# Remotely Controlled Entanglement Generation

Ferran Riera-Sàbat  Universität Innsbruck, Institut für Theoretische Physik,
Technikerstraße 21a, 6020 Innsbruck, Austria  Pavel Sekatski  University of
Geneva, Department of Applied Physics, 1211 Geneva, Switzerland  Wolfgang Dür
Universität Innsbruck, Institut für Theoretische Physik, Technikerstraße 21a,
6020 Innsbruck, Austria

###### Abstract

We consider a system of multiple qubits without any quantum control. We show
that one can mediate entanglement between different subsystems in a controlled
way by adding a (locally) controlled auxiliary system of the same size that
couples via an always-on, distant-dependent interaction to the system qubits.
Solely by changing the internal state of the control system, one can
selectively couple it to selected qubits, and ultimately generate different
kinds of entanglement within the system. This provides an alternative way for
quantum control and quantum gates that does not rely on the ability to switch
interactions on and off at will and can serve as a locally controlled quantum
switch where all entanglement patterns can be created. We demonstrate that
such an approach also offers an increased error tolerance w.r.t. position
fluctuations.

##  1 Introduction

Quantum control is a crucial feature to generate entanglement and realize
devices that can process quantum information. In a typical setup, local
control of individual systems is assisted by tunable interactions or specific
gates between selected qubits, that can be switched on and off at will. This
allows one to entangle systems in a controlled way, and establish different
kinds of entangled states that serve as a valuable resource for various tasks,
ranging from quantum computation over quantum metrology to quantum networks.
In turn, this demands advanced control of systems.

In many realistic scenarios, one does not have direct controllability and
observability of the target system. One then considers the possibility to
control such a system indirectly by bringing it in contact with an auxiliary
system that one manipulates [1, 2, 3, 4, 5]. Here, we consider the remotely
controlled generation of entanglement and entangling gates in systems without
quantum control and without tunable interactions. We show that apart from
initialization in a homogeneous product state, no additional control on system
qubits is required to produce different kinds of entangled states, including
all graph states [6, 7]. We assume that a distant-dependent, always-on
interaction couples the system qubits with an auxiliary control system of the
same size. For commuting interactions –like in e.g., dipole-dipole
interactions [8, 9, 10], and many other types of couplings– we show that
manipulating the internal state of the control system suffices to generate a
large class of entangled states among the system qubits. The fixed, intrinsic
interactions within the control system can be utilized to achieve this aim by
only locally manipulating individual control qubits. The control system can be
a selected sub-part of the total system we have access to, or a separate
device that is, e.g., brought to a surface or a crystal structure and adds
control to qubits within this system. The scheme works in case of always-on or
no interactions among the system qubits themselves.

Figure 1: An auxiliary control system (green) couples selectively to a target
system (red) and mediates an effective interaction by proper choice of the
internal state of the control system.

The key element of our method is to use the spatial dependence of
interactions, that induce different phases depending on the relative distance
between the system and control qubits. By properly adjusting the internal
state of the control device, one can make it selectively couple to several of
the system qubits. This is done by using the multi-qubit control system as a
single logical qubit, i.e., restricting its state to a two-dimensional
subspace that is chosen in such a way that interactions with all but some
selected system qubits yield a zero phase within this subspace. Such an
approach is similar in spirit as used in the design of decoherence-free
subspaces (DFS) [11, 12] and is applicable to all commuting interactions with
spatial dependence, provided the size of the control system is sufficiently
large. These selected pairwise interactions with the logical control system
can be manipulated by different means, but importantly by only operating on
the control system, to yield an effective interaction between selected qubits
in the system, in such a way that the control system is factored out. In this
sense, the control system serves to mediate interactions between the system
qubits.

This article is organized as follows. In Sec. 2 we describe the underlying
physical system and how to achieve a selective coupling of the control system
to target qubits. In Sec. 3 we show how we can control the target system, by
preparing entangled states and by cancelling its inner interactions. In Sec. 4
we discuss the different levels of control on the control system and we
illustrate the full manipulation of the state of the control system by means
of single physical qubit general operations. In Sec. 5 we provide an analysis
of the required resources for the setting in terms of single-qubit operations.
In Sec. 6 we discuss different applications for the setting and we analyse two
different physical examples based on trapped ions. In Sec. 7 we analyse the
effect of noise in the position of the target system and introduce techniques
to reduce these effects. Finally, in Sec. 8 we conclude with an outlook and an
overall discussion.

##  2 Setting

###  2.1 Physical layer

We consider an inaccessible target system of $n$ spatially distributed qubits,
i.e., $S=\\{S_{i}\\}_{i=1}^{n}$ where qubit $S_{i}$ is located at position
$\boldsymbol{r}_{i}^{S}$, and an additional fully controllable control system
of $N$ qubits, $C=\\{C_{i}\\}_{i=1}^{N}$ where $C_{i}$ is located at
$\boldsymbol{r}_{i}^{C}$, see Fig. 1. We also assume an inherent pairwise ZZ
distance-dependent interaction, i.e., qubits $i$ and $j$ at positions
$\boldsymbol{r}_{i}$ and $\boldsymbol{r}_{j}$, interact via $f_{ij}Z_{i}Z_{j}$
where $f_{ij}=J\,\text{f}\,(|\boldsymbol{r}_{i}-\boldsymbol{r}_{j}|)$ is the
coupling strength that depends on the coupling constant $J$ and the distance
between the two qubits. In particular, we consider $\text{f}(x)=x^{-1}$, but
the setting is not restricted to this particular choice — any other non-
trivial distance dependence, e.g. of the form $x^{-\alpha}$ works as well.

The Hamiltonian describing the two systems is given by

$H=H^{C}+H^{S}+H^{CS},$

where

$\displaystyle H^{C}=\sum_{1\leq i<j\leq N}f^{C}_{ij}\,Z^{C}_{i}Z^{C}_{j}$
$\displaystyle H^{S}=\sum_{1\leq i<j\leq n}f^{S}_{ij}\,Z^{S}_{i}Z^{S}_{j}$
$\displaystyle H^{CS}=\sum_{\begin{subarray}{c}1\leq i\leq N\\\ 1\leq j\leq
n\end{subarray}}f^{CS}_{ij}Z^{C}_{i}Z^{S}_{j}.$

$H^{C}$ [$H^{S}$] describes interactions within the control [target] system,
what we refer to as the self-interactions of $C$ [$S$], and $H^{CS}$ describes
interactions between qubits of different systems. In absence of self-
interaction within $S$ the ZZ-coupling can be obtained from a general XYZ-
interaction using fast control on $C$, as discussed in Appendix B.

###  2.2 Selective coupling to the target system

Using the control system $C$ as a logical qubit, one can tune the interactions
between $C$ and each of the qubits in $S$, to make it selective couple to any
qubit of the target system $S$. This is achieved by restricting the state of
the control system into a two-dimensional subspace of the form
$span\\{\left|\boldsymbol{c}\right\rangle,\left|-\boldsymbol{c}\right\rangle\\}$,
where $\left|\boldsymbol{c}\right\rangle$ is a state of the computational
basis such that the components of vector
$\boldsymbol{c}=(c_{1},\dots,c_{N})^{T}$ are given by
$Z^{C}_{i}\left|\boldsymbol{c}\right\rangle=c_{i}\left|\boldsymbol{c}\right\rangle$.
We consider non-integers values of $c_{i}$, as flipping the corresponding
qubit at some intermediate time during the evolution allows one to obtain an
arbitrary effective value $c_{i}\in[-1,1]$. More details are given at the end
of this section.

The interaction Hamiltonian between $C$ and qubit $S_{j}$ is diagonal in the
computational basis, and if we define

$\lambda_{j}\equiv\sum_{i=1}^{N}f^{CS}_{ij}c_{i}$ (1)

its eigenvalues in the logical subspace are given by $\\{\pm\lambda_{j}\\}$,
i.e.,

$\displaystyle\sum_{i=1}^{N}f^{CS}_{ij}Z^{C}_{i}Z^{S}_{j}\left|\pm\boldsymbol{c}\right\rangle_{C}\left|\bar{s}_{j}\right\rangle_{S_{j}}$
(2)
$\displaystyle=\pm\lambda_{j}\,\bar{s}_{j}\left|\pm\boldsymbol{c}\right\rangle_{C}\left|\bar{s}_{j}\right\rangle_{S_{j}}$
,

where we refer to integer vector components as $\bar{s}_{i}\in\\{-1,1\\}$.

Therefore, when the control system is prepared in the logical qubit subspace
$span\\{\left|\boldsymbol{c}\right\rangle,\left|-\boldsymbol{c}\right\rangle\\}$
the interaction Hamiltonian takes the form

$H^{CS}=\sum_{j=1}^{n}\lambda_{j}Z^{C}Z^{S}_{j},$

where $Z^{C}$ is the Pauli-Z operator acting on the logical subspace of $C$,
i.e.,
$Z^{C}\left|\pm\boldsymbol{c}\right\rangle=\pm\left|\pm\boldsymbol{c}\right\rangle$,
and $\lambda_{j}$ is the coupling strength between $C$ and qubit $S_{j}$.

The interaction pattern
$\boldsymbol{\lambda}=(\lambda_{1},\dots,\lambda_{n})^{T}$ only depends on the
spatial distribution of the qubits and the vector $\boldsymbol{c}$ that we can
freely choose by single qubit operations on $C$ (see below). In fact, if the
system $C$ is large enough, i.e. contains enough qubits, any interaction
pattern $\boldsymbol{\lambda}$ can be engineered by a proper choice of the
logical subspace. Concretely, from Eq. (1) one can see that this can be done
by setting the vector labelling the logical subspace to any vector
$\boldsymbol{c}$ fulfilling

$\boldsymbol{F}\cdot\boldsymbol{c}=\boldsymbol{\lambda},$ (3)

where $\boldsymbol{F}$ is a $n\times N$ matrix given by $F_{ij}=f^{CS}_{ij}$.
The matrix $\boldsymbol{F}$ only depends on the set of distances between
qubits on $C$ and qubits on $S$, and hence, one can ensure that
$\boldsymbol{F}$ is full-rank by properly arranging the qubits of the control
system. Note that if $C$ is of the same size as $S$, i.e., if $N=n$, Eq. (3)
can be inverted and hence any interaction pattern can be established. However,
vector $\boldsymbol{c}$ has to be divided by $\text{max}_{i}|c_{i}|$ to ensure
that $|c_{k}|\leq 1\,\forall\,k$, what scales all interaction strengths by
$\boldsymbol{\lambda}\to\boldsymbol{\lambda}/\text{max}_{i}|c_{i}|$. This
implies for a given interaction pattern, there is a maximum coupling strength
between $C$ and the qubits of $S$.

As we mentioned at the beginning of this section, we can effectively encode
the logical qubit of $C$ in the subspace given by any vector $\boldsymbol{c}$
where $c_{i}\in[-1,1]$ by flipping the qubits during the evolution. If we
initialize the control system in the subspace given by
$\bar{\boldsymbol{c}}=(1,\dots,1)$, the evolution between $C$ and $S$ at a
time $\tau$ is given by $\exp\\{-\text{i}H^{CS}\tau\\}$. Flipping qubit
$C_{i}$ at time $t_{i}<\tau$ and at $\tau$ we obtain

$\begin{gathered}X^{C}_{i}e^{-\text{i}\sum_{k,l}f^{CS}_{kl}Z_{k}^{C}Z^{S}_{l}(\tau-
t_{i})}X^{C}_{i}e^{-\text{i}\sum_{k,l}f^{CS}_{kl}Z_{k}^{C}Z^{S}_{l}t_{i}}\\\
=e^{-\text{i}\sum_{k,l}f^{CS}_{kl}c_{k}Z_{k}^{C}Z^{S}_{l}\tau}\end{gathered}$

where $\boldsymbol{c}=(1,\dots,1,c_{i},1\dots,1)^{T}$ with
$c_{i}=2(t_{i}/\tau)-1$, as the flip at $t_{i}$ turns $Z_{i}\to-Z_{i}$. Note
that as

$\left[e^{-\text{i}H^{CS}t},\,X^{C}_{i}e^{-\text{i}H^{CS}t^{\prime}}X^{C}_{i}\right]=0,$

given any vector $\boldsymbol{c}$, we can implement the evolution in the
effective logical subspace by flipping each qubit of $C$ at specific instants
of the evolution, i.e., we apply $X^{C}_{i}$ at $t_{i}=(1+c_{i})\tau/2$ for
$i=1,\cdots,N$. Observe, it is not necessary to flip again all the qubits at a
time $\tau$, as it has no effect on the operation between $C$ and $S$.
However, we need to take into account that at time $\tau$ the logical qubit in
$C$ is now implemented in $\bar{\boldsymbol{c}}=(-1,\dots,-1)$. Therefore, the
number of flips required to implement an effective logical subspace,
$\eta_{\lambda}$, is upper-bounded by $\eta_{\lambda}\leq N$.

