# Spin-squeezed GKP codes for quantum error correction in atomic ensembles

Sivaprasad Omanakuttan<EMAIL_ADDRESS>Center for Quantum Information
and Control, University of New Mexico, Albuquerque, NM, USA  Department of
Physics and Astronomy, University of New Mexico, Albuquerque, NM, USA  T.J.
Volkoff<EMAIL_ADDRESS>Theoretical Division, Los Alamos National
Laboratory, Los Alamos, NM 87545, USA

###### Abstract

GKP codes encode a qubit in displaced phase space combs of a continuous-
variable (CV) quantum system and are useful for correcting a variety of high-
weight photonic errors. Here we propose atomic ensemble analogs of the single-
mode CV GKP code by using the quantum central limit theorem to pull back the
phase space structure of a CV system to the compact phase space of a quantum
spin system. We study the optimal recovery performance of these codes under
error channels described by stochastic relaxation and isotropic ballistic
dephasing processes using the diversity combining approach for calculating
channel fidelity. We find that the spin GKP codes outperform other spin system
codes such as cat codes or binomial codes. Our spin GKP codes based on the
two-axis counter-twisting interaction and superpositions of SU(2) coherent
states are direct spin analogs of the finite-energy CV GKP codes, whereas our
codes based on one-axis twisting do not yet have well-studied CV analogs. A
state preparation scheme for the spin GKP codes is proposed which uses the
linear combination of unitaries method, applicable to both the CV and spin GKP
settings. Finally, we discuss a fault-tolerant approximate gate set for
quantum computing with spin GKP-encoded qubits, obtained by translating gates
from the CV GKP setting using the quantum central limit theorem.

##  I Introduction

Quantum error correcting (QEC) codes in which logical quantum states are
encoded by generating a comb structure in the phase space of a quantum
oscillator Gottesman  _et al._ (2001) are central to proposals for fault-
tolerant quantum computation with continuous variables (CV) Bourassa  _et al._
(2021); Larsen  _et al._ (2021); Noh and Chamberland (2020); Fukui  _et al._
(2018); Raveendran  _et al._ (2022). In the ideal (infinite energy) limit, the
code states of a single mode Gottesman-Kitev-Preskill (GKP) qubit are defined
by a superposition of quadrature eigenvectors, forming a Dirac comb with peaks
displaced by $\sqrt{2\pi}$ along a phase space direction. These CV GKP codes
protect perfectly against sufficiently small shifts in momentum and position
(within the half-period $\sqrt{\pi/2}$) although, importantly, high-
performance error correction properties of the CV GKP code extend beyond
Gaussian coherent errors. For example, the CV GKP code outperforms a variety
of bosonic codes for the case of photon loss Albert  _et al._ (2018). There
have been multiple proposals for implementations and applications of the CV
GKP codes in bosonic systems Bourassa  _et al._ (2021); Xu  _et al._ (2022);
Noh  _et al._ (2020), and low-energy code states have been experimentally
demonstrated de Neeve  _et al._ (2022). Codes similar to GKP motivated by the
specific nature of the errors in the physical system could be fundamental in
building fault-tolerant quantum computation with lower overhead, e.g., with
fewer number of physical systems comprising the logical qubits Puri  _et al._
(2020); Wu  _et al._ (2022); Cong  _et al._ (2022); Ofek  _et al._ (2016).

Similar to CV systems, atomic spin systems are potential candidates for large
scale quantum information processing due to their controllability and
potential for entanglement generation, both crucial properties for quantum
computational tasks which require a large number of qubits. Quantum
simulations with more than $200$ qubits have been experimentally demonstrated
with neutral atoms Ebadi  _et al._ (2021); Scholl  _et al._ (2021). Such large
atomic ensembles have found interesting applications in the context of quantum
metrology and large-scale entanglement generation. Proposals for QEC codes
native to spin systems utilizing conversion of the relevant error to erasure
errors and exploiting spin system symmetries have also been proposed recently
Wu  _et al._ (2022); Gross (2021). Therefore, the potential for utilizing a
collective spin degree of freedom to robustly encode a logical qubit has been
recognized. However, in this work, we fully utilize the analogies between the
overcomplete basis of SU(2) coherent states and CV coherent states, and
between atomic spin squeezing interactions and CV nonlinear interactions to
explore how the CV GKP qubit can be ported to a spin-$N/2$ system. The analogy
is a two-way street, and we further expect that some of the spin-$N/2$ codes
introduced in this work have useful CV analogues based on higher-order
effective photon-photon interactions.

Our families of spin GKP codes are built around the concept of spin squeezing
Kitagawa and Ueda (1993), which is the compact phase space analogue of
quadrature squeezing used to define finite-energy GKP states for CV systems.
The technical result that allows us to port the CV GKP codes to the spin-$N/2$
system is the quantum central limit theorem, which we phrase in terms of
asymptotic isomorphism of Lie algebras under an appropriate Hilbert space
isometry. Using this technique, the spin-$N/2$ two-axis countertwisting
interaction allows defining spin GKP codes depending on the two-body
interaction strength in a similar way as finite-energy CV GKP codes depend on
the squeezing parameter Kitagawa and Ueda (1993). We also introduce spin GKP
codes based on the one-axis twisting interaction and based on superpositions
of coherent states on a phase space grid. For both noise models, we consider,
the spin GKP codes are optimized for the best recovery performance using an
efficient semi-definite programming (SDP) approach known as the diversity
combining method Fletcher  _et al._ (2007).

