††thanks: The first two authors contributed equally

# Experimental roadmap for optimal state transfer and entanglement generation
in power-law systems

Andrew Y. Guo<EMAIL_ADDRESS>Joint Center for Quantum Information and Computer
Science, NIST/University of Maryland, College Park, MD 20742, USA  Joint
Quantum Institute, NIST/University of Maryland, College Park, MD 20742, USA
Jeremy T. Young  Institute of Physics, University of Amsterdam, 1098 XH
Amsterdam, the Netherlands  JILA, University of Colorado and National
Institute of Standards and Technology, and Department of Physics, University
of Colorado, Boulder, Colorado 80309, USA  Center for Theory of Quantum
Matter, University of Colorado, Boulder, Colorado 80309, USA  Ron Belyansky
Joint Center for Quantum Information and Computer Science, NIST/University of
Maryland, College Park, MD 20742, USA  Joint Quantum Institute,
NIST/University of Maryland, College Park, MD 20742, USA  Przemyslaw Bienias
Joint Center for Quantum Information and Computer Science, NIST/University of
Maryland, College Park, MD 20742, USA  Joint Quantum Institute,
NIST/University of Maryland, College Park, MD 20742, USA  Alexey V. Gorshkov
Joint Center for Quantum Information and Computer Science, NIST/University of
Maryland, College Park, MD 20742, USA  Joint Quantum Institute,
NIST/University of Maryland, College Park, MD 20742, USA

(August 28, 2024)

###### Abstract

Experimental systems with power-law interactions have recently garnered
interest as promising platforms for quantum information processing. Such
systems are capable of spreading entanglement superballistically and achieving
an asymptotic speed-up over locally interacting systems. Recently, protocols
developed by Eldredge _et al._ [Phys. Rev. Lett. 119, 170503 (2017)] and Tran
_et al._ [Phys. Rev. X 11, 031016 (2021)] for the task of transferring a
quantum state between distant particles quickly were shown to be optimal and
saturate theoretical bounds. However, the implementation of these protocols in
physical systems with long-range interactions remains to be fully realized. In
this work, we provide an experimental roadmap towards realizing fast state-
transfer protocols in three classes of atomic and molecular systems with
dipolar interactions: polar molecules composed of alkali-metal dimers, neutral
atoms in excited Rydberg states, and atoms with strong magnetic moments (e.g.
dysprosium). As a guide to near-term experimental implementation, we
numerically evaluate the tradeoffs between the two protocols for small system
sizes and develop methods to address potential crosstalk errors that may arise
during the execution of the protocols.

Ultracold quantum systems have recently attracted attention for their
potential for quantum information processing, and innovations in the
individual control of ultracold atoms [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] and
molecules [12, 13, 14, 15, 16, 17, 18, 19] have led to the realization of
high-fidelity quantum gates [20, 21, 22, 23, 24, 25, 26, 27, 28]. Many atomic
and molecular systems also have access to long-range interactions, which can
generate entanglement quickly between qubits at large distances. Compared to
finite-range couplings, long-range interactions allow for increased
connectivity between qubits, which can lead to speed-ups for performing
general computational tasks [29].

In particular, it was recently shown that systems with power-law
interactions—i.e. those that decay as $1/r^{\alpha}$ in the distance between
qubits—can achieve an asymptotic speed-up for _quantum state transfer_ , a
task wherein a quantum state must be transferred quickly between two distant
qubits [30, 31, 32]. Performance on the state transfer task provides a
benchmark for a system’s ability to execute nonlocal quantum gates quickly,
and also serves as a computational primitive for more complicated circuit
operations, such as quantum routing [33]. Additionally, many state-transfer
protocols often generate many-body entanglement as an intermediate step. The
generation of such many-body entangled states as the Greenberger-Horne-
Zeilinger (GHZ) and W states can be an important computational resource in
itself for applications such as sensing at the Heisenberg limit [34, 35] and
performing multi-qubit quantum gates [36].

In addition to their usefulness in generating entanglement quickly, state-
transfer protocols have also attracted significant theoretical interest due to
their ability to saturate fundamental Lieb-Robinson bounds on the rate at
which information can propagate in quantum systems [37, 38, 39, 40, 41, 29].
In particular, the state-transfer protocols in Refs. [30] and [31] have been
shown to saturate the best Lieb-Robinson bounds for power-law interactions
[41]. In addition to being asymptotically optimal, these fast entangling
protocols imply that power-law interactions are qualitatively stronger than
previously understood; indeed, while state transfer in faster-than-linear time
has been shown before using interactions with relatively small power-law
exponents (e.g. $\alpha\approx 1$ in one-dimensional [1D] systems [42]),
realizing the same state transfer times using interactions with more rapid
power-law decays (e.g. van der Waals interactions with $\alpha=6$ in 3D) would
be the first demonstration of fast information transfer using interactions
previously thought to be effectively short-range.

Figure 1: Schematic for the Eldredge state-transfer protocol. (a) Interactions
from $H(t)$ fan out the state from the initial qubit $X$ (black) to the rest
of the lattice. (b) Once the CNot gate has been completed on a “target” qubit,
it becomes a “control” qubit and part of the GHZ-like state (purple shaded
region). The remaining, “target” qubits are shaded gray according to the phase
that has accumulated due to the interacting Hamiltonian and will be added to
the GHZ state in future steps of the protocol. Afterwards, the GHZ-state
creation is reversed and the state is transferred to the final qubit, $Y$.

In this work, we present a roadmap for the experimental implementation of fast
quantum state transfer protocols using long-range interactions. We first
review the power-law interacting protocols developed by Eldredge _et al._ [30]
and Tran _et al._ [31] in Section I, and analyze both their asymptotic
runtimes as well as their numerical performance for finite system sizes. Then,
in Section II, we provide schemes for realizing these protocols in three model
power-law interacting systems: polar molecules, Rydberg atoms, and atoms with
large magnetic moments (dysprosium in particular). Our approach utilizes only
global power-law interactions and single-qubit control. In Section III, we
analyze the contribution of certain forms of crosstalk errors that may arise
in the course of the state transfer and propose a few methods to mitigate
those errors.

##  I Fast state-transfer protocols for power-law interactions

In this section, we will review the two protocols for performing fast quantum
state transfer using power-law interactions developed by Eldredge _et al._
[30] and Tran _et al._ [31]. We will summarize the protocols briefly, present
the asymptotic scaling of their state transfer times, and compare their
relative advantages in finite-size systems.

The state transfer task is given as follows. Given a system of $N$ qubits, we
wish to transfer an unknown input state $\ket{\psi}=a\ket{0}+b\ket{1}$
initially located on qubit $X$ to qubit $Y$. We assume that the intermediate
qubits (as well as qubit $Y$) are initialized in state $\ket{0}$. Then the
state transfer task is to realize the following unitary $U$:

$\displaystyle
U\bigl{(}\ket{\psi}_{X}\ket{0}^{N-2}\ket{0}_{Y}\bigr{)}\mapsto\ket{0}_{X}\ket{0}^{N-2}\ket{\psi}_{Y}.$
(1)

For the rest of this paper, we will assume that the system is arranged in a
$d$-dimensional grid of qubits of linear size $L\equiv\sqrt[d]{N}$, although
other lattice configurations are possible. The state transfer will take place
over a distance $r=\text{dist}(X,Y)$, which can be as large as
$r_{\text{max}}=L\sqrt{d}$.

The Eldredge protocol achieves state transfer by first encoding the qubit
state into a many-body entangled state that includes the initial and target
qubits (cf. Fig. 1). It does so by performing a sequence of cascaded
controlled-Not (CNot) gates, which realize the operation

$\textsc{CNot }(a\ket{0}+b\ket{1})\ket{0}=a\ket{00}+b\ket{11}.$ (2)

The CNot gate from qubit $i$ to qubit $j$ can be implemented by a Hamiltonian
$H_{ij}=h_{ij}\outerproduct{1}{1}_{i}\otimes X_{j}$ acting for time
$t=\pi/(2h_{ij})$, up to a local unitary term. In Ref. [30], a Hamiltonian
$H(t)=\sum_{ij}H_{ij}(t)$ is applied that progressively turns on/off the
interactions between pairs of qubits at different times and enables the
encoding of the state $\ket{\psi}$ into a GHZ-like state
$\ket{\text{GHZ}(a,b)}=a\ket{0}^{\otimes N}+b\ket{1}^{\otimes N}$. The
protocol starts by acting on all qubits in the lattice using a single control
qubit storing the initial state. At each time step, once the CNot is fully
complete on a target qubit, then it is added to the list of controls. At each
point in time, multiple controls may act on a single target. This process
continues until all of the qubits have been added to the GHZ state. The time
required to create the full GHZ state is the sum of these time steps. At the
end of the process, the protocol is reversed, interchanging the roles of
qubits $X$ and $Y$, to implement the full state transfer.

By using a long-range interacting Hamiltonian, the protocol developed by
Eldredge _et al._ is able to achieve a sublinear state transfer time. Assuming
that $h_{ij}=1/r_{ij}^{\alpha}$, where $r_{ij}$ is the distance between qubits
$i$ and $j$, the time required to complete the CNot on qubit $j$ being acted
on by controls indexed by $i$ is equal to

$\displaystyle
t=\frac{\pi}{2\sum_{i}h_{ij}}=\frac{\pi}{2\sum_{i}r^{-\alpha}_{ij}}.$ (3)

The scaling of the state transfer time $t_{\text{ST}}$ using the protocol
derived in Ref. [30] is given by

$t_{\text{ST}}=\begin{cases}\mathcal{O}\left\lparen 1\right\rparen&\alpha<d\\\
\mathcal{O}\left\lparen\log r\right\rparen&\alpha=d\\\ \mathcal{O}\left\lparen
r^{\min(\alpha-d,1)}\right\rparen&\alpha>d.\end{cases}$ (4)

The intuition for the various scalings are as follows: for $\alpha<d$, the
extreme non-locality of the interactions causes the state transfer velocity to
diverge in the thermodynamic limit, which implies that the state can be
transferred over arbitrary distances in constant time, so the scaling of
$t_{\text{ST}}$ is independent of $r$.

For $\alpha\in[d,d+1]$, take a block of qubits of linear size $r$ and suppose
we wish to double $r$. Then the qubit at the corner of the next block is acted
on by controlled rotations of total strength proportional to $\\{\text{\\# of
controls acting on the qubit\\}}/r^{\alpha}\propto r^{d-\alpha}$. Thus, for
$\alpha\in(d,d+1)$, the state-transfer time is proportional to $r^{\alpha-d}$,
which is sublinear.

For $\alpha=d$, the time it takes to double the linear block size is constant,
so it takes overall $\log r$ time to implement the state transfer protocol.
Lastly, for $\alpha=d+1$, the state-transfer time is linear in $r$, thus
recovering the nearest-neighbor-interaction limit.

For $\alpha>d+1$, the doubling approach thus leads to worse scaling than
nearest-neighbor interactions. Instead, the incremental approach which
immediately converts targets into controls must be used, and there is at most
a constant speedup from the power-law interactions. This incremental approach
realizes the same scaling as the doubling approach for $d\leq\alpha\leq d+1$
and is the optimal form of the Eldredge protocol for isotropic power-law
interactions. In conclusion, the Eldredge protocol yields an asymptotic speed-
up over nearest-neighbor interactions for all $\alpha<d+1$.

The protocol discovered by Tran _et al._ is able to achieve an even faster
speed-up for the state transfer task (see Fig. 2). As with the Eldredge
protocol, it uses a GHZ state as an intermediate and performs the state
transfer recursively by encoding the qubit state $\ket{\psi}$ into
progressively larger GHZ-like states until the state includes the whole
sublattice of sites between site $X$ and site $Y$ [31]. Then the protocol
reverses the GHZ-state encoding procedure to map the state of the original
qubit on site $X$ onto the qubit at site $Y$. In the process, small GHZ-like
states are merged together into larger GHZ-like states. The merge subroutine
is performed using diagonal interactions of the form
$H_{ij}=h_{ij}\outerproduct{1}{1}_{i}\otimes\outerproduct{1}{1}_{j}$, which
serve to perform a controlled-PHASE gate between qubits $i$ and $j$.