Figure 2: Schematic representation of a quantum switch controlling the flow of
information in a quantum network.

##  3 Remote control of system S

By choosing the logical subspace of the control system one can impose any
interaction pattern between $S$ and $C$. This, combined with full control of
$C$ that we assume, see Sec. 4, allows one to implement different effective
interactions within $S$, which we can use to establish entangled states. The
only requirement that we make on the target system is that it is initialized
on a polarized state in the X-direction, i.e., in the state
$\left|+\right\rangle^{\otimes n}$. We now illustrate different techniques to
induce effective interactions and generate entanglement within $S$.

As it is shown later in Sec. 3.4, interactions within systems $C$ and $S$ do
not affect our analysis, but for now, we ignore them, i.e., $H^{C}=H^{S}=0$.

###  3.1 Entanglement generation by gates sequence

(a)  (b)

(c)  (d)

Figure 3:  Different operations that can be implemented in $S$ (external
circles) by manipulating $C$ (inner circles). Solid wavy lines correspond to a
coupling between a qubit and the control system, dashed wavy lines correspond
to effective ZZ interactions and the shadow area corresponds to a multiple Z
interaction. (a) Single qubit Z rotation. (b) Two qubits ZZ-interaction by
coupling to two qubits plus an $X^{C}$-rotation. (c) Three qubits pairwise ZZ-
interaction with the same method as in (b). (d) Multiple Z interaction by
first entangling $C$ with the qubits followed by an X-rotation of $C$.

Without performing measurements, the effective interactions that we can
implement within $S$ are of Z-type. As we demonstrate below, we can implement
all diagonal unitary operations in $S$ by only controlling $C$, which suffices
to generate any state of the form

$\left|\psi\right\rangle_{S}=\frac{1}{\sqrt{2^{n}}}\sum_{\bar{s}_{1},\dots,\bar{s}_{n}=0}^{1}e^{\text{i}\theta_{1,\dots,n}}\left|\bar{s}_{1},\dots,\bar{s}_{n}\right\rangle_{S},$
(4)

where $\bar{s}_{i}\in\\{-1,1\\}$. This includes the class of local maximally
entangled (LME) states [13, 14, 15], as well as all graph and hypergraph
states. Graph states are a large class of highly entangled states [6, 7] that
are a valuable resource in measurement-based quantum computation [16, 17],
error correction [18, 19, 20, 21] and quantum communication [22], including
e.g., 2D cluster states and Greenberger–Horne–Zeilinger (GHZ) states. Any
graph state can be written as

$\left|\Psi\right\rangle=\prod_{(i,j)\in E}U_{ij}\left|+\right\rangle^{\otimes
n},$

where $E$ are the edges of the underlying graph and $U_{ij}$ are commuting,
maximally entangling phase gates that we can generate here.

One method is to selective couple $C$ with a subset of $S$, i.e.,
$S^{\prime}\subseteq S$, to implement

$U^{S^{\prime}}=e^{-\text{i}\,\frac{\pi}{4}\sum_{j\in
S^{\prime}}Z^{C}Z^{S}_{j}},$ (5)

together with local operations on $C$, to generate an effective multi-qubit Z
interaction on the selected subset $S^{\prime}$. This is accomplished by
applying the sequence

$U^{S^{\prime}}\,e^{-\text{i}\,\omega
X^{C}}U^{S^{\prime}\dagger}=e^{-\text{i}\,\omega
G^{C}Z^{S^{\prime}}_{1}\\!\cdots Z^{S^{\prime}}_{n^{\prime}}}$ (6)

and while leaving $C$ in an eigenstate of $G^{C}$, with

$G=\left(\frac{\text{i}}{2}\right)^{\\!k}[\cdots[[[X,Z],Z],Z],\cdots Z],$ (7)

where we concatenate $k$ commutators. In this case, we need to take
$k=n^{\prime}$. Note that $G\in\\{\pm X,\pm Y\\}$ depending on the size of
$S^{\prime}$ given by $n^{\prime}$. For $\omega=\pi/4$, it generates a GHZ
state in $S^{\prime}$. Applying the same method on all subsets of qubits
sequentially, and with a proper choice of induced interaction phases, one can
generate any state of the form Eq. (4) (see Appendix C). For small interaction
phases, this might however be costly, as maximally entangling gates between
$C$ and $S$ are required to produce even a small interaction, and at least
four maximally entangling gates are used to generate a maximally entangled
state in $S$. This is however not an issue for graph states, where each edge
corresponds to a $\pi/4$ interaction phase. In particular, the time required
to implement Eq. (6) is given by

$t_{g}=2t_{U}+2t_{V}+t_{\text{i}}+\frac{\pi}{4\omega},$ (8)

where $t_{U}$ is the time to implement $U^{S^{\prime}}$ which depends on
$S^{\prime}$ and the spatial distribution of the qubits, $t_{V}$ is the time
required to implement a logical Hadamard gate, and $t_{\text{i}}=t_{V}/2$ is
the time to initialize the control system in the $\left|+\right\rangle_{C}$
state, see Sec. 4. Note that two Hadamard gates are required for the logical X
rotation.

###  3.2 Entanglement generation by projective measurements

####  3.2.1 Arbitrary graph states

We can also prepare different states in the target system by first entangling
it with the control system and then performing a projective measurement of the
logical qubits in $C$, see Sec. 4.2.2. In particular, by first entangling $C$
with $S^{\prime}$ of size $n^{\prime}$, and then measuring out the control
system, we can directly implement a control-Z gate between all pairs of qubits
in $S^{\prime}$, i.e.,

$\displaystyle R^{S^{\prime}}_{\pm}\,\mathcal{M}_{g}^{C}\\!\\!:$
$\displaystyle\;\;U^{S^{\prime}}\\!\left|+\right\rangle_{C}\left|\psi\right\rangle_{S}$
(9) $\displaystyle\mapsto\prod_{(i<j)\in
S^{\prime}}\\!\\!\text{CZ}^{S}_{ij}\left|\psi\right\rangle_{S}$

where $\left|\psi\right\rangle$ is an arbitrary multipartite state,
$U^{S^{\prime}}$ is given in Eq. (5), $\mathcal{M}^{C}_{g}$ is a projective
measurement on the basis of $G$ given in Eq. (7) (with now $k=n^{\prime}-1$)
acting on $C$, and
$\\{R^{S^{\prime}}_{+}=\mathbbm{1},R^{S^{\prime}}_{-}=\bigotimes_{i\in
S^{\prime}}Z^{S}_{i}\\}$ is a correction operation that depends on the outcome
of $\mathcal{M}_{g}$. Note we can implement $R^{S^{\prime}}_{-}$ by
establishing the interaction pattern $\lambda_{i}=\lambda$ if $S_{i}\in
S^{\prime}$ and $\lambda_{i}=0$ otherwise, and letting evolve $C$ and $S$ for
a time $t=\pi/(2\lambda)$ while keeping the control system in the
$\left|0\right\rangle_{C}$ state.

Repeating this method, we can prepare an arbitrary graph state in $S$ by
sequentially creating each edge of the graph state. The time required to
implement a control-$Z$ gate between two qubits on $S$ is given by

$t_{m}=t_{U}+t_{\text{i}}+t_{\mathcal{M}},$ (10)

where $t_{\mathcal{M}}$ is the time required to implement measurement
$\mathcal{M}^{C}_{g}$. Therefore, the time to prepare an arbitrary graph state
is given by $|E|\,t_{m}$, where $|E|$ is the number of edges.

####  3.2.2 Decoherence-free GHZ states

By performing projective measurements in the logical X basis on the control
system, we also can prepare GHZ states in $S$ in the computation basis, i.e.,
of the form
$(\left|\bar{\boldsymbol{s}}\right\rangle+\left|-\bar{\boldsymbol{s}}\right\rangle)/\sqrt{2}$,
up to random (but known) single-qubit X-gates. These states are of particular
interest and once prepared they are not affected by the self-interaction in
$S$, and hence they can be stored and used at later times without the need to
cancel self-interaction resulting from $H^{S}$. In turn, states can only be
manipulated further in a restricted way as some of the coherence among $S$ is
destroyed.

There are different ways to obtain GHZ states on the computational basis. Here
we introduce a general method that consists in iterating a Bell state
generation procedure. The steps are the following:

  1. 1.

Prepare a GHZ state between the logical qubit in $C$ and qubits
$\\{S_{1},S_{2}\\}$, by applying $U^{S^{\prime}}$. The final state is given by

$\displaystyle
U^{S^{\prime}}\left|+\right\rangle_{C}\left|+\,+\right\rangle_{S_{1}S_{2}}$
$\displaystyle=\frac{1}{2}\Big{[}\left|-\right\rangle\big{(}\left|00\right\rangle-\left|11\right\rangle\big{)}+\text{i}\left|+\right\rangle\big{(}\left|01\right\rangle+\left|10\right\rangle\big{)}$
$\displaystyle\Big{]}.$

  2. 2.

Perform a projective measurement $\mathcal{M}_{x}^{C}$ to the logical qubit in
$C$. The state of $S^{\prime}$ collapses to

$\displaystyle R^{S_{1}}_{\pm}$
$\displaystyle\mathcal{M}_{x}^{C}\\!\\!:\,U^{S^{\prime}}\left|+\right\rangle_{C}\left|+\,+\right\rangle_{S_{1}S_{2}}$
$\displaystyle\mapsto\frac{1}{\sqrt{2}}\left|\pm\right\rangle_{C}\left(\,\left|\bar{\boldsymbol{s}}_{\pm}\right\rangle_{S_{1}S_{2}}+\left|-\bar{\boldsymbol{s}}_{\pm}\right\rangle_{S_{1}S_{2}}\right),$

where $\bar{\boldsymbol{s}}_{\pm}=(1,\pm 1)$. Note that using the control
system we can tune the relative phase in the state but not change vector
$\bar{\boldsymbol{s}}$ which depends on the measurement outcome.

  3. 3.

Add qubit $S_{3}$ to the entangled state by first applying
$U^{S^{\prime\prime}}$ where $S^{\prime\prime}=\\{S_{1},S_{3}\\}$, i.e.,

$\displaystyle
U^{S^{\prime\prime}}\frac{1}{\sqrt{2}}\left|+\right\rangle_{C}\big{(}\left|\bar{\boldsymbol{s}}\right\rangle_{S^{\prime}}+$
$\displaystyle\left|-\bar{\boldsymbol{s}}\right\rangle_{S^{\prime}}\big{)}\left|+\right\rangle_{S_{3}}$
$\displaystyle=\frac{1}{2}\Big{[}\left|-\right\rangle\big{(}\left|\bar{\boldsymbol{s}},0\right\rangle-$
$\displaystyle\left|-\bar{\boldsymbol{s}},1\right\rangle\big{)}$
$\displaystyle+\text{i}\left|+\right\rangle\big{(}$
$\displaystyle\left|\bar{\boldsymbol{s}},1\right\rangle+\left|-\bar{\boldsymbol{s}},0\right\rangle\big{)}\Big{]},$

and next performing measurement $\mathcal{M}^{C}_{x}$ followed by
$R^{S_{1}}_{\pm}$. Qubits $S_{1}$, $S_{2}$ and $S_{3}$ end up in a state of
the from
$(\left|\bar{\boldsymbol{s}}^{\prime}\right\rangle+\left|-\bar{\boldsymbol{s}}^{\prime}\right\rangle)/\sqrt{2}$
where $\bar{\boldsymbol{s}}^{\prime}$ depends on the two measurement outcomes.

  4. 4.

Note step 3 is independent of the size of $S^{\prime}$ and therefore it can be
iterated to sequentially add more qubits to the entangled state and prepare an
arbitrarily large GHZ state.