One of the significant challenges in using GKP states for quantum computation,
regardless of whether one considers the spin-$N/2$ or CV setting, is that the
preparation of the code states is not straightforward. Multiple approaches to
GKP state generation have been proposed using the quantum phase estimation
algorithm and using Kerr interaction. In this work, we propose to use the
recently developed linear combination of unitaries method Berry  _et al._
(2015); Low and Chuang (2019); Holmes  _et al._ (2022), which is now a
prominent module for quantum algorithms. This scheme is applicable, in
principle, to general grid states, although the size of the grid is limited by
the controllability of an ancilla degree of freedom. To implement universal,
fault-tolerant quantum computation with spin-$N/2$ GKP-encoded qubits, we
require rotation-resistant gates, and we develop these gates using the quantum
central limit theorem to port the displacement resistant gates from the CV GKP
setting to the spin-$N/2$ GKP setting. The fault-tolerant approximate Clifford
group generators are composed of Hamiltonians at most quadratic in angular
momentum operators of two spin-$N/2$ systems. Assuming that these generators
can be implemented with high fidelity, the magic states sufficient for
universal quantum computation can be generated using a similar approach as in
Baragiola  _et al._ (2019).

The remainder of this article is organized as follows. In Section II we
discuss how the quantum central limit theorem motivates spin analogues of the
CV GKP codes. In Section III we introduce a family of spin GKP code states
using both two-axis countertwisting and one-axis twisting atomic interactions,
and we compare the free parameters that appear in these codes to parameters
that define finite-energy CV GKP codes. In Section IV we consider two kinds of
error channels relevant to atomic ensembles and compare the recovery
performance of our various versions of spin GKP codes to that of other
spin-$N/2$ bosonic codes. The main result is that the spin GKP codes perform
better than other codes over a large range of noise strengths for both of the
channels we consider. In Section V we present an algorithm to prepare the GKP
states based on the linear combination of unitaries technique and discuss
potential schemes for creating spin GKP states using the light-matter
interaction. Lastly, in Section VI we describe how one could construct a
universal gate set for the spin GKP codes following again the quantum central
limit theorem. Discussions and possible future directions are in Section VII.

The appendices are devoted to important technical details. In App.A, we
describe the diversity combining approach which allows us to efficiently
calculate the optimal recovery channel. App.B provides details about the
stochastic relaxation channel and App.C discusses numerical benchmarking
details for optimal recovery with respect to the stochastic relaxation and
isotropic ballistic dephasing channels. In App.D, the one-axis ballistic
dephasing channel is discussed in order to show that other collective spin QEC
codes can outperform spin GKP codes when the noise is known to be directional.
Lastly, in App.E, we analyze the optimal recovery of various finite-energy CV
GKP codes under photodetection noise, in order to show that different well-
motivated versions of the GKP code can have different recovery performances.

##  II Motivation from quantum central limit theorem and spin squeezing

In this work, we provide several schemes to encode a logical qubit in a
collection of $N$ qubits which differ fundamentally from the current
experimental approaches for fault-tolerant quantum computation which require
individual addressing of each qubit. Specifically, our logical states can be
viewed as symmetric atomic ensembles in which the $N$ qubits are prepared in
the symmetric subspace and addressed by global operations. This system is
mathematically equivalent to a spin-$j$ system where $j=N/2$. In this section,
we describe the motivation of the encoding schemes in detail.

The logical state encodings we detail in Section III are motivated by
combining spin squeezing of a symmetric $N$-qubit ensemble with the GKP scheme
for encoding logical qubits in CV systems. Finite-energy (i.e., physical)
versions of the ideal CV GKP code are constructed by forming approximate
eigenvectors of the stabilizer operators which are CV displacement operators.
These stabilizers define the phase space lattice structure of the code
projector. In a spin-$N/2$ representation of SU(2), as occurs in, e.g., a
system of $N$ indistinguishable bosonic atoms distributed among two orthogonal
modes, a strictly canonical approach based on stabilizers would lead to the
replacement of idealized CV quadrature eigenstates in the original CV GKP code
by idealized position eigenstates $\ket{(\theta,\phi)}$ on the Riemann sphere
Albert  _et al._ (2020). However, if one wants to work with physical states of
the symmetric $N$-qubit system and make use of realizable interactions such as
spin squeezing, it is unclear how to construct logical states defining a code
projector associated with a lattice-like phase space distribution allowing for
isotropic correction of small rotation errors. The approach taken in this work
generates the characteristic comb structure of CV GKP logical states in a
system of $N$ bosons by taking superpositions of rotated spin-squeezed states
and utilizing the asymptotics of these states with respect to atom number $N$
to make a connection with the phase space structure of CV GKP codes. This
connection has its roots in the Cushen-Hudson quantum central limit theorem.

