#  Universal terminal for mobile edge-quantum computing

Mohammadsadegh Khazali  Institute for Quantum Optics and Quantum Information
of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria  School of
Physics, Institute for Research in Fundamental Sciences (IPM), Tehran
19395-5531, Iran

###### Abstract

To bring the quantum computing capacities to the personal edge devices, the
optimum approach is to have simple non-error corrected personal devices that
offload the computational tasks to scalable quantum computers via edge servers
with cryogenic components and fault-tolerant schemes. Hence the network
elements deploy different encoding protocols. This article proposes quantum
terminals that are compatible with different encoding protocols; paving the
way for realizing mobile edge-quantum computing. By accommodating the atomic
lattice processor inside a cavity, the entangling mechanism is provided by the
Rydberg-Fermi cavity-QED technology. The gate operates by the Fermi scattering
of a Rydberg electron from the plaquette atoms hosting the physical qubits.
Therefore, different arrangements of logical-qubits derive the central atom
over distinguished eigenstates, featuring photon emission at the early or late
times distinguished by quantum interference. Applying an entanglement-swapping
gate on two emitted photons would make the far-separated qubits entangled
regardless of their encoding protocols. This gate provides a universal
photonic interface for clustering the processors and connecting them with the
quantum memories and quantum cloud that is compatible with different encoding
formats.

##  I introduction

Complicated quantum algorithms require large-scale fault-tolerant quantum
processors. With the current noisy devices, the errors scale with the qubit
numbers to the limit that fails the operation of complicated tasks. The
correction capabilities come with encoding the logical qubits in multiple
physical qubits and are protected by error-correction codes Got97 ; Got99 .
Quantum operations on the logic level are performed by an overload of
operations at the level of physical qubits and require costly techniques such
as complex optimization Her17 ; Han18 , magic state distillation Bra05 ,
transversal gates Got97  and lattice surgery Pou17 ; Gut19 ; Hor12 . This
would dramatically increase the required computational power and capacity of
the quantum processors. Considering the laboratory equipment for laser cooling
and cryogenic environment, fault-tolerant quantum computation is not
accessible on personal devices. To bring the quantum advantage to society, the
optimized cost-benefit approach is to have small-scale non-error corrected
mobile devices that offload the computational tasks to the error-corrected
servers Xu22 . Hence the elements of the quantum cloud deploy different
encoding protocols. A plug-&-play approach for clustering and connecting these
devices requires universal terminals that are compatible with all encoding
schemes.

This paper proposes a gate that entangles a single photonic qubit with a
logical-qubit encoded in 4-atoms Mar21

${|{0}\rangle}_{L}=\frac{{|{0000}\rangle}+{|{1111}\rangle}}{\sqrt{2}},\quad\,{|{1}\rangle}_{L}=\frac{{|{0101}\rangle}+{|{1010}\rangle}}{\sqrt{2}}.$
(1)

The extension to other encoding protocols e.g. 6-qubit Sha08  is discussed.
Applying a projective Bell-state measurement (PBM) on two flying qubits makes
the qubits in stationary units entangled regardless of their encoding scheme,
see Supp Supp . This universal photonic interface is valuable for clustering
the fault-tolerant processors Mon14 , connecting them with the quantum
memories Hes16 ; Kav13 ; Mon14 , and the quantum cloud Xu22 ; Jia22 ; Lok22 ;
Zha22 ; LokSen22 ; Kris22 ; Ngu22 ; Nguy22 ; Khat22 ; Roh22 ; cac20 ; calef20
; Chehimi22 ; Meter22 ; Dah22 .

Figure 1: Universal terminal for long-distance entanglement of logical qubits
with different encoding formats. (a) The auxiliary atom at the center of the
plaquette is emitting a photon at the early or late time, conditioned on the
Rydberg-Fermi sensing of the logical qubit encoded on plaquette atoms, see
Fig. 2,4. Subsequent PBM of photons, entangles far-separated logical qubits
Supp . (b) Level scheme of the auxiliary atom in the 4-qubit encoding scheme
used in the left cavity (see Fig. 4 for the right cavity). The lambda
configuration consists of atom-laser and atom-cavity couplings and is
responsible for photon emission. The interaction-induced level-shift of the
Rydberg state depends on the plaquette’s spin-state $S_{p}$. The
$|0_{L}\rangle$ logical state is associated with $S_{p}=0$ and 4, making
$\Omega_{r}$ laser out of resonance with the Rydberg level. Hence, the Raman
transition from $|s\rangle$ to $|e\rangle$ would generate a single photon at
the $|\text{early}\rangle$ time. On the other hand, $|1_{L}\rangle$ logical
state accommodates two plaquette atoms in Rydberg wave-function $S_{p}=2$,
making the $\Omega_{r}$ laser in-resonance with the Rydberg level.
Consequently, destructive interference inhibits the transition to $|e\rangle$
and therefore blocks the early photon emission. The following exclusive
$\Omega_{s}$ pulse shown in (c) leads to $|\text{late}\rangle$ photon emission
in case of $|1_{L}\rangle$ state.

