Access to this full-text is provided by IOP Publishing.
Content available from Quantum Science and Technology
This content is subject to copyright. Terms and conditions apply.
Quantum Sci. Technol. 7(2022) 015008 https://doi.org/10.1088/2058-9565/ac2d39
OPEN ACCESS
RECEIVED
8 June 2021
REVISED
10 September 2021
ACCEPTED FOR PUBLICATION
6 October 2021
PUBLISHED
11 November 2021
Original content from
this work may be used
under the terms of the
Creative Commons
Attribution 4.0 licence.
Any further distribution
of this work must
maintain attribution to
the author(s) and the
title of the work, journal
citation and DOI.
PAPER
Universal quantum computation and quantum error correction
with ultracold atomic mixtures
Valentin Kasper1,∗, Daniel González-Cuadra1, Apoorva Hegde2,AndyXia
2,
Alexandre Dauphin1, Felix Huber1,EberhardTiemann
3, Maciej Lewenstein1,4,
Fred Jendrzejewski2and Philipp Hauke5
1ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860
Barcelona, Spain
2Universität Heidelberg, Kirchhoff-Institut für Physik, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
3Institut für Quantenoptik, Leibniz Universität Hannover, 30167 Hannover, Germany
4ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain
5INO-CNR BEC Center and Department of Physics, University of Trento, Via Sommarive 14, I-38123 Trento, Italy
∗Author to whom any correspondence should be addressed.
E-mail: valentin.kasper@icfo.eu
Keywords: ultracold atoms, quantum computation, atomic mixtures, quantum error correction
Abstract
Quantum information platforms made great progress in the control of many-body entanglement
and the implementation of quantum error correction, but it remains a challenge to realize both in
the same setup. Here, we propose a mixture of two ultracold atomic species as a platform for
universal quantum computation with long-range entangling gates, while providing a natural
candidate for quantum error-correction. In this proposed setup, one atomic species realizes
localized collective spins of tunable length, which form the fundamental unit of information. The
second atomic species yields phononic excitations, which are used to entangle collective spins.
Finally, we discuss a finite-dimensional version of the Gottesman– Kitaev–Preskill code to protect
quantum information encoded in the collective spins, opening up the possibility to universal
fault-tolerant quantum computation in ultracold atom systems.
Quantum information processing is expected to show fundamental advantages over classical approaches
to computation, simulation, and communication [1]. To exploit these advantages, both the efficient
creation of entanglement and the correction of errors are essential [2,3]. There have been spectacular
advancements in the field of synthetic quantum devices, which allow for controlled many-body
entanglement [4–6] or the implementation of quantum error correction (QEC) [7–10]. However, it is still a
challenge to design a quantum computation platform that efficiently realizes both features simultaneously.
Here, we propose to use a mixture of two ultracold atomic species to implement a universal quantum
computer, permitting long-range entangling gates as well as QEC (see figure 1). One atomic species forms
localized collective spins, which can be fully controlled and represent the basic units of information. The
second species is used for the generation of pairwise entanglement of the collective spins, resulting in a
universal gate set. Moreover, we illustrate how to encode a logical qubit in the collective spin, how to detect
and correct errors, and present a universal gate set on the logical qubits. Altogether, this opens a new
possibility for quantum computation with error correction in ultracold atomic systems, presenting a
significant step toward fault-tolerant quantum computation.
The proposal of this work complements existing platforms such as nuclear magnetic resonance [11],
nitrogen-vacancy centers in diamonds [12], photonics [13], silicon based qubits [14], superconducting
qubits [15],andtrappedions[16]. One key challenge in many of these platforms is to reduce the technical
overhead when synthesizing long-range gates out of short range gates, as it happens, e.g. in superconducting
qubit experiments. While trapped ion systems provide efficient long-range entangling gates, they face
technical challenges in establishing full control beyond one hundred ions [17,18]. Furthermore, many
implementations of QEC demand a considerable number of physical qubits to store one logical qubit, and
© 2021 The Author(s). Published by IOP Publishing Ltd
Quantum Sci. Technol. 7(2022) 015008 VKasperet al
Figure 1. Quantum computation with ultracold atomic mixtures. (a) Proposed experimental platform for the case of one
dimension: Bosonic atoms A (yellow) are trapped via optical potentials (red tweezers) and immersed in a one dimensional
quasi-condensate which is formed by bosonic atoms B (blue). (b) and (c) One/two-dimensional spin–phonon model: Two
hyperfine levels of several A atoms form collective spins (yellow arrows), which interact via phonon excitations of B atoms (green
lines).
additional control qubits for the encoding, decoding, and correcting processes [19]. Whereas the challenges
of efficient long-range entanglement and QEC appear daunting, ultracold atomic mixtures may provide a
solution for both.
Ultracold atoms have become a major quantum simulation platform to solve problems in the field of
condensed-matter [20–22] and high-energy physics [23–27]. Yet, even though universal quantum
information processing with ultracold atom systems was investigated conceptually, the experimental
implementations remain elusive [28–30]. The realization of gates with high-fidelity and the ability to apply
sequences of several gates is particularly challenging. However, recent progress in systems of atoms trapped
in optical lattices or optical tweezers has made it possibletorealizelargequantummany-bodysystems[31]
with fast, high-fidelity entangling gates [32,33]. Simultaneously, multi-component Bose–Einstein
condensates were used to form large collective spins [34–38] and spatially distribute entanglement via
expansion [39]. Ultracold atomic mixtures allow for other entanglement mechanisms such as phonon
induced interactions [40–43], or particle mediation [29]. The effect of phonons on a single atomic species
has already been experimentally studied in the context of polaron physics [44,45].
We first introduce the fundamental unit of information of our platform: collective spins of controllable
length. They are realized by condensing bosonic atoms into a single spatial mode of an optical tweezer. The
remaining degrees of freedom are two internal states of the atoms, which can be described using a collective
spin. The length of the collective spin is controlled by the number of trapped atoms enabling access to the
qubit, qudit, and continuous variable regime. The collective occupation of the internal degrees of freedom
constitute the computational basis. By tuning the length of the collective spin, quantum information
processing can be done with qubits, qudits or continuous variables. The atoms within a tweezer can be
reliably prepared in a fully polarized state, which then acts as the initial state of the computation. A single
collective spin can be fully controlled locally via linear operations such as Raman coupling, and non-linear
operations such as the interaction between the atoms in different hyperfine states.
In order to create entanglement between the collective spins, we employ a second atomic species, which
forms a bosonic bath with phononic excitations. The contact interaction between the two atomic species
gives rise to the exchange of phononic excitations between the collective spins. The resulting long-range
interaction permits an efficient generation of entanglement between distant spins. The operations on a
single collective spin in combination with their pairwise entanglement forms a universal gate set. The gate
speed is expected to be much faster than the decoherence time in the platform suggested here. Altogether,
this platform fulfills DiVinvenzo’s criteria for quantum computation [46].
Moreover, we illustrate a scheme for QEC based on the stabilizer formalism. The method encodes a
logical qubit into superpositions of states within the higher dimensional Hilbert space of the collective spin.
