# Electron cloud design for Rydberg multi-qubit gates

Mohammadsadegh Khazali  Institute for Quantum Optics and Quantum Information
of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria  Wolfgang
Lechner  Institute for Theoretical Physics, University of Innsbruck, A-6020
Innsbruck, Austria  Parity Quantum Computing GmbH, A-6020 Innsbruck, Austria

###### Abstract

We propose a quantum information processing scheme in an optical lattice,
using the Rydberg electron’s Fermi scattering from ground-state atoms in a
spin-dependent lattice as a source of interaction. Instead of relying on
Rydberg pair potentials, the interaction is controlled by engineering the
electron cloud of a sole Rydberg atom. The absence of multiple excited Rydberg
atoms mitigates different competing infidelity criteria, eliminates unwanted
cross-talks, and allows operations in dense atomic lattices. Here we
specifically propose the direct implementation of two prominent multi-qubit
gates i.e. the stabilizer-phase operator and the Toffoli gate. The features in
the new scheme are of special interest for the implementation of quantum
optimization and error correction algorithms.

The recent experimental progress in development of highly controllable quantum
systems using Rydberg atoms enabled a parameter regime where fully functional
quantum information processing algorithms are in reach. While individual
control of atoms and their interactions has been achieved, a next step is the
development of complex interactions between multiple quantum bits (qubits)
which are of great interest for applications ranging from quantum optimization
to error correcting codes.

A particular application of near term quantum devices is solving optimization
problems using a gate model approach. The parity architecture LHZ15 ; End21 ,
translates a problem with all to all connectivity to a simple nearest-neighbor
problem-independent interaction. Hence, the quantum approximate optimization
algorithm (QAOA) Far14 ; Far16  implementation would be simplified to
programming single-qubit operations, as well as applying problem independent
four-body stabilizer-phase gate

$U_{\square}(\theta)=e^{i\theta\prod_{i\in p}\sigma_{z}^{(i)}}$ (1)

over the atoms on a plaquete. The other prominent multi-qubit gate is Toffoli,
playing a pivotal role in quantum error correction Cor98 ; Sch11 , fault
tolerant QC Den01 ; Pae13  and Shore’s algorithm Sho95 .

The realization of multi-qubit gates with the concatenation of one- and two-
qubit gates Mot04 ; Lec20  results in a significant overhead Mas03 ; She09 .
The long-range many-body Rydberg interaction is vastly used in different
quantum operations Saf10 ; Ada19 ; Jak00 ; Bro20 ; Ise10 ; Wil10 ; Gra19 ;
Kha19 ; Kha15 ; Kha17 ; Khaz21 ; Khaz16 ; Gr017 ; khaz2020rydberg ; khaz2021 ;
Kha16  including direct operation of multi-qubit gates Lev19 ; Kha20 ; Ise11 ;
Gla17 . The fidelity of these direct Rydberg multi-qubit gates are limited by
the competing requirement of the presence (absence) of inter- (intra-)
component interaction. In other words, interaction in dipolar-Rydberg schemes
requires Rydberg population in all the interacting qubits, and thus
unavoidable intra-component cross talks, reducing the operation fidelity.

Figure 1: Ryd-Fermi interaction in Spin-dependent lattice. (a) The atoms in
$|0\rangle$ and $|1\rangle$ qubit states are trapped in lattices shifted along
$z$, distinguished by red and blue circles. Depending on whether the central
atom is excited from $|0\rangle$ or $|1\rangle$ state, Rydberg electron’s last
lob would mainly scatter from the plaquet atoms in $|0\rangle$ or $|1\rangle$
spin-state. (b) Tuning the trapping laser between $5P_{3/2}$ and $5P_{1/2}$,
the polarizability of qubit states $|0\rangle$ and $|1\rangle$ are given by
distinguished circularly polarized lights $\varepsilon_{-}$ and
$\varepsilon_{+}$ respectively. (c) The same fields, trap the
$|p\rangle=|5P_{1/2},1/2\rangle$ electronic state with $\varepsilon_{0}$ and
$\varepsilon_{+}$ elements. (d) Counter propagating linearly polarized lights
with relative polarization of $2\theta$, forms two distinguished optical-
lattice of $\varepsilon_{-}$ and $\varepsilon_{+}$ displaced by $D=2\theta/k$.
(e) The Raman transition in a dual spin/spatial encoded qubit would be
modified by the Frank-Condon factor.

In this article we introduce a versatile toolbox to engineer multi-qubit
operations by designing the Rydberg electronic cloud with respect to the
lattice geometry. A spin-dependent lattice Bri00 ; Dut98 ; Man03 ; Kum18  is
used to harvest the Fermi-scattering of the Rydberg electron Gre00 ; Gaj14
from neighboring lattice sites as a source of coherent interaction for the
desired gates. The spin-dependent geometrical shift of atoms accommodates them
inside or outside of the Rydberg electron’s wave-function, providing a spin-
dependent nearest-neighbor interaction appropriate for gate operations. This
double-encoding of qubits in internal and external degrees of freedom
distinguishes the gate from previous proposals without spatial encoding of the
qubit. At very short inter-atomic distances below 50nm, the Rydberg-Fermi
interaction could be used for spin-dependent operation Nie16 . However such
short lattice constants are not realizable in the experiment. The dual
spin/spatial encoding proposed in this article provides strong qubit dependent
interaction within the present optical lattices.