In modern terminology, part of the original Cushen-Hudson quantum central
limit theorem Cushen and Hudson (1971) states that the characteristic function
$\text{tr}\rho^{\otimes M}e^{ix\bar{q}_{M}+iy\bar{p}_{M}}$ of the multimode
canonical operators $\bar{q}_{M}:={1\over\sqrt{M}}\sum_{j=1}^{M}q_{j}$,
$\bar{p}_{M}:={1\over\sqrt{M}}\sum_{j=1}^{M}p_{j}$ in the $M$-mode CV state
$\rho^{\otimes M}$ converges pointwise on $\mathbb{R}^{2}$ to the Gaussian
characteristic function $\text{tr}e^{ixq+ipy}\rho_{G}$ of the single mode CV
state $\rho_{G}$, the Gaussification of $\rho$ Lami  _et al._ (2019). This
convergence was characterized in Ref.Becker  _et al._ (2021). Physically, this
theorem says that quantum fluctuations of quadratures of $M$ CV modes can be
described by a Gaussian quantum state. Subsequently, inspired by mathematical
treatments of laser theory, Giri and von Waldenfels extended this part of the
Cushen-Hudson central limit theorem to the free Lie algebra generated by an
associative algebra Giri and von Waldenfels (1978). For the purposes of the
present work, it is sufficient to have convergence of (spin-squeezed) SU(2)
coherent states in the spin-$N/2$ representation to (quadrature squeezed)
Heisenberg-Weyl coherent states of a single CV modes. Accardi and Bach
utilized the argument of Giri and von Waldenfels (1978) applied to spin
generators to prove precisely this convergence in Theorems 4.2 and 6.2 of
Accardi and Bach (1987).

To explicitly state how we use the limits of Accardi and Bach (1987), we first
relate the orthonormal basis of Dicke states for the symmetric subspace of $N$
qubits to the Fock basis of a single mode CV system using an isometry $V$.
Specifically, we define $V:\ket{N-k,k}\rightarrow\ket{k}$, where the Dicke
state $\ket{N-k,k}:={1\over\sqrt{{N\choose
k}}}\sum_{\text{Ham}(x)=k}\ket{x_{1}}\cdots\ket{x_{N}}$, $x_{j}\in\\{0,1\\}$.
For large $N$, we have the following relations between the Schwinger boson
spin operators $J_{+}:=a^{\dagger}_{0}a_{1}$, $J_{-}=J_{+}^{\dagger}$,
$J_{z}={1\over 2}(a^{\dagger}_{0}a_{0}-a_{1}^{\dagger}a_{1})$ under this
isometry:

$\displaystyle
Ve^{{\alpha\over\sqrt{N}}J_{-}-{\overline{\alpha}\over\sqrt{N}}J_{+}}\ket{N,0}\rightarrow
e^{\alpha a^{\dagger}-\overline{\alpha}a}\ket{0}$ $\displaystyle Ve^{{z\over
N}J_{+}^{2}-{\overline{z}\over N}J_{-}^{2}}\ket{N,0}\rightarrow
e^{za^{2}-\overline{z}a^{\dagger 2}}\ket{0}$ $\displaystyle Ve^{i\theta
J_{z}}\ket{N-k,k}=e^{-i\theta a^{\dagger}a}\ket{k}$ (1)

where the limits are in the weak topology on the Hilbert space of the harmonic
oscillator and the last equality is only valid in the projective Hilbert
space. These relations imply that a $1/\sqrt{N}$-scaled spin rotation
generator about an axis in the $xy$-plane is asymptotically equivalent to the
generator of CV displacement $\alpha a^{\dagger}-\overline{\alpha}a$, and that
the $1/N$-scaled generator of two-axis countertwisting (TACT) is
asymptotically equivalent to the generator of quadrature squeezing. Note that
straightforwardly applying a Holstein-Primakoff transformation to the left
hand sides of the first two lines of (1) results in $N$-dependent algebraic
functions of the creation and annihilation operators in the exponents on the
right hand side of (1). Such parameterized generators do not naturally appear
in finite-energy approximations to the ideal CV GKP code. In Section III, our
first spin-$N/2$ GKP code tactgkp is defined by pulling back (using the
isometry $V$) a finite-energy CV GKP code defined by superposing displaced
quadrature squeezed states with Gaussian weights. This is the form of the
finite-energy CV GKP code originally proposed in Gottesman  _et al._ (2001).

The equivalence of defining ideal CV GKP codes from a superposition of
displaced quadrature eigenvectors or from a lattice superposition of CV
coherent states motivates a spin-$N/2$ code that consists of an approximately
Gaussian weighted superposition of SU(2) coherent states. Our spin-$N/2$ codes
spinGKP and unigkp are defined in (7) and (9), respectively, and both codes
asymptote to a square lattice finite-energy CV GKP code in the
$N\rightarrow\infty$ limit. The optimal recovery properties of these codes
must be analyzed in addition to those of tactgkp because the equivalence of
their respective CV asymptotics does not imply that the spin-$N/2$
counterparts exhibit equivalent optimal recovery properties under spin system
noise channels. We consider realistic noise channels that apply to atomic
ensembles independent of $N$, viz., the noise channels we consider are actual
decoherence processes for atomic ensembles, not simply pullbacks of the
commonly considered CV bosonic attenuation channel by $V$. Further, the
comparison is especially relevant due to the fact that an ensemble of $O(100)$
atoms represents the present state-of-the-art system size for spin squeezing.
For CV GKP codes, optimization of the defining lattice is known to give order-
of-magnitude higher channel fidelity for small noise parameters Albert  _et
al._ (2018). In the present work, we focus on schemes for generating logical
states of a spin-$N/2$ atomic ensemble possessing comb-like distributions of
$J_{x}$ or $J_{y}$, while leaving the question of optimization of the phase
space “lattice” of the corresponding code projectors as a direction for future
investigation (“lattice” in quotations because the phase space is spherical).
We expect that even for finite $N$, such optimization of the codes by
utilizing results on packing of spherical caps or spherical ellipses will
further increase the error recovery performance.

