Qudit entanglers using quantum optimal control

In the Lie algebraic approach to quantum control, we consider a Hamiltonian of the form $H[\mathbf{c}(t)]=H_{0}+\sum_{j=1}^{k}c_{j}(t)H_{j}$, where $\mathbf{c}(t)=\{c_{j}(t)\}$ is the set of time-dependent classical control waveforms, and $H_{0}$ is called the drift Hamiltonian. The system is said to be “controllable” if the set of Hamiltonians, $\{H_{0},H_{1},H_{2},\ldots,H_{k}\}$, are generators of the desired Lie algebra, e.g., $\mathfrak{su}{(d)}$. Then $\exists\hskip 2.84544pt\mathbf{c}(t)$    such that $U[\mathbf{c},T]=\mathcal{T}\left[\exp\left(-i\int_{0}^{T}H[\mathbf{c}(t)]dt\right)\right]=U_{\mathrm{tar}}$ for any target unitary in desired Lie Group, e.g., $U_{\mathrm{tar}}\in\mathrm{SU}(d)$. In addition, we require $T\geq T_{*}$, where $T_{*}$ is known as the “quantum speed limit time,” which sets the minimal time needed for the system to be fully controllable.

We consider here open-loop control determined by a well-defined Hamiltonian of the general form,

where $H^{(i)}(t)$ are time-dependent Hamiltonians acting on the individual subsystems, and $H_{\mathrm{ent}}$ is the interaction that entangles them. Here we include the time dependence in the Hamiltonian that acts on the individual system as these will be generally easier to implement experimentally. In this formulation, $H_{\mathrm{ent}}=H_{0}$, is the drift Hamiltonian. However, one could in principle include time dependence in the entangling Hamiltonian as well and this may achieve faster gates.

In the digital, Lie group approach to quantum control, we consider a family of unitary maps in the desired group that are easily implementable, $U(\lambda_{j})$, where $\{\lambda_{j}\}$ are the parameters that specify the unitary matrices at our disposal. The relevant Lie group of interest here is $\mathrm{SU}(d^{2})$, the group of two-qudit unitary matrices in $d^{2}$ dimensions, where the overall phase is removed. The system is controllable if $\forall U_{\mathrm{tar}}\in\mathrm{SU}(d^{2})$, $\exists\{\lambda_{i}\}$ such that $\prod_{j=1}^{k}U(\lambda_{j})=U_{\mathrm{tar}}$. Similar to the Lie algebraic quantum control approach, the goal is to find $\{\lambda_{j}\}$ through numeric optimization, e.g., via gradient-based methods.

For the case of two-qudit gates, a controllable Lie group structure is given as,

where $U_{1,2}\in\mathrm{SU}(d)$ and $U_{\mathrm{ent}}=\exp(-iH_{\mathrm{ent}}t)\notin\mathrm{SU}(d)\otimes\mathrm{SU}(d)$. Thus, we can achieve the target gate to the desired fidelity by intertwining a sequence of local $\mathrm{SU}(d)$ gates and the available entangling interaction in alternating layers of single qudit gates and entangling gates, as shown in Fig. 1(b). This approach is similar to the construction based on Givens rotation Ringbauer et al. (2022). Here, the possibility of accessing arbitrary local $\mathrm{SU}(d)$ gates makes this protocol very powerful. A schematic comparison of both these approaches is shown in Fig. 1.

To make these ideas concrete, we consider the implementation of entangling gates in neutral atoms using the strong van der Waals interactions between atoms in high-lying Rydberg states. We use the Rydberg dressing paradigm in which one adiabatically superposes the Rydberg state into the ground states to introduce interactions between dressed ground states  Johnson and Rolston (2010); Keating et al. (2015); Jau et al. (2016); Zeiher et al. (2016, 2017); Borish et al. (2020). Rydberg dressing has been studied with multiple applications including the dynamics of interacting spin models  Zeiher et al. (2016, 2017); Borish et al. (2020) as well as to prepare metrologically-useful states  Kaubruegger et al. (2019). Entanglement between neutral atoms via Rydberg dressing has been theoretically proposed for creating qubit entangling gates  Keating et al. (2015); Mitra et al. (2020, 2022) and experimentally implemented  Jau et al. (2016); Martin et al. (2021); Schine et al. (2022).