##### Comparing Rydberg-Fermi with Rydberg-Dipolar schemes –

In Rydberg-dipolar schemes the interaction between two or multiple atoms
requires Rydberg population in every single atom. This results in competing
requirements i.e. to preserve the lattice trapping against strong dipolar
interaction, to overcome or preserve the blockade, and to not exciting the
neighboring Rydberg levels Saf05 ; Ise11 ; Zha12 ; The16 ; Kha20 . With our
new approach, interactions of two or multiple atoms are carried out by
exciting a single atom to the Rydberg state. The system thus operates at a
different regime of energy hierarchy without the mentioned rivalry in Rydberg-
dipolar systems. In terms of scalability, this allows significant compressing
of the quantum processor (i.e. the atomic lattice) size with inter-atomic
distances equivalent to half Leroy radius. From the operation side,
eliminating the competing population rotation errors mentioned above and
limiting the Rydberg decaying population to a single atom makes the Rydberg-
Fermi scheme appealing for single-step multi-qubit operations. Furthermore,
the absence of intra-component interaction eliminates the unwanted phase
errors in multi-qubit gates.

##  I Results

###  I.1 Scheme

In a two dimensional lattice shown in Fig. 1a, atoms in spin-states
${|{0}\rangle}$, ${|{1}\rangle}$ are trapped in shifted lattices distinguished
by red and blue. The gate operations are carried out by exciting the centred
atom in a plaquette to the Rydberg level. Depending on whether the central
atom is excited from ${|{0}\rangle}$ or ${|{1}\rangle}$ state, the Rydberg
electron’s last lob would mainly scatter from the plaquette atoms in
${|{0}\rangle}$ or ${|{1}\rangle}$ spin-lattice, respectively. This effect
provides the desired qubit dependent interaction.

The spin-dependent lattice along $z$ direction is formed by counter-
propagating linearly polarized light as depicted in Fig. 1d. Introducing a
relative shift between the fields’ polarizations of $2\theta$, the total
electric field can be written in terms of the sum of right and left circularly
polarized lights $E=E_{0}\exp(-i\nu
t)(\varepsilon_{+}\sin(kz+\theta)+\varepsilon_{-}\sin(kz-\theta))$. 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.
1b, the trap 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
$V_{\pm}=\alpha|E_{0}|^{2}\sin(kz\pm\theta)$. The hyperfine qubit states
$|0\rangle=|F=1,m_{F}=1\rangle$ and $|1\rangle=|F=2,m_{F}=2\rangle$ experience
$V_{0}=(V_{+}+3V_{-})/4$ and $V_{{|{1}\rangle}}=V_{+}$.

##### Qubit-rotation in the spin-dependent lattice –

Having spatially shifted spin-dependent lattices, provides dual encoding of
the qubit. A Raman transition coherently transfer atoms from one internal
state to the other, thereby causing hopping between the two Wannier-functions
Jak98 ; Jak03 ; Maz10 , see Fig. 1e. The polarizability of intermediate
electronic-level ${|{5P_{1/2}1/2}\rangle}$ would be given by the
$\varepsilon_{\\{0,+\\}}$. Hence by tuning $\theta$ it is possible to confine
the intermediate state between the two qubit-dependent lattices. The effective
Raman Rabi frequency in the spin dependent lattice would be
$\tilde{\Omega}=\sum_{n}f_{0n}\Omega_{0}f_{n1}\Omega_{1}/\Delta_{n}$ with the
Frank-Condon factors $f_{in}=\int w_{i}^{*}({\bf x}-{\bf l_{i}})w_{p_{n}}({\bf
x}-{\bf l_{p}})\text{d}{\bf x}$ quantifying the overlapping of the qubit
$i={|{0,1}\rangle}_{n=0}$ and intermediate ${|{p}\rangle}_{n}$ state’s Wannier
function $w_{p_{n}}$ in the $n$th motional state (see App. A for the
derivation). The centre of the qubit-dependent Wannier-functions $l_{j}$s are
separated by $l_{1}-l_{0}=D$. Considering the lifetime of the intermediate
state, the coherent Raman transition would effectively result in
$\Delta\gg\omega_{tr}$ and hence $\tilde{\Omega}\approx\Omega_{Ra}F$ with
effective Frank-Condon factor of $F=\sum_{n}f_{0n}f_{n1}$. The qubit-rotation
is performed in the regime of $\tilde{\Omega}\ll\omega_{\text{tr}}$ to avoid
exciting the motional Bloch bands.

##### Ryd-Fermi Interaction –

The interaction between the Rydberg electron and the ground state atom is a
Fermi type pseudo potential Fer ; Eil17 ; Eil19 ,

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

with r and R being the positions of the Rydberg electron and the 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 the neighboring
ground state atom khu02 . The electron wave-vector $k(R)$ is defined by the
kinetic energy of the Rydberg electron at energy $E=-1/2n^{2}$ when it
collides with a ground state atom, i.e. $k^{2}(R)/2=E+1/R$. Having limited
number of ground state atoms $i$ in the Rydberg electron orbit, the
interaction would add up to $iV_{\text{RF}}$ Gaj14 .

Exciting $|nD_{5/2},5/2\rangle$ Rydberg state would confine electron wave-
function close to the 2D lattice plane comparing to the isotropic
${|{nS}\rangle}$ orbital. Here the quantization axis (defined by the
propagation direction of polarized exciting lasers) is perpendicular to the
atomic lattice plane. This would preserve the symmetry of Rydberg wave-
function experienced by the neighboring qubits.