Lastly, we will also introduce in Section III a class of spin-$N/2$ GKP codes
that are not asymptotically equal to finite-energy CV GKP codes as
$N\rightarrow\infty$. The construction of these codes is enabled by the
existence of spin squeezing generators that do not involve $J_{\pm}$. For
example, instead of utilizing the TACT generator, one can seek to generate the
characteristic GKP comb structure of the code projector by utilizing the one-
axis twisting (OAT) interaction $J_{z}^{2}$. Our OAT-based spin-$N/2$ codes
are oatGKP and oatGKP(uni), which differ from each other by the form of the
amplitudes of the rotated OAT parent state. The OAT interaction is
mathematically analogous to the Kerr interaction in a single-mode CV system,
with both interactions generating multipronged cat states for specific
constant interaction times. We emphasize that all of the codes in Section III
define finite energy single-mode CV states in the $N\rightarrow\infty$ limit.

For the codes $\\{\texttt{tactgkp},\texttt{spingkp},\texttt{oatgkp}\\}$, we
define amplitudes on the rotated spin-squeezed states that asymptotically
approach the Gaussian amplitudes in finite-energy CV GKP codes. While the
exact form of such amplitudes can be informed by specific aspects of the
experimental state generation protocol, for simplicity we define the
amplitudes by utilizing the deMoivre-Laplace formula in the following form,
which holds for $x\in\mathbb{R}$:

$\displaystyle{\Gamma(N+1)\over 2^{N}\Gamma({N\over 2}+x+1)\Gamma({N\over
2}-x+1)}$ $\displaystyle\sim\sqrt{2\over\pi N}e^{-2x^{2}/N}.$ (2)

##  III Spin-$N/2$ GKP codes

In this section, we provide definitions for the spin-$N/2$ GKP codes whose
quantum error correction properties we explore in this work. The kinematic
structure of the code states is motivated by considering a comb-like structure
in phase space, similar to that in CV GKP code states. The CV GKP codes are
designed to allow correction of small displacements of the $q$ and $p$
quadratures, and the spin-$N/2$ GKP states are analogously designed to allow
correction of small $\mathrm{SU}(2)$ rotations generated by $J_{y}$ and
$J_{x}$. We find in Section IV that the phase space structure of the
spin-$N/2$ code states allows high-fidelity optimal recovery with respect to
specific noise channels and also noise-agnostic recovery.

The logical states of our spin-$N/2$ qubits will be written as
$\ket{\mu_{S}}$, where $\mu$ is the qubit orthonormal basis index and $S$ is a
list of real parameters. Later in the paper, when we fix the parameters of a
certain spin-$N/2$ GKP code for various purposes, we will omit $S$ and simply
label the logical states as $\ket{\mu_{L}}$ with $L$ standing for “logical”.
The first spin-$N/2$ GKP code strictly converges to a finite-energy CV GKP
code by combining the quantum central limit theorem and the deMoivre-Laplace
theorem. This code is labeled tactgkp and has the exact expression

$\ket{\mu_{\lambda,z}}\propto\left[\sum_{t=1}^{T}\Gamma(N+1)\left(\frac{e^{-2i(\sqrt{2\pi}t+\mu\sqrt{\pi\over
2}){J_{y}\over\sqrt{N}}}}{C^{N,t}_{\lambda,\mu}C^{N,t}_{-\lambda,\mu}}+\frac{e^{2i(\sqrt{2\pi}t-\mu\sqrt{\pi\over
2}){J_{y}\over\sqrt{N}}}}{C^{N,t}_{\lambda,-\mu}C^{N,t}_{-\lambda,-\mu}}\right)+\Gamma(N+1)\frac{e^{-2i\mu\sqrt{\pi\over
2}{J_{y}\over\sqrt{N}}}}{C^{N,0}_{\lambda,\mu}C^{N,0}_{-\lambda,\mu}}\right]e^{{z\over
2N}(J_{+}^{2}-J_{-}^{2})}\ket{N,0}$ (3)

where to lighten the notation we use,

$C^{N,t}_{\lambda,\mu}=\Gamma\left({N\over 2}+\lambda\sqrt{\pi
N}\left(t+{\mu\over 2}\right)+1\right),$ (4)

with $\mu\in\\{0,1\\}$ and $\ket{N,0}=\ket{J,J_{z}=J}$ is the spin coherent
state with maximal $J_{z}$ eigenvalue. By considering the spectral projections
of the operator $J_{x}/\sqrt{N}$, which is asymptotically equivalent to the CV
operator $q$ under the isometry $V$ in (1), one can obtain a probability
distribution analogous to the $q$-quadrature distribution of a CV GKP code. In
Fig. 1, we show this distribution for the case of $N=100$, $T=10$, and
$\delta=0.3$.