We study here encoding a qudit in the spin of ⁸⁷Sr. In the ground state there is neither orbital nor spin angular momentum in the electrons, $J=0$, and only nuclear spin, $I=9/2$, giving us ten possible levels in which to encode our qudit, labeled from $\ket{0}=\ket{m_{I}=9/2},\ket{1}=\ket{m_{I}=7/2},\cdots,\ket{9}=\ket{m_{I}=-9/2}$. The nuclear spin is highly isolated from the environment and thus serves as a robust memory for quantum information. In Omanakuttan et al. (2021) we studied how we could implement single qudit gates in these systems through a combination of a laser-induced tensor light-shift and radio frequency (rf)-induced Larmor precession. We generalize to the two-qudit case here.

To implement entangling two-qudit control, we will make use of the excitation to the $5sns\hskip 2.84544pt^{3}S_{1}$ Rydberg series from the metastable $5s5p\hskip 2.84544pt^{3}P_{J}$ first excited states. The choice of energy levels depends on practical considerations. By choosing the metastable state $5s5p\hskip 2.84544pt^{3}P_{2}$, the total electron angular momentum gives rise to a large magnetic dipole moment, which substantially increases the speed of gates, but this comes at the cost of increased sensitivity to background magnetic fields and residual tensor light shifts from the trapping lasers. By choosing the metastable ${}^{3}{P}_{0}$ clock state, one retains the robustness of having $J=0$ at the cost of much slower gates, since the magnetic coupling is now solely to the nuclear spin. We consider here coherently transferring qudits from the ${}^{1}{S}_{0}$ ground state to the $F=9/2$ state hyperfine states of the ${}^{3}{P}_{2}$ manifold, which provides for faster and more flexible control Trautmann et al. (2022), putting technical noise aside.

To achieve the entangling interaction, we consider Rydberg dressing, generalizing the mechanism discussed in Jau et al. (2016); Keating et al. (2015); Mitra et al. (2020). The AC Stark shift (light shift) associated with a dressed state when a laser is tuned near a Rydberg resonance is modified for two atoms because of the Rydberg blockade. The deficit between the two-atom light shift and twice the one-atom light shift determines the entangling energy Keating et al. (2015). For the case of qudits, the same physics holds, but now with a multilevel structure and a spectrum of entangling energies. When the spectrum is nonlinear, the system is controllable.

Fig. 2 depicts the basic scheme. Those levels of the qudit that we chose to participate in the gate are excited from the ground ${{}^{1}S_{0}}$ to the first excited ${{}^{3}P_{2}}$ state. The Rydberg states in ⁸⁷Sr have well-resolved hyperfine splitting. We consider UV dressing laser near the resonance between the ${{}^{3}P_{2}}$, $F=9/2$ hyperfine manifold and the ${{}^{3}S_{1}}$, $F^{\prime}=11/2$ Rydberg hyperfine states. In the presence of a bias magnetic field, due to the difference in the g-factors, the two manifolds will be differently Zeeman shifted. The different magnetic sublevels that define the qudit will thus be differently detuned to the Rydberg magnetic sublevels. Due to this and the Clebsch-Gordan coefficients associated with the different transitions, each sublevel will be differently dressed (equivalently, there is a tensor light shift). When two atoms are dressed, the effect of the Rydberg blockade modifies the spectrum as discussed above.

An example of two sublevels (one from each atom) is shown in Fig. 2(b). Diagonializing this atom-laser Hamiltonian under the approximation of a perfect Rydberg blockade yields the representation

where the tilde indicates dressed states,

and $E^{ij}$ are the light shifts originating from these interactions. The spectrum of the entangling Hamiltonian shown in Fig. 2(c) gives us insight into the controllability of the system. In the chosen order, the spectrum reveals the structure of $10$ quadratic potentials arising from a combination of the tensor light shift and Rydberg blockade. This nonlinearity makes the Hamiltonian controllable; further details are discussed in Appendix (B).

The time-dependent Hamiltonian necessary for the Lie algebraic control can be chosen as phase-modulated Larmor precession, $H_{\mathrm{mag}}=-\bm{\mu}\cdot\mathbf{B}(t)$, with $\bm{\mu}=g_{F}\mu_{B}\mathbf{F}$ the magnetic dipole vector operator, and where $\mathbf{B}(t)=B_{\parallel}\mathbf{e}_{z}+B_{T}\real\left[(\mathbf{e}_{x}+i\mathbf{e}_{y})\mathbf{e}^{-i\left(\omega_{\text{rf}}t+\phi(t)\right)}\right]$. Defining the auxilary subspace, $a$, for the levels in hyperfine manifold $\{5s5p\hskip 2.84544pt^{3}P_{2},\hskip 2.84544ptF=9/2\}$ and the subspace, $r$, for the levels $\{5sns\hskip 2.84544pt^{3}S_{1},\hskip 2.84544ptF^{\prime}=11/2\}$ in the Rydberg hyperfine manifold, we have $g_{F}(r)/g_{F}(a)\approx 2$. Thus defining the Zeeman shift $\omega_{0}=g_{F}(a)B_{\parallel}$, the Larmor precession frequency $\Omega_{\mathrm{rf}}=g_{F}(a)B_{T}$, and choosing rf drive on resonance in the $a$-manifold, $\omega_{\mathrm{rf}}=\omega_{0}$, in the co-rotating frame at $\omega_{0}$, the Hamiltonian is