Figure 2: Outline of the Tran protocol. We start with qubit $X$ in an unknown
state $\ket{\psi}=a\ket{0}+b\ket{1}$, while the rest of the qubits start in
state $(\ket{0}+\ket{1})/\sqrt{2}$. In step 1, either nearest-neighbor
interactions or the Eldredge protocol are used to create a GHZ state on
neighboring qubits. Then, in step 2, controlled-phase interactions are used to
merge together neighboring GHZ states. The last three steps (steps 3-5) serve
to transform the state into the desired GHZ-like state by moving the
entanglement into a single qubit, performing a single-qubit rotation, and then
reentangling the qubits. This final GHZ-like state is fed back into the
protocol recursively, after which the protocol is implemented in “reverse” in
order to implement the state transfer from qubit $X$ onto qubit $Y$.

The scaling of the state transfer time $t_{\text{ST}}$ of the Tran protocol is
given as follows:

$t_{\text{ST}}=\begin{cases}\mathcal{O}\left\lparen\log^{\kappa_{\alpha}}r\right\rparen&\alpha\in(d,2d)\\\
\mathcal{O}\left\lparen e^{\gamma\sqrt{d\log r}}\right\rparen&\alpha=2d\\\
\mathcal{O}\left\lparen
r^{\min(\alpha-2d,1)}\right\rparen&\alpha>2d.\end{cases}$ (5)

Due to the protocol’s recursive nature, it can transfer a state in time that
scales faster than any polynomial in $r$ for all $\alpha<2d$. For $\alpha=2d$,
the state-transfer time is superlogarithmic—but still subpolynomial—and for
$\alpha\in(2d,2d+1)$, the time is sublinear and proportional to
$r^{\alpha-2d}$. The intuition for the latter scaling is as follows: consider
two blocks of GHZ-like states of linear size $r$ situated a distance $r$
apart. The time required to merge the two blocks scales inversely with the
combined interaction strength, which is proportional to $\\{\text{\\# of
interacting pairs of qubits\\}}/r^{\alpha}\propto r^{2d-\alpha}$. For more
details on the implementation of the protocol, see Appendix A.

Thus, for all $\alpha<2d+1$, the Tran protocol allows for even more
significant speed-ups over nearest-neighbor systems than the Eldredge
protocol. This regime of $\alpha$ includes dipolar interactions in 2D and 3D
and van der Waals interactions in 3D. For dipolar interactions in 3D, both the
Eldredge and Tran protocols realize exponential speedups. However, since the
Eldredge protocol still retains a polynomial speedup over the Tran protocol
for $\alpha=d$, in the rest of the paper, we shall focus on dipolar
interactions only in 2D.

###  I.1 Comparison of state-transfer protocols

Figure 3: (a) Comparison of the scaling of the state-transfer times for the
two protocols described in Section I.1 as a function of the characteristic
length $r$, using dipolar interactions in 2D ($\alpha=3$ and $d=2$). The
hybrid protocol consists of a single layer of the recursive Tran protocol on
top of the base Eldredge protocol, and the crossover point where it improves
over the pure Eldredge protocol occurs at a characteristic length of
$r_{*}=244$. (b) The scaling of the optimal values of $m=r/r_{1}$ for the
hybrid protocol as a function of $r$, as determined numerically using a
dynamic programming algorithm. The curve begins at the characteristic length
$r_{*}=244$, when the first layer of the Tran protocol is applied. The
stepwise transition for $r\approx 3700$ indicates the transition from a single
recursive step to two. The jaggedness in the curve arises due to the fact that
the $m$ values scale inversely with $r_{1}$, which can only take on discrete
values by assumption.

While the state-transfer protocol developed by Tran _et al._ in Ref. [31] is
asymptotically faster than the one presented by Eldredge _et al._ in Ref. [30]
for all $\alpha>d$, it is less efficient to implement for small system sizes.
This is due to the recursive step needing to call a subroutine that creates
smaller GHZ states a total of three times. Instead of using the Tran protocol
for small $r$, it would be faster to use the Eldredge protocol for $r$ up to a
fixed size, and then utilize the recursive step of the former protocol to
achieve larger GHZ states. In the rest of this subsection, we numerically
determine the optimal crossover point when a hybrid protocol consisting of
performing the Eldredge protocol followed by a single step of the Tran
protocol surpasses the pure Eldredge protocol for dipole-dipole interactions
on a 2D lattice.

To implement the Eldredge protocol described in Section I, specifically the
incremental approach which immediately converts target qubits into control
qubits, we numerically calculate the state transfer times for system sizes up
to $r\leq 110$, and then use a linear fit to extrapolate to larger values of
$r$ (since the asymptotic runtime of the protocol was shown to obey the
scaling $t\propto r$ for $\alpha\geq d+1$). We use these state transfer times
as the benchmark to compare the hybrid protocol.

To implement the hybrid protocol, we apply a recursive approach known as
_bottom-up dynamic programming_ : first, we choose a fixed value of $r=r_{0}$
below which we will use the Eldredge protocol to create a GHZ state of that
size. Then, for all larger $r$, we recursively calculate the time $t(r;r_{1})$
required to create a GHZ state of size $r$ by assembling it from smaller GHZ
states of size $r_{1}<r$. By minimizing $t(r;r_{1})$ over all values of
$r_{1}$, we obtain the optimal creation time $t(r)$ for a GHZ state of size
$r$ using a single step of the Tran protocol to get from $r_{1}$ to $r$. We
note that this protocol could require multiple recursive steps, since creating
a GHZ-state of size $r_{1}$ could itself potentially require recursive steps,
thus leading to the need to apply multiple layers of the Tran protocol.

In Fig. 3, we show the transition point above which a single step of the
hybrid protocol can create a GHZ state faster than the pure Eldredge protocol.
For $\alpha=3$ on a 2D lattice, this critical point occurs at $r_{*}=244$,
which corresponds to a lattice size of $N=r_{*}^{2}=59,538$. We note that this
is an estimate of the true crossover point, since we did not exactly calculate
the exact state transfer time for the pure Eldredge protocol beyond $r=110$
due to the numerical cost of the algorithm (with runtime scaling as
$\mathcal{O}\left\lparen r^{6}\right\rparen$).

In Fig. 3, we show the scaling of the optimal choice of the ratio of the sizes
of the GHZ states in the recursive step of the protocol, which we define to be
equal to $m\equiv r/r_{1}$ for a given choice of $r$. For one step of the
hybrid protocol, $m$ ranges from 5 to 13. Asymptotically, the analysis of the
runtime of the protocol predicts that $m$ will scale as $m\propto
r^{2d/\alpha-1}=r^{1/3}$ [31].

In order to recover the optimal asymptotic scaling of the state-transfer time
for the hybrid protocol, it will in general be necessary to use the recursive
step of the hybrid protocol multiple times. For a fixed $r_{0}$, the number of
times the recursive step is called will scale as
$n=\mathcal{O}\left\lparen\log\log(r)\right\rparen$, as one can show by
applying the relation $r_{1}=r/m=r^{2/3}$ recursively $n$ times to get the
equation $(r^{2/3})^{n}=r_{0}$ and solving for $n$.

In summary, using a hybrid of both the Tran and Eldredge protocols to create
an intermediate GHZ state allows for faster state-transfer times than using
either of those protocols alone. Our estimate of the crossover point suggests
that the performance of the hybrid protocol surpasses that of the Eldredge
protocol for system sizes exceeding roughly 60,000 qubits. As such, the speed-
up for the hybrid protocol could be potentially be realized in larger-scale
quantum devices.

##  II Proposal for experimental realization of state-transfer protocols

Before explaining the details of how to implement the state-transfer protocol
for particular systems, we will first discuss the features that apply to all
of the following systems.

In the state-transfer protocol, one generically has three types of qubits:
control qubits, target qubits, and uninvolved qubits. The uninvolved qubits
can generally be prepared by storing them in states (e.g., two hyperfine
ground states for neutral atoms) that do not interact with the other two types
of qubit or each other, which we discuss in further detail below. For the
control and target qubits, we must engineer a Hamiltonian with long-range
Ising interactions between control and target qubits. However, the control
qubits must not interact with themselves, and similarly the target qubits must
not interact with themselves, neither of which occurs generically, so these
interactions must be engineered.

To do so, we adopt the spin-echo-like approach from Ref. [30]. This approach
relies on the ability to realize a Hamiltonian of the form

$H_{\text{int}}=\frac{1}{2}\sum_{i\neq j}J_{ij}Z_{i}Z_{j},$ (6)

as well as its negative $H^{\prime}_{\text{int}}\equiv-aH_{\text{int}}$, where
$Z_{i}$ denotes the Pauli-Z matrix for the $i$th qubit and $a>0$. By evolving
under $H_{\text{int}}$ for a time $t$, applying a Pauli-X gate to either the
control qubits or the target qubits, and evolving under $-aH_{\text{int}}$ for
a time $t/a$, followed by another Pauli-$X$ gate on the same set of qubits,
the evolution from the control-control and target-target interactions is
undone while the control-target interactions remain, as desired. In light of
this, most of the following discussion will be centered around how to realize
both signs of the Ising interaction. Often, the engineered interactions
include an inhomogeneous longitudinal field term. The corresponding evolution
due to this term can be easily corrected via single-qubit gates.

The necessary local control can be achieved in tweezer arrays of Rydberg atoms
[1, 2, 3, 4], polar molecules [12, 13, 14, 15], and very recently dysprosium
[5] or by using quantum gas microscopes for polar molecules [16, 17, 18, 19]
and dysprosium [6, 7, 8]. As far as magnetic atoms go, we will focus on
dysprosium in this paper, but similar advances have been made for quantum gas
microscopes of erbium [9] as well as for mixtures of dysprosium and erbium
[10]. Additionally, when utilizing $1/r^{3}$ dipole-dipole interactions to
realize the desired Ising interactions, they will be proportional to
$1-3\cos^{2}\theta_{ij}$, where $\theta_{ij}$ is the polar angle of
$\mathbf{r}_{i}-\mathbf{r}_{j}$ for particles $i$ and $j$ relative to the
quantization axis. Since the protocol relies on all interactions having the
same sign, for dipole-dipole interactions we shall restrict our focus to 2D
with $\theta=\pi/2$. Moreover, the Eldredge protocol can already be utilized
with an exponential speedup in 3D. In the case of Rydberg atoms, isotropic vdW
interactions can emerge, so we also discuss how to realize the desired
positive and negative interactions for this case in 3D.

Finally, we note that it is important to be able to turn off the interactions
for individual qubits as necessary. There are two main ways that this can be
done. First, the qubit states can be transferred to non-interacting states.
For example, when considering Rydberg systems, the Rydberg states can be
coherently transferred to non-interacting ground states via an optical drive.
In some cases, there may not be any fully non-interacting states which can be
utilized during the protocol. In these cases, one can utilize similar spin-
echo techniques as discussed above in order to remove the unwanted
interactions. This can be achieved as follows. First, during the positive
interaction evolution, each target qubit requires an interaction time $t_{i}$.
However, without non-interacting shelving states, each control qubit will
interact with the target qubits for a time
$t_{\text{max}}\equiv\max_{i}t_{i}$. To remove the excess interactions felt by
an individual target qubit, a $\pi$ pulse is applied to the corresponding
target qubit after time $t_{i}+(t_{\text{max}}-t_{i})/2$, and the remaining
evolution time of $(t_{\text{max}}-t_{i})/2$ reverses the interactions,
leading to an effective interaction time of $t_{i}$. The same approach is used
for the negative interaction evolution, thus removing all unwanted
interactions.

###  II.1 Polar molecules

In the following subsections, we provide further details regarding the
experimental implementation of the state-transfer procedures described in
Section I. First, we will describe how to perform state transfer using dipole-
dipole interactions in a system of polar alkali-metal dimers.

We will assume the molecules are arrayed in a 2D lattice with a single
molecule per lattice site and negligible tunneling of molecules. The single
molecule Hamiltonian is

$H_{0}=B_{0}\mathbf{N}^{2}-d_{0}E,$ (7)

where $B_{0}$ is the rotational constant of the electronic and vibrational
ground state of the molecule, $E$ is the electric field strength, and
$d_{p}=\mathbf{\hat{e}_{p}}\cdot\mathbf{d}=d\sqrt{4\pi/3}Y_{p}^{1}(\theta,\phi)$
($p=0,\pm 1$) is a component of the dipole operator $\mathbf{d}$ where
$\mathbf{\hat{e}_{0}}=\mathbf{\hat{z}},\mathbf{\hat{e}_{\pm
1}}=\mp(\mathbf{\hat{x}}\pm i\mathbf{\hat{y}})/\sqrt{2}$, $d$ is the permanent
dipole moment of the molecule, and $Y_{p}^{1}$ are spherical harmonics.

