Time-Optimal Two- and Three-Qubit Gates for Rydberg Atoms

Consider $n$ atoms treated as three-level systems: A qubit is stored in states $\ket{0}$ and $\ket{1}$, which can be taken to be hyperfine states of the ground state manifold. Additionally, a Rydberg state $\ket{r}$ is used to mediate the interactions between the qubits. This work focuses on the cases $n=2$ and $n=3$ to describe two- and three-qubit gates.

The level scheme for $n=2$ is shown in Fig. 1(a). The states $\ket{1}_{j}$ and $\ket{r}_{j}$ of the $j$-th atom are coupled by a laser with time-dependent Rabi frequency $\Omega_{j}(t)=|\Omega_{j}(t)|e^{i\varphi_{j}(t)}$. The Rabi frequency $\Omega_{j}(t)$ is taken to be complex, encoding both the amplitude $|\Omega_{j}(t)|$ and the phase $\varphi_{j}(t)$ of the laser. For this reason, no additional detuning of the laser is included, since the laser detuning $\Delta_{j}(t)$ and the laser phase are related through $\Delta_{j}=\mathrm{d}\varphi_{j}/\mathrm{d}t$. In any experiment, the maximal achievable Rabi frequency $|\Omega_{j}|$ is limited by the laser power and waist diameter. To include this into the model, a maximum Rabi frequency $\Omega_{\mathrm{max}}$ is introduced and only pulses with $|\Omega_{j}(t)|\leq\Omega_{\mathrm{max}}$ are considered for the rest of the paper.

The inter-atomic interaction term $B_{jk}\ket{rr}_{jk}\leftidx{{}_{jk}}{\!\bra{rr}}{}$ shifts the energy of the atoms $j$ and $k$ that are prepared in the Rydberg state, with $B_{jk}$ called the blockade strength. This results in the Hamiltonian (with $\hbar=1$)

	$\displaystyle H(t)=$	$\displaystyle\sum_{i=j}^{n}\frac{\Omega_{j}(t)}{2}\ket{1}_{j}\leftidx{{}_{j}}{\!\bra{r}}{}+\mathrm{h.c.}$		(1)
	$\displaystyle+$	$\displaystyle\sum_{j<k}B_{jk}\ket{rr}_{jk}\leftidx{{}_{jk}}{\!\bra{rr}}{}.$

The Hamiltonian (1) ignores any Rydberg-laser induced light shifts on $\ket{0}$, $\ket{1}$ and $\ket{r}$. These light shifts can be cancelled by adjusting the laser frequency and by applying additional single qubit rotations around the $z$ axis after the pulse.

In previous studies, blockade strengths up to $B_{jk}/{2\pi}=3$GHz have been considered [32, 38]. In Appendix A we provide a concrete example of how a blockade strength of $B_{jk}/2\pi=180$MHz can be achieved in experiments, taking fully into account the dipole-dipole interaction between all relevant Rydberg states. The corresponding gate pulse errors for blockade strengths between $100\mathrm{MHz}<B_{jk}/2\pi<3\mathrm{GHz}$ are calculated in Sec. 7. We note that the coupling between the electronic and the motional degree of freedom [58] is not treated explicitly in Eq. (1). However, we expect that the gate errors arising due to this coupling are mitigated by the short gate durations of the time-optimal pulses.

For $q\in\{0,1\}^{n}$ we denote by $P_{q}$ the projector onto the space spanned by all states such that atom $j$ is in state $\ket{0}$ iff $q_{j}=0$. For example, at $n=2$ we have $P_{00}=\ket{00}\bra{00}$, $P_{01}=\ket{01}\bra{01}+\ket{0r}\bra{0r}$ and $P_{11}=\ket{11}\bra{11}+\ket{1r}\bra{1r}+\ket{r1}\bra{r1}+\ket{rr}\bra{rr}$ (Here and in the following, a bra or a ket vector without a subscript means that this vector describes the state of all atoms, e.g. $\ket{00}=\ket{00}_{12}$). Since the lasers only couple $\ket{1}_{j}$ to $\ket{r}_{j}$, $H$ is block-diagonal with respect to the $P_{q}$, i.e.

$$H=\sum_{q}H_{q}\qquad H_{q}=P_{q}HP_{q}.$$

(2)

This block-diagonality allows for solving the time evolution of a computational basis state $\ket{q}$ by solving the Schrödinger equation for $H_{q}$ instead of $H$, which reduces the dimension of the relevant Hilbert space. This will simplify the use of GRAPE and PMP methods in Secs. 3 and 4.