In (3), there are three $N$-indexed sequences that contribute to the large $N$
convergence of the code states to CV GKP code states: 1. the sequence of
$J_{y}$ rotations converges to a displacement of CV quadrature $q$, 2. the
sequence of TACT states converges to a $q$-quadrature squeezed state, and 3.
the sequence of coefficients of each $J_{y}$ rotation converges to a Gaussian
weight. The convergence of the first and second sequences follows from the
quantum central limit theorem, according to which the $2/\sqrt{N}$ prefactor
of the rotation generators causes convergence of the rotations about $J_{y}$
to CV displacements of $q$ with the same parameter, while the $1/N$ prefactor
of the TACT interaction causes convergence of spin squeezing with interaction
time $z=\tanh^{-1}{d-1\over d+1}$ to CV squeezing of the canonical $q$
quadrature with squeezing parameter $z$ and covariance matrix ${1\over
2}\text{diag}(d^{-1},d)$. For consistency with common convention in the CV
literature, we use $\lambda=\delta$ and $d=\delta^{-2}$, which allows the code
states to be parametrized with a single parameter, but one can also choose the
value of $d$ independently from $\lambda$ based on minimization of the overlap
between the code states or based on best optimal recovery for a specific noise
channel. Lastly, the convergence of the coefficient of each $J_{y}$ rotation
to a Gaussian weight follows from (2). Specifically, the three coefficients
asymptote to $e^{-2\pi\lambda^{2}(t\pm{\mu\over 2})^{2}}$ and
$e^{-{\pi\lambda^{2}\over 2}\mu^{2}}$, respectively, up to a $\mu$-independent
prefactor. The asymptotically Gaussian amplitudes suggest that the fidelity
between code states should be insensitive to contributions from $t\geq
1/2\pi\lambda^{2}$ in the sum index, and we empirically find that taking $T=5$
is appropriate for the range of $\lambda$ that gives a low fidelity between
codewords.

Figure 1: Probability distribution of the codewords in the $q$ basis. The
probability distribution of the tactgkp for $N=100$ , $T=10$, and $\delta=0.3$
is shown in the basis of eigenvectors of $J_{x}/\sqrt{N}$, which asymptotes to
the CV quadrature $q$ under isometry (1). Similar to the CV GKP codes one can
see the well-separated combs which are central to the error correction
properties of the grid state codes.

In a spin-$N/2$ system, finite-energy considerations do not demand that a
superposition of discretely rotated spin-squeezed states have decaying
coefficients with respect to changing the rotation index from zero. Therefore,
it is valid to consider a simplified version of tactgkp defined solely as a
superposition of rotated two-axis countertwisted SU(2) coherent states
Kitagawa and Ueda (1993). We refer to this code as tactgkp(uni) for its
uniform amplitudes on the rotated spin-squeezed states (which become more
distinguishable as $N$ is increased), and the code states are explicitly given
by

$\ket{\mu_{z}}\propto\sum_{t=-T}^{T}e^{2i(\sqrt{2\pi}t-\mu\sqrt{\pi\over
2}){J_{y}\over\sqrt{N}}}e^{{z\over 2N}(J_{+}^{2}-J_{-}^{2})}\ket{N,0}$ (5)

with $\mu\in\\{0,1\\}$ giving the two code space and

$z=\tanh^{-1}{\delta^{-2}-1\over\delta^{-2}+1}$ (6)

is the squeezing parameter. Unlike the three-parameter tactgkp, the
tactgkp(uni) code states are a function of only the TACT interaction time $z$
and the superposition cut off $T$ and one could optimize both of these two in
order to get the best performance. The $T$ parameter can be determined by
fixing the particle number and noise model and optimizing the channel fidelity
over $z$ for each $T$.

For single-mode CV GKP codes, the connection between a Gaussian-weighted
superposition of quadrature squeezed states and a Gaussian-weighted
superposition of coherent states on a lattice in phase space $\mathbb{R}^{2}$
can be made due to the Gaussian nature of quadrature squeezed states Albert
_et al._ (2018). Similar to the CV GKP code defined by a superposition of CV
coherent states with mean vectors on a square grid in phase space (66), we
define a spin-$N/2$ code called spingkp

$|\mu_{\delta}\rangle\propto\sum_{t\in\\{-T,\ldots,T\\}^{\times
2}}\frac{\Gamma(N+1)}{D^{N,t_{1},t_{2}}_{\delta,\mu}D^{N,t_{1},t_{2}}_{-\delta,\mu}}e^{-2i(\sqrt{2\pi}t_{1}+\mu{\sqrt{\pi\over
2}}){J_{y}\over\sqrt{N}}}e^{2i\sqrt{\pi\over
2}t_{2}{J_{x}\over\sqrt{N}}}|N,0\rangle$ (7)

where to lighten the notation we use,

$D^{N,t_{1},t_{2}}_{\delta,\mu}=\Gamma\left({N\over 2}+{\delta\over
2}\sqrt{N\pi((2t_{1}+\mu)^{2}+t_{2}^{2})}+1\right),$ (8)

with $\mu\in\\{0,1\\}$. If one removes the amplitudes, one gets the code
unigkp which depends only on the superposition cutoff $T$, and is given by

$|\mu\rangle\propto\sum_{t\in\\{-T,\ldots,T\\}^{\times
2}}e^{-2i(\sqrt{2\pi}t_{1}+\mu{\sqrt{\pi\over
2}}){J_{y}\over\sqrt{N}}}e^{2i\sqrt{\pi\over
2}t_{2}{J_{x}\over\sqrt{N}}}|N,0\rangle.$ (9)

