User:Quantal/sandbox/Entagling Gates in Trapped Ion Quantum Computers

Entangling gates in trapped ion quantum computers are quantum logic operations involving two or more qubits that are implemented in an ion trap quantum computer. In general these operations entangle the states of the involved qubits, inducing correlations between them that can not be reproduced classically. The operation of these gates relies on the coupling of the ions internal degree of freedom corresponding to the qubit to the collective motional degrees of freedom of the ions in the trap. In this sense the motion of the ions is an information bus between the qubits. An advantage of these trapped ion gates is their capability of interacting spatially distant qubits due to the range of the Coulomb interaction.

Preliminaries
We will describe the normal modes of trap via the Hamiltonian $$H_{osc} = \sum_n \hbar \omega_n(\hat{a}^\dagger_n \hat{a}_n + \frac{1}{2}) $$ while the qubit degrees of freedom of the ions are described by the Hamiltonian $$ H_{qubit} = \sum_i \frac{\hbar \omega_{eg}}{2}\hat{\sigma}_z^i $$.

When an ion is illuminated by a laser of frequency $$ \omega_L $$ a coupling can be induced between the internal state of the ion and the motional degrees of freedom. For simplicity here, considering just one ion denoted by $$ i $$ being illuminated, the interaction Hamiltonian is given by

$$ H_{int} =\sum_n \frac{\hbar \Omega_n}{2} (\hat{\sigma}_+^i + \hat{\sigma}_-^i)(e^{-i(k_L \hat{z}_n -\omega_L t + \phi)} +e^{+i(k_L \hat{z}_n -\omega_L t + \phi)}) $$

where $$ \hat{z}_n = z_n(\hat{a}_n + \hat{a}_n^\dagger) $$, $$ z_n = \sqrt{\frac{\hbar}{2 m \omega_n}} $$ and $$\Omega_n$$ is the nth Rabi frequency. It is useful to define the Lamb-Dicke parameter as $$ \eta_n = k_L z_n $$. All of the operations outlined in this article will assume the the Lamb-Dicke limit defined by $$ \eta_n << 1 $$. In this regime transitions between motional states that change the occupation number by more than one are strongly suppressed. Going into the interaction picture defined by $$ H_0 = H_{osc} + H_{qubit} $$ and discarding the fast oscillating terms $$ e^{i(\omega_{eg} + \omega_L)} $$ the interaction Hamiltonian in this frame is given by

$$ H_{int}' = U(H_0) H_{int} U^\dagger(H_0) = \sum_n \frac{\hbar \Omega_n}{2}[\hat{\sigma}_+^i\text{exp}(i(\eta_n(\hat{a}_n e^{-i\omega_n t} + \hat{a}_n^\dagger e^{+i\omega_n t}))-\delta t + \phi) + h.c. ] $$

where $$ \delta = (\omega_L - \omega_{eg}) $$. If $$ \delta $$ is sufficiently close to an integer multiple of one of the $$ \omega_n $$'s, the other modes can be ignored. This can be seen by invoking the Lamb-Dicke limit and expanding each $$ \text{exp}(i(\eta_n(\hat{a}_n e^{-i\omega_n t} + \hat{a}_n^\dagger e^{+i\omega_n t}))) $$ in a Taylor series, combining with $$ e^{-i\delta t} $$ and throwing away all but the terms with the slowest (or no) oscillations in the entire Hamiltonian.

Cirac-Zoller Cold Ion Gate
One of the first proposals for implementing quantum logic gates in ion traps was introduced in 1994 by J.I. Cirac and P. Zoller. The operation assumes that the ions have been cooled to their motional ground states (hence the "cold"). In this gate $$ \delta $$ is taken to be the negative of the center of mass mode frequency $$ \omega_0 $$ so that to first order in the Lamb-Dicke regime the interaction Hamiltonian is

$$ H_{int} = \eta_0 \frac{\hbar \Omega_0}{2}(\hat{\sigma}_+^i \hat{a}_0e^{-i\phi} +\hat{\sigma}_-^i \hat{a}^\dagger_0e^{+i\phi}) $$.