Because any quantum gate has to map a state in the computational subspace $\mathcal{C}=\mathrm{span}\{\ket{q}|q\in\{0,1\}^{n}\}$ back to $\mathcal{C}$ and $H$ is block diagonal with respect to the $P_{q}$, the only quantum gates that can be implemented with Eq. (1) are phase gates.

Given laser pulses $\Omega_{1},...,\Omega_{n}$ of duration $T$, we denote by $U(t)=\tau\exp\left(-i\int_{0}^{t}H(t^{\prime})\mathrm{d}t^{\prime}\right)$ the time evolution operator. The pulse is said to implement a $(\xi_{q})_{q\in\{0,1\}^{n}}$ phase gate if

$$U(T)\ket{q}=e^{i\xi_{q}}\ket{q}.$$

(3)

Note that since $H_{0...0}=0$ only phase gates with $\xi_{0...0}=0$ can be implemented. A phase gate on $n=2$ qubits is, up to single qubit gates, a CZ gate if $\xi_{11}-\xi_{01}-\xi_{10}=\pi$. A phase gate on $n=3$ qubits is, up to single qubit gates, a C₂Z gate if $\xi_{111}-\xi_{001}-\xi_{010}-\xi_{100}=\pi$ and $\xi_{011}=\xi_{001}+\xi_{010}$ and all permutations of the second equation hold.

The averaged fidelity $F$ of a pulse aiming to implement a $(\xi_{q})_{q}$ phase gate is defined by

$$F=\int_{\mathcal{C}}\mathrm{d}\psi\left|\left\langle\psi|U_{0}^{\dagger}U(T)|\psi\right\rangle\right|^{2}$$

(4)

where $U_{0}=\sum_{q}e^{i\xi_{q}}\ket{q}\bra{q}$ is the desired phase gate and the integral is taken over all normalized states in $\mathcal{C}$ and with respect to the Haar measure on the unit sphere $\mathbb{S}^{2^{n}-1}$. The integral can be evaluated to [59]

	$\displaystyle F=\frac{1}{2^{n}(2^{n}+1)}\Big{(}$	$\displaystyle\left|\sum_{q}e^{-i\xi_{q}}\braket{q}{U(T)}{q}\right|^{2}$		(5)
		$\displaystyle+\sum_{q}\left|\braket{q}{U(T)}{q}\right|^{2}\Big{)}.$

Interestingly, the fidelity can be found by propagating a single (unnormalized) state – namely $\ket{\psi(0)}=\sum_{q}\ket{q}$ – under $H(t)$, since by the bock-diagonality of $H$ it holds that $\braket{q}{U(T)}{q}=\braket{q}{\psi(T)}$. The so called gate error $1-F$ is used in the following as a quality measure for a pulse aiming to implement a $(\xi_{q})_{q}$ phase gate. In Sec. 3, GRAPE will be used to time-optimally steer $\ket{\psi(0)}$ to $U_{0}\ket{\psi(0)}$ by minimizing the gate error, which then provides a time-optimal pulse to implement the phase gate by considering the evolution of a single state only.

For the specific case of a CZ gate as a phase gate the Bell state fidelity is commonly used instead of the averaged fidelity [58, 22, 21, 38] . The Bell state fidelity measures the quality of a pulse by the fidelity between the state obtained by applying the pulse to $\ket{++}=(\ket{00}+\ket{01}+\ket{10}+\ket{11})/2$ and the desired output state $CZ\ket{++}$. It is thus given by $F_{\mathrm{bell}}=|\braket{00}{\psi_{00}}+\braket{01}{\psi_{01}}+\braket{10}{\psi_{10}}-\braket{11}{\psi_{11}}|^{2}/16$, which can be generalized to arbitrary phase gates as

$$F_{\mathrm{bell}}=4^{-n}\left|\sum_{q}e^{-i\xi_{q}}\braket{q}{U(T)}{q}\right|^{2}$$

(6)

Comparing Eq. (5) and Eq. (6) shows that the averaged fidelity puts more emphasis on errors leading outside of the computational subspace than the Bell state fidelity. In this work we use the averaged fidelity. We expect that the same pulses can be obtained by optimizing the Bell state fidelity instead.