###  3.3 Entanglement generation by control rotation

A third method for implementing an effective ZZ-interaction between two qubits
consists in simultaneously coupling $C$ to the two qubits, and applying an
additional field in the X-direction on the logical qubit in $C$. At a certain
time $\tau$, the state of $C$ factors out, while an effective ZZ-interaction
between the two qubits in $S$ is generated. The evolution generated by

$H_{x}=\lambda Z^{C}\left(Z^{S}_{1}+Z^{S}_{2}\right)+\omega X^{C},$

of duration $\tau$ results in the global unitary transformation

$\displaystyle e^{-\text{i}\,\tau H_{x}}$
$\displaystyle=\cos\\!\left(\tau\sqrt{\lambda^{2}\left(Z_{1}^{S}+Z_{2}^{S}\right)^{2}+\omega^{2}}\,\right)\mathds{1}^{C}$
$\displaystyle-\text{i}\,\frac{\sin\\!\left(\tau\sqrt{\lambda^{2}\left(Z_{1}^{S}+Z_{2}^{S}\right)^{2}+\omega^{2}}\,\right)}{\sqrt{\lambda^{2}\left(Z_{1}^{S}+Z_{2}^{S}\right)^{2}+\omega^{2}}}H_{x}.$

The eigenvalues of $Z_{1}^{S}+Z_{2}^{S}$ are $2,0$ and $-2$. For any state
$\left|\psi\right\rangle_{S_{1}S_{2}}$ of the target system with
$\big{(}Z_{1}^{S}+Z_{2}^{S}\big{)}\left|\psi\right\rangle_{S_{1}S_{2}}=0$, we
have
$e^{-\text{i}H_{x}\tau}\left|+\right\rangle_{C}\\!\left|\psi\right\rangle_{S_{1}S_{2}}=e^{-\text{i}\,\omega\tau}\left|+\right\rangle_{C}\left|\psi\right\rangle_{S_{1}S_{2}}$
since the control qubit is prepared in an eigenstate of $X^{C}$. For a state
$\left|\psi\right\rangle_{S_{1}S_{2}}$ with
$\big{(}Z_{1}^{S}+Z_{2}^{S}\big{)}\left|\psi\right\rangle_{S_{1}S_{2}}=\pm
2\left|\psi\right\rangle_{S_{1}S_{2}}$, we tune the interaction time to
$\tau=\pi/\sqrt{\omega^{2}+4\lambda^{2}}$ guaranteeing that
$\sin\big{(}\tau\sqrt{\lambda^{2}(\pm 2)^{2}+\omega^{2}}\,\big{)}=0$ and
$e^{-\text{i}H_{x}\tau}\left|+\right\rangle_{C}\\!\left|\psi\right\rangle_{S_{1}S_{2}}=-\left|+\right\rangle_{C}\left|\psi\right\rangle_{S_{1}S_{2}}$.
Combining the two observations one concludes that the resulting evolution
reads

$e^{-\text{i}H_{x}\tau}\\!\left|+\right\rangle_{C}\\!\left|\psi\right\rangle_{S_{1}S_{2}}\\!\\!=\\!e^{-\text{i}\phi
Z^{S}_{1}Z^{S}_{2}}\\!\left|+\right\rangle_{C}\\!\left|\psi\right\rangle_{S_{1}S_{2}}\\!,$
(11)

where $\left|\psi\right\rangle$ is an arbitrary two-qubits state and
$\phi=\frac{\pi}{2}\left(1-\omega/\sqrt{\omega^{2}+4\lambda^{2}}\,\right)$ is
an arbitrary phase. That is, for any desired interaction phase $\phi$, one can
find $\lambda,\omega$ and the corresponding time $\tau$, to generate an
effective interaction phase between the two target qubits in $S$, while
keeping the control system decoupled.

The same method allows one to simultaneously generate a maximally entangling
pairwise interaction between three qubits, i.e., to implement
$\exp\\{-\text{i}\,\frac{\pi}{4}\sum_{i,j=1}^{3}Z^{S}_{i}Z^{S}_{j}\\}$. For
this we couple $C$ equally to the three qubits, set
$\omega=\lambda\sqrt{5}/\sqrt{3}$, and the state is prepared at time
$\tau=\sqrt{3}\pi/(2\lambda\sqrt{2})$ where $\tau\sqrt{\lambda^{2}(\pm
1)^{2}+\omega^{2}}=\pi$ and $\tau\sqrt{\lambda^{2}(\pm
9)^{2}+\omega^{2}}=2\pi$. See Appendix D for details.

Note in Eq. (11) we cannot implement an arbitrary interaction pattern by
sequentially applying the evolution in integer subspaces as explained in Sec.
2.2, as the evolution generated in this case does not fulfil Eq. (2).
Therefore, the evolution should be implemented employing Hamiltonian
simulation techniques, i.e., we use the Trotter formula to reproduce the
desired evolution

$\lim_{k\to\infty}\left[e^{-\text{i}\lambda
Z^{C}\big{(}Z^{S}_{1}+Z^{S}_{2}\big{)}\tau/k}e^{-\text{i}\omega
X^{C}\tau/k}\right]^{k}=e^{-\text{i}H_{x}\tau}.$

When the time evolution is split in such a way, the $\exp\\{-\text{i}\lambda
Z^{C}\big{(}Z^{S}_{1}+Z^{S}_{2}\big{)}\\}$ interaction can be implemented by
flipping the qubits of the control system as described in Sec. 5. This shows
that the method can be implemented even if only local control on the $C$ is
available. In this case, however, it is highly demanding in terms of number of
operation cost. In contrast, it is efficient in scenarios where direct global
control on $C$ is available.

More methods to generate entanglement are detailed in Appendix E and F. This
includes sequences of three gates to directly establish a Bell pair or other
techniques from Hamiltonian simulation where the alternating application of
non-commuting pairwise interactions between $C$ and two different qubits leads
to a three-qubit interaction in a second order.

###  3.4 Cancellation of self-interactions

So far we have ignored interactions between qubits of the same ensemble, i.e.,
$H^{C}=H^{S}=0$. Even though self-interactions are present in our physical
model, they do not affect our analysis. $H^{S}$ commutes with $H^{CS}$, which
means at any time, one can use $C$ to apply a correction operation that
generates an effective pairwise ZZ interaction in $S$ that cancels all
inherent self-interactions. Note that in principle $H^{S}$ can be stronger
than the coupling $H^{CS}$ in which case self-interactions cannot be cancelled
dynamically. However, for a fixed time the phases induced by self-interactions
need only to be cancelled modulo $2\pi$, this can always be done for a long
enough time. One may also use the self-interactions to speed up the
entanglement generation between qubits, rather than cancelling it. Hence, in
the case of self-interactions, the desired entanglement state is established
at a fixed time only, and further operations are required to maintain it.

On the other hand, in $C$, we do not need any correction operation since it is
not affected by its self-interactions. Restricting its state into a logical
sub-space of the form
$span\\{\left|\boldsymbol{c}\right\rangle,\left|-\boldsymbol{c}\right\rangle\\}$
we are also making it insensitive to $H^{C}$, i.e., any pair of states
$\left|\boldsymbol{c}\right\rangle$ and $\left|-\boldsymbol{c}\right\rangle$
have the same eigenvalue for $H^{C}$. Therefore, within the subspace, the
evolution generated by $H^{C}$ just yields a global phase.

##  4 Control of the logical system C

One of the requirements of our setting is to have full control of the logical
qubit in $C$. In this section, we discuss the scenario where full control is
directly assumed, and then we also show that actually, control of its
individual physical qubits suffices to achieve a full manipulation of the
logical qubit (unitaries and measurements).

###  4.1 Global control

We first consider the situation where the control system $C$ is given by a
small quantum processor of $N$ qubits, for which full control is ideally
assumed and any desired operation on the control system can be realized by a
sequence of elementary single- and two-qubit gates. Notice that the
restriction to a particular two-dimensional subspace as outlined above
actually limits the required operations. Under this assumption, effective spin
values can be obtained, and any of the methods to remotely generate
entanglement between the system qubits described above is applicable,
including the one described in Sec. 3.3. That is, our methods allow one to
extend the control of the small-scale quantum processor to remote systems.

###  4.2 Local control

In the following, we show how we can implement any single logical qubit gate
and arbitrary projective logical measurements by using the inherent physical
qubit-qubit interaction and applying single physical qubit operations on the
control system.

####  4.2.1 Logical single qubit gates

Some gates on the logical subspace can be decomposed as a combination of
single physical qubit gates. For instance, an arbitrary Z-rotation in the
Bloch sphere, $R_{z}(\phi)=e^{-\text{i}\phi Z/2}$, can be applied to the
logical qubit by individually rotating the qubits of $C$, i.e.,

$R^{C}_{z}(\varphi)=R^{C}_{z,1}(\varphi_{1})\otimes\cdots\otimes
R^{C}_{z,N}(\varphi_{N}),$

where $\varphi=\sum_{i}\varphi_{i}/c_{i}$. Notice that an operation on a
single physical qubit also suffices to obtain the desired logical Z-rotation.

However, in general, a single qubit gate on the logical qubit is an entangling
gate between the physical qubits, and hence it requires entangling physical
qubit operations to be implemented. Therefore, we generally use the self-
interaction term with local control of the physical qubits to control the
logical qubit. In particular, a single qubit gate, $V$, on the logical qubit
can be implemented as

$V^{C}=\Bigg{(}\prod_{i=2}^{N}\text{CX}^{C}_{1\to
i}\Bigg{)}\,V^{C}_{1}\,\Bigg{(}\prod_{j=2}^{N}\text{CX}^{C}_{1\to j}\Bigg{)}.$
(12)

As we explain below, by performing fast flips of the physical qubits in the
control system, we can isolate the interaction between any pair of physical
qubits in $C$. This combined with single physical qubit gates, we can
implement a control-X gate between any pair of physical qubits in $C$. In this
way, single qubit control suffices to implement any gate on the logical qubit.

In the protocols shown in Sec. 3 we need the control system in the
$\left|+\right\rangle_{C}$ state. In this case, we can initialize the control
system by first measuring each physical qubit in the $Z$ basis to obtain the
$\left|0\right\rangle_{C}$, and then apply a Hadamard gate to the logical
qubit using Eq. (12), as
$\text{H}_{\text{ad}}\left|k\right\rangle=(\left|0\right\rangle+(-1)^{k}\left|1\right\rangle)/\sqrt{2}$.
Note this procedure would require $2N-2$ physical qubit control gates.
However, as the control system is known to be in the
$\left|\boldsymbol{0}\right\rangle_{C}$ state, we also can prepare the
$\left|+\right\rangle_{C}$ state by first encoding the logical qubit in the
subspace given by $\bar{\boldsymbol{c}}=\boldsymbol{1}=(1,\dots,1)$ and then
applying

$\left[\prod_{i=2}^{N}\text{CX}^{C}_{1\to
i}\right]\text{H}_{\text{ad},1}^{C}\left|\boldsymbol{1}\right\rangle_{C}=\frac{\left|\boldsymbol{1}\right\rangle+\left|-\boldsymbol{1}\right\rangle}{\sqrt{2}}=\left|+\right\rangle_{C}.$
(13)

Note that this initialization procedure only requires $N-1$ physical qubit
control gates, and hence, it is twice faster to implement than the logical
Hadamard gate. Finally, the effective logical subspace is established by
performing flips during the interaction as described in Sec. 2.2.

In order to make use of the self-interactions in the control system, $H^{C}$,
while implementing a control gate between a pair of physical qubits in $C$,
the control qubits must leave the logical qubit subspace which is designed to
be insensitive to $H^{C}$. During this process, in order to avoid the control
qubits interacting with the target system and between themselves in an
uncontrolled way, we set an effective null spin value for all pairwise
interactions within $C$ and interactions with external systems by flipping
them at certain times.

We show how we can “turn off” interactions within $C$ by detailing the
procedure. First, we divide the ensemble into two subsets of the same size
$\Omega_{0}$ and $\Omega_{1}$, e.g., $\Omega_{0}=\\{C_{1},\dots,C_{N/2}\\}$.
Note, we can cancel out all interactions between the two sets at time $2t$ by
flipping the qubits in $\Omega_{0}$ at $t$, i.e.,

$\begin{gathered}W^{(1)}\equiv X^{C}_{1}\cdots
X^{C}_{N/2}\,e^{-\text{i}H^{C}t}X^{C}_{1}\cdots
X^{C}_{N/2}\,e^{-\text{i}H^{C}t}\\\
=e^{-\text{i}\big{(}\sum_{k<l\leq\frac{N}{2}}f^{C}_{kl}Z^{C}_{k}Z^{C}_{l}+\sum_{\frac{N}{2}<k<l}f^{C}_{kl}Z^{C}_{k}Z^{C}_{l}\big{)}\,2t}.\end{gathered}$

As $W^{(1)}$ only contains interaction within $\Omega_{0}$ and within
$\Omega_{1}$, we split $\Omega_{0}=\Omega_{00}\cup\Omega_{01}$
($\Omega_{1}=\Omega_{10}\cup\Omega_{11}$). We can iterate the previous step
but now flipping the qubits in $\Omega_{00}\cup\Omega_{10}$ what cancels out
half of the interactions in $W^{(1)}$, i.e.,

$W^{(2)}\equiv\\!\Bigg{[}\prod_{i\in\Omega_{00}\cup\Omega_{10}}\\!\\!\\!X^{C}_{i}\Bigg{]}W^{(1)}\Bigg{[}\prod_{j\in\Omega_{00}\cup\Omega_{10}}\\!\\!\\!X^{C}_{j}\Bigg{]}W^{(1)}.$
(14)

We proceed by iterating this step, i.e., applying twice the previous evolution
$W^{(j)}$ with an X flip to half of the qubits of each previous subset between
the concatenation. In the $k=\left\lceil\log_{2}N\right\rceil$ step each
subgroup $\Omega$ contains a single qubit and, hence, all interactions are
cancelled, i.e., $W^{(k)}=\mathbbm{1}$. Note that $k$ steps correspond to
concatenate $2^{k}(\leq 2N)$ times the gate $\exp\\{-\text{i}H^{C}t\\}$, where
between each application of the evolution we perform $N$ flips. However, as
one can see in Eq. (14), half of the flips overlap with the next step, and we
do not have to count them. Also, we have to add an extra simultaneous flip to
all of the qubits of $C$ at $2^{k}t/2$ to cancel out interactions of $C$ with
$S$. Observe the global flip does not affect the interactions within $C$.
Therefore, the total number of flips to “turn off” the interactions of $C$ is
given by $N+2^{\left\lceil\log_{2}N\right\rceil-1}N$.