Further confinement can be obtained by exciting superposition of Rydberg
levels. The spatial constructive (destructive) interference of Rydberg wave-
functions over the position of plaquet qubits (elsewhere) could further
confine the electron and hence enhances the interaction. The desired Rydberg
superposition could be controlled by the polarization angles
$\theta_{\\{R,B\\}}$ of the two linearly polarized lights $\Omega_{\\{R,B\\}}$
used for Rydberg excitations, see Fig. 2. These fields are propagating along
the $z$ direction. The linear polarized light could be expressed in terms of
circular polarizations
$\Omega_{j}=(\exp(\text{i}\theta_{j})\Omega_{j}^{+}+\exp(-\text{i}\theta_{j})\Omega_{j}^{-})/\sqrt{2}$.
Adjusting $\theta_{R}-\theta_{B}=\pi/2$ the transition to
${|{nD_{j},m_{j}=1/2}\rangle}$ would be canceled by destructive interference.
Hence the excited state would be
$\exp(\text{i}(\theta_{R}+\theta_{B})){|{nD_{5/2},5/2}\rangle}/\sqrt{2}+\exp(-\text{i}(\theta_{R}+\theta_{B}))({|{nD_{5/2},-3/2}\rangle}+{|{nD_{3/2},-3/2}\rangle})/2$.
The closely separated states
$E_{{|{66D_{3/2}}\rangle}}-E_{{|{66D_{5/2}}\rangle}}=40$kHz are not
distinguishable by strong lasers used for deterministic excitation of Rydberg
control atom, hence both states would get excited. The polarization angles
would act as a controlling parameter to rotate the interaction maxima e.g.
$\theta_{R}+\theta_{B}=\pi,0$ in Fig. 2c,d. Here for the sake of presentation
only the s-wave scattering part of $V_{RF}$ is plotted, which is the dominant
term at the desired last lob in Rb atoms. Further discussions can be found in
App. B. The exciting laser’s polarization and propagation direction could act
as a controlling knob to program different interaction connectivities among
neighboring lattice sites Clover .

Figure 2: Superposition of Rydberg states. (a) The desired Rydberg
superposition could be controlled by the polarization angles $\theta_{R,B}$ of
the two linearly polarized fields $\Omega_{R,B}$ propagating along the $z$
direction. (b) The dashed-line transitions would experience destructive
interference when $\theta_{R}-\theta_{B}=\pi/2$, living the superposition of
$(\text{e}^{\text{i}(\theta_{R}+\theta_{B})}|m_{j}=5/2\rangle+\text{e}^{-\text{i}(\theta_{R}+\theta_{B})}|m_{j}=-3/2\rangle)/\sqrt{2}$
Rydberg states. (c,d) The Ryd-Fermi s-wave scattering profile upon the
presence of ground state atom at each point of the space. The excited Rydberg
superposition state would be further confined around the position of plaquete
atoms and hence enhances the interaction. Data at the centre is cropped out.
The polarization angles would act as a controlling parameter to rotate the
interaction maxima e.g. $\theta_{R}+\theta_{B}=\pi,0$ in (c,d). This would be
a further controlling knob to program different interaction connectivities.

Another approach in enhancing the interaction to loss ratio is harnessing the
p-wave near resonance scattering from Cs atoms, available at smaller electron
kinetic energies at larger interatomic distances comparing to the resonance
position in Rb Boo15 . Fig. 3 plots the potential energy curves (PEC) of
$|46D_{5/2},5/2\rangle+|5S\rangle$ coupled with the neighbouring Rydberg
states $|43H+42H+47P+48S\rangle+|5S\rangle$ under Ryd-Fermi interaction of Eq.
2. The hydrogen manifolds are shown by ${|{nH}\rangle}$ containing $2<l<n$.
The maximum of the p-wave interaction are at the anti-nodes of the Rydberg
electron wave-function.

Figure 3:  PEC with S- and P-wave scattering in Cs atoms. Here the coupling of
the Rydberg state $|46D_{5/2},5/2\rangle$ with the neighbouring states
$|43H+42H+47P+48S\rangle$ is considered under Eq. 2. Interaction strength is
plotted (a) across radial direction with $\theta=\pi/2$ and (b) in $\rho-z$
dimension in cylindrical coordinate where $z$ is along the quantization axis.
Below the third node, the interaction is cropped out. (a) The solid-line and
dashed-line shows the total and exclusive s-wave scattering respectively. The
black and white circles in (b) depicts the confining area of the $|0\rangle$
and $|1\rangle$ qubit states of a plaquete lattice site, while the Cs+ core is
at the origin.

###  I.2 Implementation of multi-qubit gates

This chapter proposes the implementation of multi-qubit gates in both weak and
strong regime of Rydberg-Fermi interaction discussed in Rb and Cs
respectively.

####  I.2.1 Parallelized gate

The weak Ryd-Fermi interaction in Rb atomic lattice could be used for the
implementation of the fan-out gate. The parallelized gate operator

$U_{g}=|0\rangle_{c}\langle 0|\otimes\text{II}+|1\rangle_{c}\langle
1|\otimes\prod\limits_{i=1}^{4}\sigma^{i}_{x}$ (3)

is an essential element in realizing the stabilizer-phase gates Wei10 , see
Methods for detailed discussion. The presence of the Rydberg control atom at
the centre of plaquete would form a conditional detuning over the Raman
transition of surrounding target qubits in Fig. 1a. In a given plaquete, the
qubit-rotation of target atoms under the Fermi scattering of control atom’s
Rydberg electron $V_{\text{RF}}$ would be governed by

$\displaystyle H=\sum\limits_{l\in
p}[\tilde{\Omega}(a^{\dagger}_{l_{0}}a_{l_{1}}+\text{h.c.})+\delta
a^{\dagger}_{l_{1}}a_{l_{1}}+V_{\text{RF}}a^{\dagger}_{l_{1}}a_{l_{1}}a^{\dagger}_{c_{1}}a_{c_{1}}]\quad\,$
(4)