A logical qubit is a highly entangled two-dimensional subspace in the larger
Hilbert space of multiple physical qubits. To avoid the costly techniques used
for logical operations Her17 ; Han18 ; Bra05 ; Got97 ; Pou17 ; Gut19 ; Hor12 ,
system-specific properties could be deployed to significantly reduce the
number of operations on the physical qubits and hence the errors incurred
during execution. Laser excited Rydberg atoms are an ideal example where the
long-range interactions provide the possibility of simultaneous operations on
multiple qubits Saf10 ; Kha20 ; Ise11 ; Gla17 ; Dla21 ; Kha16 ; Kha18 ;
khaz2020rydberg , with the bonus opportunities in quantum optics Ada19 ; Kha19
; KhaRev21 ; Kha22 . However, fast multi-qubit operation requires Rydberg
population in all interacting qubits. This would affect the Rydberg stabilizer
operations Gla17  due to unwanted cross-talk between physical qubits with
demolishing effects on the logical encoding.

The proposed gate scheme is based on cavity-QED photon emission from an
auxiliary centered atom, conditioned on the logical-qubit that is encoded on
the surrounding plaquette atoms. The single-step logical operation is
performed by Fermi scattering of the central atom’s Rydberg electron from the
plaquette atoms Gre00 ; Gaj14 ; Khaz21Fermi  in a spin-dependent lattice Dut98
; Sol11 ; Kar09 ; Lee07 ; Jak99 ; Bre99 ; Bri00 ; Man03 ; Mandel03 ; Kum18 .
Therefore, the logical-qubit determines which eigenstate the system would
follow, either containing or excluding the atom-cavity coupling. As a result,
the logical-qubit would get entangled with the time-bin photonic qubit emitted
by the central atom, generating the entangled state

$({|{\text{early}}\rangle}{|{\text{0}_{L}}\rangle}+{|{\text{late}}\rangle}{|{\text{1}_{L}}\rangle})/\sqrt{2}.$
(2)

Subsequent PBM on two photons in time-bin basis Rie05 ; Mar03 ; Hal07 ; Tak09
; Sun17 ; Boa18 ; Jin19 , entangles the stationary-qubits regardless of their
encoding schemes. The time-bin qubit is an ideal choice due to intrinsic
robustness against phase fluctuations Rie05 , with new advances in high-
fidelity operations Bow21 .

Deploying system-specific properties, the current scheme operates on all
physical qubits in a single step, circumventing the costly techniques used for
logical operations Her17 ; Han18 ; Bra05 ; Got97 ; Pou17 ; Gut19 ; Hor12 .
Remarkably, the current proposal operates via a sole auxiliary Rydberg atom,
eliminating the unwanted cross-talk of physical qubits over the logical
operation. Furthermore, the Rydberg-Fermi interaction provides a molecular
type potential that eliminates the frozen gas regime requirement of the
Rydberg dipolar schemes and also operates at much shorter interatomic
distances, addressing the scalability. Localizing the Rydberg population on
one atom in this scheme, closes the collective line broadening problems Gol16
; Zei16 . Unlike the majority of Rydberg dipolar schemes that operates via a
$\pi$-gap-$\pi$ scheme, the proposed Ryd-Fermi gate does not leave any Rydberg
population unprotected from the laser, closing the major decoherence source
Mal15 ; Gra19 . Suppressing the short-lived Rydberg population, high-
fidelities of 99.6% is predicted.

##  II scheme

The setup consists of cesium 133Cs atomic lattice quantum processor
accommodated in a cavity. Fig. 1a, is a simplified figure with a single
plaquette and a central atom in two edge devices of the network. The logical
information in two devices is encoded in different 4- and 6-qubit basis. The
system atoms are placed on square (hexagonal) plaquette consisting of physical
qubit states ${|{0}\rangle}={|{6S,F=3}\rangle}$ and
${|{1}\rangle}={|{6S,F=4}\rangle}$, while auxiliary atom responsible for
conditional photon emission is at the center of the plaquettes with the
electronic level scheme presented in Fig. 1b (and Fig. 4b in case of 6-qubit
encoding). The lower lambda configuration is responsible for on-demand single-
photon emission, similar to Kuh99 ; Gog10 . The $\Lambda$ transition would be
(allowed) prohibited due to the quantum interference in case the blue laser is
(off-resonance) in-resonance with the Rydberg transition. The tuning of the
blue laser is determined by the Fermi scattering of the Rydberg electron from
the plaquette atoms, designed to make the photon emission conditioned on the
logical qubit. In the lower lambda system the transition between
${|{s}\rangle}={|{6S_{1/2},F=3}\rangle}$ and
${|{p}\rangle}={|{7P_{3/2}}\rangle}$ states is derived by $\Omega_{s}$ laser
while the ${|{p}\rangle}$ to
${|{e}\rangle}=\\{{|{6S_{1/2},F=4}\rangle}\,\text{or}\,{|{5D_{3/2}}\rangle}\\}$
is governed by the Jaynes-Cummings interaction with the coupling constant $g$
and the cavity mode with frequency
$\omega_{c}\approx\\{450,\,\text{or}\,1300\\}$nm Microwave . The transitions
are detuned from the intermediate state by $\Delta$. The state basis
${|{i,n}\rangle}$ in Fig. 1b includes the central atom’s electronic state
${|{i}\rangle}$ and cavity number state ${|{n}\rangle}$. Over a Raman $\pi$
pulse, the transition from ${|{s,0}\rangle}$ to ${|{e,1}\rangle}$ would
generate a photon in the cavity. The decay of the cavity mode causes single-
photon emission and the system settles itself in the state ${|{e,0}\rangle}$.