We will encode the computational states in the states $|N,M\rangle$, which are
eigenstates of the angular momentum operator $\mathbf{N}^{2}$ and its
projection $N_{z}$ on the quantization axis in the electronic and vibrational
ground state of the molecule. Defining $|\phi_{N,M}\rangle$ as the states
which are adiabatically connected (as we turn on $E$) to $|N,M\rangle$, we
consider in scheme (a) (see Fig. 4) dressing the $|\phi_{0,0}\rangle$ and
$|\phi_{1,0}\rangle$ states to engineer the $|0\rangle$ state and use
$|\phi_{2,-2}\rangle$ as the $|1\rangle$ state. The $|\phi_{0,0}\rangle$ and
$|\phi_{1,0}\rangle$ states will be dressed via a resonant microwave drive
with $\pi$ polarization and Rabi frequency $\Omega_{\pi}$, defining
$|0\rangle=(|\phi_{0,0}\rangle+|\phi_{1,0}\rangle)/\sqrt{2}$. Although our
method can be generalized to off-resonant drives, we will focus on a resonant
drive since it provides the strongest interactions in the absence of an
electric field, but we discuss the case of off-resonant drive in Appendix C.
Since the electric field does not generate a transition dipole moment between
$|0\rangle$ and $|1\rangle$, no flip-flop interactions are introduced. We note
that, although our formalism will consider a nonzero electric field, our
method works without issue in the absence of an electric field as well.
Additionally, throughout, we will assume that the drive strength is always
sufficiently strong that the hyperfine structure of the states being dressed
can be neglected.

Figure 4: Schemes for realizing dipole-dipole Ising interactions using polar
molecules, where $J_{z}$ denotes the strength of the Ising interactions. The
polarization of the drive determines whether (a) positive $J_{z}$ or (b)
negative $J_{z}$ is realized. Each approach works both in the presence or
absence of a DC electric field, although the interactions realized in (b)
exhibit positive $J_{z}>0$ if $d|E|/B_{0}\gtrsim 1.7$.

The relevant dipole moments of interest are
$\mu_{0}=\langle\phi_{0,0}|d_{0}|\phi_{0,0}\rangle$,
$\mu_{1}=\langle\phi_{1,0}|d_{0}|\phi_{1,0}\rangle$,
$\mu_{2}=\langle\phi_{2,-2}|d_{0}|\phi_{2,-2}\rangle$, and
$\mu_{01}=\langle\phi_{0,0}|d_{0}|\phi_{1,0}\rangle$. In units of
$(4\pi\varepsilon_{0})^{-1}$, the corresponding dipole-dipole interactions
between $|0\rangle$ and $|1\rangle$ are (see Appendix C)

$\begin{split}H_{\text{int}}=\frac{1}{2}\sum_{i\neq
j}\frac{1}{r_{ij}^{3}}&\left[(V_{0}+V_{1}-2V_{01})Z_{i}Z_{j}\right.+\\\
&\left.(V_{0}-V_{1})(Z_{i}+Z_{j})+(V_{0}+V_{1}-2V_{01})\right]\end{split}$
(8a) $\displaystyle V_{0}=\frac{(\mu_{0}+\mu_{1})^{2}+2\mu_{01}^{2}}{16},\quad
V_{1}=\frac{\mu_{2}^{2}}{4},$ (8b) $\displaystyle
V_{01}=\frac{(\mu_{0}+\mu_{1})\mu_{2}}{8},$ (8c)

which has an Ising interaction proportional to $V_{0}+V_{1}-2V_{01}>0$ for all
$d|E|/B_{0}$. There are two considerations to avoid leakage to other states.
First, there are the resonant flip-flop interactions present in the absence of
an electric field. As long as $\Omega_{\pi}\gg H_{\text{dd}}$, these will be
made off-resonant through the dressed energies. Second, when the electric
field is turned on, additional potentially near-resonant interactions emerge
due to the induced dipole moments, such as flip-flop interactions between the
$|\phi_{1,0}\rangle$, $|\phi_{1,1}\rangle$ states. Here, the dressed energies
will typically keep these interactions off-resonant, although care has to be
taken when $d|E|\approx|\Omega_{\pi}|$, as the energy shifts due to the
electric field could induce resonances in combination with the dressed
energies.

In scheme (b), see Fig. 4, to realize negative interactions, we instead
consider dressing the $|\phi_{0,0}\rangle$ and $|\phi_{1,1}\rangle$ states to
engineer the $|0^{\prime}\rangle$ state and continue to use
$|\phi_{2,-2}\rangle$ as the $|1\rangle$ state. The $|\phi_{0,0}\rangle$ and
$|\phi_{1,1}\rangle$ states will be dressed via a resonant microwave drive
with $\sigma_{+}$ polarization and Rabi frequency $\Omega_{\sigma}$, defining
$|0^{\prime}\rangle=(|\phi_{0,0}\rangle+|\phi_{1,1}\rangle)/\sqrt{2}$. Again,
we will focus on the case of a resonant drive, which realizes the strongest
interactions in the absence of an electric field. Like with scheme (a),
although our formalism assumes a nonzero electric field, scheme (b) also works
in the absence of an electric field. However, we note that at sufficiently
strong electric fields, the sign of the interactions can become positive.

The additional relevant dipole moments of interest are
$\mu_{1}^{\prime}=\langle\phi_{1,1}|d_{0}|\phi_{1,1}\rangle$ and
$\mu_{0,1}^{\prime}=\langle\phi_{0,0}|d_{-}|\phi_{1,1}\rangle$. Once again, in
units of $(4\pi\varepsilon_{0})^{-1}$, the corresponding dipole-dipole
interactions between $|0^{\prime}\rangle$ and $|1\rangle$ are

$\begin{split}H_{\text{int}}=\frac{1}{2}\sum_{i\neq
j}\frac{1}{r_{ij}^{3}}&\left[(V_{0}^{\prime}+V_{1}-2V_{01}^{\prime})Z_{i}Z_{j}\right.+\\\
&\left.(V_{0}^{\prime}-V_{1})(Z_{i}+Z_{j})+(V_{0}^{\prime}+V_{1}-2V_{01}^{\prime})\right]\end{split}$
(9a) $\displaystyle
V_{0}^{\prime}=\frac{(\mu_{0}+\mu_{1}^{\prime})^{2}-\mu_{01}^{\prime
2}}{16},\quad V_{1}=\frac{\mu_{2}^{2}}{4},$ (9b) $\displaystyle
V_{01}^{\prime}=\frac{(\mu_{0}+\mu_{1}^{\prime})\mu_{2}}{8},$ (9c)

which has an Ising interaction proportional to
$V_{0}^{\prime}+V_{1}-2V_{01}^{\prime}$, which is less than 0 for for
$d|E|/B_{0}\lesssim 1.7$. This value is modified by off-resonant drives,
increasing or decreasing its value depending on the relative superposition of
the two states being dressed (see Appendix C). Like for scheme (a), leakage to
other states can be avoided with sufficiently strong $\Omega_{\sigma}$
compared to $H_{\text{dd}}$ and by ensuring the energy shifts due to the
electric field do not induce any new resonances.

Thus, by switching between the two schemes and simultaneously transferring the
$|0\rangle$ population to the $|0^{\prime}\rangle$ state, it is possible to
implement the state-transfer protocols, although it is important to avoid
unwanted interactions when switching between the schemes and states. Provided
the relevant time-scales are fast enough relative to the interactions, one can
transfer the states by adiabatically turning $\Omega_{\pi}$ off while
adiabatically turning $\Omega_{\sigma}$ on in a STIRAP-like process, which
simultaneously changes between the two schemes and changes the sign of the
interactions as long as $d|E|/B_{0}\lesssim 1.7$.

At the end of the state-transfer protocol, it is important to shelve the
qubits in non-interacting states. A natural approach is to adiabatically turn
off the DC electric and microwave dressing fields. By adiabatically detuning
the dressing field simultaneously, this ensures that the $|0\rangle$ or
$|0^{\prime}\rangle$ state will be transferred to either of the two states
which are dressed, while $|1\rangle$ becomes $|2,-2\rangle$. Alternatively,
one can keep the fields on by storing the qubits in pairs of states which
possess the same electric dipole moment, and the resulting interactions will
only generate a global phase on each individual qubit. In this case, all qubit
states must be transferred simultaneously. For example, this might be achieved
by storing the qubit in $|\phi_{N,\pm M}\rangle$ pairs of states. Nuclear spin
states provide another natural means of realizing this while utilizing a
single $|\phi_{N,M}\rangle$ state. Although we refer to these as non-
interacting states, in reality they are states which exhibit state-
insensitive, identity interactions both with each other and with the states
$|0\rangle$ and $|1\rangle$ (or $|0^{\prime}\rangle$ and $|1\rangle$) used in
the protocol, although the strength of the identity interactions may be
different.

Non-interacting states can in principle be prepared for use during the
protocols. For example, in the absence of an electric field, any states
without a transition dipole moment with the dressed states can be utilized as
long as the two chosen states have identity interaction with each other, like
nuclear spin states as above. This is more difficult in the presence of an
electric field, as the permanent dipole moments will typically cause
interactions with the $|0\rangle,|0^{\prime}\rangle,$ and $|1\rangle$ states.
By choosing parameters such that the induced dipole moments of $|0\rangle$ and
$\ket{1}$ (or $|0^{\prime}\rangle$ and $\ket{1}$) are the same, while the two
non-interacting states have the same induced dipole moment and identity
interactions with each other, the interactions with the non-interacting qubits
become state-insensitive, leading only to a global phase. When the induced
dipole moment of $|0\rangle$ and $\ket{1}$ are not the same, the interactions
with the non-interacting states lead to $Z$ rotations in $|0\rangle$ and
$\ket{1}$, which must be corrected (likewise for $|0^{\prime}\rangle$ and
$\ket{1}$).

Now, we determine when these single qubit corrections can be avoided for the
two schemes. When $E\neq 0$, for scheme (a), this requires $\mu_{2}$ be
between $\mu_{0}$ and $\mu_{1}$ while for scheme (b), this requires $\mu_{2}$
be between $\mu_{0}$ and $\mu_{1}^{\prime}$. While this is possible for scheme
(a) with the proper choice of drive and detuning (changing the relative
contributions of $\mu_{0},\mu_{1}$ to the induced dipole moment of
$|0\rangle$), it is not possible for scheme (b). However, if $E=0$, this is
automatically satisfied in both cases. Hence there are no single qubit
corrections needed when there is no electric field.

#### Asymmetric blockade

In this section, we discuss the realization of microwave-dressed asymmetric
blockade for polar molecules, which was developed for Rydberg atoms in Ref.
[43]. This provides a means of realizing the desired control-target
interactions without needing to engineer both signs of interactions to
eliminate the control-control and target-target interactions. Here, we discuss
how this approach can be extended to polar molecules. Aside from a few
subtleties which we discuss further below, the original analysis for the
Rydberg atoms is largely the same, so we leave several of these details to
Appendix C.

In contrast to the microwave-dressing approaches above, we consider dressing
three states with two different drives, one which is $\pi$-polarized and one
which is $\sigma_{\pm}$-polarized, with no electric field. We denote two of
the three resulting dressed states $|c\rangle,|t\rangle$, which are used as
controls or targets, respectively. Through proper tuning of the drives, these
states exhibit asymmetric blockade: no control-control or target-target
interactions but strong diagonal dipolar control-target interactions. This is
achieved by counterbalancing the respective dipolar interactions associated
with the two driven transitions, which differ in their overall sign due to the
polarization of the transition, as utilized in the previous dressing schemes
to realize both signs of interactions. The combination here eliminates the
necessity to realize both positive and negative interactions because the
unwanted interactions between controls and targets are already zero. Thus if
we use the $|c\rangle$ ($|t\rangle$) state to encode the $|0\rangle$ state for
the control (target) molecules and an additional non-interacting state (e.g.,
from $N=4$) to encode $|1\rangle$ for both, we realize the interactions

$H_{\text{int}}=\sum_{i\in\\{C\\},j\in\\{T\\}}\frac{C_{3}}{4r_{ij}^{3}}(1-Z_{i})(1-Z_{j}),$
(10)

where $C_{3}$ is the coefficient describing the strength of the dipolar
$c$-$t$ interactions and $\\{C\\},\\{T\\}$ define the sets of control and
target atoms, respectively.