Here $a^{(\dagger)}_{l_{i}}$ annihilates (creates) the Wannier-state of $l$th
target qubit in the plaquete, centred at $l_{i}$ with $i\in\\{0,1\\}$ defining
the spin-dependent trap. The Raman lasers, detuned by $\delta$ would only get
in resonance in the presence of control Rydberg atom excited from the
population in the $|1_{c}\rangle$ qubit dependent Wannier state. Fig. 4a
presents the qubit rotation probability $P_{\text{rot}}$ of target atoms as a
function of $\frac{\delta+V_{\text{RF}}}{\tilde{\Omega}}$. The operation steps
are as following: 1- We initially apply a $\pi$ pulse on $|1\rangle_{c}\langle
r|$ transition. 2- Then we apply a Raman $\pi$ pulse on the target qubits
$|1\rangle_{t}\langle 0|$ detuned by
$\delta=(\omega_{L0}-\omega_{L1})-(\omega_{1}-\omega_{0})=-V_{\text{RF}}$. 3-
Return back the Rydberg control atom to the ground state.

##### Case study:

The 420nm lattice wavelength is considered dressing ground state $Rb$ to
${|{6P}\rangle}$ state for trapping along the $x-y$ plane lattice. To
accommodate the plaquete atoms within the last lob of central Rydberg atom,
every other atom in the trap should be removed. This arrangement helps in
further confinement of the ground motional state in the short wavelength
lattice TrapK , while benefiting from wider Rydberg lobs over larger inter-
atomic distances. In the $x-y$ plane the trap potential of $U/2\pi=10$MHz,
confines atoms to the half width at 1/e maximum of $\sigma_{x,y}=12.5$nm.
Further confinement of atoms is possible by sub-wavelength trapping inspired
by Yan18 ; Aga06 . To achieve a spin-dependent trap perpendicular to the plane
(along the $z$) the 800nm laser could be used for dressing ${|{5S}\rangle}$ to
${|{5P}\rangle}$ with the level scheme of Fig. 1b, with $U/2\pi=10$MHz,
$\sigma_{z}=17$nm. We consider a spin-dependent trap separation of
$D=150(200)$nm over which the ${|{p}\rangle}$ trap would be confined between
the qubit trapps, see Fig. 1e. Considering the $U_{tr}/\omega_{tr}=9(14)$
motional states of intermediate trap, the effective Frank-Condon factor would
be $F=0.5(0.25)$ FrankCondon . In this system, exciting the central atom
(control qubit) to $|46D_{5/2},5/2\rangle$, the scattering energy of Rydberg
electron over the spin-dependent Wannier state of a neighboring target atom
would averaged to ${\langle{1}|}V_{RF}{|{1}\rangle}=2$MHz and
${\langle{0}|}V_{RF}{|{0}\rangle}=0.5(0.05)$MHz for D=150(200)nm.

##### Gate fidelity:

The main sources of errors in quantifying the C-NOT4 gate’s operation are
spontaneous emission of the Rydberg level, and population rotation errors.
Here we discuss the errors averaged over qubit configurations. The average
spontaneous emission error is given by
$E_{sp}=\frac{1}{2}(\frac{2\pi}{2\Omega_{Ry}}+\frac{2\pi}{\tilde{\Omega}})\Gamma_{Ry}$,
where the Rydberg depopulation rates $\Gamma_{Ry}$ can be found in Bet09 . The
population rotation blocking leakage error of each target atom would be
quantified by off-resonant transition
$\frac{\tilde{\Omega}^{2}}{4\delta^{2}}$. The control atom’s rotation is due
to the unwanted excitation of neighboring accessible Rydberg levels with
$\delta_{r}$ energy separation summing up to an average error of
$1/2\frac{\Omega_{Ry}^{2}}{4\delta_{r}^{2}}$. Also non-deterministic
excitation of Rydberg atom due to qubit-dependent level-shift caused by
Rydberg-Fermi interaction should be overcome by strong exciting laser tuned to
the middle of the spectrum leading to average error of
$1/2^{5}\sum\limits_{j=0}^{4}\left(\begin{array}[]{c}4\\\
j\end{array}\right)\frac{(j-2)^{2}V_{RF}^{2}}{\Omega_{Ry}^{2}}$. The optimum
Rabi frequency of $\tilde{\Omega}=370(500)$kHz, $\Omega_{Ry}/2\pi=10$MHz and
$\delta_{r}=25$GHz, $V_{RF}=1.3(2)$MHz results in the average gate fidelity of
$F=0.94(0.96)$. Using circular state would enhance the fidelity by reducing
the spontaneous emission and providing a stronger interaction allowing faster
operations, see Supp. C.

####  I.2.2 Toffoli gate

The strong resonant Ryd-Fermi interaction in Cs atomic lattice could ideally
be used for the operation of Toffoli gate Ck-NOT with $k=4,6$ in square and
triangular lattices. The gate would operate with a single $2\pi$ rotation of
the central target atom between ${|{1}\rangle}-{|{r}\rangle}$ states,
conditioned on the value of Ryd-Fermi level-shift determined by control
plaquete atoms. The operation Hamiltonian would be