Fermi scattering of central atom’s Rydberg electron from plaquette atoms in
spin-dependent lattice results in an effective level-shift quantified by the
plaquette spin $S_{p}=\sum_{j\in p}\sigma_{00}^{(j)}$, with
$\sigma_{00}={|{0}\rangle}\\!{\langle{0}|}$ being the projective operator and
$j$ goes over the plaquette atoms, see Fig. 1b. Having logical qubit
${|{0}\rangle}_{L}=({|{0000}\rangle}+{|{1111}\rangle})/\sqrt{2}$, the blue
laser would be out of resonance with the interaction-shifted Rydberg level.
Hence, the first $\Omega_{s}$ pulse in Fig. 1c derives an early photon
emission ${|{\text{early}}\rangle}$. For the other logical qubit
${|{1}\rangle}_{L}=({|{0101}\rangle}+{|{1010}\rangle})/\sqrt{2}$, two
plaquette atoms would be in Rydberg electron wave-function. Consequently, the
blue laser would get in resonance with the Rydberg level, blocking the photon
emission over the first $\Omega_{s}$ pulse. The remaining ${|{s,0}\rangle}$
population would then get transferred over the second $\Omega_{s}$ pulse (not
accompanied by the Rydberg lasers $\Omega_{r}$) and hence emits a late photon
${|{\text{late}}\rangle}$, leading to the desired state of Eq. 2. The time
ordering of the pulses is shown in Fig. 1c, and the physics behind the
blocking and transmission are discussed below. In an entanglement swapping
station, the emitted photons from distinct sources would undergo a C-NOT gate
followed by projective Bell state measurement. This would make the two far-
separated plaquettes entangled in their own logical basis Supp .

Figure 2: Rydberg-Fermi interaction in spin-dependent lattice. (a) In a 2D
lattice with a single atom per site, applying a qubit-dependent lattice-shift
along $z$ makes each atom in a spatial superposition of being in red and green
sites where the components are controlled by the internal electronic qubit-
states $|0\rangle$ and $|1\rangle$. (b) Counter propagating linearly polarized
lights along $z$ direction with relative polarization shift of $\theta$ forms
two standing-waves of $\pi^{-}$ and $\pi^{+}$ circular polarizations. (c)
Tunning the trapping laser between $6P_{3/2}$ and $6P_{1/2}$, the
polarizability of qubit states $|0\rangle$ and $|1\rangle$ are given by
distinguished circularly polarized lights $\pi^{-}$ and $\pi^{+}$
respectively, resulting to spin-dependent trapping. The cross-sections of
$|45D_{5/2},5/2\rangle$ Rydberg wave-function along (d) xy and (e) xz are
plotted. The position of the qubit states $|0\rangle$ and $|1\rangle$ are
marked by the red and green circles. (e) Note to the spin-dependent overlap of
plaquette atoms with the Rydberg electron wave-function.

##  III Rydberg-Fermi interaction

To apply a qubit-dependent Rydberg-fermi interaction, the lattice undergoes
spin-dependent shift along the $z$ direction. Hence, the Rydberg electron of
the central atom would only scatter from the plaquette atoms in qubit state
${|{0}\rangle}$. This would cause a level-shift that depends on the plaquette
spin $S_{p}=\sum_{j\in p}\sigma_{00}^{(j)}$.

The spin-dependent lattice shift along $z$ direction is formed by counter-
propagating linearly polarized lights as depicted in Fig. 2b. Having a
relative shift between the fields’ polarizations $2\theta$, the total electric
field could be written in terms of the sum of right and left circularly
polarized lights $E=E_{0}\exp(-i\nu
t)(\pi^{+}\sin(kz+\theta)+\pi^{-}\sin(kz-\theta))$. To make a spin-dependent
lattice-shift, the spin polarizabilities should be linked to different
circular polarization components of lights Dut98 . To cancel the
polarizabilities with unwanted light elements shown by dashed lines in Fig.
2c, the trapping laser must be tuned between $P_{3/2}$ and $P_{1/2}$ states so
that the ac-Stark shifts of these two levels cancel each other. As a result
the $m_{j}=\pm 1/2$ levels of the ground state would be trapped by
$V_{\pm}=\alpha|E_{0}|^{2}\sin(kz\pm\theta)$ respectively. The hyperfine qubit
states $|0\rangle=|F=3,m_{F}=3\rangle$ and $|1\rangle=|F=4,m_{F}=4\rangle$
would experience $V_{{|{0}\rangle}}=(V_{+}+3V_{-})/4$,
$V_{{|{1}\rangle}}=V_{{|{s}\rangle}}=V_{+}$.

The Rydberg laser $\Omega_{r}$ is exciting the auxiliary atom at the centere
of the desired plaquette, see Fig. 2. Applying a spin-dependent lattice shift
perpendicular to the 2D lattice would result in dual spin/spatial encoding of
the plaquette qubits. Thus, exciting the central atom to Rydberg level, the
electron wave-function would exclusively overlap with plaquette qubits in
spin-state ${|{0}\rangle}$, see Fig. 2e. The resulting Rydberg-Fermi
interaction caused by scattering of Rydberg electron from a single neutral
atom would be quantified by Fer ; Eil17 ; Eil19 ,

$V_{\text{RF}}=(2\pi\frac{\tan(\delta^{s})}{k(R)}-6\pi\frac{\tan(\delta^{p})}{k^{3}(R)}\stackrel{{\scriptstyle\leftarrow}}{{\nabla}}_{{\bf
r}}.\stackrel{{\scriptstyle\rightarrow}}{{\nabla}}_{{\bf r}})\delta({\bf
r}-{\bf R})$ (3)

with r and R being the positions of the Rydberg electron and a ground state
atom with respect to the ionic core, and $\delta^{\\{s,p\\}}$ are the triplet
s- and p-wave scattering phase shift of Rydberg electron from a neighboring
ground state atom khu02 . Since $S_{p}$ presents the number of plaquette atoms
in the Rydberg wave-function of central atom, the effective level-shift caused
by the Fermi scattering would be $S_{p}V_{\text{RF}}$ Gaj14 , see Fig. 1b.