The operation requires the additional capability of coupling to an auxiliary excited state, which we will denote $$ | e_1 \rangle $$ as opposed to the excited state corresponding to logical $$ 1 $$ which we will denote $$ |e_0\rangle $$. Similarly define the operators $$ \sigma_\pm^{i,0} $$ and $$ \sigma_\pm^{i,1} $$ as the raising and lowering operators that couple the ground state to $$ |e_0\rangle $$ and $$ |e_1\rangle $$ respectively. The polarization of the addressing lasers can be modified to choose which of these states is coupled to. Define the $$ k\pi $$ pulse unitary:

$$ U_i^{k,q}(\phi) = \text{exp}(-ik\frac{\pi}{2}(\hat{\sigma}_+^{i,q}\hat{a}_0e^{-i\phi} + h.c.))$$

which is implemented by irradiating the ith ion with a laser with polarization such that it couples to the excited state $$q$$ for a time $$ t_k = \frac{k\pi}{\Omega_0 \eta_0} $$. This unitary leaves $$|g\rangle_i|0\rangle $$ unaffected while mapping:

$$\begin{align} \end{align}$$
 * g\rangle_i|1\rangle \rightarrow \cos(k\pi/2)|g\rangle_i|1\rangle -ie^{i\phi}\sin(k\pi/2)|e_q\rangle_i|0\rangle \\
 * e_q\rangle_i|0\rangle \rightarrow \cos(k\pi/2)|e_q\rangle_i|0\rangle -ie^{i\phi}\sin(k\pi/2)|g\rangle_i|1\rangle\\

where the second ket is the motional Fock state of the COM mode. If the center of mass mode is in its ground state the unitary operation to perform a CZ gate on ions $$i$$ and $$j$$ is then: $$U_i^{1,0}(0)U_j^{2,1}(0)U_i^{1,0}(0)$$.

This is illustrated below:

$$\begin{array}{ccccccc} & U_i^{1,0}(0) && U_j^{2,1}(0) && U_i^{1,0}(0)& \\
 * g\rangle_i|g\rangle_j|0\rangle & \rightarrow & |g\rangle_i|g\rangle_j|0\rangle & \rightarrow & |g\rangle_i|g\rangle_j|0\rangle& \rightarrow & |g\rangle_i|g\rangle_j|0\rangle\\
 * g\rangle_i|e_0\rangle_j|0\rangle & \rightarrow & |g\rangle_i|e_0\rangle_j|0\rangle & \rightarrow & |g\rangle_i|e_0\rangle_j|0\rangle & \rightarrow & |g\rangle_i|e_0\rangle_j|0\rangle\\
 * e_0\rangle_i|g\rangle_j|0\rangle & \rightarrow & -i|g\rangle_i|g\rangle_j|0\rangle & \rightarrow & i|g\rangle_i|g\rangle_j|0\rangle & \rightarrow & |e_0\rangle_i|g\rangle_j|0\rangle\\
 * e_0\rangle_i|e_0\rangle_j|0\rangle & \rightarrow & -i|g\rangle_i|e_0\rangle_j|0\rangle & \rightarrow & -i|g\rangle_i|e_0\rangle_j|0\rangle & \rightarrow & -|e_0\rangle_i|e_0\rangle_j|0\rangle

\end{array}$$

The key to the gate is that the $$U_j^{2,1}(0)$$ operation puts a minus sign on $$-i|g\rangle_i|g\rangle_j|0\rangle $$ while leaving the other states unaffected.

Original Molmer-Sorenson Hot Ion Gate
A down side of the scheme of Cirac and Zoller is that it assumes that the ions have been cooled to their motional ground states. In 1999 A. Sorenson and K. Molmer introduced a scheme for implementing a gate that has the same effect regardless of the Fock state of the mode that the gate couples to. Crucially this means that that the gate can be performed if the motional modes are in a thermal state (hence the "hot"). The absence of dependence on the motional state can be seen in a perturbative approach. Consider a trap where two ions are being simultaneously irradiated with lasers of frequency $$\omega_{eg} \pm \delta $$ where $$ \delta $$ is sufficiently close to the center of mass mode of the trap that we can ignore the other modes. Note that both ions are being irradiated by lasers of both frequencies i.e. two lasers shining on each ion. In this case the interaction Hamiltonian (in the rotating wave approximation) will be:

$$ H_{int} = \sum_{j,l} \frac{\hbar \Omega}{2}(\sigma_+^je^{i(\eta_l(\hat{a}_0 + \hat{a}^\dagger_0)-\omega_lt)} + h.c.) $$