where $F_{i}^{a},F_{i}^{r}$ are the spin angular momentum operators in the respective subspaces along axis $i\in\{x,y,z\}$.

As the $H_{\mathrm{mag}}$ acts on the laser-dressed states defined in Eq. (6), which are superpositions of $a$ and $r$ states that have different $g$-factors, one needs to find the action of the magnetic interaction in the dressed basis. Due to the nonlinearity, the action of the rf-magnetic driving on the dressed states is no longer simple Larmor precession. Considering a global rf-magnetic interaction, the $H_{\mathrm{mag}}$ acts on both qudits as

Thus in the dressed basis, the Hamiltonian is $H(t)=\widetilde{H}\left[\phi(t)\right]+H_{\mathrm{ent}}$, where the action of the magnetic field in the dressed basis is given by the Hamiltonian,

By modulating the phase $\phi(t)$ one can generate any target unitary gate.

As discussed in Sec. IIc, one can implement an arbitrary entangling gate through a combination of Rydberg dressing and phase-modulated Larmor precession driven by rf-fields. Because our control Hamiltonian is symmetric with respect to the exchange of the qudits, we consider here symmetric gates, with global control. We seek, through numerical optimization, the time-dependent rf-phase, $\phi(t)$. To achieve this we employ the well-known GRAPE algorithm Khaneja et al. (2005). To implement GRAPE, we discretize the control waveform, $\phi(t)$, and numerically maximize the fidelity by gradient ascent. We choose here a piecewise constant parameterization (as in  Anderson (2013)) and write the control waveform as a vector $\mathbf{c}=\{\phi(t_{j})/\pi\;|\;j=1,\dots,n\}$ where $t=j\Delta t$ and $n=T/\Delta t$. The waveform is thus a series of square rf-pulses with constant amplitude and phase over the duration $\Delta t$.

The minimum number of elements in the control vector $\mathbf{c}$ is determined by the number of parameters needed to specify the target isometry. A $K$-dimensional partial isometry is defined by the $K$ columns in a $D\times D$-dimensional unitary matrix. Hence, to find the number of free parameters for a $K$-dimensional isometry one can count the number of parameters needed to specify $K$ orthonormal vectors uniquely in a $D$-dimensional vector space. This is given by

where in the first line, we subtracted one from the parameter count in since the overall phase of the isometry is neglected. Eq. (13) recovers well-known limits. When $K=1$ and $D=d$, $n_{\mathrm{min}}=2d-2$, which is the number of free parameters needed to specify a pure state in a $d$-dimensional Hilbert space. When $K=D=d$, $n_{\mathrm{min}}=d^{2}-1$, which is the number of free parameters needed to specify a special unitary map in $d$-dimensions.

In the Lie algebraic protocol for designing entangling gates, the control Hamiltonian, as well as the target unitary matrices, are symmetric under the exchange of qudits. In this case, one can work in the symmetric subspace for two qudits. Using the hook length formula Frame et al. (1954), the dimension of the symmetric subspace of the total vector space and isometry is,

Thus, using Eq. (13), we find the number of free parameters required for the two-qubit entangling unitary given in Table 1.

Proof-of-principle numerical examples of waveforms that generate the CPhase gate are given in Fig. 3. The figure gives the $\phi(t)$ as a piecewise constant function of time, obtained using the GRAPE algorithm. We consider prime-dimensional qudits, the cases of most interest in quantum algorithms. Fig. 3(a) shows the case of the $k=3$, a qutrit encoded in $d=10$. The total time is $T=50\pi/\Omega_{\mathrm{rf}}$, which is divided into $700$ intervals for the quantum control. Fig. 3(b) shows an example waveform for the case of $k=5$. Here, the total time is $T=240\pi/\Omega_{\mathrm{rf}}$, divided into $1600$ intervals. Similarly, Fig. 3(c) shows the case of $k=7$ in our $d=10$ level system. The total time is $T=400\pi/\Omega_{\mathrm{rf}}$, divided into 2500 time intervals. This controllable Hamiltonian can also be used to generate other two-qudit gates. The qudit generalization of the M$\o$lmer-S$\o$rensen gate, as is given in the Appendix C.