$\displaystyle H_{\text{Tof}}=$
$\displaystyle\Omega_{\text{Ry}}(a^{\dagger}_{{|{r_{t}}\rangle}}a_{{|{1_{t}}\rangle}}+\text{h.c.})+\Delta
a^{\dagger}_{{|{r_{t}}\rangle}}a_{{|{r_{t}}\rangle}}$
$\displaystyle+V_{\text{RF}}a^{\dagger}_{{|{r_{t}}\rangle}}a_{{|{r_{t}}\rangle}}\sum\limits_{i\in
p}a^{\dagger}_{{|{1_{ci}}\rangle}}a_{{|{1_{ci}}\rangle}}$

where $\Omega_{\text{Ry}}$ and $\Delta$ are the Rabi frequency and detuning of
the Rydberg exciting laser acting on the target qubit. The last term would sum
over the control qubits around the centred target atoms and apply qubit
dependent Rydberg-Fermi interaction. Adjusting the laser to
$\Omega_{\text{Ry}}\ll V_{\text{RF}}$ and $\Delta=-4V_{\text{RF}}$, the $2\pi$
rotation of the target atom would be conditioned on the presence of
${|{1_{c}1_{c}1_{c}1_{c}1_{t}}\rangle}$ resulting to C4-Z operation.
Sandwiching the target atom with Hadamard gates results in the desired Toffoli
operation. The Rydberg electron’s wave-packet dynamics are in the ps range.
Therefore, over the MHz population-rotation range of target atom, the presence
of $iV_{RF}$ level-shifts conditioned on the number of control atoms $i$ in
spin ${|{1}\rangle}$ are guaranteed. The Rydberg-Fermi interaction allows
Toffoli and C-Z gate operations with a single 2$\pi$ rotation, leaving no
Rydberg population unprotected from the laser. This would significantly reduce
the experimentally demanding conditions in reducing laser noise Lev18  for
schemes with $\pi$-Pause-$\pi$ Rydberg exciting pulses.

Case study: The in-plane $x-y$ Cs trap is formed by dressing ground state to
${|{10P}\rangle}$ states with 350nm UV laser. Applying strong trap potential
of $U_{trap}/2\pi=20$MHz with Wannier function half width at 1/e maximum
$\sigma_{x,y}=8.7$nm, accommodates the atom within single lob of $V_{RF}$, see
Fig. 3. For spin-dependent trap perpendicular to the plane (along the $z$) the
450nm laser could be used for dressing ${|{6S}\rangle}$ to ${|{7P}\rangle}$
with the level scheme of Fig. 1b,c, with $U/2\pi=2$MHz, $\sigma_{z}=17$nm,
$D_{z}=150$nm. In this system, exciting the central atom (target qubit) to
$|46D_{5/2},5/2\rangle$, the average spin-dependent interaction would be
${\langle{1}|}V_{RF}{|{1}\rangle}=354$MHz and
${\langle{0}|}V_{RF}{|{0}\rangle}=10$kHz. In the presence of the central
Rydberg atom, the confinement of ${|{1}\rangle}$ state in the last lob of
$V_{RF}$ potential is considered. This contrast of spin-dependent interaction
allows fast Rydberg excitation of $\Omega_{\text{Ry}}=35$MHz in Toffoli gate.

The main sources of errors in quantifying the C4-NOT gate’s operation are
spontaneous emission of the Rydberg level, and population rotation errors. The
average spontaneous emission error is given by
$E_{sp}=\frac{2\pi}{4\Omega_{Ry}}\Gamma_{Ry}$. Off-resonant Rydberg excitation
results to blockade leakage adding up to the average error of
$E_{r1}=\frac{1}{2^{5}}\sum\limits_{j=1}^{4}\left(\begin{array}[]{c}4\\\
j\end{array}\right)\frac{\Omega_{Ry}^{2}}{4j^{2}V_{RF}^{2}}$, where $j$ is the
number of control atoms in ${|{1}\rangle}$ qubit-dependent lattice. The other
rotation error comes from populating the neighboring accessible Rydberg
states. For ${|{46D_{5/2},5/2}\rangle}$ target state, the first accessible
state in the red (blue) side is off resonant by 83GHz (77GHz) Neighbour ,
which is far from the range of the $V_{RF}$ and $\Omega_{Ry}$, leading to
negligible error. Hence, the effective fidelity of 99.87% is expected from the
gate operation.

####  I.2.3 Direct implementation of stabilizer-phase gate

With minor laser adjustment, the above mentioned setup of Toffoli gate could
be used for direct implementation of the stabilizer-phase gate. Here the
central atom is an auxiliary element. By tunning the Rydberg exciting laser
close to the energy level of the odd spins i.e. $\Delta\approx-
V_{RF},-3V_{RF}$, the $2\theta$ phase from the Rydberg excitation would be
conditioned on the odd parity of plaquete spin. We then apply a $-\theta$
Stark shift phase on the central atom. The central qubit returns to its
initial ground state after the complete operation. Therefore, effectively
factors out from the dynamics of the spin system leaving the desired operation
of Eq. 1. The exciting phase $\theta$ is controlled by
$\frac{\delta^{\prime}}{\Omega_{Ry}}$ where $\delta^{\prime}$ is the laser
frequency offset from the targeted energy
$\delta^{\prime}=\\{\Delta+V_{RF},\Delta+3V_{RF}\\}$, see Fig. 4b.

Figure 4:  Gate operations (a) The spin-rotation probability of plaquete atoms
in parallelized gate, as a function of effective laser detuning
$\delta+V_{RF}$ under the dynamics of Eq. 4. (b) Controlling the stabilizer
phase $\theta$ by tuning the frequency of Rydberg exciting laser acting on
auxiliary atom.

###  I.3 Decoherence

Rydberg-Fermi scattering could enhance the decoherence rate in the Bose
Einstein condensate (BEC) Bal13 . This decoherence only occurs when the
attractive Rydberg-Fermi potential, moves the two interacting atoms to a very
small separation (of order 2 nm), where the binding energy of the molecules
can ionize the Rydberg electron and form a $Rb_{2}^{+}$ molecule Nie15 .
Without the mass transport, step-wise decay or ionization of the Rydberg atom
is ruled out by the quantization of Rydberg state ( as has been discussed and
experimentally tested in Bal13 ). This is because the small molecular binding
energy reported here is orders of magnitude smaller than the closest Rydberg
levels for the range of principal numbers applied here. The occurrence of ion-
pair formation is also highly unlikely in this system Nie15 . We argue that
confining the atoms by an optical-lattice can prevent the described mass
transport and completely close the Rydberg molecule loss channels.