We further explain how to prepare, detect and correct specific errors by using additional control qubits.
Finally, we present a universal gate set for the logical qubits.
This work is organized as follows. In section 1, we explain how one atomic species forms a collective
spin and how to generate arbitrary unitaries on such a single object. In section 2, we discuss the interaction
between the collective spins and a phonon bath, and how this can be used to entangle two collective spins,
2
Quantum Sci. Technol. 7(2022) 015008 VKasperet al
which then can be employed for universal quantum computation. In section 3, we discuss the state
preparation and readout of the collective spins. In section 4, we lie out the experimental characteristics of
this quantum computation platform. Finally, in section 5we discuss how to implement QEC.
1. Collective spins
In this section, we discuss the fundamental unit of information of our proposal: a collective spin with
controllable length. In order to realize such a collective spin we consider bosonic atoms of type A, which are
localized at the local minima yof the optical potential, e.g. arrays of optical tweezer or an optical lattice (see
figure 1(a)). The annihilation (creation) operators of the atoms on site yare am(y)anda†
m(y), where m
indicates the two internal states (0, 1) of the atom. The local confining potential is chosen such that there is
no hopping. Hence, the atom number per site NA(y)=a†
1(y)a1(y)+a†
0(y)a0(y)isconserved[47]andthe
atoms form a collective spin via the Schwinger representation:
Lz(y)=1
2[a†
1(y)a1(y)−a†
0(y)a0(y)], (1a)
L+(y)=a†
1(y)a0(y), (1b)
L−(y)=a†
0(y)a1(y), (1c)
with Lx(y)=[L+(y)+L−(y)]/2andLy(y)=(−i)[L+(y)−L−(y)]/2. The conservation of atom number
per site determines also the magnitude of the angular momentum =NA/2. The eigenstates of the Lz
operator are denoted by |mwith mbeing an integer or half-integer ranging from m=−,...,.By
defining the computational basis |j≡|−+jwith j=0, ...,NAa collective spin can be interpreted as a
qudit with dimension d=NA+1. Moreover, we can access the qubit (NA=1), qudit (NA>1), and
continuous variable (NA→∞) regime by tuning the atom number per site. In particular, the Hilbert space
dimension of a single collective spins scales linearly with the numbers of atoms.
In order to realize unitary operations acting on a single collective spin, we consider the time evolution
generated by the Hamiltonian
HA=
yχ(y)L2
z(y)+Δ(y)Lz(y)+Ω(y)Lx(y).(2)
As detailed in appendix A, this Hamiltonian can be implemented by a two component Bose gas localized in
optical tweezers, where the first term corresponds to the interaction between the atoms, the second is due to
the presence of the magnetic field, and the third term to a Rabi coupling between the hyperfine states. The
interactions between the atoms are described by L2
z(y)=1
4n2
1(y)+n2
0(y)−2n0(y)n1(y),whichinvolve
both spin-states, but no spin-changing collisions.
The Hamiltonian in equation (2) can generate all unitary operations on a single collective spin since all
couplings χ(y), Δ(y), and Ω(y) can be switched on and off independently on each site in ultracold atom
systems [48]. For the F=1 hyperfine manifold, the coupling χ(y)=0 can be achieved by hosting the
atoms in the magnetic substates mF=1andmF=−1, while χ(y)=0 when the atoms are in the magnetic
substates mF=1andmF=0. The switching of χ(y) was illustrated in the context of mixture experiments
in a recent work using Raman transitions [49]. A great advantage of using laser addressing is that the
magnetic field does not have to be controlled at the μm scale, which is the typical separation of optical
tweezers. Similarly, the couplings Ω(y)andΔ(y) can be tuned via Raman coupling in the same way as in
the high-fidelity implementations with trapped ions [50].
For the qubit case (d=NA+1=2), the operators Lxand Lztogether with the identity 1suffice to
obtain all U(2) transformations [51]. For qudits (d=NA+1>2) the additional non-linear operation L2
z)
is required to approximately synthesize any unitary U(d) via Trotterization. Namely, using
e−iˆ
Aδteiˆ
Bδteiˆ
Aδte−iˆ
Bδt=e[ˆ
A,ˆ
B]δt2+Oδt3, we can engineer all possible commutators of Lx,Lz,andL2
zand
higher order commutators. These commutators span the d-dimensional Hermitian matrices with the basis
{Mi}d2
i=1, which in turn allows to synthesize all unitary matrices. In appendix Bwe give an explicit
construction of the enveloping algebra of Lx,Lz,andL2
zfor d=3, which allows one to generate each
element of U(3). As detailed in reference [52], the construction given in the appendix can be generalized to
d>3. In this way, the operators Lx,Lz,andL2
zprovide universal control over each single collective spin. In
the next section, we consider the entanglement of several collective spins to achieve exponential growth of
the Hilbert space.
3
Quantum Sci. Technol. 7(2022) 015008 VKasperet al
2. Unitary operations on two spins
The entanglement of different collective spins can be achieved by the exchange of delocalized phonons. The
phonons are the Bogoliubov excitations of a weakly interacting condensate of atomic species B in n
dimensions, where we consider n=1, 2 (see figure 1). The purely phononic part of the Hamiltonian is
given by
HB=
k
ωkb†
kbk,(3)
where bkis the annihilation operator of a phonon mode at wave number kwith components ki=π
Lmiand
miis a positive integer. The phonon dispersion at low momenta is linear ωk≈c|k|,wherec=˜
gBnB/MB
is the speed of sound determined by the interaction strength ˜
gBof the B atoms with mass MBand the
density nB, see appendix Cfor more details.
By immersing the collective spins into the phonon bath, we induce interactions between the phonons
and the spins rooted in the contact interactions of the two atomic species A and B. The spin–phonon
interaction can be described by the Hamiltonian
HAB =
y,k¯
gk(y)+δgk(y)Lz(y)(bk+H.c.), (4)
with the explicit forms of the coupling constants ¯
gk(y)andδgk(y). A detailed derivation of equation (4)is
given in appendix D. This interaction is similar to the phonon–ion interactions in trapped ion systems [16]
or photon–atom interactions [53]. The first term of equation (4) leads to a constant polaronic shift
[45,54], which can be absorbed by redefining the phonon operators. Consequently, we focus here on the
second term, which can be used to generate entanglement between the spins.
Since Lz(y) is a conserved quantity for Ω(y)=0 it is possible to decouple the phonons and spins in
equation (4) by shifting the phonon operators (see appendix E). Eliminating the phonons leads to an
effective long-range spin–spin interaction
HI=−
x,y
g(x,y)Lz(x)Lz(y), (5)
where we introduced the coupling g(x,y) between the spins after a proper redefinition of χ(y)andΔ(y).