Figure 5: (a) Example of dressing scheme used to realize asymmetric blockade
in polar molecules. The dressed $|0\rangle,|1\rangle$ states are composed of
the molecular states denoted by solid lines. The remaining dashed levels
contribute to the vdW interactions. (b) Dipolar c-t interaction coefficient
($C_{3}$, solid blue line), vdW c-c interaction coefficient ($C_{6}^{(c)}$,
dashed green line), and vdW t-t interaction coefficent ($C_{6}^{(t)}$, dashed
orange line) as functions of $\mathcal{C}$. Values of $C_{3}$ are in units of
$d^{2}/(4\pi\varepsilon_{0})$ and $C_{6}$ in units of
$d^{4}/[\Omega_{\pi}(4\pi\varepsilon_{0})^{2}]$.

Here, we consider the particular example of Fig. 5(a), dressing states
$|0,0\rangle,|1,0\rangle,$ and $|2,1\rangle$ with the rotating-frame
Hamiltonian

$\displaystyle H_{\text{mw}}=$ $\displaystyle\leavevmode\nobreak\
\Omega_{\pi}(|0,0\rangle\langle 1,0|+|1,0\rangle\langle
0,0|)\leavevmode\nobreak\ +$ (11) $\displaystyle\leavevmode\nobreak\
\Omega_{\sigma}(|2,1\rangle\langle 1,0|+|1,0\rangle\langle
2,1|)+\leavevmode\nobreak\ $ $\displaystyle\leavevmode\nobreak\
-\Delta_{\pi}|0,0\rangle\langle 0,0|-\Delta_{\sigma}|2,1\rangle\langle 2,1|,$

where $\Omega,\Delta$ denote the Rabi frequencies and detunings of the
corresponding drives. The resulting dressed states take the form

$|c\rangle\propto c_{\pi}|0,0\rangle+|1,0\rangle+c_{\sigma}|2,1\rangle,$ (12a)
$|t\rangle\propto t_{\pi}|0,0\rangle+|1,0\rangle+t_{\sigma}|2,1\rangle,$ (12b)

in addition to a third dressed state which we will not use. For this choice of
states, the maximal dipolar interaction coefficient between $|c\rangle$ and
$|t\rangle$ is $C_{3}=0.07\frac{d^{2}}{4\pi\varepsilon_{0}r^{3}}$, which has
an associated $c_{\pi}^{\text{max}}$ coefficient that uniquely defines the
other coefficients. However, $|c\rangle,|t\rangle$ are degenerate in the
rotating frame of the drives when this maximal interaction is realized, which
can lead to additional unwanted dipolar interactions (e.g., $|cc\rangle\langle
tt|$). By working with $c_{\pi}$ close to $c_{\pi}^{\text{max}}$ but not equal
to it, which we quantify via $\mathcal{C}\equiv c_{\pi}/c_{\pi}^{\text{max}}$,
this degeneracy can be lifted while still realizing asymmetric blockade with a
comparable interaction strength, see Fig. 5(b).

An important consideration is the emergence of $1/r^{6}$ vdW
$|cc\rangle\langle cc|,|tt\rangle\langle tt|$ interactions with coefficients
$C_{6}^{(c)},C_{6}^{(t)}$, respectively, which can lead to errors. For Rydberg
atoms, there were two degrees of freedom that could be utilized to tune the
vdW interactions: $\mathcal{C}$ and the overall scale of $H_{\text{mw}}$. The
former modifies the dressed states while the latter modifies the dressed state
energies. However, for polar molecules the overall scale of $H_{\text{mw}}$
only determines the overall scale of the resulting vdW interactions. This is
because unlike Rydberg atoms which have many internal Rydberg states
contributing to the vdW interactions, for polar molecules the transition
energies involved differ on the order of GHz, several orders of magnitude
larger than feasible dipolar interactions and microwave dressing fields.
Hence, the only contributions to the vdW interactions arise from the typically
resonant dipole-dipole interactions (in the absence of external fields) that
are made off-resonant by the dressing field, which thus defines the overall
scale of the vdW interactions. In the example we consider here, these involve
all of the $N=0,1,2$ states.

In Fig. 5(b), we plot both the dipolar coefficient $C_{3}$ and the vdW
coefficients $C_{6}^{(c/t)}$ as a function of $\mathcal{C}$ in units of
$d^{2}/(4\pi\varepsilon_{0})$ and
$d^{4}/[\Omega_{\pi}(4\pi\varepsilon_{0})^{2}]$, respectively. First, we note
that there are several divergences in the $C_{6}$ coefficients. These are a
consequence of pair states composed of $|0\rangle$ or $|1\rangle$ becoming
degenerate with other dressed pair states. Since this leads to leakage outside
of the computational states, these parameters must be avoided. Additionally,
we see that, for the example considered, while one of the $C_{6}$ passes
through zero for certain values of $\mathcal{C}$, the other does not.
Nevertheless, we remark that the overall scale that relates the strength of
the vdW interactions to dipolar interactions is given by
$d^{2}/(\Omega_{\pi}4\pi\varepsilon_{0}r^{3})$, i.e., the ratio of the dipolar
interactions to the dressing fields. As before, our dressing scheme already
requires $\Omega_{\text{mw}}\gg H_{\text{dd}}$, so this is already small to
begin with, but we wish to minimize it as much as possible. If we assume
$\Omega_{\pi}$ to be on the order of MHz and
$d^{2}/(4\pi\varepsilon_{0}r^{3})$ to be on the order of kHz, this ensures
that any vdW interactions are at least 1000 times smaller. Furthermore, we
remark that, if an additional dressing field were included, this would likely
provide sufficient tunability to fully eliminate both vdW interactions, but we
leave this direction to future work.

###  II.2 Rydberg atoms

In this section, we consider the implementation of the state-transfer protocol
using Rydberg atoms, which will share several similarities with the
implementation in polar molecules. In order to do so, as shown in Fig. 6, we
will consider encoding $|0\rangle,|1\rangle$ in linear combinations of four
different states $|g\rangle$, $|s\rangle$, $|p_{\pi}\rangle$, and
$|p_{\sigma}\rangle$, corresponding to a ground state, a Rydberg $s$ state,
and two Rydberg $p$ states, respectively. In the case of the latter two
states, $0/+$ correspond to the polarization of the drive needed to couple the
$s$ state to the corresponding $p$ state. Additionally, we will assume that
the $p$ states have similar values of the principal quantum number $n$
compared to the $s$ state, leading to strong dipole-dipole interactions
between $s$ states and $p$ states. While Ref. [30] also realized the desired
form of interactions for Rydberg atoms via a weak electric field, the scheme
discussed here allows for much stronger interactions, providing faster state
transfer with fewer errors due to dissipation.

In order to realize $H_{\text{int}}$, we will consider dressing the
$|s\rangle$ state with the $|p_{\pi}\rangle$ state with a strong resonant
microwave field. In the absence of any external fields, these two states
interact according to the dipole-dipole interaction

$H_{\text{dd}}=\sum_{i\neq
j}\mu_{0}^{2}\frac{1-3\cos^{2}\theta_{ij}}{r_{ij}^{3}}|s_{i}p_{j,\pi}\rangle\langle
p_{i,\pi}s_{j}|,$ (13)

where $r_{ij}$ is the distance between atoms $i$ and $j$, $\theta_{ij}$ is the
angle the displacement vector makes with the quantization axis, and
$\mu_{0}=\langle p_{\pi}|d_{0}|s\rangle$ is the corresponding transition
dipole moment. Like for polar molecules, we have neglected additional resonant
interaction terms involving other states because the introduction of a strong
microwave field will cause these to become strongly off-resonant due to the
resulting energy shifts. Additionally, since we are considering dipolar state
transfer only in $d=2$, we shall consider $\theta_{ij}=\pi/2$ throughout.

Figure 6: Dressing scheme for realizing (left) $H_{\text{int}}$ and (right)
$H_{\text{int}}^{\prime}$ via microwave dressing of Rydberg states, with
$|1\rangle=(|s\rangle+|p_{\pi}\rangle)/\sqrt{2}$ and
$|1^{\prime}\rangle=(|s\rangle+|p_{\sigma}\rangle)/\sqrt{2}$. In each case, we
require $\Omega_{0/+}\gg H_{\text{dd}}$ so that we may work in the microwave-
dressed basis and make additional flip-flop interactions (e.g. those between
$\ket{s}$ and the dashed-line states) off-resonant.

Using the strong resonant microwave field, we may encode $|0\rangle=|g\rangle$
and $|1\rangle=(|s\rangle+|p_{\pi}\rangle)/\sqrt{2}$. In the limit of a strong
microwave Rabi frequency $\Omega_{\text{mw}}\gg H_{\text{dd}}$, we may express
the dipole-dipole interactions in the $|0\rangle,|1\rangle$ basis as

$H_{\text{int}}=\frac{1}{2}\sum_{i\neq
j}\frac{\mu_{0}^{2}}{8}\frac{1}{r_{ij}^{3}}(1-Z_{i})(1-Z_{j}).$ (14)

In order to realize $H^{\prime}_{\text{int}}\equiv-aH_{\text{int}}$, we will
use the same approach as for $H_{\text{int}}$. However, rather than dressing
the $|s\rangle$ state with $|p_{\pi}\rangle$, we instead resonantly dress it
with $|p_{\sigma}\rangle$. In this case, the corresponding dipole-dipole
interaction is

$H_{\text{dd}}^{\prime}=\sum_{i\neq
j}-\frac{\mu_{+}^{2}}{2}\frac{1}{r_{ij}^{3}}|s_{i}p_{j,\sigma}\rangle\langle
p_{i,\sigma}s_{j}|,$ (15)

where $\mu_{+}=\langle p_{\sigma}|d_{+}|s\rangle$ and we neglect the other
terms due to the energy shifts caused by the strong microwave field once
again. Note that, aside from a constant factor, the dipole-dipole interaction
here only differs by a sign from $H_{\text{dd}}$. As such, if we now encode
$|1^{\prime}\rangle=(|s\rangle+|p_{\sigma}\rangle)/\sqrt{2}$ and again take
the limit of a strong microwave Rabi frequency $\Omega_{\text{mw}}\gg
H_{\text{dd}}$, the resulting interactions are

$H_{\text{int}}^{\prime}=-\frac{1}{2}\sum_{i\neq
j}\frac{\mu_{+}^{2}}{16}\frac{1-3\cos^{2}\theta}{r_{ij}^{3}}(1-Z_{i})(1-Z_{j}),$
(16)

which differs from $H_{\text{int}}$ by a negative constant factor.

Like for polar molecules, by transferring the $|1\rangle$ population to the
$|1^{\prime}\rangle$ state while also changing the polarization of the
microwave field, it is possible to implement the state-transfer protocols via
Rydberg atoms. Note that when the polarization of the drive is changed, one
must ensure that the qubit state is temporarily not encoded in
$|s\rangle,|p_{\pi}\rangle,$ or $|p_{\sigma}\rangle$. Alternatively, provided
the relevant time-scales are fast enough relative to the interactions, one can
transfer the states by adiabatically turning $\Omega_{\pi}$ off while
adiabatically turning $\Omega_{\sigma}$ on, which simultaneously changes
between the two schemes and changes the sign of the interactions.
Additionally, to avoid leakage out of the computational basis, like with the
polar molecules we utilize the fact that $\Omega\gg H_{\text{dd}}$, so the
dressed energies will make these leakage processes off-resonant.

As discussed in the polar molecules section, it is also possible to realize
the desired control-target interactions without the spin-echo approach by
engineering asymmetric blockade interactions. This is achieved by turning on
both drives $\Omega_{\pi/\sigma}$ with detunings $\Delta_{\pi/\sigma}$ and
using different principal quantum numbers $n$ for $|p_{\pi/\sigma}\rangle$.
With a proper choice of parameters, two of the resulting dressed states
$|c\rangle,|t\rangle$ interact with each other but not themselves, realizing
asymmetric blockade. Thus by forming the control qubits with $|c\rangle$ and
the target qubits with $|t\rangle$ as the logical $|1\rangle$ states and the
ground state as $|0\rangle$ for both, the desired pure control-target
interactions can be realized. The full details of engineering this form of
interactions for Rydberg atoms be found in Ref. [43].