##  IV entangling gate

This section discusses the main physics behind the conditional photon emission
that entangles the stationary logical and flying qubits. In the case of
${|{1_{L}}\rangle}$, the physical qubits’ arrangement in the qubit-dependent
lattice would accommodate two plaquette atoms in the Rydberg wave-function of
the central atom, making the $\Omega_{r}$ laser in-resonance with the shifted
Rydberg level $\delta_{\text{RF}}({|{1_{L}}\rangle})=0$. Having
${|{0_{L}}\rangle}$, the plaquette’s arrangement contains either $S_{p}=0$ or
4 atoms in the Rydberg wave-function, resulting in the effective detuning of
$\delta_{\text{RF}}({|{0_{L}}\rangle})=2V_{\text{RF}}$. The Hamiltonian of the
central atom in the electronic basis is given by

$\displaystyle H=$
$\displaystyle\Omega_{s}/2(\hat{\sigma}_{sp}+\text{h.c.})+\Omega_{R}/2(\hat{\sigma}_{rp}+\text{h.c.})$
$\displaystyle+g/2(\hat{\sigma}_{ep}+\text{h.c.})+{\Delta}\hat{\sigma}_{pp}+\delta_{RF}\hat{\sigma}_{rr}\quad\quad\quad$

where $\hat{\sigma}_{\alpha\beta}={|{\alpha}\rangle}\\!{\langle{\beta}|}$. To
simplify the analytic discussion here the large detuning regime
$\Delta\gg\Omega_{\\{s,r\\}},g$ and equal couplings $g=\Omega_{s}$ are
considered. After adiabatic elimination of the intermediate state
${|{p}\rangle}$, the Hamiltonian is represented in new basis
${|{\pm}\rangle}=({|{s}\rangle}\pm{|{e}\rangle})/\sqrt{2}$, and
${|{r}\rangle}$ as

$\displaystyle
H/\epsilon=\lambda^{2}{|{+}\rangle}\\!{\langle{+}|}+(1-\delta){|{r}\rangle}\\!{\langle{r}|}+\lambda({|{+}\rangle}\\!{\langle{r}|}+\text{h.c.})\quad\quad\quad$
(5)

where $\lambda=\Omega_{s}/\Omega_{r}$ is the dimensionless Rabi frequency and
$\delta=\delta_{RF}/\epsilon$ is the Rydberg-Fermi interaction scaled by
$\epsilon=\Omega_{r}^{2}/4\Delta$. In the regime of $\lambda,\delta\ll 1$ the
lower bright
${|{b}\rangle}=[(1-\delta){|{+}\rangle}-\lambda{|{r}\rangle}]/\sqrt{(1-\delta)^{2}+\lambda^{2}}$
state, with the energy $2\delta\lambda^{2}$ would interfere with the dark
state ${|{d}\rangle}={|{-}\rangle}$. Having ${|{1_{L}}\rangle}$, the Rydberg
laser would be in resonance $\delta=0$. Hence, the system would follow the new
dark state
${|{D}\rangle}=({|{d}\rangle}+{|{b}\rangle})/\sqrt{2}=({|{s}\rangle}-\lambda{|{r}\rangle})/\sqrt{1+\lambda^{2}}$
featuring destructive interference that blocks the transition to
${|{e}\rangle}$ state. A Ryd-Fermi induced level-shift of
$\delta=\delta_{\text{RF}}/(\Omega_{r}^{2}/4\Delta)\gtrsim 1$ lifts the
${|{d}\rangle}$ and ${|{b}\rangle}$ states degeneracy, the system would
exclusively follow the dark state ${|{d}\rangle}$ and emits an early photon,
see Fig. 3a.

Figure 3: The operation regime of entangling gate. The operation is simulated
under the non-hermitian Hamiltonian $\tilde{H}$ of Eq. 7. (a) With
$|0_{L}\rangle$, the detuning $\delta\gtrsim 1$ caused by Ryd-Fermi
interaction lifts the interference of dark and bright states, allowing early
photon emission. (b) With $|0_{L}\rangle$ logical state
($\delta_{RF}=2V_{RF}$), the optimum early emission occurs at
$\Gamma_{c}=\Omega_{\text{eff}}$. Larger cavity leakage would reduce the
coherence, scatter the population out of the dark state, and reduce the
emission probability. (c) Having $|1_{L}\rangle$ logical state
($\delta_{RF}=0$), the early photon emission would be blocked for
$\Omega_{r}/\Omega_{s}>3$ by more than 99%. Applied parameters are
$\delta_{RF}/2\pi=130$MHz, $\Delta/2\pi=33$MHz, $\Omega_{r}/2\pi=40$MHz,
$\max(\Omega_{s})/2\pi=10$MHz, $g/2\pi=8.5$MHz, $\Gamma=\Omega_{\text{eff}}$,
$\sigma\Gamma=10$.