Slightly adapting the procedure just described, we can isolate the interaction
between any pair of qubits by flipping the two qubits simultaneously between
each concatenation. This keeps cancelling the interactions with the rest of
the qubits but leaves a clean interaction between the pair. The interaction
plus an extra single-qubit operation allows us to implement a control-X gate.
Therefore, as in Equations (12) and (13), $N-1$ and $2N-2$ control gates are
performed respectively, the number of flips required to initialize the control
system, $\eta_{\text{i}}$, and to implement a general single logical qubit
gate, $\eta_{V}$, are bounded by

$\displaystyle\eta_{\textit{i}}\leq N(N^{2}-1)$ $\displaystyle\eta_{V}\leq
2N(N^{2}-1)$

where we used that $2^{\left\lceil\log_{2}N\right\rceil-1}\leq N$. This
implies that the required spin flips in system $C$ to perform arbitrary
logical operations scale as $O(N^{3})$, i.e. polynomial in the number of
qubits.

####  4.2.2 Logical projective measurements

Any projective measurement of the logical qubit can be effectively performed
by individually measuring each of the physical qubits in $C$. As shown in
[23], one can always distinguish between two orthogonal states by local
operations. Given two $N$-qubit orthogonal states $\left|\psi\right\rangle$
and $\left|\psi^{\perp}\right\rangle$ we can write them as

$\begin{gathered}\big{|}\psi\big{\rangle}_{C}=\big{|}u\big{\rangle}_{C_{1}}\big{|}v\big{\rangle}_{C_{2}\dots
C_{N}}+\big{|}u^{\perp}\big{\rangle}_{C_{1}}\big{|}w\big{\rangle}_{C_{2}\dots
C_{N}}\\\
\big{|}\psi^{\perp}\big{\rangle}_{C}=\big{|}u\big{\rangle}_{C_{1}}\big{|}v^{\perp}\big{\rangle}_{C_{2}\dots
C_{N}}+\big{|}u^{\perp}\big{\rangle}_{C_{1}}\big{|}w^{\perp}\big{\rangle}_{C_{2}\dots
C_{N}}\end{gathered}$

where $\\{\left|u\right\rangle,\big{|}u^{\perp}\big{\rangle}\\}$ is an
orthonormal basis, $\\{\left|v\right\rangle,\big{|}v^{\perp}\big{\rangle}\\}$
and $\\{\left|w\right\rangle,\big{|}w^{\perp}\big{\rangle}\\}$ are two pairs
of (non-normalized) orthogonal states. After measuring qubit $C_{1}$ in the
basis $\\{\left|u\right\rangle,\big{|}u^{\perp}\big{\rangle}\\}$, the problem
reduces to distinguishing between the orthogonal states $\left|v\right\rangle$
and $\big{|}v^{\perp}\big{\rangle}$ (or $\left|w\right\rangle$ and
$\big{|}w^{\perp}\big{\rangle}$) of the remaining $N-1$ qubits. After
repeating the same procedure to all the parts, one distinguishes between the
two original states.

Note that this measurement does not project the state of the control system
into $\left|\psi\right\rangle$ or $\big{|}\psi^{\perp}\big{\rangle}$ but in
some other (known) state. However, if $C$ is entangled with $S$, we can
collapse the state of $S$ in the same way as the actual projective logical
measurement, i.e., if $\mathcal{M}_{\Psi}$ is a concatenation of single
physical measurements that distinguishes between $\left|\psi\right\rangle$ and
$\big{|}\psi^{\perp}\big{\rangle}$ then

$\displaystyle\mathcal{M}_{\Psi}^{C}\\!:$
$\displaystyle\;\gamma_{\alpha}\left|\psi\right\rangle_{C}\left|\alpha\right\rangle_{S}+\gamma_{\beta}\big{|}\psi^{\perp}\big{\rangle}_{C}\left|\beta\right\rangle_{S}$
$\displaystyle\mapsto$
$\displaystyle\begin{cases}\;\big{|}\tilde{\psi}\big{\rangle}_{C}\left|\alpha\right\rangle_{S}&\text{with
prob. }p=|\gamma_{\alpha}|^{2}\vspace{0.03in}\\\
\;\big{|}\tilde{\psi}^{\perp}\big{\rangle}_{C}\left|\beta\right\rangle_{S}&\text{with
prob. }1-p=|\gamma_{\beta}|^{2},\end{cases}$

where $\left|\alpha\right\rangle$ and $\left|\beta\right\rangle$ are two
arbitrary $n$-qubit states and $\big{|}\tilde{\psi}\big{\rangle}$ and
$\big{|}\tilde{\psi}^{\perp}\big{\rangle}$ are two random $N$-qubit product
states that depend on the outcome of the $N$ single-qubit measurements
performed to implement $\mathcal{M}_{\Psi}$. The choice of
$\left|\psi\right\rangle$ and $\left|\psi^{\perp}\right\rangle$ within the
logical subspace hence allows one to effectively perform the corresponding
two-outcome measurement, i.e., an arbitrary projective measurement on the
logical qubit.

##  5 Resources cost

Here we analyze the resources in terms of the number of flips, $\eta$,
required for our setting. We call $\eta_{\lambda}$ the number of flips
required to implement the evolutions in different effective logical subspaces,
and $\eta_{l}$ the number of flips required in the control of the logical
qubit, i.e., $\eta=\eta_{l}+\eta_{\lambda}$. Recall that we denote by $n$ the
number of target qubits in $S$ and by $N$ the number of qubits of the control
system $C$.

Entanglement generation by gates sequence. In Sec. 3.1 and in Appendix C, we
show how we can prepare any state of the form Eq. (4) by applying at most
$2^{n}-1$ gate sequences of the form Eq. (6). Therefore, as for each subset
$S^{\prime}$ we need to apply $U^{S^{\prime}}$ and $U^{S^{\prime}\dagger}$,
and in Sec. 3, we showed that for implementing $U^{S^{\prime}}$ at most $N$
flips are required, the total number of flips to implement the target
interaction fulfils $\eta_{\lambda}\leq 2N(2^{n}-1)$. On the other hand,
$\eta_{l}\leq 2\eta_{V}(2^{n}-1)+\eta_{\text{i}}$, as first we need to
initialize the control system in the $\left|+\right\rangle_{C}$ state and two
Hadamard gates are needed to implement $\exp\\{-\text{i}\omega X^{C}\\}$ what
is used in each gate sequence. Therefore, this method requires at most
$\mathcal{O}(2^{n}N^{3})$ spin flips. We remark that for the preparation of
graph states, at most $n(n-1)/2$ phase gates are required. Hence, in this
case, the total number of flips is reduced to $\mathcal{O}(n^{2}N^{3})$.

Entanglement generation by projective measurements. To analyse the method
detailed in Sec. 3.2, we consider the preparation of an arbitrary graph state
$\left|G\right\rangle$ with $|E|$ edges and $V$ vertices, see Eq. (9). We
showed for each edge we need to implement one gate of the form
$U^{S^{\prime}}$, and hence $\eta_{\lambda}\leq|E|N$. Also in this case we
need to prepare the control system to the $\left|+\right\rangle_{C}$ state $E$
times, as after each measurement the state of the control system is collapsed,
and hence $\eta_{l}=|E|\eta_{\text{i}}$. Therefore, this method requires at
most $\mathcal{O}(|E|N^{3})$ spin flips. Again, since the number of edges is
upper bounded by $n(n-1)/2$, the required flips are polynomial in $n$ and $N$.

Cancellation of self-interactions. To cancel the effects of $H^{C}$ at a
specific time, we need to implement an interaction between each pair. The
method introduced in Sec. 3.1 corresponds to sequentially implementing an
effective interaction between each of the $n(n-1)/2$ pairs of qubits in $S$.
Therefore, if we use the method introduced in Sec. 3.1, $\eta_{\lambda}\leq
Nn(n-1)$ and $\eta_{l}=\eta_{V}n(n-1)+\eta_{\text{i}}$. The total number of
flips is given by $\mathcal{O}(n^{2}N^{3})$.

##  6 Applications and examples

The method we present above is completely general and can be utilized in
different setups. One can also envision applications for different quantum
processing tasks.

###  6.1 Applications

In the context of quantum computation, our approach can be used as a novel
design principle to realise a quantum processor in a setup where local control
is available, while interactions between qubits are not tunable and cannot be
manipulated selectively. Half of the qubits serve as a control system to
mediate interactions and realize two and multi-qubit gates between the
remaining qubits that form the quantum register. One can envision that the
whole system consists of the same type of qubits, but it is also conceivable
that the control system and the quantum register correspond to different
types, e.g., a well-controllable NV center with some surrounding nuclear
spins, or a system of a few trapped ions that is brought close to a crystal or
a surface to control interactions within atoms there. Notice that a self-
interaction among the system qubits, i.e., the quantum register, is not
required - only that the qubits that couple to the control system. Here we
have shown that controlling the internal state of $C$ suffices to mediate any
diagonal unitary operations on $S$. Notice that our technique can also be
combined with a more standard approach of controlling the coupling $H^{CS}$
directly, e.g., by moving the control system. The combination of the two
control techniques might allow to increase the coupling strengths, or reduce
the size of $C$.

For quantum networks, a crucial element is a quantum switch [24, 25] that
allows one to selectively transmit quantum information between input and
output ports, Fig. 2. Our approach provides this functionality in an
entanglement-based way, where any desired entanglement pattern (that can be
used to transmit quantum information via teleportation between the ports) can
be generated.

###  6.2 Examples: Trapped ions

(a)

(b)

(c)

Figure 4: Schematic representation of the different distributions of the ions.
Central circles (green) represent the qubits of the control system $C$,
meanwhile, the circles of the extremes (red) correspond to the qubits of the
target system $S$.

Several experiments have been performed using systems of trapped ions with
effective long-ranged, distant-dependent Ising interactions that are induced
by spin-dependent optical dipole forces implemented by an applied laser field
[8, 9, 10], where $f_{ij}=J|\boldsymbol{r}_{i}-\boldsymbol{r}_{j}|^{-\alpha}$
and $\alpha\approx 1$. This setup can also be used to demonstrate our
approach, which we illustrate with two examples.

The minimal setting is a linear string of four ions, where $C$ is given by the
two ions in the middle, while $S$ consists of the two ions at the ends of the
chain, see Fig. 4a. The internal state of $C$ allows one to couple selectively
to either (or both) of the two other ions, and also to generate entanglement
among the two outer qubits. Considering $\alpha=1$ and a constant distance $d$
between qubits, we can couple $C$ to only one of the qubits by setting the
control system into the logical sub-space given by
$\boldsymbol{c}=(1,\,-0.5)^{T}$ with an interaction coupling of
$\lambda_{1}=2.25f^{S}_{12}$, $\lambda_{2}=0$, or $C$ can couple
simultaneously to both qubits with the logical sub-space given by
$\boldsymbol{c}=(1,\,1)^{T}$ with an interaction coupling of
$\lambda_{1,\,2}=4.5f^{S}_{12}$, where $f^{S}_{12}=J/(3d)$ is the coupling
strength of the inherent ion-ion interaction between the qubits of $S$. If
there is a null interaction between qubit $s_{1}$ and qubit $S_{2}$, the time
required to prepare a Bell state in the target system with the methods
introduced in Sec. 3.1 and Sec. 3.2 are given in Equations (8) and (10), where
here correspond to

$\displaystyle t_{g}$ $\displaystyle=\frac{\pi
d}{3J}+\frac{\pi}{4\omega}+\frac{5\pi d}{4J}$ $\displaystyle t_{m}$
$\displaystyle=\frac{\pi d}{6J}+\frac{\pi d}{4J}+t_{\mathcal{M}}.$

This example can easily be scaled up to chains of size $4n$, with $n$ ions at
the borders forming $S$, and the remaining $2n$ ions in the middle forming
$C$. See Appendix G for details.

The second example is a 2D cross setting as illustrated in Fig. 4b, where a
control system of four ions serves as an entanglement switch between the four
outer qubits. Considering $\alpha=1$ and a constant distance $d$ between
qubits in each axis, we can simultaneously couple $C$ to selected spins with a
coupling strength $\lambda_{1,\,2}\approx 1.8f^{S}_{12}$ for adjacent and
$\lambda_{1,\,3}=1.3f^{S}_{13}$ for diametrically opposed pairs of qubits.
Equally coupling $C$ to all the qubits the coupling strength is given by
$\lambda_{1,2,3,4}\approx 6.5\,\langle f^{S}_{ij}\rangle$. If there is a null
interaction between qubit $S_{1}$ and qubit $S_{2}$, the time required to
prepare a Bell state in the target system with the methods introduced in Sec.
3.1 and Sec. 3.2 are given in Equations (8) and (10), where here correspond to

$\displaystyle t_{g}$ $\displaystyle=\frac{2\pi
d}{3.6J\sqrt{2}}+\frac{15\sqrt{2}\pi d}{2J}+\frac{\pi}{4\omega}$
$\displaystyle t_{m}$ $\displaystyle=\frac{\pi
d}{3.6J\sqrt{2}}+\frac{3\sqrt{2}\pi d}{2J}+t_{\mathcal{M}}.$

See Appendix H for details.