#### Rydberg vdW interactions in 3D

We briefly discuss how to use $1/r^{6}$ vdW interactions for state transfer in
3D. While this will not be as fast as using $1/r^{3}$ dipole-dipole
interactions ($t\sim e^{3\sqrt{3\log L}}$ for vdW interactions with the Tran
protocol vs. $t\sim\text{polylog}(L)$ for the Tran and Eldredge protocols with
dipolar interactions), some aspects are less complicated since the vdW
interactions will have no significant angular dependence. Moreover, it
provides a natural means of experimentally demonstrating that vdW interactions
are long-range in three dimensions.

One option to realize the desired sign change in the vdW interactions is by
inducing a Förster resonance for an $s$ state via an external magnetic field.
Provided one keeps the atoms sufficiently far apart outside of the dipole-
dipole interaction regime, where the interaction causes pairs of $s$ states to
hybridize with pairs of $p$ states, the sign of $C_{6}$ will be opposite on
either side of the Förster resonance [44].

Alternatively, microwave drive approaches can be used here as well. In this
case, we would use an asymmetric blockade approach as in Ref. [43] applied to
vdW interactions. Ref. [45] discusses how a microwave-dressed superposition of
Rydberg $s$ states can experience no vdW interactions. By engineering two such
states with two pairs of Rydberg $s$ states, only the interactions between the
two states remains. By separating these two pairs of $s$ states from each
other in $n$, the flip-flop interactions will fall off rapidly compared to the
diagonal vdW interactions as in Ref. [46], leaving only the desired Ising
interactions.

Figure 7: Level structure and dressing schemes for bosonic dysprosium atoms.
We show only a subset of the full $J=8$ manifold. In all three schemes, a far-
detuned optical field is included, while a magnetic field is also applied in
(b) and (c). (a) Dressing scheme for realizing positive interactions. (b)
Dressing scheme for realizing negative interactions. (c) Dressing scheme for
realizing non-interacting states.

###  II.3 Magnetic atoms (dysprosium)

In this subsection, we consider the implementation of the state-transfer
protocol in a two-dimensional lattice of ultracold dysprosium atoms. For
concreteness, we consider bosonic isotopes of dysprosium (such as 162Dy,
164Dy), so that the nuclear spin is zero. We choose to encode the
computational states $\ket{0},\ket{1}$ in the Zeeman manifold of the
electronic ground state of the dysprosium. The interactions between a pair of
dysprosium atoms $i,j$ are given by the following magnetic dipole-dipole
interactions—written in units of $\mu_{0}(g_{J}\mu_{B})^{2}/4\pi$, where
$g_{J}$ is the Landé $g$ factor, $\mu_{B}$ is the Bohr magneton, and $\mu_{0}$
is the vacuum magnetic permeability (not to be confused with the dipole
transition moment used previously) [47]:

$\begin{split}H_{ij}=\frac{1}{r_{ij}^{3}}\Big{[}(&1-3\cos^{2}\theta_{ij})[J_{i}^{z}J_{j}^{z}-\frac{1}{4}(J_{i}^{+}J_{j}^{-}+J_{i}^{-}J_{j}^{+})]\\\
&-\frac{3}{4}\sin^{2}\theta_{ij}[e^{-2i\phi_{ij}}J_{i}^{+}J_{j}^{+}+\text{H.c}]\Big{]}.\end{split}$
(17)

Here, $J=8$ is the electronic angular momentum of the ground state, and
$\theta_{ij},\phi_{ij}$ are the polar and azimuthal angles of the displacement
vector between sites $i$ and $j$ with respect to the quantization axis, which
is set by a far-detuned optical field. This field additionally breaks the
degeneracy of the Zeeman levels $\ket{M}_{J},-8\leq M\leq 8$, which can be
utilized to limit leakage out of the computational basis. Since the
corresponding energy shifts are quadratic in $M$, in some cases it might also
be necessary utilize a linear energy shift in $M$ via a Zeeman shift, further
breaking degeneracies.

In the following, we will describe example dressed states that can realize the
desired forms of interactions. While we will not discuss precise dressing
schemes to prepare these dressed states, the same principles will hold as for
polar molecules and Rydberg atoms. Specifically, the dressing field must be
stronger than the resulting dressed interactions. Additionally, it is
important to ensure that the interactions do not lead to leakage out of the
computational basis. While typically we expect the dressing to ensure any such
processes will be off-resonant, we will discuss which leakage processes need
to be considered. As always, we restrict our focus to $\theta_{ij}=\pi/2$,
which here allows us to neglect the $J_{i}^{+}J_{j}^{+}$ terms in the
Hamiltonian as well.

The simplest way of realizing the diagonal interactions that we require is to
make use of the $J_{i}^{z}J_{j}^{z}$ term in Eq. 17, setting
$\ket{0}\equiv\ket{-m}_{J},\ket{1}\equiv\ket{m}_{J}$ with
$\absolutevalue{m}\geq 2$, as shown in Fig. 7(a). The choice $m=8$ maximizes
the interaction strength and yields the Hamiltonian

$H_{\text{int}}=\frac{1}{2}\sum_{i\neq j}\frac{64}{r_{ij}^{3}}Z_{i}Z_{j}$ (18)

Flip-flop interactions can be avoided by once again using an off-resonant
optical field. Neither a magnetic field nor a dressing field is necessary in
this case since we utilize the Zeeman levels directly.

Figure 8: Scheme for implementing different colors in 2D. Each plaquette only
interacts with other plaquettes of the same “color,” e.g. plaquettes labeled A
(green) only talk to each other. For $n$ different colors, the distance
between plaquettes of the same color scales as $R=\sqrt{n}L$, where $L$ is the
size of the individual plaquette. In subfigures (a) and (b), we depict the
Tran protocol generating a GHZ state with characteristic length $r=6L$ using
$n=4$ and $n=9$ colors respectively; the distances between plaquettes of the
same color will be $R=2L$ and $R=3L$ respectively. (c) The spin-echo pulse
sequence that cancels out interactions between plaquettes with different
colors. Vertical lines between the $+$’s and $-$’s denote the spin-echo
pulses. For $n$ colors, the protocol will in general require $2^{n-1}$
distinct pulses.

In order to realize the reverse Hamiltonian, i.e the same Hamiltonian but with
opposite sign, we will make use of the flip-flop interactions in Eq. 17. Using
microwave and/or optical dressing, we can choose, for example, the following
dressed states: $\ket{0^{\prime}}\equiv(\ket{-4}_{J}+\ket{4}_{J})/\sqrt{2}$,
and
$\ket{1^{\prime}}\equiv\frac{1}{\mathcal{N}}(a\ket{8}_{J}+b\ket{7}_{J}+c\ket{-8}_{J})$,
where the $a,b,c$ amplitudes are tunable via the dressing parameters and
$\mathcal{N}$ is a normalization constant [Fig. 7(b)]. This choice clearly
yields $\bra{0^{\prime}}J_{z}\ket{0^{\prime}}=0$ and by properly tuning
$a,b,c$ (namely, setting
$\absolutevalue{b}^{2}=8(\absolutevalue{c}^{2}-\absolutevalue{a}^{2})/7$) we
can also ensure that $\bra{1^{\prime}}J_{z}\ket{1^{\prime}}=0$. In this case,
only the flip-flop interactions from Eq. 17 contribute, for which the only
nonzero matrix element is
$\bra{1^{\prime}1^{\prime}}H_{ij}\ket{1^{\prime}1^{\prime}}$. This realizes
the new interaction Hamiltonian

$H^{\prime}_{\mathrm{int}}=-\frac{1}{2}\sum_{i\neq
j}\frac{2\absolutevalue{ab}^{2}}{r_{ij}^{3}\absolutevalue{\mathcal{N}}^{4}}(1-Z_{i})(1-Z_{j}),$
(19)

with the same $Z_{i}Z_{j}$ interactions as before, but with the opposite sign.

Finally, we describe how to realize non-interacting states. We cannot use the
bare Zeeman levels, since they will always have non-trivial interactions,
particularly due to the $J_{i}^{z}J_{j}^{z}$ term in Eq. 17. Thus, we propose
to use the dressed states
$\ket{0^{\prime\prime}}\equiv|0\rangle_{J},\ket{1^{\prime\prime}}\equiv(\ket{-2}_{J}+\ket{2}_{J})/\sqrt{2}$
(which can be realized using microwave or optical dressing) to encode the
computational $\ket{0^{\prime\prime}},\ket{1^{\prime\prime}}$ states, as shown
in Fig. 7(c). Due to the choice of states, both have $\langle J_{z}\rangle=0$
and are not coupled via the flip-flop interactions. We could not have used
$(\ket{-2}_{J}-\ket{2}_{J})/\sqrt{2}$ as the $|0^{\prime\prime}\rangle$ state
(which could be achieved with no dressing in the absence of a Zeeman shift)
because the Ising interactions would induce resonant flip-flop interactions in
the computational basis. Thus dressing is necessary to realize non-interacting
states.

In schemes (b) and (c), it is important in to ensure that the bare $H_{ij}$
interactions do not induce leakage to states outside of the computational
basis. For example in (b), the flip-flop interactions can take
$|7\rangle_{J}|8\rangle_{J}\to|8\rangle_{J}|8\rangle_{J}$ or
$|{-4}\rangle_{J}|4\rangle_{J}\to|{-3}\rangle_{J}|5\rangle_{J}$, while the
Ising interactions could take
$|0^{\prime}\rangle|0^{\prime}\rangle\to(|{-4}\rangle_{J}-|4\rangle_{J})(|{-4}\rangle_{J}-|4\rangle_{J})/2$.
While in (a) these are automatically off-resonant due to the far-detuned
optical field or 0 due to the choice of states, similar processes exist for
(c). Although these interactions may or may not be resonant in the undressed
basis, this can change in the dressed basis. In light of this, it is important
that any dressed states which have overlap with the bare Zeeman levels
involved in these interaction have dressed energies which ensure that any
potential leakage is off-resonant. In some cases, the dressing fields will be
sufficient to ensure that they are off-resonant; however, in cases where this
is not the case, for example due to the $M\to-M$ symmetry, the Zeeman shift
(in addition to the far-detuned optical field) provides another degree of
tunability which simultaneously removes any accidental resonances from the
symmetry.

##  III Crosstalk interactions

One concern in the implementation of the protocol by Tran _et al._ [31] is
that, in the process of simultaneously creating GHZ states on multiple
registers of qubits, crosstalk interactions may occur between qubits in
different clusters. Specifically, while the spin-echo procedure is able to
eliminate interactions within and between registers of control (target)
qubits, the control qubits in one register may interact with target qubits in
another register, leading to undesirable crosstalk interactions. In this
section, we analyze the extent of these crosstalk interactions and propose a
few methods to address them.

Recall from Section I that, in the Tran protocol, in order to encode a qubit
into a generalized GHZ state of characteristic length $r$, it is first
necessary to create $m^{d}$ GHZ states (including one generalized GHZ state)
with characteristic length $r_{1}\equiv r/m$ in parallel, where $m$ is some
$\alpha$-dependent factor. In Fig. 8, we illustrate the set-up using qubits
arrayed in a 2D grid. We take $L=r_{1}$ to be the characteristic length of one
of the $m^{2}$ ($d=2$) plaquettes $C_{1},C_{2},\dots,C_{m^{2}}$—each
containing $L^{2}$ qubits. Then, crosstalk interactions may arise due to the
$L^{4}$ pairwise couplings present between qubits in different plaquettes.

Using a variation of the spin-echo protocol with additional single-qubit
$\pi/2$ pulses, it’s possible to cancel the interactions between controls and
targets in different registers as follows. We label the $m^{d}$ registers with
$n$ different “colors” and apply single-qubit pulses that cancel the crosstalk
interactions between the qubits in registers with different colors. The pulse
sequence for the protocol is shown for $n=4$ in Fig. 8(c). The sequence of
single-qubit pulses will lead to alternating signs of the interaction
Hamiltonians and lead to cancellations between registers with different
colors. However, interactions will still occur between qubits in registers
with the same color. To estimate the crosstalk error incurred by these
interactions, we perform the following analysis.

