}|)t}\left|11\rangle\hspace{-0.07cm}\langle 02\right| && \hspace{1cm} && (\text{CZ}_{02})\\ &+2e^{-i(\Delta +|\eta_{F}|-|\eta_{T}|)t}\left|21\rangle\hspace{-0.07cm}\langle 12\right|+\text{H.C.}\Big\}. \end{aligned} \end{equation}\]

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 two-qubit parametric control Hamiltonian for a description of the physical parameters appearing in this Hamiltonian. Implementation of target two-qubit operations are discussed below for various subspaces of interest.

iSWAP subspace

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

\[\begin{align} H_{c,\text{iSWAP}}(t) = \frac{1}{2} \Lambda(t) e^{+i \xi(t)} |10\rangle\hspace{-0.07cm}\langle 01| + \text{H.C.} \end{align}\]

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

\[\begin{align} U_{\text{iSWAP}}(\theta,\xi) &= \exp \left[-i \frac{\theta}{2} \left( e^{+i \xi} |10\rangle\hspace{-0.07cm}\langle 01| + e^{-i \xi} |01\rangle\hspace{-0.07cm}\langle 10| \right) \right], \hspace{1cm} \theta = \Lambda\tau. \end{align}\]

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

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

where we have restricted attention to the relevant \((4\times 4)\) iSWAP subspace, spanned by the eigenstates

\[\begin{align} \left|00\right\rangle, \hspace{0.25cm} \left|10\right\rangle, \hspace{0.25cm} \left|01\right\rangle, \hspace{0.25cm} \left|11\right\rangle. \end{align}\]

Mølmer-Sørensen drive

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

\[\begin{aligned} H_{c}(t) &= \gamma_\text{MS}(t) C_\text{MS} +\text{H.C.} \end{aligned}\]

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

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

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

\[\begin{aligned} U_{c}(t) = \exp \Bigg( i\sum_{\mu,\nu=1}^{N}\varphi_{\mu \nu}(t) \hat{S}^{(\mu)}_{x} \hat{S}^{(\nu)}_{x} \otimes \mathbb{I} \Bigg) \exp \Bigg( -i \mathcal{H}_{1}(t) \Bigg). \end{aligned}\]

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

\[\begin{aligned} &\text{entangling phase:} && && \varphi_{\mu\nu}(t) &&=&& \text{Im}\left[ \sum_{k=1}^M\int_{0}^{t}dt_1 \int_{0}^{t_1}dt_2 \kappa_{k}^{(\mu)}(t_1) \left(\kappa_{k}^{(\nu)}(t_2)\right)^{*} \right]\\ &\text{first-order Magnus term:} && && \mathcal{H}_{1}(t) &&=&& i\sum_{\mu=1}^{N} \hat{S}^{(\mu)}_{x} \otimes \sum_{k=1}^{M} \left( \alpha^{(\mu)}_{k}(t) \hat{a}^\dagger_{k} - \left( \alpha^{(\mu)}_{k}(t)\right)^{*} \hat{a}_{k} \right) \end{aligned}\]

where we define

\[\begin{aligned} &\text{geometric phasor:} && && \kappa_{k}^{(\mu)}(t) &&=&& -i\Omega(t)e^{-i\phi(t)}\eta_{k}^{(\mu)}e^{-i\delta_{k} t}, \\ &\text{phase-space trajectory:} && && \alpha_{k}^{(\mu)}(t) &&=&& \int_{0}^{t}dt'\kappa_{k}^{(\mu)}(t'). \end{aligned}\]

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

\[\begin{aligned} U_\text{spin-mode}(t) \equiv \exp \Bigg( \sum_{\mu=1}^{N} \hat{S}^{(\mu)}_{x} \otimes \sum_{k=1}^{M} \left( \alpha^{(\mu)}_{k}(t) \hat{a}^\dagger_{k} - \left( \alpha^{(\mu)}_{k}(t)\right)^{*} \hat{a}_{k} \right) \Bigg) \end{aligned}.\]

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.

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