Explicitly, the coupling between the spins is given by
g(x,y)=g
k
(uk+vk)2
ωk
e−1
4|k|2σ2
A
n
i=1
sin(kixi)sin(kiyi).(6)
The overall prefactor
g=nB2/Ln(˜
g1
AB −˜
g0
AB)2(7)
is determined by the inter-species interaction ˜
g0
AB and ˜
g1
AB, where the ~ indicates the renormalization
according to the optical potential. Further, we introduce the Bogoliubov amplitudes ukand vk,whichare
determined by performing the Bogoliubov approximation in a box potential of length L[55,56]. We further
approximate the Bogoliubov mode functions by sin(kixi). The length scale σAis the harmonic oscillator
length of the tweezer confining the A atoms, which provides a cutoff for the momentum sum in
equation (6). For more details we refer to appendix D.
Performing the momentum sum numerically, the resulting interaction between two spins for one and
two dimensions is illustrated in figures 2and 3. Positioning the spins appropriately within the bath allows
one to tune the strength and sign of the spin–spin interaction. The effective interaction scales as nB
independent of the dimension, as long as only the linear part of the dispersion ωkcontributes to the sum of
equation (6).
In order to entangle two specific spins, the ability to deliberately switch on and off the interactions
g(x,y) is necessary. A first possibility is to physically move the optical tweezers such that there is no overlap
of the collective spins with the phonons. This approach is similar to the shuttling approach in trapped ions
[57].
The second approach, is similar to the optical shelving used in trapped ions [58]. The interaction
between the collective spins is proportional to the scattering length difference, g(x,y)∝˜
g1
AB −˜
g0
AB2.The
first term describes the interaction strength between A atoms in mF=1andBatomsinmF=0, while the
second term describes the interaction between atoms of both species being in mF=0. To shelve spins, we
can coherently transfer the mF=0 component of A atoms into the mF=−1 state, while the atoms in the
4
Quantum Sci. Technol. 7(2022) 015008 VKasperet al
Figure 2. Spin– spin interaction in one dimension: one collective spin is located at position y1, while the other collective spin is
located at position x1. The interaction between spins at x1and y1decays toward the boundary of the box and for specific
positions changes its sign. For our proposal, the collective spins do not overlap, because of the small spatial extent of the tweezer
(see appendix Afor details). Hence, we focus on the region with |x1−y1|σA,whereσAis the length scale of the confining
harmonic oscillator. (Inset) The phonon-mediated interaction decays when moving away from the collective spin and becomes
zero at the boundary due to the box potential (Dirichlet boundary conditions). The sign of the interaction can be tuned by
changing the relative position of the collective spins. The colored dashed lines correspond to the cuts in the outer figure.
Figure 3. Spin–spin interaction in two dimension: the collective spin is immersed in a two dimensional bosonic bath and
located at the center of the box (a) or at the lower left corner (b). Similarly to the one-dimensional case the interaction is
long-ranged, decays when moving away from the impurities, and vanishes at the boundary of the system. The interaction can be
tuned by choosing the relative position of the two collective spins.
mF=1 component are left unaltered. The resulting interaction is strictly zero since
g(x,y)∝˜
g−1
AB −˜
g1
AB2=0.
By tuning the interaction time between the spins, using two additional single qudit gates and adding a
phase allow for synthesizing a controlled-Zgate [59] between two collective spins
CZ(x,y)=ei2π
dLz(x)Lz(y)ei2π
dLz(x)ei2π
dLz(y)ei2π2
d,(8)
which completes the universal gate set in the multi-spin system [60,61]. The quality of the local spin
addressing and the magnetic field stability [62] will determine the fidelity of the gates.
3. State preparation and detection
A universal quantum computer requires a reliable state-preparation and readout. Using the external
magnetic field one can prepare all A atoms in one hyperfine component, which corresponds to a fully
polarized collective spin along the quantization axis |ψ=|d⊗...|d. In case the particle number per site
is probabilistic, one can perform post-selection to fix NA[63–65]. If one uses instead an optical lattice to
create the collective spins, one can deterministically control the atom number by preparing first a
Mott-insulating state, which nowadays can be prepared with almost unit filling [26]andasubsequent
merging of a fixed number of wells. For the preparation as well as for the detection one has to ensure single
counting statistics per site, which can be achieved through fluorescence imaging [34].
When working in the large collective spin regime, one can efficiently prepare condensates through
evaporative cooling in the optical dipole trap [66]. This results in an atomic cloud of a few hundred to
thousand atoms of low entropy in each dipole trap. In case the A atoms are not in the motional ground
state, one can perform Raman sideband cooling [67]. For larger collective spins, one can use homodyne
detection to map out large collective spins [68].
5
Quantum Sci. Technol. 7(2022) 015008 VKasperet al
4. Experimental characteristics
Having established all the necessary ingredients for a quantum information platform, we discuss details of a
possible experimental realization as well as main sources of errors. The experiments can rely on standard
experimental tools employed for cold atoms [20]. For concreteness, we focus on a mixture of 39K(species
A) forming the collective spins and 23Na (species B) forming the phonon bath.
The collective spins can be realized through the trapping of a few atoms in optical tweezers distanced by
a few micrometers and with motional ground state extension σA≈100 nm. With this confinement and
working with ≈10 atoms per tweezer leads to three body loss between the species [69]onatimescaleof
∼1 Hz. The typical single qubit gates can then be performed with a Rabi frequency of up to a few kHz with
above 99%fidelity [38]. In order to ensure the coherent evolution one has to stabilize the magnetic field,
which for state of the art experiments is possible up to ∼10 Hz. Readout reliability through fluorescence
imaging differ between schemes but can reach >99%fidelity [70–72].
Assuming a box potential for species B [55] in a tube or a slab geometry with linear dimension
L≈100 μm we expect approximately 50 collective spins in a one-dimensional geometry or 2500 collective
spins in a two dimensional geometry. In both cases, all-to-all coupling can be achieved through phonons of
a (quasi-)condensate of approximately NB≈300 ×103atoms for the 1D bath and NB≈3×106for the 2D
bath, with a transverse confinement of ω⊥≈2π×440 Hz, such that the condensate has a chemical
potential of μB/h≈7.7 kHz in 1D and μB/h≈1.9 kHz in 2D [22]. This implies typical lifetimes for single
atom losses that can be of several seconds up to a few minutes, setting an upper limit on the length of the
quantum circuit [25].
The collective spins are coupled to the phonons through contact interaction according to the scattering
length aAB
0=756a0,aAB
1=2542a0and aAB
2=−437a0. The couplings gm
AB can then be obtained by
calculating the corresponding Clebsch–Gordon coefficients [73]. The speed of the entangling gates is then
determined by the strength of the interaction between species A and B given by equation (6), as we
illustrated in figures 2and 3. For example, the typical gate speed of the CZ gate connecting two qudits at a
distance of 5 μm can be estimated to be of the order of 50 Hz in the one-dimensional case, see figure 2.
Using an optical lattice for the collective spins, 20 ×20 lattice sites are within the current experimental
feasibilities and allow all qudits to interact with the phonons. The coupling constants and time scales of the
main decoherence channels indicate, that the proposed setup is a realistic route for large-scale quantum
information processing with neutral atoms, exploiting already available state-of-the-art technology.