While a large cavity decay rate $\Gamma_{c}$ enhances the emission rate and
narrows the emission probability profile, it reduces the coherence. In the
regime of $\lambda=\Omega_{s}/\Omega_{r}\ll 1$ the steady-state of the master
equation could be obtained perturbatively as
$\rho=\rho_{0}+\lambda\rho_{1}+\lambda^{2}\rho_{2}$. The desired transition
coherence is given by

$\rho_{se}=\frac{\Omega_{s}g}{4\Delta}\frac{2i\delta_{RF}}{\Gamma_{c}(\Omega_{r}^{2}/4\Delta-\delta_{RF})}.$
(6)

Cavity decay rates $\Gamma_{c}$ larger than effective transition Rabi
frequency $\Omega_{\text{eff}}=\Omega_{s}g/4\Delta$ reduces the coherence
between ${|{s}\rangle}$ and ${|{e}\rangle}$. Since an incoherent superposition
of these states does not exclusively project into the dark state
${|{d}\rangle}$, some of the population would transfer to bright states
reducing the operation fidelity, see Fig. 3b. Finally, the
$\lambda=\Omega_{s}/\Omega_{r}\ll 1$ condition is required to block the early
transmission of photons in the presence of ${|{1_{L}}\rangle}$ as numerically
evaluated in Fig. 3c.

To study the Photon emission, Eq. IV must be modified to the non-hermitian
version

$\tilde{H}=H-\text{i}\Gamma_{c}\hat{\sigma}_{ee},$ (7)

where $\Gamma_{c}$ is the cavity leakage rate. The gate operation under this
Hamiltonian is quantified in Fig. 3. In the regime that
$\Delta,\delta_{RF}\gg\Omega_{\\{r,s\\}},g$, the adiabatic elimination of
Rydberg and intermediate $p$ state would simplify the dynamics to a two-level
system with an effective Rabi frequency of
$\Omega_{\text{eff}}=\frac{\Omega_{s}g}{4{\Delta}}(1+\frac{\Omega_{r}^{2}}{4{\Delta}\delta_{RF}})$
and an effective detuning of
$(\frac{\Omega_{s}^{2}}{4\Delta}-\frac{g^{2}}{4\Delta})(1+\frac{\Omega_{r}^{2}}{4{\Delta}\delta_{RF}})+\text{i}\Gamma_{c}/2$.
Choosing close coupling strengths $\Omega_{s}\approx g$ to facilitate the
desired transition, the photon flux out of the cavity would be given by
$\Phi(t)=\Gamma_{c}|c_{e,1}|^{2}=\Gamma_{c}\frac{\Omega_{\text{eff}}^{2}}{\tilde{\Omega}^{2}}e^{-\Gamma_{c}t}\sin^{2}(\tilde{\Omega}t)$
where $\tilde{\Omega}=\sqrt{\Omega_{\text{eff}}^{2}+\Gamma_{c}^{2}/4}$.

##  V Implementation

The implementation of the scheme encounters a 340nm in-plane $xy$ optical
lattice addressing $6S-10P$ transition with trap potential of $U_{\text{trap-
xy}}/2\pi=5$MHz. The spin-dependent trapping along $z$ direction is formed by
870nm laser with $U_{\text{trap-z}}/2\pi=2$MHz, see Fig. 2c. Targeting the
central atom to $|45D_{5/2},5/2\rangle$ and considering the spin-dependent
lattice shift of $D_{z}=150$nm, the average Rydberg-Fermi interaction with a
single plaquette atom in qubit-state ${|{0}\rangle}$ would be
${\langle{0}|}V_{RF}{|{0}\rangle}=2\pi\times 65$MHz Supp . The atoms in the
other qubit-dependent lattice experience negligible interaction of
${\langle{1}|}V_{RF}{|{1}\rangle}=2\pi\times 1$kHz. The optimum realistic
parameters in driving of the central atom includes $\Omega_{r}/2\pi=40$MHz,
max$(\Omega_{s})/2\pi=10$MHz, $g/2\pi=8.5$MHz, $\Gamma_{c}/2\pi=750$kHz Kuh99
, $\Delta/2\pi=33$MHz and $\delta_{\text{RF}}({|{0_{L}}\rangle})/2\pi=130$MHz.
The time interval of a Gaussian pulse
$\Omega_{s}(t)=\Omega_{s}\exp(-t^{2}/\sigma^{2})$ must be long enough 1- to
fulfill the adiabaticity condition and preserve the dark state $\sigma\gg
g^{-1}$; 2- to ensure the photon emission considering the cavity decay-time
$\sigma\gg\Gamma^{-1}$; and 3- to perform the population rotation
$\sigma\gg\Omega_{\text{eff}}^{-1}$. Considering the spontaneous emission of
the intermediate $\gamma_{p}/2\pi=1$MHz Toh19  and the Rydberg state
$\gamma_{r}/2\pi=4.5$kHz Bet09  and $\sigma=3.5\mu$s, the operation fidelity
of
${|{0}\rangle}({|{{0_{L}}}\rangle}+{|{{1_{L}}}\rangle})/\sqrt{2}\rightarrow({|{\text{early}}\rangle}{|{0_{L}}\rangle}+{|{\text{late}}\rangle}{|{1_{L}}\rangle})/\sqrt{2}$
for the mentioned set of parameters under the numerical simulation would be
Fid=99.6%. Further nuances regarding single-site addressing, perseverance of
ground motional state, and Rydberg-molecule decoherence channel are discussed
in Supp Supp .