Consider in the worst case that all qubits in plaquettes with the same color
will interact with each other during a single GHZ-creation step. Then,
multiply the total strength of this crosstalk interaction by the number of
colors and the time required to create the GHZ state, and then sum this
quantity over the total number of recursive steps required to implement the
full protocol. This leads to the following bound on the total crosstalk error
[full details in Appendix B]:

$\displaystyle\varepsilon_{\text{tot}}=\begin{cases}\mathcal{O}\left\lparen\frac{r^{2d-\alpha}}{n^{\alpha/d-1}}\text{polylog}(r)\right\rparen&\text{if
}d<\alpha<2d,\\\ \mathcal{O}\left\lparen\frac{e^{{\gamma}\sqrt{\log
r}}}{n}\text{polylog}(r)\right\rparen&\text{if }\alpha=2d,\\\
\mathcal{O}\left\lparen\frac{1}{n^{\alpha/d-1}}\text{polylog}(r)\right\rparen&\text{if
}2d<\alpha\leq 2d+1.\end{cases}$ (20)

As such, the number of colors required in order to achieve a fixed amount of
crosstalk error [$\varepsilon_{\text{tot}}=\mathcal{O}\left\lparen
1\right\rparen$] scales as

$\displaystyle n=\begin{cases}\Omega\left\lparen
r^{d(2d-\alpha)/(\alpha-d)}\text{polylog}(r)\right\rparen&\text{if
}d<\alpha<2d,\\\ \Omega\left\lparen e^{\gamma\sqrt{\log
r}}\text{polylog}(r)\right\rparen&\text{if }\alpha=2d,\text{and}\\\
\Omega\left\lparen\log(r)^{d/(\alpha-d)}\right\rparen&\text{if }2d<\alpha\leq
2d+1,\end{cases}$ (21)

where the $\Omega(\cdot)$ denotes an asymptotic lower bound. Intuitively, for
smaller $\alpha$, the interactions are stronger, so more colors are required
to cancel out the error. In particular, for $\alpha\leq 2d$, the number of
colors required to achieve a fixed crosstalk error scales super-
logarithmically in the state-transfer distance.

In order to cancel the errors using the spin-echo procedure, the number of
spin pulses required will scale exponentially in the number of colors.
Specifically, for color $i\in[n]$, a total of $2^{i-1}$ for $i\geq 3$ ($i=1,2$
require 0 and 2, respectively) single-qubit pulses will be required to provide
the alternating signs required for the interactions [Fig. 8(c)]. As such, for
$\alpha<2d$, the fact that $n$ scales at least polynomially in $r$ will lead
to an exponential scaling of the number of pulses with $r$. In the worst case,
when performing a single step of the hybrid protocol, the number of colors
will be at most equal to the total number of GHZ plaquettes, i.e. $m^{d}$. For
$d=2$, we determined in Section I that the crossover point where the hybrid
protocol surpasses the pure Eldredge protocol was for characteristic length
$r=244$ and $m=5$. For this value, the spin-echo procedure would in the worst
case require $2^{m^{2}-1}\approx 1.6\times 10^{7}$ pulses. As such, an
approach based solely on spin-echo pulses would likely be inefficient at
mitigating crosstalk errors.

Another approach to eliminating crosstalk between qubits in different
registers would be to use multiple different level encodings. This approach
would only require a number of encodings proportional to the number of colors
required to achieve a fixed crosstalk error, leading to a resource cost that
scales only polynomially in the state transfer distance $r$.

As an example, for Rydberg atoms, these different encodings could be realized
through the use of multiple states with different principal quantum numbers to
encode the corresponding $|1\rangle$ states for each color. Provided the
principal quantum numbers are sufficiently different, the interactions between
them are minimal, although the number of colors can become limited due to the
large variation in the vdW coefficients $C_{6}$ as a function of the principal
quantum number.

As another example, the asymmetric blockade approach, developed in Ref. [43]
and discussed in Sections II.1 and II.2, could be generalized to eliminate the
interactions between different colors through the use of additional dressing
fields, whose number will scale at most polynomially in the number of colors
(since the corresponding constraints on the interactions scale polynomially).
Used in conjunction with the spin-echo-based crosstalk mitigation strategy,
these encoding-based approaches could potentially eliminate the crosstalk
error using far fewer—indeed, exponentially so—spin-echo pulses. Concretely,
in the example discussed above with $m=5$, generating a pair of colors using
state engineering would reduce the number of pulses to $2^{m^{2}/2-1}\approx
2900$—i.e. four orders of magnitude fewer than without state engineering. If
five colors are realized, then the number of pulses drops to 16, which is well
within the realm of experimental feasibility.

In summary, we’ve outlined two methods to eliminate the crosstalk interactions
between control and target qubits in different registers incurred during the
Tran protocol: the first by using additional spin-echo pulses, and the second
by encoding qubits in different energy levels. We observed that the latter
method is able to mitigate crosstalk errors using a number of encodings that
only scales polynomially in the state transfer distance. We envision that a
combination of the two methods will help experimentalists to manage the
crosstalk interactions that occur throughout the Tran protocol in a thorough
and efficient way.

##  IV Conclusions and Outlook

In conclusion, we’ve provided a roadmap for the experimental realization of
fast state-transfer protocols in atomic and molecular systems. The approaches
we presented only require single-qubit control in addition to global power-law
interactions, which can be realized using electric and magnetic dipole-dipole
interactions and van der Waals interactions present in polar molecules,
Rydberg atoms, and atoms with large magnetic moments. For these types of
interactions, the protocols are able to realize a superballistic—and,
sometimes, even a superpolynomial—state transfer, as opposed to ballistic
state transfer for finite-range interactions.

We also addressed certain issues that may arise throughout the course of
implementing these protocols, namely the potential for crosstalk errors that
could occur in the process of creating multiple many-body entangled states in
parallel. We presented a few methods that can mitigate these errors. One
interesting further direction would be to look into whether the protocols
themselves could be modified to not suffer from the crosstalk errors at all.

Finally, we note that these protocols can be quite sensitive to noise. This is
largely due to their use of the GHZ state as an intermediate entangled state,
which is highly sensitive to single-qubit decoherence. Other protocols that
utilize more robust intermediate states could potentially provide higher noise
tolerance. For example, the protocol in Ref. [32] achieves state transfer with
the same scaling in time as the Eldredge protocol, but uses an intermediate W
state, which is known to have high robustness to single-qubit noise. However,
existing protocols using intermediate W states have typically required the use
of flip-flop interactions [38, 32], for which the spin-echo procedure for
selectively implementing control-target interactions doesn’t apply.
Additionally, since the protocol in Ref. [32] relies on uniform coupling,
inhomogeneities in the interactions—whether from the power-law form or angular
dependence—lead to fundamental errors in the state transfer that, unlike for
the Eldredge and Tran protocols, cannot be eliminated through shelving sites
in non-interacting states or through spin-echo techniques. We leave the
investigation of the experimental realization of such W-state-based protocols
to future work.

###### Acknowledgements.

We would like to thank Su-Kuan Chu, Minh Tran, Alicia Kollar, Andrew Lucas,
Adam Ehrenberg, and Yifan Hong for useful discussions. A.Y.G., R.B., P.B., and
A.V.G. were supported in part by the AFOSR MURI program, the DoE ASCR Quantum
Testbed Pathfinder program (awards No. DE-SC0019040 and No. DE-SC0024220),
AFOSR, NSF QLCI (award No. OMA-2120757), DoE ASCR Accelerated Research in
Quantum Computing program (award No. DE-SC0020312), NSF STAQ program, and
DARPA SAVaNT ADVENT. Support is also acknowledged from the U.S. Department of
Energy, Office of Science, National Quantum Information Science Research
Centers, Quantum Systems Accelerator. J.T.Y. was supported by the NIST NRC
Research Postdoctoral Associateship Award, NIST, and the NWO Talent Programme
(project number VI.Veni.222.312), which is (partly) financed by the Dutch
Research Council (NWO).

##  Appendix A Review of optimal state-transfer protocol by Tran _et al._

We first describe the setting of the problem and the main result of Ref. [31]
in this section. For simplicity, we consider a $d$-dimensional hypercubic
lattice $\Lambda$ and a two-level system located at every site of the lattice.
The protocol generalizes straightforwardly to all regular lattices. Without
loss of generality, we assume that the lattice spacing is one. We consider a
power-law interacting Hamiltonian $H(t)=\sum_{i,j\in\Lambda}h_{ij}(t),$ where
$h_{ij}(t)$ is a Hamiltonian supported on sites $i,j$ such that, at all times
$t$ and for all $i\neq j$, we have $\norm{h_{ij}}\leq
1/\textrm{dist}({i,j})^{\alpha}$, where $\textrm{dist}({i,j})$ is the distance
between $i,j$, $\norm{\cdot}$ is the operator norm, and $\alpha\geq 0$ is a
constant. We use $\ket{\text{GHZ}(a,b)}_{S}$ to denote the GHZ-like state over
sites in $S\subseteq\Lambda$:

$\displaystyle\ket{\text{GHZ}(a,b)}_{S}\equiv
a\ket{\bar{0}}_{S}+b\ket{\bar{1}}_{S},$ (22)

where $\ket{\bar{x}}_{S}\equiv\bigotimes_{j\in S}\ket{x}_{j}$ $(x=0,1)$ are
product states over all sites in $S$ and $a,b$ are complex numbers such that
$\absolutevalue{a}^{2}+\absolutevalue{b}^{2}=1$. In particular, we use
$\ket{\text{GHZ}}$ to denote the symmetric state $a=b=1/\sqrt{2}$.

Given a $d$-dimensional hypercube $C\subseteq\Lambda$ of length $r\geq 1$, we
consider the task of encoding a possibly unknown state $a\ket{0}+b\ket{1}$ of
a site $c\in C$ into the GHZ-like state $\ket{\text{GHZ}(a,b)}_{C}$ over $C$,
assuming that all sites in $C$, except for $c$, are initially in the state
$\ket{0}$. Specifically, we construct a time-dependent, power-law interacting
Hamiltonian $H(t)$ that generates
$U(t)=\mathcal{T}\exp\left(-i\int_{0}^{t}dsH(s)\right)$ satisfying

$\displaystyle U(t)\ (a\ket{0}+b\ket{1})_{c}\ket{\bar{0}}_{C\setminus
c}=a\ket{\bar{0}}_{C}+b\ket{\bar{1}}_{C}$ (23)

at time

$\displaystyle t(r)\leq
K_{\alpha}\times\begin{cases}\log^{\kappa_{\alpha}}r&\text{if }d<\alpha<2d,\\\
e^{\gamma\sqrt{\log r}}&\text{if }\alpha=2d,\\\ r^{\alpha-2d}&\text{if
}2d<\alpha\leq 2d+1.\end{cases}$ (24)

Here, $\gamma=3\sqrt{d}$, $\kappa_{\alpha}=\log(4)/\log(2d/\alpha)$, and
$K_{\alpha}$ are constants independent of $t$ and $r$. Additionally, by
reversing the unitary in Eq. 23 to “concentrate” the information in
$\ket{\text{GHZ}(a,b)}$ onto a different site in $C$, we can transfer a
quantum state from $c\in C$ to any other site $c^{\prime}\in C$ in time $2t$.

The key idea of the protocol from [31] is to recursively build the GHZ-like
state in a large hypercube from the GHZ-like states of smaller hypercubes. For
the base case, we note that hypercubes of finite lengths, i.e. $r\leq r_{0}$
for some fixed $r_{0}$, can always be generated in times that satisfy Eq. 24
for some suitably large (but constant) prefactor $K_{\alpha}$. Assuming that
we can encode information into a GHZ-like state in hypercubes of length
$r_{1}$ in time $t_{1}$ satisfying Eq. 24, the following subroutine encodes
information into a GHZ-like state in an arbitrary hypercube $C$ of length
$r=mr_{1}$ containing $c$—the site initially holding the state information
$a,b$. Here $m$ is an $\alpha$-dependent number that we choose to be
proportional to $r_{1}^{2d/\alpha-1}$ for $\alpha<2d$; to
$\exp{\frac{\gamma}{2d}\sqrt{\log r_{1}}}$ for $\alpha=2d$; and to $r_{1}$ for
$\alpha>2d$.