The waveforms found here are a proof-of-principle set of square pulses and are not intended to be taken as the best choice for experimental implementation. In practice, one can design and optimize for much smoother waveforms using well-known techniques by imposing additional constraints on bandwidth and slew rate. Alternatively, one can optimize in the Fourier domain or in any other complete basis of functions using the techniques of gradient optimization of analytic controls (GOAT) Machnes et al. (2018).

In the Lie group control protocol discussed in Sec.IIc we parameterize the target unitary map as

The control parameters $\{\lambda_{i}\}$ consist of the set of times $\{t_{i}\}$ and the $2(d^{2}-1)$ parameters $\vec{\alpha}^{(j)}$, $\vec{\beta}^{(j)}$, which specify each of the local $\mathrm{SU}(d)$ unitary maps. We can parameterize these according to

where $\Lambda$ is the generalized Gell-Mann matrices that span the Lie algebra $\mathfrak{su}(d)$. The matrices can be categorized as,

The task of the numerical optimization, thus, is to find the set of times of the entangling interaction $\{t_{j}\}$, and the expansion coefficients of the Gell-Mann matrices $\{\alpha_{i}^{(j)}\}$ and $\{\beta_{i}^{(j)}\}$. We denote this whole set of parameter as $\{\lambda_{j}\}=\{t_{j},\vec{\alpha}^{(j)},\vec{\beta}^{(j)}\}$.

We define one layer of the control as consisting of a pair of local SU$(d)$ gates followed by the entangling Hamiltonian for a time $t_{j}$. The total number of free parameters for a CPhase gate is $d^{2}(d^{2}+1)/2$, as follows from Eq. (14) for a symmetric gate in $\mathrm{SU}(d^{2})$. Thus, the minimum number of layers required to obtain the CPhase gate is given by

The numerical results for the minimum number of layers needed in the system are given in Table 2 for the cases of $d=3,5,$ and $d=7$. In practice, we find that one needs more than this minimum number of layers to implement the target unitary gate with high fidelity. This improves the optimization landscape for gradientascent Larocca et al. (2018).

For our case under study, we choose the same entangling Hamiltonian as we used in the Lie algebraic approach given in Eq. (5). However, unlike that approach, we interleave the entangling interaction with local single-qudit SU($d$) gates. Implementation of this requires another layer of optimization. As we do not have access to native Hamiltonians proportional to the Gell-Mann matrices, to implement local qudit gates we can employ local SU($d$) optimal control Omanakuttan et al. (2021). From a practical perspective, this might be implemented directly in the ${}^{3}P_{2}$ manifold, either through a combination of tensor-light shift and rf-driven Larmor precession similar to Omanakuttan et al. (2021), or alternatively through a combination of microwave-driven Rabi oscillations between different hyperfine levels in ${}^{3}P_{2}$ and rf-driven Larmor procession as in Anderson (2013). In either case, optimal control can be used to find the relevant experimental waveform that generates the desired local $\mathrm{SU}(d)$ gates.

In this analysis, we included locally addressable control on each qudit. Though the CPhase gate is symmetric under exchange, we find that this symmetry breaking is necessary for effective optimization of this parameterization, similar to that seen in Choquette et al. (2021). An alternative protocol is to employ symmetric global control of the local unitaries, $\vec{\alpha}^{(j)}=\vec{\beta}^{(j)}$, but to reverse the sign of the entangling Hamiltonian $H_{\mathrm{ent}}\to-H_{\mathrm{ent}}$ in alternating layers. This allows for effective optimization, and the corresponding result is given in Table (2).

In a closed quantum system, quantum optimal control employing either the Lie algebraic or the Lie group approaches can be used in principle to implement any qudit entangling gate to any desired fidelity. In our numerical optimization, we took the target infidelity to be $10^{-3}$. In the absence of decoherence, we could achieve that target in a reasonable time for $d\leq 5$. For $d=7$, more time is required. However, the fundamentally achievable fidelity is limited by decoherence associated with the particular physical platform. For the system at hand, decoherence occurs due to the finite lifetime of the Rydberg states, which predominantly leads to leakage and loss outside the computational basis. In that case, we can model the gate as generated by a non-Hermitian effective Hamiltonian, $H_{\mathrm{eff}}[c(t)]$, where the Hermitian part is the control Hamiltonian and the anti-Hermitian represents decay out of the Rydberg states. The fidelity of interest is given by

where $V_{\mathrm{eff}}[\mathbf{c},T]=\mathcal{T}\left[\exp\left(-i\int_{0}^{T}H_{\mathrm{eff}}[\mathbf{c}(t)]dt\right)\right]$. Here the decay amplitude from a dressed state is $\gamma_{\mathrm{decay}}^{ij}=|C_{r_{i}j}|^{2}\Gamma_{r_{i}}+|C_{ir_{j}}|^{2}\Gamma_{r_{j}}$, which in turn gives the effective Hamiltonian as