where $$ \omega_1 = \omega_{eg} + \delta  $$ and $$ \omega_2 = \omega_{eg} - \delta   $$. Taking $$ \eta_1 = \eta_2 $$ and going into the interaction picture and throwing away fast oscillating terms we are left with the Hamiltonian:$$ H'_{int} = \frac{\hbar\Omega\eta}{2}(\hat{\sigma}_+^1\hat{a}_0^\dagger + \sigma_+^2\hat{a}_0^\dagger + \hat{\sigma}_+^1\hat{a}_0 + \sigma_+^2\hat{a}_0 + h.c.) $$

where we have designated the ions being addressed as $$1$$ and $$2$$. Effective Rabi frequencies can be calculated for the transitions $$ |gg\rangle|n\rangle \rightarrow |gg\rangle|n\rangle $$ and $$|eg\rangle|n\rangle \rightarrow |ge\rangle|n\rangle $$ via

$$(\frac \tilde{\Omega}{2})^2 = \frac{1}{\hbar^2} |\sum_m \frac{\langle ee|\langle n|H'_{int}|m\rangle \langle m|H'_{int}|gg\rangle|n\rangle}{E_{ggn} +\hbar\omega_i - E_m}|^2 = \frac{1}{\hbar^2} |\sum_m \frac{\langle ge|\langle n|H'_{int}|m\rangle \langle m|H'_{int}|eg\rangle|n\rangle}{E_{egn} + \hbar\omega_i -E_m}|^2$$

where $$ \omega_i $$ is the frequency of the laser addressing the ion which is excited to state $$ |m\rangle$$. Destructive interference between paths between possible intermediary states cancels out the $$ n $$ dependence. As an example of interfering paths take $$|gg\rangle|n\rangle \rightarrow |eg\rangle|n-1\rangle \rightarrow |ee\rangle|n\rangle $$ and $$|gg\rangle|n\rangle \rightarrow |ge\rangle|n+1\rangle \rightarrow |ee\rangle|n\rangle $$. The amplitudes for these paths are $$\frac{\eta^2 \Omega^2}{4(\delta-\omega_0)}(n+1) $$ and $$\frac{\eta^2 \Omega^2}{4(\omega_0-\delta)}(n)$$. The $$n$$ dependence exactly cancels. Adding up all of the terms yields $$ \tilde{\Omega} = -\frac{(\Omega\eta)^2}{(\omega_0-\delta)} $$ while the Rabi frequency for the $$|eg\rangle \rightarrow |ge\rangle $$ is $$ \frac{(\Omega\eta)^2}{(\omega_0-\delta)} $$. Thus the gate maps the basis vectors:

$$ \begin{align} \end{align} $$
 * gg\rangle \rightarrow \cos(\frac{\tilde{\Omega}t}{2})|gg\rangle + i\sin(\frac{\tilde{\Omega}t}{2})|ee\rangle \\
 * ee\rangle \rightarrow \cos(\frac{\tilde{\Omega}t}{2})|ee\rangle + i\sin(\frac{\tilde{\Omega}t}{2})|gg\rangle \\
 * eg\rangle \rightarrow \cos(\frac{\tilde{\Omega}t}{2})|eg\rangle - i\sin(\frac{\tilde{\Omega}t}{2})|ge\rangle \\
 * ge\rangle \rightarrow \cos(\frac{\tilde{\Omega}t}{2})|ge\rangle - i\sin(\frac{\tilde{\Omega}t}{2})|eg\rangle \\

Unfortunately this analysis is not complete as leftover entanglement between the internal states and the motional modes can remain at the end of the operation whicj can reduce the fidelity of the gate. Many quantum control strategies have been devised to counteract this and will be described in the next section

Decoupling Schemes
As noted in the previous section left over entanglement between motional bosonic modes and the internal qubit degrees of freedom can persist in the method of Molmer and Sorenson. In order to combat this many quantum control techniques that dynamically modulate the amplitude and phase of the addressing lasers have been developed.