Unlike quadrature squeezing in a single-mode CV system, spin squeezing in a
spin-$N/2$ system can be realized in many ways depending on the two-body
interaction that is implemented. For example, one-axis twisting Kitagawa and
Ueda (1993) is one of the fundamental interactions exploited in atomic systems
for quantum simulation and quantum metrology. This interaction can be used to
create GKP code states

$\ket{\mu_{\lambda,z}}\propto\left[\sum_{t=1}^{T}\Gamma(N+1)\left(\frac{e^{-2i(\sqrt{2\pi}t+\mu\sqrt{\pi\over
2}){J_{y}\over\sqrt{N}}}}{C^{N,t}_{\lambda,2\mu}C^{N,t}_{-\lambda,2\mu}}+\frac{e^{-2i(\sqrt{2\pi}t-\mu\sqrt{\pi\over
2}){J_{y}\over\sqrt{N}}}}{C^{N,t}_{\lambda,-2\mu}C^{N,t}_{-\lambda,-2\mu}}\right)+\Gamma(N+1)\frac{e^{-2i\mu\sqrt{\pi\over
2}{J_{y}\over\sqrt{N}}}}{C^{N,0}_{\lambda,2\mu}C^{N,0}_{-\lambda,2\mu}}\right]e^{-i\delta
J_{x}^{2}}\ket{N,0}$ (10)

where $\mu\in\\{-{1\over 2},{1\over 2}\\}$. Although the combinatorial
prefactors go to Gaussians as $N\rightarrow\infty$, for the present code there
is no a priori relation between the parameters $\delta$ and $\lambda$ that can
be imposed as was done for tactgkp with the aim of convergence to a finite
energy CV GKP code. Therefore, we analyze two reduced parameter versions of
(10): oatGKP having unparameterized, nonuniform amplitudes

$\ket{\mu_{\delta}}\propto\left[\sum_{t=1}^{T}{N\choose{N\over
2}+t}\left[e^{-2i(\sqrt{2\pi}t+\mu\sqrt{\pi\over
2}){J_{y}\over\sqrt{N}}}+e^{2i(\sqrt{2\pi}t-\mu\sqrt{\pi\over
2}){J_{y}\over\sqrt{N}}}\right]+{N\choose N/2}e^{-2i\mu\sqrt{\pi\over
2}{J_{y}\over\sqrt{N}}}\right]e^{-i\delta J_{x}^{2}}\ket{J,J_{\mathrm{z}}=J}$
(11)

and oatGKP(uni) with uniform amplitudes on rotated one-axis twisted states

$\displaystyle\ket{\mu_{\delta}}\propto$
$\displaystyle\;\sum_{t=-T}^{T}e^{2i\left(\sqrt{2\pi}t-\mu\sqrt{\frac{\pi}{2}}\right)\frac{J_{\mathrm{y}}}{\sqrt{N}}}e^{-i\delta
J_{x}^{2}}\ket{J,J_{\mathrm{z}}=J}$ (12)

where $\mu\in\\{1/2,-1/2\\}$ for both codes. We use the oatgkp and oatgkp(uni)
defined in Eq. (11) and Eq. (12) in this article. Note that the
$\delta\rightarrow 0$ limit ($\delta\rightarrow 1$ limit) corresponds to
infinite interaction time (zero interaction time) for tactgkp (oatgkp).

Taking the logical index as $\mu\in\\{1/2,-1/2\\}$ instead of
$\mu\in\\{0,1\\}$ effectively shifts the code states in phase space and, for
finite $N$, has implications for optimal recovery performance with respect to
the noise channels considered in this work. In fact, a similar point arises
when requiring CV GKP codes states to have exactly the same number of photons
in expectation (see Appendix E for a short review of square grid CV GKP codes,
equal-energy versions of these codes, and their recovery properties). For the
equal-energy CV GKP codes, we observed slightly reduced optimal recovery
performance in the CV case.

Figure 2: Husimi Q function. The spin Husimi Q function,
$Q(\alpha)=\frac{1}{\pi}\langle\alpha|\hat{\rho}|\alpha\rangle$ where
$\ket{\alpha}$ is the spin-$N/2$ coherent state, for the code states
considered in this article. For the spin version of the GKP codes we have
derived in this article, we can see the comb-like structure originating in the
phase space similar to the comb-like structure in phase space for CV GKP
codes. The parameters used for the simulation of spingkp is $\delta=0.2$ and
$T=5$, for tactgkp(uni) is $\delta=0.3$ and $T=2$, for unigkp is $T=2$, and
for oatgkp is $\delta=0.04$ and $T=5$.

The Husimi Q representation is one of the well-known quasiprobability
distributions used to represent the phase space structure of a quantum state
of a CV system. A spin analog of the Husimi Q representation is defined by the
analogy of CV coherent states to SU(2) coherent states Agarwal (2012). In Fig.
2, we show the spin-$N/2$ Husimi Q function
$Q(\alpha)=\frac{1}{\pi}\langle\alpha|\hat{\rho}|\alpha\rangle$ where
$\ket{\alpha}$ is a spin coherent state, for the main code states considered
in this article. For the spin GKP codes, a distorted comb-like structure is
seen to appear in spherical phase space, with TACT interaction producing
longitudinal alignment of the rotated spin-squeezed states and OAT interaction
producing rotated spin-squeezed states aligned on twisted axes.