5. Quantum error correction
QEC allows one to mitigate the effects of a noisy environment and faulty operations on the information
stored in quantum states. The main idea of QEC is to embed quantum information in a Hilbert space of
larger dimension, enabling a distribution of information that leads to resilience against noise. For this
purpose, one can use the combined Hilbert space of multiple qubits, while an alternative approach is to use
a single system that has a larger Hilbert space. A way to realize the latter goes back to a seminal work of
Gottesmann, Kitaev, and Preskill (GKP) [74], which proposed encoding a qubit into a harmonic oscillator.
The finite-dimensional version of the GKP code encodes a qubit into a collective spin. This code is based
on a set of commuting operators, the stabilizer set, which can be measured simultaneously. The quantum
information is then encoded into the joint (+1)-eigenspace of the stabilizer set. Errors acting on the
encoded information may lead to a change in one or few stabilizer measurements, which helps in detecting
and correcting errors.
The formalism of the finite-dimensional GKP code rests on the generalized Pauli operators Xand Z
defined by
X|jd=|(j+1) mod dd,(9a)
Z|jd=ωj|jd.(9b)
These operators obey the relation ZX =ωXZ,withω=exp(2πi/d). In particular, two operators of the
form XαZβand XγZδcan commute with each other, if their exponents α,β,γ,andδare chosen
appropriately. Such operators can then be used to form the stabilizer group of a code.
To illustrate the idea behind the finite dimensional GKP code, we choose d=8 corresponding to seven
atoms in an optical tweezer. In order to generate the stabilizer group, we choose the operators
S1=X4, (10a)
S2=Z4.(10b)
6
Quantum Sci. Technol. 7(2022) 015008 VKasperet al
Notice that S1and S2commute. Their joint (+1)-eigenspace defines a two-dimensional subspace, which can
be used to encode a logical qubit. A basis for this code space is
|0=1
√2(|08+|48), (11a)
|1=1
√2(|28+|68).(11b)
A state is then encoded as |ψL=α|0+β|1and one can detect all errors XaZbthat have |a|,|b|1.
However, from the fact that X|¯
0=X−1|¯
1we infer that some of these errors can only be detected, but not
corrected, because the action of Xand X−1on the logical states cannot be distinguished. More explicitly,
this can be seen from the conditions for QEC [75].
Experimentally, the two logical states |¯
0,|¯
1are prepared by using a control qubit as shown in
figure 4(b): the collective spin and the qubit are initially prepared in the product state |02|08. Applying a
Hadamard gate onto the control qubit leads to 1
√2(|02+|12)|08. Next, we employ a conditional CX4gate
defined as
CX4|j2|k8=(𝟙⊗X4j)|j2|k8, (12)
which results in 1
√2(|02|08+|12|48). A second Hadamard gate on the control qubit yields
1
√2|02|0+|12(Z|0).(13)
After measuring the Zeigenvalue of the control qubit, the collective spins is projected onto either |0or
Z|0.Inthelattercase,anadditionalZgate is applied. In this way, the system is deterministically prepared
in |0. In an experiment the circuit depicted in figure 4(b) can be realized as follows: the circuit contains
Hadamard gates, which acts on a qubit, a Z-gate, which acts on a qudit, a CX4gate acting on a qudit and
qubit, and a local readout. The Hadamard gate on the qubit is realized by performing a π/2 pulse locally.
The Z-gate on the qudit is performed by changing the detuning of the microwave. The CX4is not a native
gate described in section 2. However, since the gates generated by Lz,Lx,L2
z,andtheCZ gate are universal,
we can decompose the CX4gate into a judiciously chosen sequence of local gates and two-body gates. The
local gates correspond to local optical control, microwave pulses, and local collision of atoms. The two-body
CZ-gate corresponds to phonon entanglement, which can be switched off via optical shelving.
The encoding (11) allows one to detect certain errors affecting the logical qubit. Consider errors of the
form XaZbacting on |ψL. From the commutation relation S1XaZb=ω−4bXaZbS1and S2XaZb=ω4aXaZbS2,
one obtains
S1XaZb|ψL=ω−4bXaZb|ψL, (14a)
S2XaZb|ψL=ω4aXaZb|ψL.(14b)
Thus an error may move the logical qubit out of the (+1)-eigenspace of S1and S2. In particular, this
happens for all non-trivial XaZbwith |a|,|b|1. The circuit in figure 4(c) can then be used to detect when
this happens. Likewise, it can be shown that the same circuit also detects any linear combinations of such
errors.
To also correct errors, one has to use a larger collective spin. Using d=18, the stabilizers S1=X6and
S2=Z6yield a code with basis [74]
|0=1
√3|018 +|618 +|1218, (15a)
|1=1
√3|318 +|918 +|1518.(15b)
The circuit given in figure 4(c) can now not only detect, but also correct all errors XaZbwith |a|,|b|1.
Then, the encoded states can also be manipulated through a set of universal gates. To be specific, we return
tothecaseofd=8. The logical gates X=X2and Z=Z2act like the usual spin 1/2 Pauli matrices on |0
and |1,i.e.
¯
X|0=|1,¯
Z|0=|0, (16a)
¯
X|1=|0,¯
Z|1=−|1, (16b)
such that ¯
X2=¯
Z2=1and ¯
X¯
Z+¯
Z¯
X=0.
7
Quantum Sci. Technol. 7(2022) 015008 VKasperet al
Figure 4. Error correction: (a) experimental implementation with a tweezer setup. The phonons of the bosonic bath couple a
tweezer with NAatoms (providing the d=NA+1-dimensional Hilbert space to encode a logical qubit) to a tweezer with one
atom (providing a control qubit required for state preparation and error detection). (b) Preparation of the logical state |0given
in equation (11) using a control qubit, Hadamard gates, a CX4gate, and a Zrotation conditioned on the outcome of the
measurement on the control qubit. (c) Error detection and correction on an encoded state |ψL.TheGKPcoded=8with
S1=X4and S2=Z4can detect the errors Zand X, and no feedback of the measurement result is needed. However, the feedback
becomes essential for the correction of errors e.g. for the GKP code in d=18 with S1=X6and S2=Z6.
As explained in section 1, we can implement all single-qubit logical gates through Trotterization, which
allows to synthesize
¯
R(α,β,γ,δ)=eiαeiβ¯
Zeiγ¯
Xeiδ¯
Z.(17)
Moreover, two logical qubits can be entangled by using a logical controlled Zgate,
C¯
Z(x,y)=eiπ¯
Z(x)¯
Z(y), (18)
which can be also synthesized via Trotterzation using the gates generated by Lx,Lzand L2
z,andthe
entangling gate CZ(x,y) given in equation (8). Together, the operations ¯
R(α,β,γ,δ)andC¯
Zthen generate
a universal set of logical gates [60].
In summary, we have constructed an explicit example of a quantum error-detecting code for d=8,
described the experimental preparation of its logical states, and provided a universal gate set to manipulate
the stored quantum information. The presented formalism will also work in larger dimensions, allowing for
the detection and correction of a larger set of errors [74]. The here proposed platform with a suitable
quantum error-correcting scheme, such as finite GKP codes, may allow one to go beyond the abilities of
noisy intermediate-scale quantum (NISQ) technologies.