Note the effective coupling strength between spins can be increased using a
larger control system that mediates interactions. In turn, suppressing some
interactions and selectively coupling to specific spins reduces the coupling
strength.

##  7 Position noise in S

An important feature of our setting is that it still can be used in the
presence of noise in the position of the qubits, at the price of a slightly
reduced fidelity of the prepared states. More importantly, the effects of
position noise can be reduced by increasing the size of $C$. Moreover, with a
larger control system, we can implement techniques introduced in [26], which
allow us to just assume that each qubit is located within a certain region
instead of knowing its position exactly. This reduces the effective coupling,
yet leads to higher fidelity of the prepared states.

###  7.1 Noise model

First, we analyse how noise in the position of the target qubits affects our
setting. We consider that the position of each qubit is given by a normal
probability distribution, i.e., the probability of finding qubit $S_{i}$ in a
two-dimensional region $R$ is given by

$\int_{R}\rho_{i}\left(\boldsymbol{r}\right)\mathrm{d}\boldsymbol{r},$

with

$\rho_{i}\left(\boldsymbol{r}\right)=\frac{1}{2\pi\sigma^{2}}\,e^{-\frac{\left(\boldsymbol{r}-\bar{\boldsymbol{r}}^{S}_{i}\right)^{2}}{2\sigma^{2}}},$

where $\bar{\boldsymbol{r}}^{S}_{i}$ is the expectation value of its position,
and $\sigma^{2}$ is the variance of the distribution.

As the position of the qubits is not completely determined, it is convenient
to define the scalar field $\lambda(\boldsymbol{r})$ which corresponds to the
interaction coupling between a virtual qubit at position $\boldsymbol{r}$ and
$C$. It is given by

$\lambda(\boldsymbol{r})=\sum_{i=1}^{N}c_{i}\,\frac{J}{\left|\boldsymbol{r}-\boldsymbol{r}^{C}_{i}\right|}=\boldsymbol{f}(\boldsymbol{r})\cdot\boldsymbol{c},$

where

$\boldsymbol{f}(\boldsymbol{r})=\left(J\left|\boldsymbol{r}-\boldsymbol{r}^{C}_{1}\right|^{-1},\dots,J\left|\boldsymbol{r}-\boldsymbol{r}^{C}_{N}\right|^{-1}\right)$
(15)

is a vector field where component $f_{i}(\boldsymbol{r})$ corresponds to the
physical coupling strength between the virtual qubit at $\boldsymbol{r}$ and
qubit $C_{i}$.

Given a certain target interaction pattern
$\boldsymbol{\lambda}^{t}=(\lambda^{t}_{1},\dots,\lambda^{t}_{n})^{T}$, we
still can obtain the corresponding logical subspace by assuming that each
qubit of $S$ is located at its mean value and computing the interaction
matrix, i.e.,
$(\boldsymbol{F})_{ij}=J|\boldsymbol{r}^{C}_{i}-\bar{\boldsymbol{r}}^{S}_{j}|^{-1}$
and $\boldsymbol{c}=\boldsymbol{F}^{-1}\cdot\boldsymbol{\lambda}^{t}$.
However, once a logical subspace is established, the actual interaction
pattern depends on the exact location of the qubits, i.e.,
$\boldsymbol{\lambda}(\\{\boldsymbol{r}^{S}_{i}\\})$, whereby construction it
fulfils that
$\boldsymbol{\lambda}(\\{\bar{\boldsymbol{r}}^{S}_{i}\\})=\boldsymbol{\lambda}^{t}$,
and hence, $\boldsymbol{\lambda}$ is given by a probability distribution what
yields a noisy evolution of the system. The probability distribution of
$\lambda_{i}$ is given by
$\rho(\lambda_{i})=\rho_{i}(\boldsymbol{r})\,(\mathrm{d}\boldsymbol{r}/\mathrm{d}\lambda_{i})$,
and hence, an initial state $\varrho(0)$ evolves as

$\varrho(t)=\int
U\\!\left(\\{\boldsymbol{r}_{k}\\},t\right)\,\varrho(0)\;U^{\dagger}\\!\left(\\{\boldsymbol{r}_{k}\\},t\right)\,\rho_{1}(\boldsymbol{r}_{1})\cdots\rho_{n}(\boldsymbol{r}_{n})\,\mathrm{d}\boldsymbol{r}_{1}\cdots\mathrm{d}\boldsymbol{r}_{n},$
(16)

where

$U\left(\\{\boldsymbol{r}_{k}\\},t\right)=e^{-\text{i}\sum_{k=1}^{n}\lambda(\boldsymbol{r}_{k})\,Z^{C}Z^{S}_{k}\,t}.$
(17)

####  7.1.1 2D set-up: 8 ions with position fluctuation

To see how position noise affects the performance of our setting, we consider
this noisy model in the particular example of Fig. 4b without qubit $S_{4}$,
i.e., $S=\\{S_{1},S_{2},S_{3}\\}$. For that, we analyse the preparation of the
maximally entangled state

$\left|\psi\right\rangle=\frac{1}{2}\big{(}\left|00\right\rangle+e^{\text{i}\pi/2}\left|01\right\rangle+e^{\text{i}\pi/2}\left|10\right\rangle+\left|11\right\rangle\big{)}$
(18)

with qubits $S_{1}$ and $S_{2}$, and we calculate the fidelity of resulting
state $\varrho_{\psi}$ with the target state
$\left|\psi\right\rangle\\!\left\langle\psi\right|$.

We consider the method detailed in Sec. 3.1, which consist in applying the
gates sequence $U\exp\\{\text{i}\frac{\pi}{4}X^{C}\\}U^{\dagger}$ with

$U=e^{-\text{i}\,\frac{\pi}{4}Z^{C}\left(Z^{S}_{1}+Z^{S}_{2}\right)}$

to the initial state $\left|+\right\rangle_{C}\left|+++\right\rangle_{S}$. Due
to the noise, the evolution of the state is given by Equations (16) and (17),
with $n=3$, $t=\pi/(4\lambda_{\max}^{(1,2)})$, and
$\lambda(\boldsymbol{r}_{k})=\boldsymbol{f}(\boldsymbol{r}_{k})\cdot\boldsymbol{c}^{(1,2)}$.
Notice, that qubit $S_{3}$ must be also taken into account as due to the noise
in its position, it is not totally decoupled from $C$.

The resulting state is a mixed state $\varrho_{\psi}$ that correlates $C$ and
$S$. Therefore, the fidelity with the target state
$\left|\psi\right\rangle\\!\left\langle\psi\right|_{S_{1}S_{2}}$ is given by

$F=\langle\psi|\,\text{tr}_{CS_{3}}(\varrho_{\psi})\,\left|\psi\right\rangle.$

In Table 1, we show the fidelity of the resulting state for different values
of the standard deviation $\sigma$ in the position of the qubits. Observe,
that if the qubit is trapped with a small error the setting still allows one
to prepare highly entangled states. On the other hand, when the error in the
trap increases the fidelity of the prepared state drops drastically. In that
case, by increasing the number of qubits in $C$ we can reduce the effect of
noise, as we show in the next section.

$\quad\sigma\quad$ | $F$
---|---
$\quad 0.15d\quad$ | $\quad 0.918986\quad$
$\quad 0.10d\quad$ | $0.965297$
$\quad 0.05d\quad$ | $0.991552$
$\quad 0.01d\quad$ | $0.999665$
Table 1: Fidelity of the state of qubits $S_{1}$ and $S_{2}$, resulting from
the preparation of the Bell state given in Eq. (18) with the method of Sec.
3.1, for different values of the standard deviation $\sigma$ of the position
of the qubits of $S$, where the spatial distribution of the qubits is shown in
Fig. 4b with $d=d_{1}=d_{2}$.

###  7.2 Reducing noise effects

In this section, we show how we can protect our setting from the effect of
noise on the positions of the target system by increasing the number of qubits
in the control system. This is achieved by setting the effective couplings
with each qubit for a certain region instead of only for the exact positions.

As we showed above, the interaction coupling between $C$ and a virtual qubit a
position $\boldsymbol{r}$ is given by
$\lambda(\boldsymbol{r})=\boldsymbol{f}(\boldsymbol{r})\cdot\boldsymbol{c}$,
where $\boldsymbol{f}(\boldsymbol{r})$ is defined in Eq. (15). So far, to find
the logical subspace that generates a target interaction pattern
$\boldsymbol{\lambda}^{t}$, we assume that each qubit is in a particular known
position, i.e., we find the vector $\boldsymbol{c}$ that fulfils
$\lambda_{i}^{t}=\boldsymbol{f}(\bar{\boldsymbol{r}}^{S}_{i})\cdot\boldsymbol{c}\;\forall\,i$.
However, when the position of each qubit is given by a probability
distribution with a non-negligible variance, our assumption of its position
can significantly decrease the fidelity of the resulting state, as we showed
in the previous section. To reduce the impact of this kind of noise, we
enhance our setting by considering that each qubit $S_{i}$ is located in a
region $R_{i}$ around the expected value of its position
$\bar{\boldsymbol{r}}^{S}_{i}$. Then, we establish a logical subspace for $C$
that generates the target interaction coupling $\lambda_{i}^{t}$ for all
region $R_{i}$. We achieve that by implementing the two methods introduced in
[26].

Method 1. We call $R^{(n)}$ the region around a point $\boldsymbol{r}_{0}$
where the vector field $\boldsymbol{f}(\boldsymbol{r})$ can be approximated by
its $n$-order Taylor expansion, i.e., assuming a two-dimensional space for the
position of the qubits in $R^{(1)}$ we can approximate

$\boldsymbol{f}\left(\boldsymbol{r}\right)\,\approx\,\boldsymbol{f}(\boldsymbol{r}_{0})+(x-x_{0})\left(\frac{\partial\boldsymbol{f}}{\partial
x}\right)_{\\!\boldsymbol{r}_{0}}+(y-y_{0})\left(\frac{\partial\boldsymbol{f}}{\partial
y}\right)_{\\!\boldsymbol{r}_{0}},$

where $\boldsymbol{r}=(x,y)$, and hence, in $R^{(1)}$ the effective coupling
with $C$ is given by

$\lambda(\boldsymbol{r})\,\approx\,\underset{(a)}{\underbrace{\boldsymbol{f}(\boldsymbol{r}_{0})\cdot\boldsymbol{c}}}+(x-x_{0})\underset{(b)}{\underbrace{\left[\left(\frac{\partial\boldsymbol{f}}{\partial
x}\right)_{\\!\boldsymbol{r}_{0}}\cdot\boldsymbol{c}\,\right]}}+(y-y_{0})\underset{(c)}{\underbrace{\left[\left(\frac{\partial\boldsymbol{f}}{\partial
y}\right)_{\\!\boldsymbol{r}_{0}}\cdot\boldsymbol{c}\,\right]}}.$ (19)

Then, we can obtain an effective coupling $\lambda^{t}$ between $C$ and a
virtual qubit in $R^{(1)}$, by finding a vector $\boldsymbol{c}$ that fulfills
$(a)=\lambda^{t}$, and terms $(b)=(c)=0$ in Eq. (19). Therefore, when
considering multiple target qubits with a noisy position, we can assume that
each qubit is within its corresponding $R^{(1)}_{i}$ instead of in a
particular position and then the logical vector $\boldsymbol{c}$ is such that
fulfils
$\widetilde{\boldsymbol{F}}\cdot\boldsymbol{c}=\boldsymbol{\lambda}^{t}$,
i.e.,

$\begin{pmatrix}\boldsymbol{f}(\bar{\boldsymbol{r}}^{S}_{1})\\\ \vdots\\\
\boldsymbol{f}(\bar{\boldsymbol{r}}^{S}_{n})\\\
(\partial_{x}\boldsymbol{f})(\bar{\boldsymbol{r}}^{S}_{1})\\\ \vdots\\\
(\partial_{y}\boldsymbol{f})(\bar{\boldsymbol{r}}^{S}_{n})\end{pmatrix}\begin{pmatrix}c_{1}\\\
c_{2}\\\ c_{3}\\\ \vdots\\\ c_{N-1}\\\
c_{N}\end{pmatrix}=\begin{pmatrix}\lambda_{1}^{t}\\\ \vdots\\\
\lambda_{n}^{t}\\\ 0\\\ \vdots\\\ 0\end{pmatrix}$

where $dim(\widetilde{F})=3n\times n$ (where $n$ is the number of qubits in
$S$). Notice, that if $dim(\boldsymbol{c})=3n$ the system of equations can
always be solved. This means if $S$ consist of $n$ qubits, $C$ has to contain
$N=3n$ qubits. If considering three-dimensional settings $\widetilde{F}$ has
$4n$ rows and the required number of qubits in $C$ has to be given by $N=4n$,
as it has to be insensitive to the three spatial derivatives of each qubit
location.