The time-optimal pulse implementing a phase gate is not necessarily unique, as other pulses implementing the same or a related gate can be obtained by symmetry operations. Here, three of these symmetry operations are discussed: Phase shifts, time-reversal, complex conjugation.

Given a pulse $\Omega_{1},...,\Omega_{n}$ of duration $T$ implementing a $(\xi_{q})_{q}$ phase gate for a blockade strength $B_{jk}$, several other pulses $\tilde{\Omega}_{1},...,\tilde{\Omega}_{n}$ with the same duration $T$ can be constructed that implement a $(\tilde{\xi}_{q})_{q}$ phase gate using the following symmetry operations

1. a)

   Phase shifts: $\tilde{\Omega}_{j}(t)=e^{i\alpha_{j}}\Omega_{j}(t)$, $\tilde{B}_{jk}=B_{jk}$ and $\tilde{\xi}_{q}=\xi_{q}$, for constants $\alpha_{1},...,\alpha_{n}$. To prove this statement, we denote here and in the following by $\tilde{H}$ the Hamiltonian given by the pulses $\tilde{\Omega}_{1},...,\tilde{\Omega}_{n}$ and blockade strengths $\tilde{B}_{ij}$ according to Eq. (1). Further, let $\tilde{U}(t)$ be the time evolution operator under $\tilde{H}$. Then with the basis change $V=\bigotimes_{j}(\ket{0}\bra{0}+\ket{1}\bra{1}+e^{-i\alpha_{j}}\ket{r}\bra{r})$ we have $\tilde{H}=VHV^{\dagger}$. Hence $\tilde{U}(T)=VU(T)V^{\dagger}$, thus also $\braket{q}{\tilde{U}(T)}{q}=\braket{q}{U(T)}{q}=e^{i\xi_{q}}$.

2. b)

   Time reversal: $\tilde{\Omega}_{i}(t)=\Omega_{i}(T-t)$, $\tilde{B}_{ij}=-B_{ij}$ and $\tilde{\xi}_{q}=-\xi_{q}$. To prove that, we first show that another symmetry is given by $\tilde{\Omega}_{i}(t)=-\Omega_{i}(T-t)$, $\tilde{B}_{ij}=-B_{ij}$ and $\tilde{\xi}_{q}=-\xi_{q}$. To see this, note that $\tilde{H}(t)=-H(T-t)$, so the time evolution operator under $\tilde{H}$ is $\tilde{U}(t)=U(T-t)U(T)^{\dagger}$. Hence $\braket{q}{\tilde{U}(T)}{q}=\braket{q}{U(T)^{\dagger}}{q}=e^{-i\xi_{q}}$. Together with the symmetry under phase shifts, symmetry under time reversal follows.

3. c)

   Complex conjugation: $\tilde{\Omega}_{i}(t)=\Omega_{i}(t)^{*}$, $\tilde{B}_{ij}=-B_{ij}$ and $\tilde{\xi}_{q}=-\xi_{q}$. To prove this, we note that joint time reversal and complex conjugation is a symmetry, given by $\tilde{\Omega}_{i}(t)=\Omega_{i}(T-t)^{*}$, $\tilde{B}_{ij}=B_{ij}$ and $\tilde{\xi}_{q}=\xi_{q}$. To see this, note that $\tilde{H}(t)=H(T-t)^{*}$, so the time-evolution operator under $\tilde{H}$ is $\tilde{U}(t)=U(T-t)^{*}(U(T)^{*})^{\dagger}$. Hence $\braket{q}{\tilde{U}(T)}{q}=\braket{q}{(U(T)^{*})^{\dagger}}{q}=e^{i\xi_{q}}$. Together with symmetry under time reversal, symmetry under complex conjugation follows.