Step 1: We divide the hypercube $C$ into $m^{d}$ smaller hypercubes
$C_{1},\dots,C_{m^{d}}$, each of length $r_{1}$. Without loss of generality,
let $C_{1}$ be the hypercube that contains $c$. Let $V=r_{1}^{d}$ be the
number of sites in each $C_{j}$. In this step, we simultaneously encode $a,b$
into $\ket{\text{GHZ}(a,b)}_{C_{1}}$ and prepare $\ket{\text{GHZ}}_{C_{j}}$
for all $j=2,\dots,m^{d}$, which, by our assumption, takes time

$\displaystyle t_{1}\leq
K_{\alpha}\times\begin{cases}\log^{\kappa_{\alpha}}r_{1}&\text{if
}d<\alpha<2d,\\\ e^{\gamma\sqrt{\log r_{1}}}&\text{if }\alpha=2d,\\\
r_{1}^{\alpha-2d}&\text{if }2d<\alpha\leq 2d+1.\end{cases}$ (25)

By the end of this step, the hypercube $C$ is in the state

$\displaystyle(a\ket{\bar{0}}+b\ket{\bar{1}})_{C_{1}}\bigotimes_{j=2}^{m^{d}}\frac{\ket{\bar{0}}_{C_{j}}+\ket{\bar{1}}_{C_{j}}}{\sqrt{2}}.$
(26)

Step 2: Next, we apply the following Hamiltonian to the hypercube $C$:

$\displaystyle
H_{2}=\frac{1}{(mr_{1}\sqrt{d})^{\alpha}}\sum_{j=2}^{m^{d}}\sum_{\mu\in
C_{1}}\sum_{\nu\in C_{j}}\ket{1}\bra{1}_{\mu}\otimes\ket{1}\bra{1}_{\nu}.$
(27)

This Hamiltonian effectively generates the so-called controlled-PHASE gate
between the hypercubes, with $C_{1}$ being the control hypercube and
$C_{2},\dots,C_{m^{d}}$ being the target hypercubes. We choose the
interactions between qubits in Eq. 27 to be identical for simplicity. If the
interactions were to vary between qubits, we would simply turn off the
interaction between $C_{1}$ and $C_{j}$ once the total phase accumulated by
$C_{j}$ reaches $\pi$ 111Because only the total accumulated phase matters in
choosing the evolution time, we also expect the protocol to be robust against
experimental errors such as uncertainties in the positions of individual
particles: If the position of each particle is known up to a precision
$\varepsilon\ll 1$, the total worst-case error in the accumulated phase scales
as $t({r_{1}^{2d}}/{r^{\alpha}})\times(\varepsilon/r)$, with $r_{1}$ being the
length of each hypercubes and $r$ being the minimum distance between them. The
result is a relative phase error proportional to $\varepsilon/r$, which
becomes smaller and smaller as the distance between the hypercubes increases.
Moreover, we expect the relative error to be even smaller in the commonly
occurring situation when uncertainties in the positions are uncorrelated
between different particles . The prefactor $1/(mr_{1}\sqrt{d})^{\alpha}$
ensures that this Hamiltonian satisfies the condition of a power-law
interacting Hamiltonian. It is straightforward to verify that, under this
evolution, the state of the hypercube $C$ rotates to

$\displaystyle
a\ket{\bar{0}}_{C_{1}}\bigotimes_{j=2}^{m^{d}}\frac{\ket{\bar{0}}_{C_{j}}+\ket{\bar{1}}_{C_{j}}}{\sqrt{2}}+b\ket{\bar{1}}_{C_{1}}\bigotimes_{j=2}^{m^{d}}\frac{\ket{\bar{0}}_{C_{j}}-\ket{\bar{1}}_{C_{j}}}{\sqrt{2}}$
(28)

after time $t_{2}=\pi d^{\alpha/2}(mr_{1})^{\alpha}/{V^{2}}$.

The role of power-law interactions in our protocol can be inferred from the
value of $t_{2}$. Intuitively, the speed of simultaneously entangling
hypercube $C_{1}$ with hypercubes $C_{2},\dots,C_{m^{d}}$ is enhanced by the
$V^{2}=r_{1}^{2d}$ couplings between the hypercubes. However, the strength of
each coupling, proportional to $1/(mr_{1})^{\alpha}$, is suppressed by the
maximum distance between the sites in $C_{1}$ and those in
$C_{2},\dots,C_{m^{d}}$. With a small enough $\alpha$, the enhancement due to
$V^{2}$ overcomes the suppression of power-law interactions, resulting in a
small entanglement time $t_{2}$. In particular, when $\alpha<2d$, $t_{2}$
actually _decreases_ with $r_{1}$, implying that Step 2 would be faster in
later iterations if we were to keep $m$ constant.

To obtain the desired state $\ket{\text{GHZ}(a,b)}_{C}$, it remains to apply a
Hadamard gate on the effective qubit
$\\{\ket{\bar{0}}_{C_{j}},\ket{\bar{1}}_{C_{j}}\\}$ for $j=2,\dots,m^{d}$. We
do this in the following three steps by first concentrating the information
stored in hypercube $C_{j}$ onto a single site $c_{j}\in C_{j}$ (Step 3), then
applying a Hadamard gate on $c_{j}$ (Step 4), and then unfolding the
information back onto the full hypercube $C_{j}$ (Step 5).

Step 3: By our assumption, for each hypercube $C_{j}$ ($j=2,\dots,m^{d}$) and
given a designated site $c_{j}\in C_{j}$, there exists a (time-dependent)
Hamiltonian $H_{j}$ that generates a unitary $U_{j}$ such that

$\displaystyle(a\ket{0}+b\ket{1})_{c_{j}}\ket{\bar{0}}_{C_{j}\setminus
c_{j}}\xrightarrow{U_{j}}a\ket{\bar{0}}_{C_{j}}+b\ket{\bar{1}}_{C_{j}}$ (29)

for all complex coefficients $a$ and $b$, in time $t_{1}$ satisfying Eq. 25.
By linearity, this property applies even if $C_{j}$ is entangled with other
hypercubes. Consequently, backward time evolution under $H_{j}$ generates
$U_{j}^{\dagger}$, which “undoes” the GHZ-like state of the $j$th hypercube:

$\displaystyle
a\ket{\bar{0}}_{C_{j}}+b\ket{\bar{1}}_{C_{j}}\xrightarrow{U_{j}^{\dagger}}(a\ket{0}+b\ket{1})_{c_{j}}\ket{\bar{0}}_{C_{j}\setminus
c_{j}}$ (30)

for any $a,b$. In this step, we simultaneously apply $U_{j}^{\dagger}$ to
$C_{j}$ for all $j=2,\dots,m^{d}$. These unitaries rotate the state of $C$ to

$\displaystyle
a\ket{\bar{0}}_{C_{1}}\bigotimes_{j=2}^{m^{d}}\ket{+}_{c_{j}}\ket{\bar{0}}_{C_{j}\setminus
c_{j}}+b\ket{\bar{1}}_{C_{1}}\bigotimes_{j=2}^{m^{d}}\ket{-}_{c_{j}}\ket{\bar{0}}_{C_{j}\setminus
c_{j}},$ (31)

where $\ket{\pm}=(\ket{0}\pm\ket{1})/{\sqrt{2}}$.

Step 4: We then apply a Hadamard gate, i.e.

$\displaystyle\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix},$ (32)

to the site $c_{j}$ of each hypercubes $C_{j}$, $j=2,\dots,m^{d}$. These
Hadamard gates can be implemented arbitrarily fast since we do not assume any
constraints on the single-site terms of the Hamiltonian. The state of $C$ by
the end of this step is

$\displaystyle
a\ket{\bar{0}}_{C_{1}}\bigotimes_{j=2}^{m^{d}}\ket{0}_{c_{j}}\ket{\bar{0}}_{C_{j}\setminus
c_{j}}+b\ket{\bar{1}}_{C_{1}}\bigotimes_{j=2}^{m^{d}}\ket{1}_{c_{j}}\ket{\bar{0}}_{C_{j}\setminus
c_{j}}.$ (33)

Step 5: Finally, we apply $U_{j}$ again to each hypercube $C_{j}$
($j=2,\dots,m^{d}$) to obtain the desired GHZ-like state:

$\displaystyle
a\ket{\bar{0}}_{C_{1}}\bigotimes_{j=2}^{m^{d}}\ket{\bar{0}}_{C_{j}}+b\ket{\bar{1}}_{C_{1}}\bigotimes_{j=2}^{m^{d}}\ket{\bar{1}}_{C_{j}}=\ket{\text{GHZ}(a,b)}_{C}.$
(34)

At the end of this routine, we have implemented the unitary satisfying Eq. 23
in time

$\displaystyle t=3t_{1}+t_{2}=3t_{1}+\pi
d^{\alpha/2}m^{\alpha}r_{1}^{\alpha-2d}.$ (35)

Solving this recurrence relation for three different ranges of $\alpha$ yields
the scalings given in Eq. 24.

##  Appendix B Analysis of crosstalk interactions during the Tran protocol

In this section, we estimate the number of colors required to upper bound the
potential error incurred by the crosstalk interactions from creating GHZ
states in parallel during the Tran protocol. The analysis assumes that one of
the procedures outlined in Section III of the main text has already been used
to cancel out interactions between plaquettes with different color labels.

To estimate the total amount of crosstalk error in the worst case, we first
compute the total possible crosstalk interactions between all plaquettes of
the same color and multiply them by the timescales required to create the GHZ
states. We then sum over the sizes of the plaquettes required by different
steps in the protocol and multiply by the number of colors, $n$, in order to
obtain an estimate of the worst-case error incurred over the course of the
whole protocol.

For a given step of the protocol with a plaquette size of $L$, we define the
worst-case crosstalk error $\varepsilon(L)$ to be

$\displaystyle\varepsilon(L)=nt_{\text{GHZ}}(L)\|H_{\text{crosstalk}}(L,R)\|,$
(36)

where $t_{\text{GHZ}}(L)$ is the time required to create a GHZ state of size
$L$ by merging GHZ states from the previous step and
$\|H_{\text{crosstalk}}(L,R)\|=\max_{C_{i}}\sum_{C_{j}}\|H_{C_{i}C_{j}}\|$,
where $C_{i}$ and $C_{j}$ are plaquettes of the same color separated by a
distance of at least $R$, is the largest possible crosstalk interaction
between plaquettes of the same color for fixed $L$ and $R$.

We observe that
$\|H_{C_{i}C_{j}}\|\leq\frac{|C_{i}||C_{j}|}{d(C_{i},C_{j})^{\alpha}}=\frac{L^{2d}}{d(C_{i},C_{j})^{\alpha}}$,
where $d(C_{i},C_{j})$ is the distance between the sets $C_{i}$ and $C_{j}$.
Bounding the sum in $\|H_{\text{crosstalk}}(L,R)\|$ by an integral, we have:

$\displaystyle\varepsilon(L)\lesssim
nt_{\text{GHZ}}(L)\int_{1}^{r/R}d^{d}\vec{x}\frac{L^{2d}}{(R|\vec{x}|)^{\alpha}},$
(37)

where $r$ is the length scale corresponding to the final (largest) GHZ state
being created. For all $\alpha>d$, the integral can be upper bounded by a
constant times $L^{2d}/R^{\alpha}$.

The time $t_{\text{GHZ}}$ for each GHZ-creation step has the same scaling as
in Eq. 24, which we reproduce here for ease of reference:

$\displaystyle t_{\text{GHZ}}(L)\leq
K_{\alpha}\times\begin{cases}\log^{\kappa_{\alpha}}L&\text{if }d<\alpha<2d,\\\
e^{\gamma\sqrt{\log L}}&\text{if }\alpha=2d,\\\ L^{\alpha-2d}&\text{if
}2d<\alpha\leq 2d+1.\end{cases}$ (38)

For $\alpha<2d$, $m\propto L^{\lambda-1}$, where $\lambda=2d/\alpha$. To
compute the total crosstalk error, it is necessary to sum over all length
scales $L_{i}$ from $r_{0}$ to $r$. It can be observed that $L_{i}$ grows
doubly exponentially in the number of time steps $i$, due to the relation
$L_{i+1}=mL_{i}=L_{i}^{\lambda}$, where we note that $\lambda>1$. Since the
largest crosstalk error is incurred when $L_{i}=r$, the total crosstalk error
$\varepsilon_{\text{tot}}$ can be upper bounded by the maximum number of
recursive steps $i_{\text{max}}$ times the largest crosstalk error
$\varepsilon(r)$ as follows:

$\displaystyle\varepsilon_{\text{tot}}$
$\displaystyle=\sum_{i}\varepsilon(L_{i})=\sum_{i=0}^{i_{\text{max}}}\varepsilon(r_{0}^{\lambda^{i}})<i_{\text{max}}\varepsilon(r)$
(39)
$\displaystyle\lesssim\log\log(r)\log^{\kappa_{\alpha}}(r)\frac{r^{2d-\alpha}}{n^{\alpha/d-1}},$

where $i_{\text{max}}=\log\log(r)/(\log({\lambda}){\log\log(r_{0})})$. So
maintaining constant crosstalk error requires
$n\gtrsim\text{polylog}(r)r^{d(2d-\alpha)/(\alpha-d)}$ colors. This leads to
the first line in Eq. 21.