This provides us with a way of establishing an effective coupling with a
target qubit that is within $R^{(1)}$. If the position noise is small enough
to fulfil this assumption, the effect of the noise is significantly reduced.
However, if the position noise is too large it may not be enough to assume
that $S_{i}$ is within $R_{i}^{(1)}$. In this case, we assume that the qubits
are located in a larger region $R^{(n)}$ and we establish the coupling for all
that region. For that, field $\boldsymbol{f}(\boldsymbol{r})$ has to be
expanded in its Taylor series until order $n$, and in the same way, as we did
for $R^{(1)}$, we have to find a vector $\boldsymbol{c}$ that eliminates all
the contributions except the zero order, i.e.,

$\displaystyle\boldsymbol{f}(\boldsymbol{r}_{0})\cdot\boldsymbol{c}$
$\displaystyle=\lambda$
$\displaystyle\left(\frac{\partial^{k}\boldsymbol{f}}{(\partial
x)^{r}(\partial y)^{k-r}}\right)_{\boldsymbol{r}_{0}}\cdot\boldsymbol{c}$
$\displaystyle=0$

for $0\leq r\leq k$ and $1\leq k\leq n$. In this way, if one can assume that
the target qubits are located in finite regions that do not overlap, the
effect of noise can be counteracted by considering a large enough control
system. In particular, considering a two-dimensional position space the number
of constraints that vector $\boldsymbol{c}$ has to fulfil increase
quadratically with $n$, and therefore, the size of $C$.

(a)

(b)  (c)

(d)

Figure 5:  Four different ways of discretise region $R$. Black circles
represent virtual qubits.

Method 2. We also can establish an approximated effective coupling for region
$R$ by discretising it. Given a region $R$ we divided it in $k$ subregions
$R^{(i)}$, and we consider a virtual qubit $v_{i}$ in the center of each
subregion $\boldsymbol{r}_{v_{i}}$, see Fig. 5. To find a logical subspace
that couples with a target qubit within region $R$ with a certain effective
coupling $\lambda$, we just have to obtain a subspace that yields the same
coupling with all the virtual qubits, i.e.,

$\begin{pmatrix}c_{1}\\\ \vdots\\\
c_{N}\end{pmatrix}=\begin{pmatrix}\boldsymbol{f}(\boldsymbol{r}_{v_{1}})\\\
\vdots\\\
\boldsymbol{f}(\boldsymbol{r}_{v_{k}})\end{pmatrix}^{-1}\begin{pmatrix}\lambda\\\
\vdots\\\ \lambda\end{pmatrix}$

Note that with this method, to couple $C$ with $n$ target qubits, the control
system has to contain $kn$ qubits where $k$ is the number of virtual qubits in
each region.

This approach guarantees a constant coupling strength whenever the physical
qubit is at the position of one of the virtual qubits. By continuity, the
coupling strength is also approximately the same in the neighbourhood of the
positions, and hence in the whole region.

####  7.2.1 Two-dimensional example of noise reduction

$\sigma$ | $F_{0}$ | $F_{1}$ | $F_{2}$ | $F_{3}$
---|---|---|---|---
$\quad 0.25d\quad$ | $0.934582$ | $0.948641$ | $0.981530$ | $0.990489$
$0.20d$ | $\quad 0.959314\quad$ | $\quad 0.967918\quad$ | $0.993573$ | $\quad 0.996809\quad$
$0.15d$ | $0.977671$ | $0.982306$ | $\quad 0.998282\quad$ | $0.999155$
$0.10d$ | $0.990255$ | $0.992248$ | $0.999701$ | $0.999853$
Table 2: Fidelity of the prepare state of with qubits $S_{1}$ and $S_{2}$, with the target state given in Eq. (18) with the method of Sec. 3.1, for different values of the standard deviation $\sigma$ of the position of the qubits of $S$, where the spatial distribution of the qubits is shown in Fig. 4c with $d=d_{1}=d_{2}$. $F_{i}$ consider the state obtained by using $\boldsymbol{c}_{i}$. $\sigma$ | $\langle\lambda_{1,2}\rangle$ | $\boldsymbol{c}_{3}$
---|---|---
$\;\;0.25d\;\;$ | $\;\;0.0139\;\;$ | $\;\;(-0.0970,0.3703,-0.0966,0.3827,-1,0.3814,-0.0331,0.1508,-0.0324)\;\;$
$0.20d$ | 0.0136 | $(-0.0945,0.3675,-0.0941,0.3788,-1,0.3775,-0.0307,0.1504,-0.0301)$
$0.15d$ | 0.0133 | $(-0.0927,0.3654,-0.0923,0.3756,-1,0.3748,-0.0292,0.1506,-0.0287)$
$0.10d$ | 0.0131 | $(-0.0913,0.3639,-0.0911,0.3739,-1,0.3735,-0.0285,0.1513,-0.0282)$
Table 3: Logical vector used to couple to $S_{1}$ and $S_{2}$ with method 2,
where the spatial distribution of the qubits is shown in Fig. 4c with
$d=d_{1}=d_{2}$.

We analyse the effect of position noise in a particular example. We compare
the fidelity of the prepared state in the minimal size setting with different
options to reduce the noise effects. In total, we consider three models. Model
0 consists of the minimal-size control system, which is the one where $C$
contains the same number of qubits as $S$. Model 1 is given by a setting where
the number of qubits in $C$ is three times larger than in $S$. Models 2 and 3
are given by the same setting used in model 1 but by generating the effective
coupling of $C$ around a certain region $R_{i}^{(1)}$ for each qubit by using
method 1 and method 2 respectively. We denote as $F_{k}$ the prepared state
fidelity and $\boldsymbol{c}_{k}$ the logical subspace of $C$ of model $k$.

In particular, we consider a target system $S$ of 3 qubits where the qubits
are distributed as shown in Fig. 4c where we take $d_{2}=3d_{1}$. The qubits
of $C$ are also spatially distributed as shown in Fig. 4c, but for the
minimal-size setting, we only consider qubits $\\{C_{1},C_{3},C_{8}\\}$. We
then assume the preparation of the Bell state given in Eq. (18) between
$S_{1}$ and $S_{2}$ with the method detailed in Sec. 3.1.

For that, we have to generate the target interaction pattern given by
$\boldsymbol{\lambda}^{t}=(1,\,1,\,0)^{T}$. We compute the logical subspace as
$\boldsymbol{c}=\boldsymbol{F}^{-1}\cdot\boldsymbol{\lambda}^{t}$ for the mean
value in the position of the qubits. We obtain: for the minimal-size setting,
i.e., model 0,

$\boldsymbol{c}_{0}^{(1,2)}=\Big{(}1,\,1,\,-4/\sqrt{17}\Big{)}^{T}$

with $\langle\lambda_{1,2}\rangle=0.675J/d_{1}$. For model 1

$\displaystyle\boldsymbol{c}_{1}^{(1,2)}=\Big{(}1,\,1,\,1,\,1,\,0.0807,\,1,\,-1,\,-1,\,-1\Big{)}^{T}$

with $\langle\lambda_{1,2}\rangle=1.265J/d_{1}$. For model 2 we obtain the
logical subspace as
$\boldsymbol{\boldsymbol{c}}=\widetilde{\boldsymbol{F}}^{-1}\cdot\tilde{\boldsymbol{\lambda}}^{t}$,
where
$\tilde{\boldsymbol{\lambda}}^{t}=(\lambda^{t}_{1},\lambda^{t}_{2},\lambda^{t}_{3},0,\dots,0)$,
and we obtain

$\displaystyle\boldsymbol{c}_{2}^{(1,2)}=\Big{(}-0.4842,\,0.5667,\,-0.4842,\,1,$
$\displaystyle-0.1852,\,1,\,-1,\,0.8339,$ $\displaystyle\,-1\Big{)}^{T}$

with $\langle\lambda_{1,2}\rangle=0.098J/d_{1}$. For model 3 we assume a
virtual qubit in the expected value of the position, and the other two around
the first one as shown in Fig. 5a at a distance of $\sigma$. therefore, the
logical subspaces used depend on $\sigma$ and are shown in Table 3.

We compute the fidelity $F$ of the prepared state by using the four different
logical subspaces, $\boldsymbol{c}_{0}$, $\boldsymbol{c}_{1}$,
$\boldsymbol{c}_{2}$ and $\boldsymbol{c}_{3}$, see Table 2. Observe the lowest
fidelity is obtained for the minimal-size control system, i.e., if $C$ only
contains three qubits ($C_{1}$, $C_{2}$ and $C_{8}$). Then, if $C$ contains
nine qubits, a stronger coupling can be obtained and the effect of noise is
reduced, i.e., the fidelity of the output state is larger. Finally,
considering nine qubits in $C$ and the coupling for each qubit is established
in a region $R_{i}^{(1)}$, the effect of noise is strongly reduced even though
a much weaker coupling is obtained. Therefore, in this last case, the
evolution time required to produce the state is larger.

##  8 Conclusion and outlook

We have introduced a method to generate entanglement in systems without
quantum control in a remotely controlled way. The main tool is the effective
control of distance-dependent always-on interactions by the choice of the
internal state of a high-dimensional control system. By using the control
system as an effective two-level system, i.e., as a single logical qubit, we
can selectively couple it to outside systems, and use built-up entanglement to
mediate interactions, perform gates and prepare entangled states on a remote
system. While the latter has been utilized in different contexts [27], the
control of effective couplings by choice of internal states only provides new
and interesting possibilities for the design of quantum processors or
entanglement switches. In Appendix A we also compare our approach to more
restricted techniques introduced in [2, 3, 4, 5].

We point out that the techniques to deal with spatially correlated noise
processes outlined in [11, 12] in the context of distributed quantum metrology
can be utilized here as well. That is, we can use our control system also in
such a way that noise processes with spatial correlation (e.g., a constant but
fluctuating global field) can be fully suppressed in the target system while
maintaining the full functionality of our remote entanglement preparation
scheme. This only requires a slightly enlarged control system. In addition,
noise within the control system can be treated similarly. However, that is a
much more complex problem that we treat in [28].

Finally, we point out that we use our control system as an artificial two-
level system, where we can control the interactions with the surroundings by
choice of internal state. One can use a collection of such artificial two-
level systems to realize a programmable quantum simulator [29] in the presence
of some distance-dependent, always-on coupling.

This work was supported by the Austrian Science Fund (FWF) through projects
No. P30937-N27, No. P36009-N and No. P36010-N. Finanziert von der Europäischen
Union - NextGenerationEU.

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##  Appendix A Comparison with previous results on remote control

Other control device techniques have been developed in recent years. Here, we
point out the main differences with our approach, including the considered
problems, methods, and possible applications.

In our approach, we consider the controllability of an ensemble of spins
utilizing an auxiliary quantum system in the presence of an interaction with
some (arbitrary) distance dependence. Indeed, this is similar to the scenario
considered in [3-6], where criteria to add control to a remote quantum system
were studied. However, the approaches are fundamentally different as we now
explain.

In Refs. [3-6] symmetry assumptions are crucial. In these works, the dynamical
Lie algebra of the setting is analyzed to obtain controllable subspaces, by
studying the symmetries in the target system. On the other hand, our setting
uses the distance dependence of the Ising interaction between the qubits to
selectively choose the coupling of our control system. There is no need for
any underlying symmetry, and in fact, the method is also applicable for other
interactions, in particular also with other distance dependencies - or even
other types of interactions if one allows for dynamical control as we point
out below. As long as interactions show some kind of distance dependence, we
provide methods to remotely generate entanglement in a target system without
control there. In this case, we can perform any computational diagonal unitary
in our target system and therefore generate any state of the form

$\left|\psi\right\rangle=\sum_{i_{1},\dots,i_{n}=0}^{1}e^{\text{i}\,\theta_{i_{1}\cdots
i_{n}}}\left|i_{1},\dots,i_{n}\right\rangle,$

what includes any stabilizer or graph state.

To summarize, we point out the advantages of our method over previous
approaches, and novel contributions:

  * •

We use an auxiliary control system that consists of multiple qubits. Local
control of individual single qubits is sufficient to obtain not only full
control within the control system (using the always-on self-interaction) but
also to achieve remote control in an outside system that is not accessible.

  * •

We explicitly show how to remotely generate multipartite entanglement
efficiently. We explicitly provide the states and control sequences that are
required for particular settings.

  * •

Our methods are applicable in asymmetric situations and solely rely on the
distance dependence in the underlying interaction that can be arbitrary.

  * •

Our approach can be extended to XY and XYZ interactions using (fast) local
control in the control system only, as we now point out in the next section.