These three symmetry operations show that when using numerical methods to find the time-optimal pulse for a gate, several solutions are to be expected. Firstly, two time-optimal pulses can differ by an arbitrary constant phase. This degree of freedom can be eliminated by restricting the discussion to pulses where $\Omega_{j}(0)$ is real and positive, as we will do from now on. Even with this restriction there are in general two different time-optimal pulses for a given phase gate, related by joint complex conjugation and time reversal. In the special case of $B=\infty$ and for phase gates where all $\xi_{q}$ are real, which is studied in Sec. 3, there are in general even four different time-optimal pulses, because also individual time reversal or complex conjugation give pulses implementing the same gate. As will be apparent in Sec. 3, sometimes, but not always, time-optimal pulses are invariant under joint time-reversal and complex conjugation, reducing the number of distinct time-optimal pulses to two.

This work focuses primarily on so called global pulses, where $\Omega_{1}=\Omega_{2}=...=\Omega_{n}=:\Omega$. In fact, except in Sec. 3.2, all considered pulses will be global. Global pulses can be implemented with just a single global laser addressing all atoms at the same time, no single-site addressability being needed. They are thus expected to be easier to realize experimentally. When working with global pulses it will be further assumed (except briefly in Sec. 6) that all $B_{jk}$ are equal to the same blockade strength $B$.

The theoretical treatment using global pulses is simplified compared to the general case in two different ways: Firstly, now it holds that $\braket{q}{\psi(T)}=\braket{q^{\prime}}{\psi(T)}$ if $q$ and $q^{\prime}$ have the same number of 1s. As a consequence, only phase gates with $\xi_{q}=\xi_{q^{\prime}}$ can be implemented. For $n=2$ the whole gate is then determined by the evolution of $\ket{\psi(0)}=\ket{01}+\ket{11}$ under $H_{01}+H_{11}$, and the fidelity is given by

	$\displaystyle F=\frac{1}{20}$	$\displaystyle\Big{(}\left|1+2a_{01}+a_{11}\right|^{2}$		(7)
		$\displaystyle+1+2|a_{01}|^{2}+|a_{11}|^{2}\Big{)}$

with $a_{q}=e^{-i\xi_{q}}\braket{q}{\psi(T)}$. Similarly, for $n=3$ the whole gate is determined by the evolution of $\ket{\psi(0)}=\ket{001}+\ket{011}+\ket{111}$ under $H_{001}+H_{011}+H_{111}$, with

	$\displaystyle F=\frac{1}{72}$	$\displaystyle\Big{(}\left|1+3a_{001}+3a_{011}+a_{111}\right|^{2}$		(8)
		$\displaystyle+1+3|a_{001}|^{2}+3|a_{011}|^{2}+|a_{111}|^{2}\Big{)}$

The second simplification due to global pulses is a simplification of Hamiltonians $H_{q}$ if $q$ contains two or more 1s. For $n=2$ we have

	$\displaystyle H_{11}=$	$\displaystyle\frac{\Omega}{2}\Big{(}\ket{11}\bra{1r}+\ket{11}\bra{r1}+\ket{1r}\bra{rr}$		(9)
		$\displaystyle+\ket{r1}\bra{rr}\Big{)}+\mathrm{h.c.}+B\ket{rr}\bra{rr}$
	$\displaystyle=$	$\displaystyle\frac{\sqrt{2}\Omega}{2}\Big{(}\ket{11}\bra{W}+\ket{W}\bra{11}\Big{)}+\mathrm{h.c.}$
		$\displaystyle+B\ket{rr}\bra{rr}$

with $\ket{W}=(\ket{1r}+\ket{r1})/\sqrt{2}$. The rank of $H_{11}$ thus reduces from generally 4 to 3 in the global case. For $n=3$ we make the analogous simplification

$$H_{011}=\ket{0}\bra{0}\otimes H_{11}.$$

(10)