With this model for decoherence in hand, the numerical results for the Lie algebraic approach are given in Fig. 4, which shows the infidelity as a function of time for a CPhase gate for different dimension isometries. We focus here on the case of the prime dimensional qudits. In contrast to closed-system control, in the presence of decoherence, infidelity decreases at first and then increases. This is due to the fact there is an optimal time of evolution, larger than the quantum speed limit, but not too large when compared to the coherence time of the system. As expected, one needs more time as the qudit dimension increases, which in turn results in an increase in the minimum infidelity one could achieve in each of these cases as shown in Fig. 4. We obtain a maximum fidelity of 0.9985, 0.9980, 0.9942, and 0.9800 for $d=2$, $d=3$, $d=5$, and $d=7$ respectively for the CPhase gate. Note, the values of fidelity for different dimensional qudits should be considered in the context of a particular application. For example, the threshold for fault tolerance for qudits, in general, is larger for larger $d$ Anwar et al. (2014); Watson et al. (2015). For the particular scheme considered in Anwar et al. (2014), the threshold for $d=2$, $d=3$, $d=5$, and $d=7$ are close to $0.008$, $0.012$, $0.0135$, and $0.015$ respectively. Hence, the proof-of-principle fidelity obtained here is promising and can be further optimized.

In the Lie group approach, we can use the effective Hamiltonian to describe the evolution when the Rydberg dressing is employed. In this case, we have,

We neglect here any decoherence associated with the local SU($d$) gates. Thus the fidelity including the decoherence effects is given as,

A comparison of the fidelities achieved based on the Lie algebraic and Lie group approaches is given in Fig. 5 for $d=3,5,$ and $d=7$. The results suggest that the Lie algebraic protocol slightly outperforms the Lie group protocol in the presence of decoherence.

This difference in the performance can be attributed to the time spent in the Rydberg state for these two approaches, as shown in Fig. 6. Fundamentally, we can understand this from the fact that the Lie algebraic approach has more control parameters as compared to the Lie group protocol. Thus, based on the Magnus expansion Merkel (2009); Jurdjevic and Sussmann (1972); Brockett (1973), the nested commutators which are at the heart of controllability become easier to achieve. Both approaches yield high fidelities in large dimensional qudits. Nevertheless, the Lie group approach may be preferable when considering the complexity necessary for experimental control.

In general, a key experimental consideration for the successful implementation of open-loop quantum control is the effect of uncertainties in Hamiltonian parameters. These can be mitigated to some degree using the tools of robust quantum control Anderson et al. (2015); Goerz (2015); Glaser et al. (2015); Koch (2016). Such techniques are generalizations of spin-echo type composite pulses which can be useful when there is sufficient coherence time. With a detailed understanding of the dominant inhomogeneities, robust optimal control can be used to implement suitable composite waveforms for qudit entanglers on any platform.

The specific experimental foundation of this proposal is well-motivated by existing literature, particularly the work of the Jessen group Anderson (2013). One particular issue discussed above is the trap-induced differential light shifts between the ground state and excited state ${}^{3}\mathrm{P}_{2}$ manifold Trautmann et al. (2023). It will be necessary to mitigate motional dephasing arising from vector- and tensor-shifts, which induce an $m_{F}$-dependence on polarizability, thus inducing possible motional dephasing between $m_{F}$ levels. The easiest way around this problem is to operate with a linearly-polarized optical trap, with polarization vector aligned at the “magic angle” Norcia et al. (2018) and corresponding magic wavelength Ye et al. (2008) for the ${}^{1}\mathrm{S}_{0}\rightarrow{}^{3}\mathrm{P}_{2}$ transition. This allows intra-state coherence within the ${}^{3}\mathrm{P}_{2}$ $F=9/2$ (and other $F$-levels) manifold, and inter-state (i.e., optical qubit) coherence between the ${}^{1}\mathrm{S}_{0}$ and ${}^{3}\mathrm{P}_{2}$ $F=9/2$. We can also mitigate motional effects via high-fidelity ground-state cooling Kaufman et al. (2012); Thompson et al. (2013); Lester et al. (2014).