  * •

We provide a way to deal with noise and imperfections. In particular, we show
how one can deal with thermal position fluctuation in the target systems, and
still obtain entangled target states with high fidelity. Increasing the size
of the control system allows one to increase fidelity.

##  Appendix B Heisenberg interaction model

In this appendix we show how our setting still can be implemented if a general
XYZ-type interaction (including e.g. Heisenberg interaction) is given between
the qubits, i.e., the interaction between $C$ and $S$ is given by

$H^{CS}=\sum_{\begin{subarray}{c}1\leq i\leq N\\\ 1\leq j\leq
n\end{subarray}}\alpha^{CS}_{ij}\,X^{C}_{i}X^{S}_{j}+\beta^{CS}_{ij}\,Y^{C}_{i}Y^{S}_{j}+\gamma^{CS}_{ij}\,Z^{C}_{i}Z^{S}_{j}.$

The analysis can not be straightforwardly extended to deal with such kinds of
interactions directly. However, local operations in the control system suffice
to modify the interactions between $S$ and $C$ to the ZZ-coupling we consider
in the main text. This makes our method also applicable for the more general
interactions and hence to a much larger class of systems and set-ups.

The basic idea is to use fast local control to eliminate some of the
interaction of the Hamiltonian. This can be done by fast intermediate control
pulses in $C$ only, which leads to a negative sign for $XX$ and $YY$ terms,
and allows one to eliminate them by altering between the modified and
unperturbed evolution. In other words, we define

$U(t)=e^{-\text{i}H^{CS}t}$

and

$\tilde{U}(t)=Z^{C}_{1}\cdots Z^{C}_{N}\,e^{-\text{i}H^{CS}t}\,Z^{C}_{1}\cdots
Z^{C}_{N},$

and by alternating them we reproduce the pairwise ZZ interaction, i.e.,

$\lim_{k\to\infty}\left[\tilde{U}\left(t/2k\right)U\left(t/2k\right)\right]^{k}=e^{-\text{i}H_{z}^{CS}t}$

where

$H_{z}^{CS}=\sum_{\begin{subarray}{c}1\leq i\leq N\\\ 1\leq j\leq
n\end{subarray}}\gamma^{CS}_{ij}Z^{C}_{i}Z^{S}_{j}.$

##  Appendix C Entanglement generation by generating multi-qubit Z gates

Multi-qubit gates with arbitrary phases can be mediated in any subset
$S^{\prime}\subseteq S$ by manipulating $C$. Applying first a maximally
entangling operation between the qubits and the control system, given by
$U^{S^{\prime}}$, see Eq. (5), followed by an X-rotation of the logical qubit
and again by the same entangling operation, one produces a multi-qubit
interaction between the logical qubit and the qubits in $S^{\prime}$, i.e.,

$U^{S^{\prime}}\,e^{-\text{i}\,\omega X^{C}}\,U^{\dagger}=e^{-\text{i}\,\omega
G^{C}Z^{S}_{1}\\!\cdots Z^{S}_{N}},$

where if $C$ couples to the $n^{\prime}$ qubits in $S^{\prime}$, the unitary
$G$ is given in Eq. (7). Note that, $G\in\\{\pm X,\pm Y\\}$, and leaving the
state of $C$ in an eigenstate of $G^{C}$, one obtains an effective multiqubit
Z-interaction in $S$ which can, e.g., be used to generate GHZ states (up to
local unitary operations) directly by choosing $\omega=\pi/4$.

Alternatively, one can use this method on all subsets $S^{\prime}$ of qubits
sequentially. There are $2^{n}-1$ such subsets, and one can freely choose the
interaction strength $\omega_{i}$ for each subgroup. Hence, with a proper
choice of induced interaction phases $\\{\omega_{i}\\}_{i=1}^{2^{n}-1}$, one
can generate all states of the form Eq. (4). Notice that the phases
$\\{\omega_{i}\\}$ can be determined from $\\{\theta_{1,\dots,n}\\}$ of Eq.
(4) by solving a system of linear equations. These methods are however costly,
as maximally entangling gates between $C$ and $S$ are required to produce even
a small interaction, and at least four maximally entangling gates are used to
generate a maximally entangled state in $S$.

##  Appendix D Entanglement generation by control rotation: details

Setting an interaction pattern that equally couples the control system with
two qubits $S_{i}$ and $S_{j}$, and simultaneously applying a rotation to the
logical qubit in $C$ in the X-direction, one can implement an effective ZZ-
interaction between the qubits.

Such evolution is generated by

$H_{x}=\lambda Z^{C}\big{(}Z^{S}_{i}+Z^{S}_{j}\big{)}+\omega X^{C},$

and it transforms a general two qubits state
$\left|\psi\right\rangle=\sum_{i,j=0}^{1}\psi_{ij}\left|ij\right\rangle$ as

$\displaystyle
e^{-\text{i}H_{x}t}\left|+\right\rangle_{C}\left|\psi\right\rangle_{S_{i}S_{j}}$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}a_{+}\psi_{00}\left|0\right\rangle_{C}\left|00\right\rangle_{S_{i}S_{j}}+b\,\psi_{01}\left|0\right\rangle_{C}\left|01\right\rangle_{S_{i}S_{j}}+b\,\psi_{10}\left|0\right\rangle_{C}\left|10\right\rangle_{S_{i}S_{j}}+a_{-}\psi_{11}\left|0\right\rangle_{C}\left|11\right\rangle_{S_{i}S_{j}}$
$\displaystyle+\,a_{-}\psi_{00}\left|1\right\rangle_{C}\left|00\right\rangle_{S_{i}S_{j}}+\,b\,\psi_{01}\left|1\right\rangle_{C}\left|01\right\rangle_{S_{i}S_{j}}+b\,\psi_{10}\left|1\right\rangle_{C}\left|10\right\rangle_{S_{i}S_{j}}+a_{+}\psi_{11}\left|1\right\rangle_{C}\left|11\right\rangle_{S_{i}S_{j}}$
$\displaystyle\big{)},$

where

$\begin{gathered}a_{\pm}=\cos\left(\sqrt{\omega^{2}+4\lambda^{2}}t\right)-\text{i}\,\frac{(\omega\pm
2\lambda)\sin\left(\sqrt{\omega^{2}+4\lambda^{2}}t\right)}{\sqrt{\omega^{2}+4\lambda^{2}}}\\\
b=e^{-\text{i}\omega t}.\end{gathered}$

Note that $a_{\pm}=(-1)^{k}$ for $t=k\pi/\sqrt{\omega^{2}+4\lambda^{2}}$
($k\in\mathbb{N}$), and hence the control system factors-out leaving the state
of $S_{i}$ and $S_{j}$ in an entangled state, i.e.,

$e^{-\text{i}H_{x}\tau}\left|+\right\rangle_{C}\left|\psi\right\rangle_{S_{i}S_{j}}=\left|+\right\rangle_{C}\left(\psi_{00}\left|00\right\rangle_{S_{i}S_{j}}+e^{\text{i}2\phi}\psi_{01}\left|01\right\rangle_{S_{i}S_{j}}+e^{\text{i}2\phi}\psi_{10}\left|10\right\rangle_{S_{i}S_{j}}+\psi_{11}\left|11\right\rangle_{S_{i}S_{j}}\right)$

where $\phi=\frac{\pi k}{2}(1-\omega/\sqrt{\omega^{2}+4\lambda^{2}})$. Since
the final state of $S_{i}$ and $S_{j}$ can be written as $e^{-\text{i}\,\phi
ZZ}\left|\psi\right\rangle$ an arbitrary effective ZZ-interaction is generated
between $S_{i}$ and $S_{j}$.

For an initial state $\left|\psi\right\rangle=\left|+\,+\right\rangle$,
setting $\lambda=\omega\sqrt{3}/2$ and leaving the system to evolve a time
$\tau=\pi/(2\omega)$ the induced phase between the qubits is given by
$\phi=\pi/4$ and therefore the qubits end up in a maximally entangled state,
i.e.,

$e^{-\text{i}\frac{\pi}{4}ZZ}\left|++\right\rangle=\frac{1}{\sqrt{2}}\big{(}\left|00_{y}\right\rangle+\text{i}\,\left|11_{y}\right\rangle\big{)}$

where
$\left|0_{y}\right\rangle=(\left|0\right\rangle+\text{i}\,\left|1\right\rangle)/\sqrt{2}$
and
$\left|1_{y}\right\rangle=(\left|0\right\rangle-\text{i}\,\left|0\right\rangle)/\sqrt{2}$.

The same method can be used to prepare a locally unitary (LU) equivalent GHZ
state with three qubits. The evolution generated by

$H_{x}=\lambda Z^{C}\big{(}Z^{S}_{i}+Z^{S}_{j}+Z^{S}_{k}\big{)}+\omega X^{C}$

leads to

$\displaystyle
e^{-\text{i}H_{x}t}\left|+\right\rangle_{C}\left|+++\right\rangle_{S_{i}S_{j}S_{k}}$
$\displaystyle=\frac{1}{4}\big{(}a_{+}\left|0\right\rangle\left|000\right\rangle+b_{+}\left|0\right\rangle\left|001\right\rangle+b_{+}\left|0\right\rangle\left|010\right\rangle+b_{-}\left|0\right\rangle\left|011\right\rangle$
$\displaystyle+b_{+}\left|0\right\rangle\left|100\right\rangle+b_{-}\left|0\right\rangle\left|101\right\rangle+b_{-}\left|0\right\rangle\left|110\right\rangle+a_{-}\left|0\right\rangle\left|111\right\rangle$
$\displaystyle+a_{-}\left|1\right\rangle\left|000\right\rangle+b_{-}\left|1\right\rangle\left|001\right\rangle+b_{-}\left|1\right\rangle\left|010\right\rangle+b_{+}\left|1\right\rangle\left|011\right\rangle$
$\displaystyle+b_{-}\left|1\right\rangle\left|100\right\rangle+b_{+}\left|1\right\rangle\left|101\right\rangle+b_{+}\left|1\right\rangle\left|110\right\rangle+a_{+}\left|1\right\rangle\left|111\right\rangle\big{)}$

where

$\displaystyle a_{\pm}$
$\displaystyle=\cos\left(\sqrt{\omega^{2}+9\lambda^{2}}\,t\right)-\text{i}\,\frac{\omega\pm
3\lambda}{\sqrt{\omega^{2}+9\lambda^{2}}}\sin\left(\sqrt{\omega^{2}+9\lambda^{2}}\,t\right)$
$\displaystyle b_{\pm}$
$\displaystyle=\cos\left(\sqrt{\omega^{2}+\lambda^{2}}\,t\right)-\text{i}\,\frac{\omega\pm\lambda}{\sqrt{\omega^{2}+\lambda^{2}}}\sin\left(\sqrt{\omega^{2}+\lambda^{2}}\,t\right).$

Setting $\omega=\sqrt{5}\lambda/\sqrt{3}$ such that
$\sqrt{\omega^{2}+9\lambda^{2}}=2\sqrt{\omega^{2}+\lambda^{2}}$, and
$\lambda=\sqrt{3}\pi/(2\sqrt{2}t)$ such that
$t\sqrt{\omega^{2}+\lambda^{2}}=\pi$ and
$t\sqrt{\omega^{2}+9\lambda^{2}}=2\pi$ one guarantees that the complex part of
the coefficients cancels and they become $b_{\pm}=-1$ and $a_{\pm}=1$. At thus
time the control state factors out leaving the qubits in a LU equivalent graph
state, i.e.,

$\begin{gathered}e^{-\text{i}\,\frac{\pi}{4}H_{x}\tau}\left|+\right\rangle_{C}\left|+++\right\rangle_{S_{i}S_{j}S_{k}}=e^{-\text{i}\,\frac{\pi}{4}\left(Z^{S}_{i}Z^{S}_{j}+Z^{S}_{i}Z^{S}_{k}+Z^{S}_{j}Z^{S}_{k}\right)}\left|+\right\rangle_{C}\left|+++\right\rangle_{S_{i}S_{j}S_{k}}\\\
=\frac{1}{2\sqrt{2}}\left|+\right\rangle\big{(}\left|000\right\rangle-\left|001\right\rangle-\left|010\right\rangle-\left|011\right\rangle-\left|100\right\rangle-\left|101\right\rangle-\left|110\right\rangle+\left|111\right\rangle\big{)}.\end{gathered}$

##  Appendix E Entanglement generation by alternating interaction patterns

A simple way to entangle two qubits of the target system is to let $C$ and $S$
evolve sequentially under two different interaction patterns where each
pattern couples $C$ to only one of the two qubits. With the appropriate
locally (at $C$) altered patterns, a Bell state between the pair of qubits can
be generated while the state of the control system factors out, e.g.,

$\displaystyle
e^{-\text{i}\,\frac{\pi}{4}Z^{C}Z^{S}_{i}}e^{-\text{i}\,\frac{\pi}{4}Y^{C}Z^{S}_{j}}e^{-\text{i}\,\frac{\pi}{4}Z^{C}Z^{S}_{i}}\left|+\right\rangle_{C}\left|++\right\rangle_{S_{i}S_{j}}=\frac{1}{\sqrt{2}}\,\left|-\right\rangle_{C}\Big{(}\left|00_{y}\right\rangle_{S_{i}S_{j}}-\text{i}\,\left|11_{y}\right\rangle_{S_{i}S_{j}}$
$\displaystyle\Big{)}.$

where
$\left|0_{y}\right\rangle=(\left|0\right\rangle+\text{i}\,\left|1\right\rangle)/\sqrt{2}$
and
$\left|1_{y}\right\rangle=(\left|0\right\rangle-\text{i}\,\left|1\right\rangle)/\sqrt{2}$.
Notice that, the evolution generated by $\lambda Y^{C}Z^{S}_{i}$ can be
obtained by applying a change of basis of the logical qubit,

$e^{\text{i}\,\frac{\pi}{4}X^{C}}\,e^{-\text{i}\lambda
Z^{C}Z^{S}_{i}t}\,e^{-\text{i}\,\frac{\pi}{4}X^{C}}=e^{-\text{i}\lambda\,Y^{C}Z^{S}_{i}\,t}.$

By repeating the protocol on different pairs of qubits in $S$, one can
generate any graph state in up to some LU operations.