##  II Discussion

This article proposes a highly controllable multi-qubit operations, based on
engineering the electronic cloud of a single Rydberg atom with respect to the
atomic lattice. Restricting the Rydberg population to a single site eliminates
unwanted intra-component interaction, facilitating high-fidelity multi-qubit
gates. Quantifying the effects of this unwanted phase in dipolar schemes, we
argue that the new approach is a significant step in the implementation of
high fidelity single step multi-qubit operations, see App. D.

The Rydberg-Fermi interaction allows gate operations with continuous 2$\pi$
Rydberg rotation, leaving no Rydberg population unprotected from the laser.
This would significantly reduce the experimentally demanding conditions in
reducing laser noise Lev18  comparing to the Rydberg dipolar gates operating
under $\pi$-pause-$\pi$ Rydberg rotation scheme. Another advantage of having a
single Rydberg atom is closing the Anomalous broadening decoherence Zei16 ;
Gol16 .

Direct implementation of multi-qubit gates in this proposal would reduce the
operation steps and hence the accumulative errors. As an example, the
concatenated application of four-body stabilizer-phase gate with Rydberg C-NOT
gates Lec20  requires 41 laser pulses while the Ryd-Fermi direct
implementation only needs three laser pulses. Also in Toffoli, the C6-NOT gate
operation with concatenated Rydberg C-NOT gates Mas03 ; She09  requires 112
pulses, while in the presented scheme the operation requires only three
pulses.

The presented stabilizer operator could be used in quantum simulation,
Topological QC (Toric code), and in the spin ice model Wei10 . Extension of
4-body phase stabilizer gates to 3- and 6-body gates only requires reshaping
the lattice geometry.

###### Acknowledgements.

This work was supported by the Austrian Science Fund (FWF) through a START
grant under Project No. Y1067-N27 and the SFB BeyondC Project No. F7108-N38,
the Hauser-Raspe foundation, and the European Union’s Horizon 2020 research
and innovation program under grant agreement No. 817482. This material is
based upon work supported by the Defense Advanced Research Projects Agency
(DARPA) under Contract No. HR001120C0068. Any opinions, findings and
conclusions or recommendations expressed in this material are those of the
author(s) and do not necessarily reflect the views of DARPA.

##  III Methods

###  III.1 Implementing stabilizer phase-gate using parallelized gate

The implementation of the stabilizer operator $B_{p}=\prod\limits_{i\in
p}\sigma_{z}^{(i)}$ over the plaquete spins applies in three steps i.e.
$B_{p}=H^{-1}U_{g}H$ where
$H=\sum\limits_{i\in\\{p,c\\}}\exp(i\pi/2\sigma^{(i)}_{x})$ is Hadamard
applying over all the plaquete and control atoms and $U_{g}$ is the
parallelized Ryd-Fermi gate of Eq. 3. For the control qubit prepared in
${|{0}\rangle}_{c}$, the gate $B_{p}$ coherently transfers the control qubit
into the state ${|{1}\rangle}_{c}$ (${|{0}\rangle}_{c}$) for the odd (even)
parity of plaquete spin. The $A_{p}=\prod\limits_{i\in p}\sigma_{x}^{(i)}$
stabilizer would be obtained by exclusive application of Hadamard on control
atom. The desired stabilizer-phase gate of Eq. 1, would be implemented by
application of a phase shift on the control qubit, sandwiched by
applying/reverting the stabilizer operator $B_{p}$

$U_{\square}(\theta)=e^{i\theta
B_{p}}=B_{p}^{-1}e^{i\theta\sigma_{z}^{(c)}}B_{p}.$ (6)

At the end, the control atom would be factor out by transferring to the ground
state.

###  III.2 Data availability

All data needed to evaluate the conclusions in the article are presented in
the article and/or the Supplementary Figures. Additional data related to this
paper may be requested from the corresponding author.

###  III.3 Competing interests

The authors declare no competing interests.

###  III.4 Author contribution

The project is defined, derived and written by M. Khazali. The project is
benefited from the scientific advice of W. Lechner.

##  IV Appendix A: Qubit-Rotation in the spin-dependent lattice

Here we analyze the Raman assisted transition of a trapped neutral atom
between the ground vibrational state of two spin-dependent lattices centered
at $l_{0}$ and $l_{1}$. A Raman transition between atom’s internal states
${|{0}\rangle}$ and ${|{1}\rangle}$ makes the atom experience a different trap
shifted by $D$ where the initial and final vibrational wave-functions have
overlap. This scheme resembles the Franck-Condon principle in molecular
physics. Under the Born-Oppenheimer approximation, the electronic and the
nuclear motions are separated and hence the effective wave-function would be
presented as a product of the electronic wave-function and the vibrational
wave-function

$\psi_{l_{i}}({\bf R},{\bf r})=w({\bf R}-{\bf l}_{i})\psi_{e,i}({\bf R},{\bf
r})$ (7)

where ${\bf r}$ and ${\bf R}$ are addressing the electronic and centre of mass
positions. The Wannier function of the $l$th site in the $i\in\\{0,1,p\\}$
spin dependent lattices is given by $w({\bf R}-{\bf l}_{i})$. The electric
dipole transition from a state A to an excited state B is given by

$\langle\psi_{A}|({\bf R}+{\bf r})|\psi_{B}\rangle=\langle w({\bf R}-{\bf
l}_{A})|w({\bf R}-{\bf l}_{B})\rangle\langle\psi_{e,A}|{\bf
r}|\psi_{e,B}\rangle.$ (8)

Here we have the orthogonality of the electronic eigenstates but the
vibrational states are belonging to different traps and do not need to be
orthogonal. Also using the Condon approximation, the dipole transition of
electronic states is assumed independent of nuclear coordinates. In
conclusion, the dipole transition would be modified by the overlap of the wave
functions i.e. Franck-Condon factor $f=\langle w({\bf R}-{\bf l}_{A})|w({\bf
R}-{\bf l}_{B})\rangle$.