For comparison to the spin GKP codes, we consider classes of well-known
bosonic codes. The binomial code (binom) Michael  _et al._ (2016) is defined
by

$\displaystyle\ket{0_{L}}$
$\displaystyle=\frac{\ket{J=\frac{N}{2},J_{z}=\frac{N}{2}}+\ket{J=\frac{N}{2},J_{z}=\frac{N}{2}-4}}{\sqrt{2}}$
(13) $\displaystyle=\frac{\ket{N,0}+\ket{N-4,4}}{\sqrt{2}},$
$\displaystyle\ket{1_{L}}$
$\displaystyle=\ket{J=\frac{N}{2},J_{z}=\frac{N}{2}-2}$
$\displaystyle=\ket{N-2,2},$

and the single-rail encoding Kok  _et al._ (2007) rail code which is defined
by $\ket{0_{L}}=\ket{N,0}$ and $\ket{1_{L}}=\ket{N-1,1}$.

One can also define the spin-$N/2$ cat qubit codes cat as,

$\displaystyle\ket{0}_{\vec{n}}$
$\displaystyle={\ket{J,\vec{n}\cdot\vec{J}=J}+\ket{J,\vec{n}\cdot\vec{J}=-J}\over\sqrt{2}}$
(14) $\displaystyle\ket{1}_{\vec{n}}$
$\displaystyle={\ket{J,\vec{n}\cdot\vec{J}=J}-\ket{J,\vec{n}\cdot\vec{J}=-J}\over\sqrt{2}}$

where $\vec{n}$ is the specific orientation of interest defined by the
$(\theta,\phi)$ in the spherical coordinates. Also, we have the coherent state
codes coherentstate given as,

$\displaystyle\ket{0}_{\vec{n}}$ $\displaystyle=\ket{J,\vec{n}\cdot\vec{J}=J}$
(15) $\displaystyle\ket{1}_{\vec{n}}$
$\displaystyle=\ket{J,\vec{n}\cdot\vec{J}=-J}$

where, similar to cat, $\vec{n}$ is the specific orientation of interest
defined by the $(\theta,\phi)$ in the spherical coordinates. One can similarly
port other CV coherent-state-based codes from the bosonic system to the spin
system.

Unlike the CV GKP codes, the spin squeezing parameter $z$ of the spin-$N/2$
GKP codes is merely kinematic and not related to the number of particles
comprising the code states. It depends on the strength of the interatomic
interaction used to generate the code states. For CV GKP codes, the desired
code energy constrains the value of the squeezing parameter Albert  _et al._
(2018). However, fixing the relation between $z$ and $\delta$ as above for
spin-$N/2$ GKP codes, we can identify a range of valid $\delta$ based on
minimization of the overlap between code states. In Fig. 3, the overlap
between the two code states $\ket{0_{L}},\ket{1_{L}}$ is shown as a function
of $\delta$ for the spingkp and oatgkp is shown. For the case of spingkp,
there is a large domain of $\delta$ where quasi-orthogonality holds, whereas
for the case of oatgkp the domain of quasi-orthogonality is smaller and
disconnected.

Figure 3: $\delta$ parameter and code state quasi-orthogonality. Fidelity is
given as a function of $\delta$ between the two codewords for tactgkp,
spingkp, oatgkp, and oatgkp(uni) with $N=64$. We observe a wide range of
potential choice of $\delta$ giving quasi-orthogonal code states, which can be
used as an additional degree to control according to the noise channel of
interest. Compared to other codes, the codes tactgkp andspingkp exhibit a
larger domain of $\delta$ corresponding to a well-defined comb-like structure,
which is due to the different physical origin of these codewords. The tactgkp
code uses TACT to generate the comb-like structure, whereas oatgkp and
oatgkp(uni) use OAT. The line $y=1/\sqrt{N}$ is shown to indicate the $\delta$
value above which oatgkp no longer has the comb-like structure necessary for
high-fidelity recovery.

Note that if a range of energies is allowed for finite energy versions of the
CV GKP states, consideration of quasi-orthogonality could also enter into a
choice for the $\delta$ parameter.

Finally, note that in Ref.Albert  _et al._ (2018), optimization over a subset
of lattices in single-mode CV phase space $\mathbb{R}^{2}$ was performed in
order to identify the best geometric configuration for optimal recovery
properties. Similar analyses can be carried out for the spin-$N/2$ codes
introduced in this section, but we now focus on examining optimal recovery
properties under two noise channels relevant for large spin systems.