6. Conclusion and outlook
In this work, we have proposed a mixture of two ultracold atomic species as a platform for universal
quantum computation. Our proposed implementation uses long-range entangling gates and, in addition,
allows for the realization of QEC. The idea of using phonon mediated interactions for quantum
information processing is also discussed in the solid state context and made substantial progress [76,77].
The presented spin–phonon system is not only useful for the processing of quantum information but is also
interesting from a quantum many-body perspective. The many body systems realized by our platform is
similar to gauge theories coupled to a Higgs field [78–80] and hence is a promising candidate for the
investigation of topological matter. Further, the spin–phonon system proposed here may give access to the
physics of the Peierls transition [81], the Jahn–Teller effect [82] in a many-body context as well as the study
of frustrated spin models [83]. The versatility of this platform also makes it an ideal candidate for a fully
programmable quantum simulation, whose large potential has been demonstrated in Rydberg and trapped
ion systems [4,5,84,85]. For example, atomic mixtures have been proposed for the quantum simulation of
quantum chemistry problems [86].
In future work, the building blocks of our proposed platform could be exchanged ad libitum exploiting
the versatility of atomic, molecular, and atomic physics. The basic unit of information can be stored, e.g.
with spinful fermions in an optical tweezer [71] or higher dimensional spins [38,87,88]. The entanglement
bus may be substituted by more complex many-body excitations like magnons or rotons [89,90]. This
8
Quantum Sci. Technol. 7(2022) 015008 VKasperet al
exchange of the basic unit of information and the entanglement mechanism has the potential to lead to new
implementations of quantum algorithms and QEC schemes, see for example [91–94]. Thus, ultracold atom
mixtures present a promising platform to implement fault-tolerant quantum computation in a scalable
platform, paving a way beyond the abilities of NISQ technology.
Acknowledgments
The authors are grateful for fruitful discussions with J Eisert, M Gaerttner, T Gasenzer, M Gluza, S Jochim,
M Oberthaler, A P Orioli, P Preiss, H Strobel and all the members of the SynQs seminar. ICFO group
acknowledges support from ERC AdG NOQIA, State Research Agency AEI (Severo Ochoa Center of
Excellence CEX2019-000910-S, Plan National FIDEUA PID2019-106901GB-I00/10.13039/501100011033,
FPI, QUANTERA MAQS PCI2019-111828-2/ 10.13039/501100011033), Fundaci´
o Privada Cellex, Fundaci´
o
Mir-Puig, Generalitat de Catalunya (AGAUR Grant No. 2017 SGR 1341, CERCA program, QuantumCAT /
U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020), EU Horizon 2020
FET-OPEN OPTOLogic (Grant No 899794), and the National Science Centre, Poland (Symfonia Grant No.
2016/20/W/ST4/00314), Marie Sklodowska-Curie grant STREDCH No.101029393, La Caixa Junior Leaders
fellowships (ID100010434), and EU Horizon 2020 under Marie Sklodowska-Curie Grant Agreement No.
847648 (LCF/BQ/PI19/11690013, LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012). FH acknowledges the
Government of Spain (FIS2020-TRANQI and Severo Ochoa CEX2019-000910-S), Fundaci´
o Cellex,
Fundaci´
o Mir-Puig, Generalitat de Catalunya (CERCA, AGAUR SGR 1381), and the Foundation for Polish
Science through TEAM-NET (POIR.04.04.00-00-17C1/18-00). PH acknowledges support by
Q@TN—Quantum Science and Technologies at Trento, the Provincia Autonoma di Trento, and the ERC
Starting Grant StrEnQTh (Project-ID 804305). This work is part of and supported by the DFG
Collaborative Research Center ‘SFB 1225 (ISOQUANT)’. FJ acknowledges the DFG support through the
Project FOR 2724, the Emmy-Noether Grant (Project-ID 377616843) and support by the
Bundesministerium für Wirtschaft und Energie through the project ‘EnerQuant’ (Project-ID 03EI1025C).
Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.
Appendix A. Collective spin Hamiltonian
The type A atoms are placed in an external magnetic field, which allows to prepare the atoms in magnetic
substates m∈{0, 1}. Further, the atoms of type A are localized at specific sites via optical potentials (see
figure 1). The single particle physics is described by the Hamiltonian
Hm
A(x)=−2∇2
x
2MA
+VA(x)+Em
A(B), (A1)
where MAis the mass of the atoms and VA(x)is the optical potential confining the atoms. The energy shift
EA
m(B) determined by the magnetic field Bis given by
EA
m(B)=pm(B)m+qm(B)m2,(A2)
where pm(B) parametrizes the linear Zeeman shift and qm(B) the quadratic Zeeman shift. The field
operators Am(x)andA†
m(x) annihilate and create a particle at position x=(x1,...,xd)Trespectively and
fulfill canonical commutation relations. The many-body Hamiltonian is given by
HA=
mx
A†
m(x)Hm
A(x)Am(x)+
m,nx
gmn
A
2A†
m(x)A†
n(x)An(x)Am(x).(A3)
The atoms are localized at different sites y, which allows one to expand the field operators as
Am(x)=
y
ϕA(x−y)am(y)(A4)
9
Quantum Sci. Technol. 7(2022) 015008 VKasperet al
with a wave function ϕAlocalized at 0and which does not depend on the magnetic substate. Inserting the
expansion (A4)intoHAand neglecting the tunneling terms, we obtain the Hamiltonian
HA=1
2
m,n,y
˜
gmn
Aa†
m(y)a†
n(y)am(y)an(y)(A5)
with the overlap integrals
˜
gmn
A=gmn
Ax|ϕA(x−y)|4.(A6)
To be explicit, we approximate ϕA(x) by the ground state wavefunction of an isotropic harmonic oscillator
ϕA(x)=ϕA(x1)ϕA(x2)ϕA(x3)with
ϕA(xi)=(√πσA)−1/2e−1
2xi
σA2
(A7)
with the characteristic length σA. Calculating the overlap integral leads to the dimensional reduced coupling
constant
˜
gmn
A=(2π)−3/2(gmn
A/σ3
A).(A8)
Inserting the Schwinger representation of the angular momentum into equation (A5) leads to equation (2)
with the coupling constants
χ=1
2˜
g00
A+˜
g11
A−2˜
g10
A,(A9a)
Δ= 1
2(NA−1)(˜
g11
A−˜
g00
A)+E1
A−E0
A,(A9b)
where NAis the particle number on each site.
Appendix B. Enveloping algebra of U(3)
In this appendix, we demonstrate for =1thatLz,Lx,andL2
zare sufficient to generate all U(3) matrices.
We use the spin matrices
Lx=1
√2⎛
⎝
010
101
010
⎞
⎠,Ly=1
√2i⎛
⎝
010
−101
0−10
⎞
⎠,Lz=⎛
⎝
10 0
00 0
00−1⎞
⎠.