To simplify $H_{111}$, we note that the only states that can be accessed from $\ket{111}$ with a global pulse are $\ket{W_{1}}=(\ket{11r}+\ket{1r1}+\ket{r11})/\sqrt{3}$, $\ket{W_{2}}=(\ket{1rr}+\ket{r1r}+\ket{rr1})/\sqrt{3}$ and $\ket{rrr}$. It is thus sufficient to consider just the projection of $H_{111}$ onto the symmetric subspace $\mathrm{span}\{\ket{111},\ket{W_{1}},\ket{W_{2}},\ket{rrr}\}$, given by

	$\displaystyle H^{\mathrm{sym}}_{111}$	$\displaystyle=\frac{\sqrt{3}\Omega}{2}\ket{111}\bra{W_{1}}+\Omega\ket{W_{1}}\bra{W_{2}}$
		$\displaystyle+\frac{\sqrt{3}\Omega}{2}\ket{W_{2}}\bra{111}+\mathrm{h.c.}$
		$\displaystyle+B\ket{W_{2}}\bra{W_{2}}+3B\ket{rrr}\bra{rrr}.$		(11)

This reduces the rank of $H_{111}$ from generally 8 to 4 in the global case.

This discussion concludes the presentation of the basic properties of pulses implementing phase gates, and of global pulses in particular. After a brief introduction to GRAPE and the PMP in the following subsection, the theoretical discussion from this section will be used in Secs. 3 and 4 to identify the time-optimal pulses for the CZ gate and the C₂Z gate.

Gradient Ascent Pulse Engineering (GRAPE) is a quantum optimal control technique based on gradient descent. It was originally developed to design pulse sequences for NMR spectroscopy, and has been used successfully to speed up quantum gates and to increase the fidelity of pulses in the presence of imperfections [60, 61, 62, 63]. In the context of neutral atoms GRAPE has been used to design arbitrary $SU(16)$ gates in the hyperfine levels of $\leavevmode\nobreak\ {}^{133}$CS [54, 55, 56].

Given an initial state $\ket{\psi(0)}$, a pulse duration $T$ and a Hamiltonian $H$ depending on controls $u(t)$, GRAPE can be used to find the optimal controls $u(t)$, minimizing $J(\ket{\psi(T)})$, for an arbitrary objective function $J$. For this, a piecewise constant Ansatz for $u(t)$ is made, i.e. $u(t)=u_{j}$ if $t\in[j\Delta t,(j+1)\Delta t]$ and $\Delta t=T/N$, with $N$ the number of pieces. Typically, several hundreds of pieces are used. Minimizing $J(\ket{\psi(T)})$ becomes an optimization problem over $u_{0},...,u_{N-1}$. GRAPE provides a fast way to calculate all derivatives $\partial J/\partial u_{j}$, which then allows for the efficient use of a gradient descent algorithm to find a local minimum of the cost function. In our implementation, we use the Broyden–Fletcher–Goldfarb–Shanno algorithm [64, 65] as the gradient descent method.

Pontryagin’s Maximum principle (PMP) [66, 67, 68] is an optimal control technique that has been used successfully to find time-optimal [69, 70, 71, 61] or robust [72] quantum gates, and to optimize variational quantum algorithms [73]. The PMP is an analytic principle that gives a set of necessary conditions that optimal pulses have to satisfy, thereby allowing to reduce the infinite dimensional control landscape to a low-dimensional space [72]. In contrast to GRAPE, which usually uses several hundreds of variational parameters, the PMP can therefore describe optimal pulses with just a handful of parameters. On the downside, the PMP does not provide a universal algorithm to find the parameters leading to the optimal pulse. In Sec. 4 we will combine GRAPE and the PMP to describe the pulses found by GRAPE using the PMP with just 4 parameters for the CZ and 6 parameters for the C₂Z gate.