##### Effective qubit-rotation rate:

The Hamiltonian $H=H_{0}+H_{d}$ consists of the energy level of electronic
states $H_{0}$, and the dipole transitions $H_{d}$. The Hamiltonian of the
system in the rotating wave approximation and in the basis
$\\{{|{0}\rangle}_{0},{|{1}\rangle}_{0},{|{p}\rangle}_{0},{|{p}\rangle}_{1},{|{p}\rangle}_{2},...,{|{p}\rangle}_{n}\\}$
would be

$\tilde{H}=\begin{bmatrix}0&0&f_{0p_{0}}\Omega_{0}/2&f_{0p_{1}}\Omega_{0}/2&\dots&f_{0p_{n}}\Omega_{0}/2\\\
0&\delta&f_{1p_{0}}\Omega_{1}/2&f_{1p_{1}}\Omega_{1}/2&\dots&f_{1p_{n}}\Omega_{1}/2\\\
f_{0p_{0}}\Omega_{0}/2&f_{1p_{0}}\Omega_{1}/2&\Delta-1/2\omega_{tr}&0&\dots&0\\\
f_{0p_{1}}\Omega_{0}/2&f_{1p_{1}}\Omega_{1}/2&0&\Delta-3/2\omega_{tr}&\dots&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
f_{0p_{n}}\Omega_{0}/2&f_{1p_{n}}\Omega_{1}/2&0&0&\dots&\Delta-(n+1/2)\omega_{tr}\\\
\end{bmatrix}$ (9)

where $\Delta=\omega_{L0}-(\omega_{p}-\omega_{0})$,
$\delta=(\omega_{L0}-\omega_{L1})-(\omega_{1}-\omega_{0})$. The Frank-Condon
factors $f_{in}=\int w_{i}^{*}({\bf x}-{\bf l_{i}})w_{p_{n}}({\bf x}-{\bf
l_{p}})\text{d}{\bf x}$ quantify the overlapping of the qubit states
$i={|{0,1}\rangle}_{n=0}$ and intermediate ${|{p}\rangle}_{n}$ state’s Wannier
function $w_{p_{n}}$ in the $n$th motional state. The Schrödinger equation in
the interaction picture could be written in terms of coupled equations:

$\displaystyle\frac{\text{d}C_{0}}{\text{d}t}=\text{i}\sum\limits_{j=0}^{n}\frac{f_{0j}\Omega_{0}}{2}C_{p_{j}}$
(10)
$\displaystyle\frac{\text{d}C_{1}}{\text{d}t}=\text{i}\sum\limits_{j=0}^{n}\frac{f_{1j}\Omega_{1}}{2}C_{p_{j}}$
(11)
$\displaystyle\frac{\text{d}C_{p_{j}}}{\text{d}t}=\text{i}(\frac{f_{0j}\Omega_{0}}{2}C_{0}+\frac{f_{1j}\Omega_{1}}{2}C_{1}+[\Delta-(j+1/2)\omega_{tr}]C_{p_{j}})$
(12)

Under the condition that $\Delta$ is the dominant term the amplitudes
$C_{p_{j}}$ would oscillate much faster than $C_{0}$ and $C_{1}$. Hence we can
apply adiabatic elimination by substituting

$C_{p_{j}}=-\frac{f_{0j}\Omega_{0}C_{0}+f_{1j}\Omega_{1}C_{1}}{2(\Delta-(j+1/2)\omega_{tr})}$
(14)

into equations 10, 11. Hence the system would be represented by an effective
two-level system:

$\displaystyle\frac{\text{d}C_{0}}{\text{d}t}=-\text{i}\sum\limits_{j=0}^{n}\frac{f_{0j}^{2}\Omega_{0}^{2}\,C_{0}+f_{0j}\Omega_{0}f_{1j}\Omega_{1}\,C_{1}}{4(\Delta-(n+1/2)\omega_{tr})}$
(15)
$\displaystyle\frac{\text{d}C_{1}}{\text{d}t}=-\text{i}\sum\limits_{j=0}^{n}\frac{f_{0j}\Omega_{0}f_{1j}\Omega_{1}\,C_{0}+f_{1j}^{2}\Omega_{1}^{2}\,C_{1}}{4(\Delta-(n+1/2)\omega_{tr})}+\text{i}\delta
C_{1}$ (16)

Assuming that $\Delta\gg U_{tr}$ the effective Rabi frequency in the two level
system would be

$\tilde{\Omega}=\frac{\Omega_{0}\Omega_{1}}{4\Delta}\sum\limits_{j=0}^{n}f_{0j}f_{j1}$
(17)

and the effective detuning of the two-level system would be

$\delta_{\text{eff}}=\delta-\sum\limits_{j=0}^{n}\frac{f_{1j}^{2}\Omega_{1}^{2}-f_{0j}^{2}\Omega_{0}^{2}}{4\Delta}$
(18)

##  V App. B: Superposition of Rydberg states

The angular part of the Rydberg wave-function in terms of the spherical
harmonics and the spin state is given by

$\psi_{j=l\pm 1/2,m_{j}}=\frac{1}{\sqrt{2l+1}}[\sqrt{l\pm
m_{j}+1/2}Y_{l,m_{j}-1/2}\chi_{1/2,1/2}\mp\sqrt{l\mp
m_{j}+1/2}Y_{l,m_{j}+1/2}\chi_{1/2,-1/2}].$ (19)

In the presence of spin-orbit coupling, the spin up and down electrons are
distinguishable. The spin up (down) angular wave-function probability of the
Rydberg superposition state presented in Fig. 2d is plotted in Fig. 5a(b).
Exciting Rydberg atom from ${|{0}\rangle}$ (${|{1}\rangle}$) the spin up
(down) electron would be concentrated around the $z=0$ plane.