##  VI extension to six-qubit encoding

The current proposal could be extended to the six-qubit code Sha08 , with the
logical basis being the odd and even parity states of the $\bar{Z}=IZIZIZ$
logical-operator qubit . Exciting Rydberg superposition state
$R_{n,l}(r)(Y_{2}^{2}(\theta,\phi)+Y_{2}^{-1}(\theta,\phi))/\sqrt{2}$, could
realize the desired parity-photon entangling gate in a triangular lattice, see
Fig. 4a. Fermi scattering of central atom’s Rydberg electron from every other
site in a hexagonal plaquette structure is conditioned on the physical qubits
to be in ${|{0}\rangle}$ state. This would cause an effective level-shift
quantified by three atoms’ spin-number, see Fig. 4b. The two-color
$\Omega_{r}$ in Fig. 4b are tuned in-resonance with odd-parity of the
$\bar{Z}$ stabilizer. The two-color blue transitions could be obtained by a
single laser in a setup of beamsplitters and acusto-optical modulators. With
even parity of $\bar{Z}$, the blue laser would be out of resonance with the
Rydberg level. Hence, the first $\Omega_{s}$ pulse in Fig. 1c derives an early
photon emission ${|{\text{early}}\rangle}$. Having an odd number of spin-downs
in the three targeted atoms, makes one of the blue lasers in resonance with
the Rydberg level, while the other laser causes a level-shift to
${|{p}\rangle}$ state resulting in a small correction in the detuning value
$\Delta\equiv\Delta\pm\Omega_{r}^{2}/4V_{\text{RF}}$. The resonant laser would
block the transition over the first $\Omega_{s}$ pulse as discussed in Sec.
IV. The remained ${|{s,0}\rangle}$ population would then transfer over the
second $\Omega_{s}$ pulse (not accompanied with $\Omega_{r}$) and hence emits
a late photon ${|{\text{late}}\rangle}$, providing the desired state of Eq. 2
in the 6-qubit encoding.

Figure 4: Photonics interface for 6-qubit surface code. The physical qubits
are dual encoded in the spin/spatial basis of a spin-dependent lattice placed
over the hexagonal structure. (a) The logical operator $\bar{Z}=IZIZIZ$ is
implemented by exciting the auxiliary atom at the centre of hexagonal
plaquette to the Rydberg superposition state with angular part
$(Y_{2}^{2}(\theta,\phi)+Y_{2}^{-1}(\theta,\phi))/\sqrt{2}$. (b) The energy
splitting of the Rydberg state depends on the number of atoms in the Rydberg
orbitals $S_{p}=\sum_{i=2,4,6}\sigma_{00}^{(i)}$. The two-colored Rydberg
lasers $\Omega_{r}$ get in resonance with the odd parity of $\bar{Z}$,
associated with $|1\rangle_{L}$, blocking the early photon emission.

##  VII Conclusion and Outlook

This article proposes a hardware architecture and protocol for long-distance
entanglement generation between logical qubits with different encoding
protocols. The Fermi-scattering of the Rydberg electron from the plaquette
atoms in a spin-dependent lattice forms a level-shift that is quantified by
the logical qubit. This level-shift would control the central atoms coupling
with the cavity mode and hence the ordering of photon emission would be
entangled by the logical qubit. The two characteristic of the Rydberg-fermi
interaction in working at small interatomic distance and in making bound state
with restoring force allows long-time operation in ultra-dense atomic
processors, making a significant advantage over the Rydberg dipolar
counterparts. The proposed photonic quantum bus solution, promises the
scalability of fault-tolerant processors at the current NISQ-era devices. This
would resemble the integrated circuits (IC) technology of silicon-based
processors characterized by Moore’s law Moo69 . With the transition/processing
limits of the quantum satellites Dir21 ; Yua21 ; Nit22 ; Min22  and mobile
edge devices, communication via a single photon is desired. With large
communication channel bandwidth, emitted photons from the gate could be
encoded into three-photons GHZ state Ash85 ; Yos18  to correct the bit-flip,
while the photon loss could be neglected by post-selection after the PBM. From
a fundamental perspective, the presented technique of entangling multiple
physical qubits in the stabilizer basis facilitates the investigation of
phenomena, properties, and protocols that arise in quantum information over a
wide dimension of state space, while operating on the basis that is growing
polynomially.

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## Supplemental document for ”Universal terminal for mobile edge-quantum
computing”

## S1.Rydberg Fermi interaction

This section discusses, the steps in calculating the potential energy curves
(PEC) under the Rydberg-Fermi interaction Fer ; Eil17 ; Eil19

$V_{\text{RF}}=(2\pi\frac{\tan(\delta^{s})}{k(R)}-6\pi\frac{\tan(\delta^{p})}{k^{3}(R)}\stackrel{{\scriptstyle\leftarrow}}{{\nabla}}_{{\bf
r}}.\stackrel{{\scriptstyle\rightarrow}}{{\nabla}}_{{\bf r}})\delta({\bf
r}-{\bf R}).$ (8)

The coupling of the Rydberg state $|45D_{5/2},5/2\rangle$ with the
neighbouring states $|43H+45D_{3/2}+46P_{\\{1/2,3/2\\}}+47S_{1/2}\rangle$ is
considered at the position of neighbouring lattice sites $R$. Here
${|{nH}\rangle}=\sum_{l,m}{|{n,l,m}\rangle}$ represents the Hydrogen state
encountering semi degenerate orbital angular momentum numbers $2<l<n$. The
matrix elements in the manifold of coupled states are given by