In this section we give a formulation of the PMP and apply it to the problem of time-optimal pulses on Rydberg atoms. We start by formulating the PMP for the specific case of time-optimal control problems [67]: Suppose one wants to steer a real vector $x\in\mathbb{R}^{n}$ from an initial point $x_{i}$ to a final point $x_{f}$ using the differential equation $\dot{x}=f(x,u)$, where $u(t)$ describe the available controls (i.e., the Rabi frequencies $\Omega_{1},...,\Omega_{n}$ of the lasers in our case). Later, $x$ will correspond to the quantum state and $f$ will be given by Schrödinger’s equation. We call a triplet $(T,x(\cdot),u(\cdot))$ a controlled trajectory if $x(0)=x_{i}$, $x(T)=x_{f}$ and $\dot{x}(t)=f(x(t),u(t))$. The PMP gives necessary conditions on the controlled trajectory $(T,x,u)$ that minimizes $\int_{0}^{T}g(x(t),u(t))\mathrm{d}t$, where $g$ is an arbitrary cost function.

By focusing on time-optimal control problems, we can fix $g(x,u)=1$. In this case, the PMP then states that for a given time-optimal controlled trajectory $(T,x,u)$ there exist so-called costates $\lambda(\cdot)$ with values in $\mathbb{R}^{n}\setminus\{0\}$ such that at each time the following equations are satisfied

$$\dot{\lambda}=-\left\langle\lambda,\nabla_{x}f(x,u)\right\rangle$$

(12)

and

$$\left\langle\lambda(t),f(x(t),u(t))\right\rangle=\sup_{u^{\prime}}\left\langle\lambda(t),f(x(t),u^{\prime})\right\rangle,$$

(13)

where $\left\langle\cdot,\cdot\right\rangle$ denotes the scalar product. Further, if the $x$ are restricted to a submanifold $M$ of $\mathbb{R}^{n}$, the $\lambda(t)$ can be restricted to the tangent space of $M$ at $x(t)$ [74]. If the supremum in Eq. (13) is only achieved by a single value $u^{\prime}$ of the controls, the PMP allows to reduce the search for the time-optimal trajectory to the search over the initial costates $\lambda(0)$, which determine the full trajectory via Eqs. (12) and (13).

In this work, we directly apply the PMP to the Schrödinger equation $\ket{\dot{\psi}}=-iH(u)\ket{\psi}$. This is done by decomposing the state $\ket{\psi}=\ket{\psi_{R}}+i\ket{\psi_{I}}$ and the Hamiltonian $H=H_{R}+iH_{I}$ into real and imaginary parts [70, 71], which results in two real costates $\ket{\chi_{R}}$ and $\ket{\chi_{I}}$. By combining the two real costates with a complex one as $\ket{\chi}=\ket{\chi_{R}}+i\ket{\chi_{I}}$, Eq. (12) becomes the Schrödinger equation $\ket{\dot{\chi}}=-iH\ket{\chi}$ for the costates, while Eq. (13) reads [71]

		$\displaystyle\mathrm{Im}(\braket{\chi(t)}{H(u(t))}{\psi(t)})$		(14)
		$\displaystyle=\sup_{u^{\prime}}\mathrm{Im}(\braket{\chi(t)}{H(u^{\prime})}{\psi(t)}).$

The key point is that, as long as the supremum is achieved at a unique value of the controls, the initial states and costates completely determine the optimal trajectory.

Applied to time-optimal phase gates the PMP states that there are costates $\ket{\chi_{q}(t)}$ evolving under the Schrödinger equation $\ket{\dot{\chi}_{q}}=-iH_{q}\ket{\chi_{q}}$ such that

		$\displaystyle\mathrm{Im}\left(\sum_{q}\braket{\chi_{q}(t)}{H_{q}(u(t))}{\psi_{q}(t)}\right)$		(15)
		$\displaystyle=\sup_{u^{\prime}}\mathrm{Im}\left(\sum_{q}\braket{\chi_{q}(t)}{H_{q}(u^{\prime})}{\psi_{q}(t)}\right)$

where the sum can be either over all $q\in\{0,1\}^{n}$, or for global pulses with $n=2$ or $n=3$ only over $q\in\{01,11\}$ or $q\in\{001,011,111\}$, respectively.

In Sec. 4 we will use the PMP to reproduce the results found by GRAPE by extracting the initial costates from the pulses found by GRAPE. This allows us to describe the time-optimal pulses just through the initial costates, which correspond to 4 parameters for the CZ gate and 6 parameters for the C₂Z gate.