Consider the following matrices generated by commutators of Lx,Lz,andL2
z
M1=Lx,M2=Lz,M3=L2
z,
M4=i[M1,M2], M5=i[M3,M1], M6=i[M3,M4],
M7=i[M5,M1], M8=i[M5,M4], M9=i[M6,M4].(B1)
These commutators form a basis for the Lie algebra of U(3). This can be explicitly checked by constructing
achangeofbasisfromthe{Mi}9
i=1to the canonical basis of Hermitian matrices {Mi}9
i=1given by
M1=⎛
⎝
100
000
000
⎞
⎠,M2=⎛
⎝
000
010
000
⎞
⎠,M3=⎛
⎝
000
000
001
⎞
⎠,
M4=⎛
⎝
010
100
000
⎞
⎠,M5=⎛
⎝
001
000
100
⎞
⎠,M6=⎛
⎝
000
001
010
⎞
⎠,
M7=⎛
⎝
0i0
−i00
000
⎞
⎠,M8=⎛
⎝
00i
000
−i00
⎞
⎠,M9=⎛
⎝
000
00i
0−i0⎞
⎠.(B2)
10
Quantum Sci. Technol. 7(2022) 015008 VKasperet al
Appendix C. Phonon Hamiltonian
The Hamiltonian involving only the atomic species B is given by
HB=x
B†(x)H0
B(x)B(x)+gB
2x
B†(x)B†(x)B(x)B(x)(C1)
with
H0
B(x)=−2∇2
x
2MB
+VB(x).(C2)
In order to create a one- or two-dimensional phononic bath we apply the following harmonic and isotropic
confinement
VB(x)=1
2MBω2
B
d
i=n+1
x2
i.(C3)
For sufficiently low temperatures, the transversal degrees of freedom will not be excited, which will confine
the particles effectively to one (n=1) or two (n=2) dimensions respectively. This freezing out of the
transversal directions allows one to write the field operator as
B(x)=B(x1,...,xn)ϕB(xn+1,...,xd)(C4)
with the transversal wave function ϕB(xn+1,...,xd)andB(x1,...,xn) the annihilation operator in n
dimensions, and in the following we will use ˜
x=(x1,...,xn)T. The stationary Gross– Pitaevskii equation is
given by
0=−2∇2
˜
x
2MB−μB+˜
gB|φB(˜
x)|2φB(˜
x)(C5)
with the condensate φB(˜
x) fulfilling the Dirichlet boundary conditions φB(˜
x)=0for˜
x∈∂Dwith
D=[0, L]nand chemical potential μB. The coupling constant is given by
˜
gB=gBxn+1,...,xd|ϕB(xn+1,...,xd)|4,(C6)
which becomes ˜
gB=(√2πσB)−(d−n)gBfor harmonic confinement. The bulk solution of the
Gross–Pitaevskii equation can be approximated by the homogeneous function
φB(˜
x)≈μB
˜
gB
(C7)
leading to the density nB=μB
˜gB. In order to study the excitations of the bulk solution, we perform the
Bogoliubov approximation
B(˜
x)=φB(˜
x)+δB(˜
x), (C8)
where δBand δB†fulfill canonical commutation relations. The Bogoliubov Hamiltonian approximation of
HBis given by
HB=D2
2MB|∇˜
xδB|2−μB|δB|2+2˜
gB|δB|2|φB|2+˜
gB
2(δB†)2(φB)2+(δB)2(φ∗
B)2.(C9)
The equations of motions can be solved by using the mode expansion
δB(˜
x,t)=
kbkuk(˜
x)e−iωkt+b†
kv∗
k(˜
x)eiωkt, (C10)
where the sum does not include the condensate mode [95]. This expansion leads to the following
generalized eigenvalue problem
ωk10
0−1uk
vk=h(˜
x)˜
gB(φB)2
˜
gB(φ∗
B)2h(˜
x)uk
vk, (C11)
with
h(˜
x)=−2∇2
˜
x
2MB−μB+2˜
gB|φB(˜
x)|2.(C12)
11
Quantum Sci. Technol. 7(2022) 015008 VKasperet al
Solving this generalized eigenvalue problem will lead to the orthonormalization condition
Du∗
k(˜
x)uk(˜
x)−v∗
k(˜
x)vk(˜
x)=δk,k.(C13)
Since the background-field is approximately constant and because of the Dirichlet boundary conditions, we
make the ansatz
uk(˜
x)=uk2
Ln/2n
i=1
sin(kixi), (C14a)
vk(˜
x)=vk2
Ln/2n
i=1
sin(kixi), (C14b)
with ki=niπ
Land ni2 and the amplitudes are given by
u2
k=1
2εk
ωk
+1, (C15a)
v2
k=1
2εk
ωk−1, (C15b)
with εk=(2k2)/(2MB)+˜
gB|φB|2. The Bogoliubov eigenfrequencies of the excitations are given by
ωk=2k2
2MB2k2
2MB
+2μB.(C16)
The Dirichlet boundary conditions can be achieved by box potentials, which lead to a homogeneous
Bose– Einstein condensate.
Appendix D. Spin–phonon interaction
In order to derive the interaction between the phonons and the collective spins, we start from Hamiltonian
modeling the interaction between the A and B atoms
HAB =
mx
gm
AB
2A†
m(x)Am(x)B†(x)B(x).(D1)
After reducing the dimension and considering the tight confinement of the spins, i.e. equations (C4)and
(A4), we obtain the Hamiltonian
HAB =
m,y
˜
gm
AB
2a†
m(y)am(y)D|ϕA(x1−y1)...ϕ
A(xn−yn)|2B†(˜
x)B(˜
x), (D2)
where we neglected hopping of the A atoms and introduced the coupling constant
˜
gm
AB =gm
ABxn+1,...,xd|ϕB(xn+1)...ϕ
B(xd)|2|ϕA(xn+1−yn+1)...ϕ
A(xd−yd)|2,(D3)
which can be written as
˜
gm
AB =gm
ABπ(σ2
A+σ2
B)−(d−n)/2(D4)
for harmonic confinement of the A and B atoms with harmonic oscillator length scale σAand σB
respectively. Expanding the field operator B in fluctuations as in equation (C8) and neglecting terms of
order O(δB2) one obtains a Hamiltonian
HAB =H(0)
AB +H(1)
AB ,(D5)
12
Quantum Sci. Technol. 7(2022) 015008 VKasperet al
where H(0)
AB is independent of the fluctuations and H(1)
AB is linear in the fluctuations. The first contribution is
given by
H(0)
AB =
m,y
Δma†
m(y)am(y), (D6)
with the coupling constant
Δm=1
2˜
gm
ABnBD|ϕA(x1−y1)...ϕ
A(xn−yn)|2,(D7)
andsincetheharmonicoscillatorwavefunctionsarenormalizedweobtain
Δm=1
2˜
gm
ABnB.(D8)
The contribution linear in the fluctuations is given by
HAB =√nB
2
m,y
˜
gm
ABa†
m(y)am(y)D
[δB(˜
x)+H.c.]|ϕA(x1−y1)...ϕ
A(xn−yn)|2.(D9)
Inserting the mode expansion (C10)intoH(1)
AB we obtain
H(1)
AB =
m,y,k
˜
gm
AB,k(y)a†
m(y)am(y)[bk+H.c.], (D10)
where we introduced the coupling constant
˜
gm
AB,k(y)=√nB
2˜
gm
ABD|ϕA(x1−y1)...ϕ
A(xn−yn)|2[uk(˜
x)+vk(˜
x)].(D11)
Inserting (C14)and(A7)weobtainforLσBapproximately
˜
gm
AB,k(y)=˜
gm
AB√nB2
Ln/2
(uk+vk)e−1
4(kσ
A)2
×
n
i=1
sin(kiyi).(D12)
Using the Schwinger representation (see equation (1)), we obtain the interaction between the spins and the
phonons
H(1)
AB =
m,y,k
˜
gm
AB,k(y)L(y)−(−1)mLz(y)(bk+H.c.), (D13)
with L(y) being the length of the angular momentum on site y. Reshuffling terms leads to equation (4)with
the coupling constants
¯
gk(y)=L(y)[˜
g0
AB,k(y)+˜
g1
AB,k(y)], (D14a)
δgk(y)=˜
g1
AB,k(y)−˜
g0
AB,k(y).(D14b)
Appendix E. Eliminating phonons
Assuming Ω(y)=0 and given the approximations of appendices A,Cand Dthe Hamiltonian
H=HA+HB+HAB is diagonal in Lz, which allows us to treat phonons and spins separately. The
Heisenberg equation of motion for the phonons is
i∂tbk=ωkbk+δbk(E1)
of bkoperators, where we introduced the abbreviation
δbk=
y¯
gk(y)L(y)+δgk(y)Lz(y).(E2)
13
Quantum Sci. Technol. 7(2022) 015008 VKasperet al
We define a shifted annihilation operator as
βk=bk+(ωk)−1δbk.(E3)
Inserting the shifted operator in the Hamiltonian equation (4) leads to a spin–spin interaction
HI=−
x,y
g(x,y)Lz(x)Lz(y)(E4)
with
g(x,y)=
k
(ωk)−1δgk(x)δgk(y).(E5)
Inserting the explicit expression for δgk(x), we obtain equation (6).