$\displaystyle H_{nlm,n^{\prime}l^{\prime}m^{\prime}}({\bf
R})={\langle{\psi_{nlm}({\bf
R})}|}V_{\text{RF}}{|{\psi_{n^{\prime}l^{\prime}m^{\prime}}({\bf
R})}\rangle}\quad\quad\quad\quad$ (9) $\displaystyle
H_{nlm,nlm}=-\frac{Ry}{n^{*2}}$

where $Ry$ is the Rydberg constant of $Cs$ atoms and $n^{*}$ is the effective
Rydberg principal number. Diagonalizing 8000 coupled states, the energy
potential is plotted in Fig. 5. In the UV optical-lattice with
$\lambda=350$nm, the Fermi scattering of Rydberg electron from the
neighbouring lattice site would result in 400MHz level-shift of the Rydberg
level ideal for fast quantum operations.

Figure 5:  PEC with S- and P-wave scattering in Cs atoms. Here the coupling of
the Rydberg state $|45D_{5/2},5/2\rangle$ with the neighbouring states
$|43H+45D_{3/2}+46P_{\\{1/2,3/2\\}}+47S_{1/2}\rangle$ is considered under Eq.
8. Interaction strength is plotted across radial direction with
$\theta=\pi/2$.

For the p-wave scattering of Rydberg electron from the neighbouring ground
state atom, the gradient of the Rydberg wave-function
$\psi=R_{nl}(r)Y_{l}^{m}(\theta,\phi)$ at the position of the neighbouring
lattice site is required which is

$\nabla\psi(r,\theta,\phi)=\begin{bmatrix}\frac{\partial R_{nl}}{\partial
r}Y_{l}^{m}\\\ \frac{1}{r}R_{nl}\frac{\partial Y_{l}^{m}}{\partial\theta}\\\
\frac{1}{r\sin\theta}R_{nl}\frac{\partial Y_{l}^{m}}{\partial\phi}\\\
\end{bmatrix}=\quad\quad\quad\quad\quad\,\quad~{}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(10) $\begin{bmatrix}\frac{\partial R_{nl}(r)}{\partial
r}Y_{l}^{m}(\theta,\phi)\\\
\frac{1}{r}R_{nl}(r)\frac{1}{2}\sqrt{l^{2}-m^{2}}[Y_{l}^{m+1}(\theta,\phi)e^{-\text{i}\phi}-(l+m+1)Y_{l}^{m-1}(\theta,\phi)e^{\text{i}\phi}]\\\
\text{i}m\frac{\psi(r,\theta,\phi)}{r\sin(\theta)}\\\ \end{bmatrix}$

in the spherical coordinate. The radial wave-function and its derivative are
calculated numerically using Numerov technique Gal05 .

## S2. Entanglement swapping

Entangling far separated atomic qubits in the logical basis requires Bell
state projective measurement of the cavity emitted photons. The proposed
Rydberg-Fermi cQED gate in this article would provide atom-photon entanglement
in each cavity setup, see top line of Eq. 11. The state could then rearranged
in a separate logical and photonic qubit pairs in the following line

$\displaystyle{|{\psi}\rangle}=\frac{({|{1_{L}}\rangle}{|{\text{late}}\rangle}+{|{0_{L}}\rangle}{|{\text{early}}\rangle})_{1}}{\sqrt{2}}\frac{({|{1_{L}}\rangle}{|{\text{late}}\rangle}+{|{0_{L}}\rangle}{|{\text{early}}\rangle})_{2}}{\sqrt{2}}$
(11)
$\displaystyle=(|{\tilde{\phi}}^{+}_{L}\rangle|\tilde{\phi}^{+}_{p}\rangle+|{\tilde{\phi}}^{-}_{L}\rangle|\tilde{\phi}^{-}_{p}\rangle+|{\tilde{\psi}}^{+}_{L}\rangle|\tilde{\psi}^{+}_{p}\rangle+|{\tilde{\psi}}^{-}_{L}\rangle|\tilde{\psi}^{-}_{p}\rangle)/2,$

where the rotated Bell states are obtained by applying a Hadamard on the
second element i.e.
$|\tilde{\phi}^{\pm}\rangle={|{1}\rangle}_{1}{|{+}\rangle}_{2}\pm{|{0}\rangle}_{1}{|{-}\rangle}_{2}$
and
$|\tilde{\psi}^{\pm}\rangle={|{0}\rangle}_{1}{|{+}\rangle}_{2}\pm{|{1}\rangle}_{1}{|{-}\rangle}_{2}$
where $|\pm\rangle=({|{1}\rangle}\pm|0\rangle)/\sqrt{2}$. The photonic states
follow the same presentation format in the early and late basis. Applying a CZ
gate on the photonic states Lo20  written in the four rotated Bell states
would result in

$\text{CZ}_{p}{|{\psi}\rangle}=(|++\rangle_{p}|\tilde{\phi}^{+}\rangle_{L}-|--\rangle_{p}|\tilde{\phi}^{-}\rangle_{L}-|-+\rangle_{p}|\tilde{\psi}^{+}\rangle_{L}-|+-\rangle_{p}|\tilde{\psi}^{-}\rangle_{L})/2.$
(12)

Subsequent projective measurement of the photonic pair in the Bell basis Val14
guarantees the entanglement of logical qubits.