General Theory
We will consider the interaction Hamiltonian between $$N$$ qubits and $$M$$ modes:

$$ H_{int} = i\hbar \sum_i \sigma_i^x \sum_k (\gamma_{i,k}(t) \hat{a}_k^\dagger +\gamma_{i,k}^*(t)\hat{a}_k)$$

where $$ \gamma_{i,k}(t)$$ is a time dependent coupling parameter that depends on the dynamics of the addressing laser pulses. The unitary time evolution under this Hamiltonian can be computed exactly using a Magnus expansion and noting that it will exactly terminate at the second order owing the Bosonic commutation relations. Thus the time evolution operator is given by:

$$ U(\tau) = \exp(i \sum_i\phi_i(\tau)\sigma_i^x + \sum_{i,j}\chi_{ij}(\tau)\sigma_i^x\sigma_j^x)$$

where

$$ \phi_i(\tau) = \sum_k \int_0^\tau dt\;\gamma_{i,k}(t)\hat{a}_k^\dagger + \int_0^\tau dt\;\gamma_{i,k}^*(t)\hat{a}_k$$

and

$$ \chi_{ij}(\tau) = \int_0^\tau dt_1 \int_0^{t_1}dt_2 \gamma_{ik}(t_1)\gamma_{jk}(t_2) $$.

The goal of the pulse shaping scheme is to ensure that $$ \int_0^\tau dt\;\gamma_{i,k}(t) =0\;\; \forall\;i,k $$ as this will ensure that by the completion of the operation all qubit to mode entanglement will go to zero. An additional constraint is to ensure that the gate performs the desired two qubit interaction which requires $$ \chi_{ij}(\tau) = \chi_{ij}^\text{target} $$.

Amplitude Modulation Scheme
In an amplitude modulation scheme the effective Rabi frequencies for each ion $$ \Omega_{i}(t) $$ will be functions of time. Thus $$\gamma_{ik}(t) = \eta_{ik}\Omega_i(t)\sin(\mu t)e^{-i\omega_k t} $$ where $$ \mu $$ is the detuning of the red and blue sidebands. The time dependent pulse is broken up into $$ S $$ equal time segments duration $$ \frac{\tau}{S}$$ where the amplitude within each segment is held constant. Denote the Rabi frequency of the $$i^{th} $$ ion during the $$s^{th} $$ segment as $$ \Omega_{i,s} $$ then:

$$ \sum_k \int_0^\tau dt\;\gamma_{i,k}(t) = \sum_{s=1}^S \Omega_{i,s} \int_{(s-1)\frac{\tau}{S}}^{s\frac{\tau}{S}}dt\sin(\mu t)e^{-i\omega_k t} = \sum_s \Omega_{i,s} C_{k,s}^i $$

and

$$ \chi_{ij}(\tau) = \sum_s \sum_{s'} \Omega_{i,s}\Omega_{j,s'}\int_{(s-1)\frac{\tau}{S}}^{s\frac{\tau}{S}}dt_1\int_{(s'-1)\frac{\tau}{S}}^{s'\frac{\tau}{S}}dt_2\sum_k \eta_{ik} \eta_{jk}\sin(\mu t_1)\sin(\mu t_2)\sin(\omega_k(t_2-t_1)) = \sum_s \sum_{s'} \Omega_{i,s}\Omega_{j,s'}D_{ss'} $$

Thus to entangle ions $$i$$ and $$j$$ if the constraints:

$$\sum_s \Omega_{i,s} C_{k,s}^i = 0$$,

$$\sum_s \Omega_{j,s} C_{k,s}^j = 0$$

and

$$\sum_s \sum_{s'} \Omega_{i,s}\Omega_{j,s'}D_{ss'} = \chi_{ij}^\text{target}$$

can be satisfied the gate can be performed with unit fidelity. Note that for just two ions a large enough $$S$$ can always be found such that the above equations are satisfiable. When attempting to entangle more than two ions at once the constraint satisfaction problem becomes more difficult and optimization techniques must be used to find an approximate solution.

Experimental Realizations
In 2019 C. Figgat et al. demonstrated parallel gates using the above decoupling scheme in a chain of five atomic $$ ^{171}Yb^+ $$ ions, using nonlinear optimization techniques to solve for the pulse shapes. The involved ions are irradiated by counter propagating Raman beams with a beat note at the desired detuning $$ \mu $$ from the motional sidebands. When solving for the pulse sequences the gate time was set at $$ \tau = 250\mu s $$ and laser power constraints were included in the optimization process. The process was repeated for a variety of detunings $$ \mu $$ and the solution with the highest fidelity and least power was chosen. The gate fidelities were found to be between $$ 96% -99% $$ with crosstalk errors of a few percent.