##  Appendix F Entanglement generation by Hamiltonian simulation

Alternating the evolution generated by two Hamiltonians $H_{1}$ and $H_{2}$,
one can approximate the evolution generated by the commutator
$[H_{1},\,H_{2}]$ for small $t$, as
$e^{\text{i}H_{1}t}e^{\text{i}H_{2}t}e^{-\text{i}H_{1}t}e^{-\text{i}H_{2}t}\approx
e^{-[H_{1},H_{2}]\,t^{2}}$. One can use this technique to generate an
effective three-qubit interaction in second order by manipulating the control
system. Setting the two interaction patterns

$\begin{gathered}H_{1}=\sum_{i=1}^{n}\lambda_{i}X^{C}Z^{S}_{i}\\\
H_{2}=\sum_{i=1}^{n}\mu_{i}Y^{C}Z^{S}_{i}\end{gathered}$

one approximates the evolution given by the commutator

$\left[H_{1},H_{2}\right]=2\,\text{i}\sum_{i,j=1}^{n}\lambda_{i}\mu_{j}Z^{C}Z^{S}_{i}Z^{S}_{j}.$

Then by leaving the state of $C$ in an eigenstate of $Z^{C}$, an effective ZZ
pairwise interaction is generated in $S$ where the interaction coupling
between qubits $S_{i}$ and $S_{j}$ is given by
$2(\lambda_{i}\mu_{j}+\lambda_{j}\mu_{i})$.

##  Appendix G 1D set-up: 4 ions

Assume four ions trapped in a 1D line where the separation between the ions is
given by $d_{1}$, $d_{2}$ and $d_{3}$, as it is shown in Fig. 4a, and consider
the coupling between ions given by
$J|\boldsymbol{r}_{i}-\boldsymbol{r}_{j}|^{-1}$. For this particular example,
the inverse of the interaction matrix $\boldsymbol{F}$ is given by

$\boldsymbol{F}^{-1}=\frac{1}{Jd_{2}(d_{1}+d_{2}+d_{3})}\begin{pmatrix}d_{1}(d_{1}+d_{2})(d_{2}+d_{3})&-d_{1}d_{3}(d_{2}+d_{3})\\\
-d_{1}d_{3}(d_{1}+d_{2})&d_{3}(d_{1}+d_{2})(d_{2}+d_{3}).\end{pmatrix}$

In this scenario we have three possible interaction patterns:

  1. 1.

Coupling to qubit $S_{1}$:

$\boldsymbol{\lambda}^{(1)}=\lambda^{(1)}\big{(}1,\,0\big{)}^{T}.$

  2. 2.

Coupling to qubit $S_{2}$:

$\boldsymbol{\lambda}^{(2)}=\lambda^{(2)}\big{(}0,\,1\big{)}^{T}.$

  3. 3.

Coupling to qubits $S_{1}$ and $S_{2}$:

$\boldsymbol{\lambda}^{(1,2)}=\lambda^{(1,2)}\big{(}1,\,1\big{)}^{T}.$

If $d_{1}=d_{3}$, the logical sub-spaces that generate each interaction
pattern are given by

$\begin{gathered}\boldsymbol{c}^{(1)}=\boldsymbol{F}^{-1}\cdot\boldsymbol{\lambda}^{(1)}=\frac{d_{1}(d_{1}+d_{2})\lambda^{(1)}}{d_{2}(2d_{1}+d_{2})J}\begin{pmatrix}d_{1}+d_{2}\\\
-d1\end{pmatrix}\\\
\boldsymbol{c}^{(2)}=\boldsymbol{F}^{-1}\cdot\boldsymbol{\lambda}^{(2)}=\frac{d_{1}(d_{1}+d_{2})\lambda^{(2)}}{d_{2}(2d_{1}+d_{2})J}\begin{pmatrix}-d1\\\
d_{1}+d_{2}\end{pmatrix}\\\
\boldsymbol{c}^{(1,2)}=\boldsymbol{F}^{-1}\cdot\boldsymbol{\lambda}^{(1,2)}=\frac{d_{1}(d_{1}+d_{2})\lambda^{(1,2)}}{(2d_{1}+d_{2})J}\begin{pmatrix}1\\\
1\end{pmatrix}\end{gathered}$

Therefore, by setting $\max|c_{i}|=1$, we find the maximum coupling for each
case:

$\begin{gathered}\lambda^{(1)}_{\max}=\frac{d_{2}(2d_{1}+d_{2})}{d_{1}(d_{1}+d_{2})^{2}}J\\\
\lambda^{(2)}_{\max}=\frac{d_{2}(2d_{1}+d_{2})}{d_{1}(d_{1}+d_{2})^{2}}J\\\
\lambda^{(1,2)}_{\max}=\frac{2d_{1}+d_{2}}{d_{1}(d_{1}+d_{2})}J,\end{gathered}$

and if $d_{1}=d_{2}=d_{3}$ this simplifies to

$\begin{gathered}\lambda^{(1)}_{\max}=\frac{3J}{4d}\\\
\lambda^{(2)}_{\max}=\frac{3J}{4d}\\\
\lambda^{(1,2)}_{\max}=\frac{3J}{2d}.\end{gathered}$

##  Appendix H 2D set-up: 8 ions

Assume 8 ions trapped in a 2D lattice forming a cross as it is shown in Fig.
4b, where the four inter ions constitute the control system and the external
four ions correspond to the target system. Here we also consider a coupling
between ions given by $f_{ij}=J|\boldsymbol{r}_{i}-\boldsymbol{r}_{j}|^{-1}$.
Therefore, in this setting the adjacent qubits of the target system interact
via the inherent ion-ion interaction with a coupling strength given by
$f^{S}_{12}=J\sqrt{2}/(d_{1}+2d_{2})$, meanwhile, the diametrically opposed
ions interact with a coupling strength of $f^{S}_{13}=J/(d_{1}+2d_{2})$. The
average interaction coupling of the inherent ion-ion in the target system is
given by

$\left\langle
f^{S}_{ij}\right\rangle=\frac{1}{3}\,f^{S}_{13}+\frac{2}{3}\,f^{S}_{12}=\frac{1+2\sqrt{2}}{3d_{1}+6d_{2}}J.$

Due to the symmetry in the spatial distribution of the qubit of the control
and the target system, each of the possible interaction patterns is equivalent
(up to some relabeling of the qubits) to one of the following 5 cases:

  1. 1.

Coupling to all four qubits:

$\boldsymbol{\lambda}^{(1,2,3,4)}=\lambda^{(1,2,3,4)}\big{(}1,\,1,\,1,\,1\big{)}^{T}.$

  2. 2.

Coupling to qubit $S_{1}$:

$\boldsymbol{\lambda}^{(1)}=\lambda^{(1)}\big{(}1,\,0,\,0,\,0\big{)}^{T}.$

  3. 3.

Coupling to qubit $S_{1}$ and $S_{2}$:

$\boldsymbol{\lambda}^{(1,2)}=\lambda^{(1,2)}\big{(}1,\,1,\,0,\,0\big{)}^{T}.$

  4. 4.

Coupling to qubit $S_{1}$ and $S_{3}$:

$\boldsymbol{\lambda}^{(1,3)}=\lambda^{(1,3)}\big{(}1,\,0,\,1,\,0\big{)}^{T}.$

  5. 5.

Coupling to qubit $S_{1}$, $S_{2}$ and $S_{3}$:

$\boldsymbol{\lambda}^{(1,2)}=\lambda^{(1,2,3)}\big{(}1,\,1,\,1,\,0\big{)}^{T}.$

The logical sub-spaces that generate them are given by
$\boldsymbol{c}=\boldsymbol{F}^{-1}\cdot\boldsymbol{\lambda}$. If
$d_{1}=d_{2}=d$, we obtain

$\displaystyle\boldsymbol{c}^{(1,2,3,4)}$
$\displaystyle=0.36\frac{d\lambda^{(1,2,3,4)}}{J}\,\big{(}1,\
1,\,1,\,1\big{)}^{T}$ $\displaystyle\boldsymbol{c}^{(1)}$
$\displaystyle=\frac{2d\lambda^{(1)}}{13J}\,\big{(}14,\,-2\sqrt{10},\,1,\,-2\sqrt{10}\big{)}^{T}$
$\displaystyle\boldsymbol{c}^{(1,2)}$
$\displaystyle=\frac{2d\lambda^{(1,2)}}{13J}\,\big{(}7.68\,7.68\,-5.32\,-5.32\big{)}^{T}$
$\displaystyle\boldsymbol{c}^{(1,3)}$
$\displaystyle=\frac{2d\lambda^{(1,3)}}{13J}\,\big{(}15,\,-4\sqrt{10},\,15,\,-4\sqrt{10}\big{)}^{T}$
$\displaystyle\boldsymbol{c}^{(1,2,3)}$
$\displaystyle=\frac{2d\lambda^{(1,2,3)}}{13J}\,\big{(}8.68,\,1.35,\,8.68,\,-11.65\big{)}^{T}$

Then the maximum coupling for each interaction pattern is given by

$\begin{gathered}\lambda_{\max}^{(1,2,3,4)}=2.76\frac{J}{d}\\\
\lambda_{\max}^{(1)}=0.46\frac{J}{d}\\\
\lambda_{\max}^{(1,2)}=0.84\frac{J}{d}\\\
\lambda_{\max}^{(1,3)}=0.43\frac{J}{d}\\\
\lambda_{\max}^{(1,2,3)}=0.56\frac{J}{d},\end{gathered}$

while the inherent interactions between the ions are given by

$\begin{gathered}f^{S}_{12}=0.47\frac{J}{d}\\\
f^{S}_{13}=0.33\frac{J}{d}.\end{gathered}$

With the values of $\lambda^{S^{\prime}}$, we compute the time to implement
all $U^{S^{\prime}}$ what is given by $t=\pi/(4\lambda^{S^{\prime}})$:

$\begin{gathered}e^{-i\,\frac{\pi}{4}\,Z^{C}(Z_{1}^{S}+Z_{2}^{S}+Z_{3}^{S}+Z_{4}^{S})}\quad\mapsto\quad
t_{1234}=\frac{\pi d}{11.04J}\\\
e^{-i\,\frac{\pi}{4}\,Z^{C}Z_{1}^{S}}\quad\mapsto\quad t_{1}=\frac{\pi
d}{1.84J}\\\
e^{-i\,\frac{\pi}{4}\,Z^{C}(Z_{1}^{S}+Z_{2}^{S})}\quad\mapsto\quad
t_{12}=\frac{\pi d}{3.36J}\\\
e^{-i\,\frac{\pi}{4}\,Z^{C}(Z_{1}^{S}+Z_{2}^{S})}\quad\mapsto\quad
t_{13}=\frac{\pi d}{1.72J}.\\\
e^{-i\,\frac{\pi}{4}\,Z^{C}(Z_{1}^{S}+Z_{2}^{S}+Z_{3}^{S})}\quad\mapsto\quad
t_{123}=\frac{\pi d}{2.24J}.\\\ \end{gathered}$

Finally, we compute the time to implement a general logical gate in the
control system:

$t_{V}=2\left(\frac{\pi}{4f_{12}^{C}}+\frac{\pi}{4f_{23}^{C}}+\frac{\pi}{4f_{34}^{C}}\right)=\frac{3\sqrt{2}d\pi}{J},$
(20)

where we consider the following circuit

$V^{C}=\text{CX}^{C}_{3\to 4}\text{CX}^{C}_{2\to 3}\text{CX}^{C}_{1\to
2}V^{C}_{1}\text{CX}^{C}_{1\to 2}\text{CX}^{C}_{2\to 3}\text{CX}^{C}_{3\to
4}.$