## Appendix D: Preserving the ground motional state

Over the gate operation, the Rydberg-Fermi potential modifies the optical
trapping experienced by the plaquette atoms in ${|{1_{t}}\rangle}$ Wannier
states. To preserve the ground motional state, it is important to apply the
$\Omega_{s}$ adiabatically. Over the operation, plaquette atoms in
${|{1_{t}}\rangle}$ Wannier state would experience trap evolution
$U_{trap}=U_{op}+P_{r_{c}}(t)V_{RF}$ where $U_{op}$ is the optical trapp,
$P_{r_{c}}=(\frac{\Omega_{s}}{\Omega_{r}})^{2}$ is the Rydberg population of
the central atom. As long as the dynamic is adiabatic, i.e.
$\dot{\omega}_{trap}\ll\omega^{2}_{trap}$ Wit20 , the Wannier states of the
${|{1_{t}}\rangle}$ can adapt continuously and stays close to the
instantaneous ground motional state. For the setup discussed in the main text,
the $\Omega_{s}$ pulses as short as 0.5$\mu$s could be designed to preserve
the adiabaticity. This is significantly shorter than the Gaussian pulse with
$\sigma=3.5\mu$s considered in the main text. Also, note that applying the
dark-state optical-lattice with better site confinement Yav09 , allows faster
adiabatic operations.

## S3: Closing the Rydberg-molecule decoherence channel

In the Bose-Einstein condensate (BEC), the Fermi scattering of Rydberg
electron from a free ground state atom could enhance the decoherence rate
Bal13 . This decoherence is due to attractive Rydberg-Fermi potential close to
the ionic core, which moves the two interacting atoms to a very small
separation of about 2 nm, where the binding energy of the molecules can ionize
the Rydberg electron and form a $Cs_{2}^{+}$ molecule Nie15 . Confining atoms
in the lattice, the interatomic separation in this scheme is tuned to be at
the last lobe of Ryd-Fermi interaction, preserving the interatomic distance.
Without the mass transport, step-wise decay or ionization of the Rydberg atom
is ruled out. This is because the Rydberg-Fermi binding energy at 170nm
lattice constant used in this paper is orders of magnitude smaller than the
closest Rydberg levels for the principal numbers applied here. The ion-pair
formation is also highly unlikely in this system Nie15 .

##  VIII A. 3: Addressing individual sites

Considering the small lattice constant, single site addressing would be
challenging. There are two approaches to avoid laser cross talk. In the first
approach, the sub-wavelength localized population rotation via semi
interferometer techniques Aga06 ; Cho07  could be used to improve single site
addressing.

Figure 6:  Dual-species spin-dependent lattice. (a) The 870nm trapping laser
perpendicular to the lattice plane is aligned between $6P_{3/2}$ and
$6P_{1/2}$ for Cs, hence the polarizability of qubit states $|0\rangle$ and
$|1\rangle$ are given by distinguished circularly polarized lights $\pi^{-}$
and $\pi^{+}$ respectively. (b) the same polarization elements traps opposite
qubit states in Rb. (c,d) For in-plane trapping the 2D standing wave is formed
by 820nm linearly polarized light trapping both Cs and Rb at the nodes and
anti-nodes of the standing wave via blue and red detuned transitions. The
resulting qubit-dependent arrangement is depicted in (e) where the green and
red ovals present the position of $|0\rangle$ and $|1\rangle$ qubit states
respectively. (f) PEC with S- and P-wave scattering of the Rb atom’s Rydberg
electron from a ground state Cs atom. Here the coupling of the Rydberg state
$|48D_{5/2},5/2\rangle$ with the neighbouring states
$|46H+48D_{3/2}+49P_{\\{1/2,3/2\\}}+50S_{1/2}\rangle$ is considered under Eq.
8. Interaction strength is plotted across radial direction with $\theta=\pi/2$
being perpendicular to the lattice plane.

In the second approach, a dual species optical lattice of 133Cs and 87Rb is
considered Sin22 ; She21 ; Sin21 ; LHZ15 . The trapping in the lattice plane
is formed by 820nm linearly polarized light that is blue and red detuned for
133Cs and 87Rb and hence they would be stored on the nodes and antinodes of
the standing wave respectively, see Fig. 6c,d. The qubit-dependent trapping
perpendicular to the lattice plane is formed by a 870nm standing waves of left
and right circularly polarized lights as depicted in Fig. 2b of the main text.
The ${|{0}\rangle}$ qubit state of Rb (Cs) would be trapped on the antinodes
of $\pi^{-}$ ($\pi^{+}$) standing wave and ${|{1}\rangle}$ gets trapped on the
opposite polarization lattice, see Fig. 6a,b,e. The physical qubits are
encoded in the Cs atoms placed on the plaquette while the central auxiliary
atoms are Rb. This arrangement optimizes the interaction since the electron
scattering from Cs atoms features resonance at lower kinetic energies close to
last lobe Boo15 . The corresponding Rydberg-Fermi interaction over a single Cs
site would be ${\langle{0}|}V_{RF}{|{0}\rangle}/2\pi=50$MHz. Considering the
adjusted $\Omega_{s}=$428nm laser for the new Rb central atom, focusing the
laser beam to a single site with NA=0.8 microscope Bak09 , the probability of
photon emission from the neighboring Rb atoms after an $\Omega_{s}$ pulse
would be 1%. The Rydberg-Fermi interaction PEC of $|48D_{5/2},5/2\rangle$
Rydberg state targeting the new inter-atomic separation of 205nm is plotted in
Fig. 6f.

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