For $\alpha=2d$, the crosstalk error is given by

$\displaystyle\varepsilon(L)\lesssim e^{\gamma\sqrt{\log
L}}\frac{L^{2d-\alpha}}{n^{\alpha/d-1}}=\frac{e^{\gamma\sqrt{\log L}}}{n}.$
(40)

Since the scaling of $m$ is given by $m\propto e^{\frac{\gamma}{2d}\sqrt{\log
L}}$, where $\gamma=3\sqrt{d}$, we have that $m(L)>m^{\prime}$ for some
constant $m^{\prime}=O(1)$. Thus, the scaling of $L_{i}$ throughout the
recursive protocol is at least geometric, so the number of steps will be at
most $\log(r)$. So the overall crosstalk error can be bounded by

$\displaystyle\varepsilon_{\text{tot}}$
$\displaystyle=\sum_{i}\varepsilon(L_{i})=\sum_{i=0}^{i_{\text{max}}}\varepsilon\left((m^{\prime})^{i}r_{0}\right)<i_{\text{max}}\varepsilon(r)$
(41) $\displaystyle\lesssim\log(r)\frac{e^{{\gamma}\sqrt{\log r}}}{n},$

where $i_{\text{max}}=\log_{m^{\prime}}(r/r_{0})$. So to cancel out the
crosstalk error, the scaling of $n$ with $r$ must be at least
$n\gtrsim\log(r)e^{\gamma\sqrt{\log r}}$, which is faster than any logarithmic
function of $r$, and reproduces the second line of Eq. 21.

Finally, for $\alpha\in(2d,2d+1)$, the scaling of $L_{i}$ increases
geometrically with $i$ and given by $L_{i+1}=cL_{i}^{2}$ for some constant
$c$. So the crosstalk error is given by

$\displaystyle\varepsilon_{\text{tot}}$
$\displaystyle=\sum_{i}\varepsilon(L_{i})=\sum_{i=0}^{i_{\text{max}}}\varepsilon\left(c^{i}r_{0}\right)$
(42)
$\displaystyle=\sum_{i=0}^{\log_{m}\frac{r}{r_{0}}}(c^{i}r_{0})^{\alpha-2d}\frac{(c^{i}r_{0})^{2d-\alpha}}{n^{\alpha/d-1}}\lesssim\frac{\log(r)}{n^{\alpha/d-1}},$
(43)

so the scaling of the colors must be $n\gtrsim\log(r)^{d/(\alpha-d)}$,
confirming the last line in Eq. 21.

##  Appendix C Polar molecule interactions

In this section, we present the details of the derivation for the polar
molecule Ising interactions using a single dressing field at a time. At the
end of this section, we discuss the dressing necessary to realize asymmetric
blockade with polar molecules.

Figure 9: Ising interactions for scheme (a) and (b) as a function of $f_{0}$
and $d|E|/B_{0}$. The equal superposition $f_{0}=1/2$ is discussed in the main
text. In (a), the dashed line denotes when the effective dipole moments of
$|0\rangle$ and $|1\rangle$ are equal. In (b), the thick black line indicates
when the Ising interactions change sign.

First, we remark on which interaction terms remain resonant and time-
independent in the rotating frame of the microwave field. For schemes (a) and
(b), these are

$\displaystyle V_{\text{a}}=\frac{1}{r_{ij}^{3}}\big{[}$
$\displaystyle\left(\mu_{0}|\phi_{0,0}\rangle\langle\phi_{0,0}|+\mu_{1}|\phi_{1,0}\rangle\langle\phi_{1,0}|+\mu_{2}|\phi_{2,-2}\rangle\langle\phi_{2,-2}|\right)\times\left(\mu_{0}|\phi_{0,0}\rangle\langle\phi_{0,0}|+\mu_{1}|\phi_{1,0}\rangle\langle\phi_{1,0}|+\mu_{2}|\phi_{2,-2}\rangle\langle\phi_{2,-2}|\right)+\leavevmode\nobreak\
$ (44a)
$\displaystyle\mu_{01}^{2}|\phi_{0,0}\phi_{1,0}\rangle\langle\phi_{1,0}\phi_{0,0}|+\mu_{01}^{2}|\phi_{1,0}\phi_{0,0}\rangle\langle\phi_{0,0}\phi_{1,0}|\big{]},$
$\displaystyle V_{\text{b}}=\frac{1}{r_{ij}^{3}}\big{[}$
$\displaystyle\left(\mu_{0}|\phi_{0,0}\rangle\langle\phi_{0,0}|+\mu_{1}^{\prime}|\phi_{1,1}\rangle\langle\phi_{1,1}|+\mu_{2}|\phi_{2,-2}\rangle\langle\phi_{2,-2}|\right)\times\left(\mu_{0}|\phi_{0,0}\rangle\langle\phi_{0,0}|+\mu_{1}^{\prime}|\phi_{1,1}\rangle\langle\phi_{1,1}|+\mu_{2}|\phi_{2,-2}\rangle\langle\phi_{2,-2}|\right)+\leavevmode\nobreak\
$ (44b) $\displaystyle(-\mu_{01}^{\prime
2}|\phi_{0,0}\phi_{1,1}\rangle\langle\phi_{1,1}\phi_{0,0}|-\mu_{01}^{\prime
2}|\phi_{1,1}\phi_{0,0}\rangle\langle\phi_{0,0}\phi_{1,1}|)/2\big{]},$

respectively, where the flip-flop interactions in the last line of equation
equation differ in sign and magnitude.

We are interested in dressed states of the form
$|0/0^{\prime}\rangle\equiv\sqrt{f_{0}}|\phi_{0,0}\rangle+\sqrt{1-f_{0}}|\phi_{1,0/1}\rangle$
depending on the scheme, where $f_{0}$ captures the dressed state population
of the $|\phi_{0,0}\rangle$ state and the relative phase of the superposition
does not affect the interactions. We evaluate the interactions in the basis of
interest for scheme (a)

$\langle
00|V_{a}|00\rangle\propto[f_{0}\mu_{0}+(1-f_{0})\mu_{1}]^{2}+2f_{0}(1-f_{0})\mu_{01}^{2},$
(45a) $\langle
01|V_{a}|01\rangle\propto[f_{0}\mu_{0}+(1-f_{0})\mu_{1}]\mu_{2},$ (45b)
$\langle 11|V_{a}|11\rangle\propto\mu_{2}^{2},$ (45c)

and for scheme (b)

$\langle
0^{\prime}0^{\prime}|V_{b}|0^{\prime}0^{\prime}\rangle\propto[f_{0}\mu_{0}+(1-f_{0})\mu_{1}^{\prime}]^{2}-f_{0}(1-f_{0})\mu_{01}^{\prime
2},$ (46a) $\langle
0^{\prime}1|V_{b}|0^{\prime}1\rangle\propto[f_{0}\mu_{0}+(1-f_{0})\mu_{1}^{\prime}]\mu_{2},$
(46b) $\langle 11|V_{b}|11\rangle\propto\mu_{2}^{2},$ (46c)

where the proportionality corresponds to the $r_{ij}$-dependence we have
dropped for simplicity. These terms correspond to the $V_{0},V_{1},V_{01}$
terms (and their primed counterparts) in the main text up to a factor of 4.

In Fig. 9, we plot the strength of the Ising interactions as a function of
$f_{0}$ and $d|E|/B_{0}$. We illustrate the parameters where the effective
dipole moments of $|0\rangle,|1\rangle$ are equal for scheme (a) and the
region where negative Ising interactions are realized for scheme (b). In the
absence of an electric field, the interactions are strongest for both schemes
when $f_{0}=1/2$, corresponding to an equal superposition. For scheme (a), the
interactions can in principle be increased at sufficiently large electric
fields $d|E|/B_{0}\gtrsim 2$. For scheme (b), the strongest negative
interactions occur at $E=0$, and increasing the electric field only shifts the
interactions towards positive values.

### Asymmetric blockade

In this section, we present some of the details for the dressing parameters
needed to realize asymmetric blockade. This is largely reproduced from Ref.
[43], so the discussion of the dressing parameters applies to Rydberg atoms as
well, so long as the drives are not sufficiently strong to cause coupling to
additional states.

First, we define $\mu_{\pi}$, $\mu_{\sigma}$ to be the transition dipole
moments associated with the driven $\pi$ and $\sigma$ transitions,
respectively, with the ratio $\mathcal{M}\equiv\mu_{\pi}/\mu_{\sigma}$.
Additionally, we have the relation
$c_{\pi}=\mathcal{C}/\sqrt{|2\mathcal{M}^{2}-1|}$. For a given choice of
$\Omega_{\pi}$, the other dressing parameters are

$\Omega_{\sigma}=\sqrt{2}\mathcal{M}\frac{1+c_{\pi}^{2}(1-2\mathcal{M}^{2})}{1-c_{\pi}^{2}(1-2\mathcal{M}^{2})}\Omega_{\pi},$
(47a)
$\Delta_{\pi}=\frac{1-2c_{\pi}^{2}+\mathcal{C}^{4}}{c_{\pi}(1-c_{\pi}^{2}(1-2\mathcal{M}^{2}))}\Omega_{\pi},$
(47b)
$\Delta_{\sigma}=\frac{1-4c_{\pi}^{2}\mathcal{M}^{2}+\mathcal{C}^{4}}{c_{\pi}(1-c_{\pi}^{2}(1-2\mathcal{M}^{2}))}\Omega_{\pi},$
(47c)

which are naturally all proportional to $\Omega_{\pi}$. At $\mathcal{C}=1$,
the dipolar c-t interactions are maximized with a value of

$C_{3}^{\text{max}}=\min\left(\frac{\mu_{\pi}^{2}}{\mu_{\sigma}^{2}/2},\frac{\mu_{\sigma}^{2}/2}{\mu_{\pi}^{2}}\right)(\mu_{\pi}^{2}-\mu_{\sigma}^{2}/2),$
(48)

although, as discussed in the main text, the two dressed states
$|c\rangle,|t\rangle$ are degenerate when this is realized.

Next, we discuss the vdW interactions in the context of polar molecules. As
indicated by the discussion in the main text, the vdW interactions for polar
molecules are less complicated than for Rydberg atoms. This is because the
transition energy differences are typically on the order of GHz, much larger
than the dressing and interaction scales. As a result, only the resonant
dipolar interactions [e.g., the flip-flop terms in Eq. (44)] are relevant to
the vdW interactions, so a Floquet approach is not needed to deal with the
multiple rotating frames as it is for Rydberg atoms.

In our example, these interactions are the flip-flop interactions between the
$N=0$ and $N=1$ states as well as the flip-flop interactions between the $N=1$
and $N=2$ states, which we capture via a dipolar interaction Hamiltonian
$H_{\text{dd}}$. Furthermore, the drives also couple additional transitions,
such as $|1,1\rangle\to|2,2\rangle$, among others. In light of this, we
utilize the same rotating frame for all $N=2$ states as the $|2,1\rangle$
state, including the uncoupled states $|2,-2\rangle,|2,-1\rangle$, which
allows us to write $H_{\text{dd}}$ in a fully time-independent form. As a
result, in the dressed basis, we have the usual expression for the vdW
interactions:

$V_{\text{vdW}}^{(c)}=\sum_{k\neq cc}\frac{|\langle
k|H_{\text{dd}}|cc\rangle|^{2}}{2E_{c}-E_{k}},$ (49)

where $k$ are all other dressed state pairs in the basis and $E_{k}$ the
corresponding dressed state energies in the rotating frame.

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  * Note [1] Because only the total accumulated phase matters in choosing the evolution time, we also expect the protocol to be robust against experimental errors such as uncertainties in the positions of individual particles: If the position of each particle is known up to a precision $\varepsilon\ll 1$, the total worst-case error in the accumulated phase scales as $t({r_{1}^{2d}}/{r^{\alpha}})\times(\varepsilon/r)$, with $r_{1}$ being the length of each hypercubes and $r$ being the minimum distance between them. The result is a relative phase error proportional to $\varepsilon/r$, which becomes smaller and smaller as the distance between the hypercubes increases. Moreover, we expect the relative error to be even smaller in the commonly occurring situation when uncertainties in the positions are uncorrelated between different particles.