ORCID iDs
Val e nt in Ka s per https://orcid.org/0000-0001-7687-663X
Daniel González-Cuadra https://orcid.org/0000-0001-7804-7333
Apoorva Hegde https://orcid.org/0000-0001-9996-9126
Andy Xia https://orcid.org/0000-0002-7486-1589
Alexandre Dauphin https://orcid.org/0000-0003-4996-2561
Felix Huber https://orcid.org/0000-0002-3856-4018
Maciej Lewenstein https://orcid.org/0000-0002-0210-7800
Fred Jendrzejewski https://orcid.org/0000-0003-1488-7901
Philipp Hauke https://orcid.org/0000-0002-0414-1754
References
[1] Acín A et al 2018 New J. Phys. 20 080201
[2] Horodecki R, Horodecki P, Horodecki M and Horodecki K 2009 Rev. Mod. Phys. 81 865
[3] Terhal B M 2015 Rev. Mod. Phys. 87 307
[4] Bernien H et al 2017 Nature 551 579
[5] Zhang J, Pagano G, Hess P W, Kyprianidis A, Becker P, Kaplan H, Gorshkov A V, Gong Z-X and Monroe C 2017 Nature 551 601
[6] Arute F et al 2019 Nature 574 505
[7] Nigg D, Müller M, Martinez E A, Schindler P, Hennrich M, Monz T, Martin-Delgado M A and Blatt R 2014 Science 345 302
[8] Flühmann C, Nguyen T L, Marinelli M, Negnevitsky V, Mehta K and Home J P 2019 Nature 566 513
[9] Campagne-Ibarcq P et al 2020 Nature 584 368
[10] Andersen C K, Remm A, Lazar S, Krinner S, Lacroix N, Norris G J, Gabureac M, Eichler C and Wallraff A 2020 Nat. Phys. 16 875
[11] Vandersypen L M K, Steffen M, Breyta G, Yannoni C S, Sherwood M H and Chuang I L 2001 Nature 414 883
[12] Doherty M W, Manson N B, Delaney P, Jelezko F, Wrachtrup J and Hollenberg L C L 2013 Phys. Rep. 528 1
[13] Knill E, Laflamme R and Milburn G J 2001 Nat ure 409 46
[14] Watson T F et al 2018 Nature 555 633
[15] Krantz P, Kjaergaard M, Yan F, Orlando T P, Gustavsson S and Oliver W D 2019 Appl. Phys. Rev. 6021318
[16] Bruzewicz C D, Chiaverini J, McConnell R and Sage J M 2019 Appl. Phys. Rev. 6021314
[17] Nigmatullin R, Ballance C J, de Beaudrap N and Benjamin S C 2016 New J. Phys. 18 103028
[18] Pagano G et al 2018 Quantum Sci. Technol. 4014004
[19] Campbell E T, Terhal B M and Vuillot C 2017 Natu re 549 172
[20] Bloch I, Dalibard J and Zwerger W 2008 Rev. Mod. Phys. 80 885
[21] Cirac J I and Zoller P 2012 Nat . Phy s. 8264
[22] Lewenstein M, Sanpera A and Ahufinger V 2012 Ultracold Atoms in Optical Lattices: Simulating Quantum Many-Body Systems
(Oxford: Oxford University Press)
[23] Görg F, Sandholzer K, Minguzzi J, Desbuquois R, Messer M and Esslinger T 2019 Na t. Phys. 15 1161
[24] Schweizer C, Grusdt F, Berngruber M, Barbiero L, Demler E, Goldman N, Bloch I and Aidelsburger M 2019 Nat. Phy s. 15 1168
[25] Mil A, Zache T V, Hegde A, Xia A, Bhatt R P, Oberthaler M K, Hauke P, Berges J and Jendrzejewski F 2020 Science 367 1128
[26] Yang B, Sun H, Ott R, Wang H-Y, Zache T V, Halimeh J C, Yuan Z-S, Hauke P and Pan J-W 2020 Natu re 587 392
[27] Bañuls M C et al 2020 Eur. Phys. J. D74 165
[28] Jaksch D 2004 Contemp. Phys. 45 367
[29] Brickman Soderberg K-A, Gemelke N and Chin C 2009 New J. Phys. 11 055022
[30] Saffman M, Walker T G and Mølmer K 2010 Rev. Mod. Phys. 82 2313
[31] Omran A et al 2019 Science 365 570
[32] Yang B, Sun H, Huang C-J, Wang H-Y, Deng Y, Dai H-N, Yuan Z-S and Pan J-W 2020 Science 369 550
[33] Madjarov I S et al 2020 Nat. Phy s. 16 857
[34] Strobel H, Muessel W, Linnemann D, Zibold T, Hume D B, Pezzè L, Smerzi A and Oberthaler M K 2014 Science 345 424
[35] Lücke B, Peise J, Vitagliano G, Arlt J, Santos L, T´
oth G and Klempt C 2014 Phys. Rev. Lett. 112 155304
[36] Bookjans E M, Vinit A and Raman C 2011 Phys.Rev.Lett.107 195306
[37] Pezzè L, Smerzi A, Oberthaler M K, Schmied R and Treutlein P 2018 Rev. Mod. Phys. 90 035005
[38] Stamper-Kurn D M and Ueda M 2013 Rev. Mod. Phys. 85 1191
[39] Kunkel P, Prüfer M, Strobel H, Linnemann D, Frölian A, Gasenzer T, Gärttner M and Oberthaler M K 2018 Science 360 413
14
Quantum Sci. Technol. 7(2022) 015008 VKasperet al
[40] Klein A and Fleischhauer M 2005 Phys. Rev. A71 33605
[41] Klein A, Bruderer M, Clark S R and Jaksch D 2007 New J. Phys. 9411
[42] Diehl S, Micheli A, Kantian A, Kraus B, Büchler H P and Zoller P 2008 Nat. Phys . 4878
[43] Ramos T, Pichler H, Daley A J and Zoller P 2014 Phys.Rev.Lett.113 237203
[44] Scelle R, Rentrop T, Trautmann A, Schuster T and Oberthaler M K 2013 Phys.Rev.Lett.111 070401
[45] Rentrop T, Trautmann A, Olivares F A, Jendrzejewski F, Komnik A and Oberthaler M K 2016 Phys. Rev. X6041041
[46] DiVincenzo D P 2000 Fortschr. Phys. 48 771
[47] If not stated otherwise we will assume the same number of atoms at all site y, whereas different atom number per sites are in
principle possible.