Figure 5:  The spin (a) up and (b) down part of the angular wave-function
probability of the Rydberg state in Fig. 2d is plotted. The quantization axis
$z$ is perpendicular to the 2D optical lattice plane along x-y direction.

##  VI App. C: Realization with Circular state

The recent advances in fast transition to Rydberg circular states Sig17 ;
Car20 , would make them ideal target for Rydberg-Fermi gates’ application. The
ponderomotive potential of focused Laguerre-Gauss (LG) beams, enables site
addressing in exciting circular states Car20 . Exciting the circular state
${|{66C}\rangle}$, the electron would be confined at the position of
neighbouring qubits at 210nm resulting to $V_{RF}=20$MHz. It simultaneously
minimizes the wave-function overlap with the ionic-core and hence results to
enhanced lifetime of the order of minutes Ngu18 .

In an alternative approach (inspired by Coh21 ) one can encode the central
atom qubits in circular states and then use transition between
${|{1}\rangle}={|{nC}\rangle}$ circular state to a distinct principal number
circular state ${|{(n+14)C}\rangle}$ to make the qubit dependent Ryd-Fermi
interaction. The interaction of low qubits would be compensated with global
dynamical decoupling (DD) sequences such as WAHUHA Wau68 .

##  VII App. D: Comparing Rydberg-dipolar and Rydberg-Fermi gates

One of the advantages of implementing multi-qubit operations with the proposed
Rydberg-Fermi interaction comparing to the conventional Ryd-dipolar schemes is
in the absence of intra-component interaction. To quantify the effects of this
unwanted phase on gate performance, the phase sensitive form of fidelity is
used

$\text{Fid}=\text{Tr}(|M+MM^{\dagger}|)/2n,$ (20)

where $M=U_{id}^{\dagger}U_{gate}$, with $U_{id}$ and $U_{gate}$ representing
the ideal and realistic gate operations. The Hilbert space dimension is given
by $n$.

Evaluating the fidelity of Rydberg-dipolar Toffoli gate C4-NOT proposed in
Ise11 , with the phase dependent definition of fidelity in Eq. 20 reveal the
effects of unwanted phase. In Ise11 , control and target atoms are getting
excited to $|60S\rangle$ and $|60P\rangle$, with optimum laser couplings of
$\Omega_{c}/2\pi=180$MHz, $\Omega_{t}/2\pi=0.8$MHz and lattice separation of
4$\mu$m. Simulating the gate operation under the Schrödinger equation,
encountering spontaneous emission and population rotation errors in addition
to the infidelity encountered by unwanted phases leads to the average
infidelity of 5% quantified by Eq. 20. Importantly, large phase dependent
infidelities would occur for specific qubit configurations with large Rydberg
population e.g. ${|{1111}\rangle}_{c}{|{1(0)}\rangle}_{t}$ experience 52%
(12%) infidelity. This is while the phase independent conventional definition
leads to 2% average infidelity. In the proposed Ryd-Fermi approach the absence
of intra component interaction, reduced rotation errors at lower principal
number and reduced Rydberg population results to 0.13% average infidelity.
This comparison suggests the importance of Ryd-Fermi scheme as an alternative
for conventional Ryd-dipolar approach. Notably, the assigned fidelity is
bellow the 1.4% infidelity threshold for surface error correction codes.

The implementation of Rydberg-dipolar parallelized gate Mul09  is also
sensitive to the intra-component interaction. Using the parameters of Ref.
Mul09 , in the absence of Rydberg control atom, each of target atoms would
follow the dark state
${|{D}\rangle}=(\Omega_{c}{|{A}\rangle}-\Omega_{p}{|{R}\rangle})/N$ with the
Rydberg population of $P_{R}=(\frac{\Omega_{p}}{\Omega_{c}})^{2}$ on each
target atom. For the abbreviation we refer the readers to Mul09  for the
scheme explanation and parameter definitions. With phase independent
definition of fidelity, Ref. Mul09  predicts the gate operation to be
insensitive to strong intra-component interaction $V_{jk}$, see Fig. 6a.
However using the parallelized gate for the implementation of the stabilizer
operator Wei10  makes the operation phase sensitive. Applying phase dependent
definition of fidelity Eq. 20, the dipolar Rydberg gate is significantly
sensitive to $V_{jk}$ as shown in Fig. 6b. This comparison shows the value of
Rydberg-fermi gates in the implementation of quantum simulation and
optimization algorithms.

Figure 6:  Effects of unwanted intra-component interaction on parallelize gate
operation. (a) Neglecting the phase error, Ref. Mul09  quantifies gate
operations in making three qubit GHZ state for different
$x=\sqrt{2}\Omega_{p}/\Omega_{c}$ . The figure suggest high fidelity in the
presence of strong intra-componenet interaction among target atoms $V_{jk}$.
Using this gate for implementing the stabilizer operator Wei10  makes the
operation phase sensitive. (b) Applying the phase-dependent definition of
fidelity in Eq. 20, this dipolar Rydberg gate Mul09  is significantly
sensitive to $V_{jk}$. The proposed Ryd-Fermi scheme in this article is an
alternative with no cross talk among target atoms.

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