[48] Weitenberg C, Endres M, Sherson J F, Cheneau M, Schauß P, Fukuhara T, Bloch I and Kuhr S 2011 Nature 471 319
[49] Farolfi A, Zenesini A, Cominotti R, Trypogeorgos D, Recati A, Lamporesi G and Ferrari G 2021 Phys. Rev. A104 023326
[50] Wineland D J et al 2003 Phil.Trans.R.Soc.A361 1349
[51] Nielsen M A and Chuang I L 2013 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)
[52] Giorda P, Zanardi P and Lloyd S 2003 Phys. Rev. A68 062320
[53] Chang D E, Douglas J S, González-Tudela A, Hung C-L and Kimble H J 2018 Rev. Mod. Phys. 90 31002
[54] Grusdt F and Demler E 2015 arXiv:1510.04934
[55] Lopes R, Eigen C, Navon N, Cl´
ement D, Smith R P and Hadzibabic Z 2017 Phys.Rev.Lett.119 190404
[56] Rauer B, Erne S, Schweigler T, Cataldini F, Tajik M and Schmiedmayer J 2018 Science 360 307
[57] Kaushal V et al 2020 AVS Quantum Sci. 2014101
[58] Riebe M et al 2004 Nature 429 734
[59] The definition of a d-dimensional CZ gate is given by CZ|p|q=ωpq |p|q.
[60] Lloyd S 1995 Phys. Rev. Lett. 75 346
[61] Bartlett S D, de Guise H and Sanders B C 2002 Phys. Rev. A65 52316
[62] Hesse A, Köster K, Steiner J, Michl J, Vorobyov V, Dasari D, Wrachtrup J and Jendrzejewski F 2021 New J. Phys. 23 023037
[63] Endres M et al 2016 Science 354 1024
[64] Barredo D, de L´
es´
eleuc S, Lienhard V, Lahaye T and Browaeys A 2016 Science 354 1021
[65] Wang Y, Shevate S, Wintermantel T M, Morgado M, Lochead G and Whitlock S 2020 npj Quantum Inf. 654
[66] Muessel W, Strobel H, Linnemann D, Hume D B and Oberthaler M K 2014 Phys.Rev.Lett.113 103004
[67] Kaufman A M, Lester B J and Regal C A 2012 Phys. Rev. X2041014
[68] Gross C, Strobel H, Nicklas E, Zibold T, Bar-Gill N, Kurizki G and Oberthaler M K 2011 Nature 480 219
[69] Schulze T A, Hartmann T, Voges K K, Gempel M W, Tiemann E, Zenesini A and Ospelkaus S 2018 Phys. Rev. A97 023623
[70] Boll M, Hilker T A, Salomon G, Omran A, Nespolo J, Pollet L, Bloch I and Gross C 2016 Science 353 1257
[71] Bergschneider A, Klinkhamer V M, Becher J H, Klemt R, Palm L, Zürn G, Jochim S and Preiss P M 2019 Nat. Phy s. 15 640
[72] Covey J P, Madjarov I S, Cooper A and Endres M 2019 Phys.Rev.Lett.122 173201
[73] Kawaguchi Y and Ueda M 2012 Phys. Rep. 520 253
[74] Gottesman D, Kitaev A and Preskill J 2001 Phys. Rev. A64 012310
[75] Knill E and Laflamme R 1997 Phys. Rev. A55 900
[76] Mirhosseini M, Sipahigil A, Kalaee M and Painter O 2020 Nature 588 599
[77] Chamberland C et al 2020 Building a fault-tolerant quantum computer using concatenated cat codes (arXiv:2012.04108)
[78] Fradkin E and Shenker S H 1979 Phys. Rev. D19 3682
[79] González-Cuadra D, Zohar E and Cirac J I 2017 New J. Phys. 19 063038
[80] Van Damme M, Halimeh J C and Hauke P 2020 arXiv:2010.07338
[81] González-Cuadra D, Bermudez A, Grzybowski P R, Lewenstein M and Dauphin A 2019 Nat. Commun. 10 2694
[82] Porras D, Ivanov P A and Schmidt-Kaler F 2012 Phys.Rev.Lett.108 235701
[83] Nevado P and Porras D 2016 Phys. Rev. A93 13625
[84] Kokail C et al 2019 Nature 569 355
[85] Henriet L, Beguin L, Signoles A, Lahaye T, Browaeys A, Reymond G-O and Jurczak C 2020 Quantum 4327
[86] Argüello-Luengo J, González-Tudela A, Shi T, Zoller P and Cirac J I 2020 Phys. Rev. Res. 2042013
[87] Aikawa K, Frisch A, Mark M, Baier S, Rietzler A, Grimm R and Ferlaino F 2012 Phys.Rev.Lett.108 210401
[88] Chalopin T, Satoor T, Evrard A, Makhalov V, Dalibard J, Lopes R and Nascimbene S 2020 Nat. Phys. 16 1017
[89] Fukuhara T, Schauß P, Endres M, Hild S, Cheneau M, Bloch I and Gross C 2013 Nature 502 76
[90] Chomaz L et al 2018 Nat. Phy s. 14 442
[91] Grimsmo A L, Combes J and Baragiola B Q 2020 Phys. Rev. X10 11058
[92] Noh K and Chamberland C 2020 Phys. Rev. A101 12316
[93] Albert V V, Covey J P and Preskill J 2020 Phys. Rev. X10 31050
[94] Gross J A 2020 arXiv:2005.10910
[95] Rogel-Salazar J, New G H C, Choi S and Burnett K 2001 J. Phys. B: At. Mol. Opt. Phys. 34 4617
15
Available via license: CC BY 4.0
Content may be subject to copyright.