##  IV Optimal code performance under ensemble errors

We now define two noise channels and compare the optimal recovery performance
of the spin-$N/2$ GKP codes discussed in the previous section. The optimal
recovery properties of the codes are analyzed according to the following
standard procedure: 1. two logical states $\ket{0_{L}}$ and $\ket{1_{L}}$ are
defined according to the $\mu$ parameter for the various encodings discussed
in the previous section, 2. a noise channel $\mathcal{N}_{\xi}$ of interest is
applied for a subset of valid noise strengths $\xi$, and 3. the optimal
recovery performance is quantified according to the optimal channel fidelity

$\displaystyle
F_{\mathcal{N}_{\gamma}}=\max_{\mathcal{R}}\langle\psi_{\rho}|(\mathcal{R}\circ\mathcal{N}_{\xi})\otimes\mathbf{1}\left(\ket{\psi_{\rho}}\bra{\psi_{\rho}}\right)|\psi_{\rho}\rangle$
(16)

where $\ket{\psi_{\rho}}$ is a purification of the state $\rho={1\over
2}P_{\text{code}}$,
$P_{\text{code}}=\ket{0_{L}}\bra{0_{L}}+\ket{1_{L}}\bra{1_{L}}$. With the code
states being in a Hilbert space of dimension $N+1$, one can show that (16) can
be solved using an SDP with a positive $(N+1)^{2}\times(N+1)^{2}$ matrix
variable Fletcher  _et al._ (2007). Even using efficient SDP solvers like
SeDuMi Sturm (1999), this optimization is computationally expensive for
$N=O(10)$. An alternative and more computationally efficient approach is
obtained by considering the encoding and noise channel as a single noise
channel from a qubit system to the spin-$N/2$ system and the recovery channel
as a recovery from the spin-$N/2$ system to the qubit system. This “diversity
combining” approach Fletcher  _et al._ (2007) allows us to analyze optimal
recovery for spin-$N/2$ GKP codes with $N=O(10)$ quickly on a laptop computer
and is discussed in more detail in Appendix A.

In the first two subsections of this section, we apply the diversity combining
method to a scenario in which one has perfect encoding, recovery, and decoding
and the only error we assume is the one arising from the specific noise
channel considered. Thus the fidelity (16) that we calculate provides a
theoretical bound on the performance of these codes for a specific noise
channel. Note that the corresponding optimal recovery channels obtained do not
necessarily correspond to an experimentally feasible design. Lastly, in the
third subsection, we introduce a noise-agnostic recovery that is intended to
work for any weak noise channel or compositions thereof.

The noise channels considered in Section IV.1, IV.2 may not prove to be the
most relevant ones for the experimental implementations, however, to our
knowledge, there is not a widely agreed upon set of typical noise models for
atom ensemble-based control, unlike the situation in, e.g., quantum optics.
Thus, instead of considering more general noise models, we find it instructive
to focus on two noise channel corresponding to different physical processes
(atom loss and dephasing, respectively) and understand the optimal recovery
performance of the spin-$N/$ GKP codes under these decohering processes.

###  IV.1 Stochastic relaxation channel

The single-mode quantum photon detection process

$\rho\mapsto\mathcal{N}_{\gamma}(\rho):=\sum_{\ell=0}^{\infty}{(1-e^{-\gamma})^{\ell}\over\ell!}e^{-{\gamma\over
2}a^{\dagger}a}a^{\ell}\rho a^{\dagger\ell}e^{-{\gamma\over 2}a^{\dagger}a}$
(17)

is a quantum channel describing random photon loss in both free-space and
waveguide-based photonics Ueda (1989). We consider a spin-$N/2$ analog of the
above process, which describes the stochastic relaxation of a two-level atom
system to a uniform ground state. To get the atomic version, define
orthonormal internal states $\ket{0}$ and $\ket{1}$ and assume that $\ket{0}$
has lower energy. Analogous to the no-count photonic process defined in
Ref.Ueda (1989), the no-count atomic detection process is
$\mathcal{S}^{(0)}_{\tau}(\rho_{t})=e^{-{\lambda\tau\over
2}(N-2J_{z})}\rho_{t}e^{-{\lambda\tau\over 2}(N-2J_{z})}$ (note that
$N-2J_{z}=a_{1}^{\dagger}a_{1}$, so the no-count process results in an
exponential decay of the amplitudes for $\ket{N-m,m}$ with $m>0$). This
process is analogous to the backaction photon decay in (17). Analogous to the
one-count, instantaneous photon detection process, we define an instantaneous
relaxation that transfers an atom from $\ket{1}$ to $\ket{0}$, and one can
describe its action on the rank-1 operators
$\ket{N-m,m}\bra{N-m^{\prime},m^{\prime}}$ as

$\displaystyle{}\mathcal{S}^{(1)}(\ket{N-m,m}\bra{N-m^{\prime},m^{\prime}})$
$\displaystyle{}={\lambda
J_{+}\ket{N-m,m}\bra{N-m^{\prime},m^{\prime}}J_{-}\over\sqrt{(N-m+1)(N-m^{\prime}+1)}}.$
(18)

The prefactor can be interpreted as an occupation-dependent scaling of the
probability of an atom decaying from $\ket{1}$ to $\ket{0}$. Using the fact
that $e^{{\lambda\over 2}J_{z}}J_{+}^{k}e^{-{\lambda\over
2}J_{z}}=e^{{k\lambda\over 2}}J_{+}^{k}$, we obtain the full stochastic
relaxation channel

$T_{t}(\rho)=\sum_{\ell=0}^{N}{(1-e^{-\lambda
t})^{\ell}\over\ell!\lambda^{\ell}}e^{-{\lambda t\over
4}(N-2J_{z})}(\mathcal{S}^{(1)})^{\ell}(\rho)e^{-{\lambda t\over
4}(N-2J_{z})}$ (19)

which one can verify to be completely positive and trace-preserving. The
trace-preserving property is shown explicitly in Appendix B.

The stochastic relaxation channel is a generalization of the amplitude damping
channel for the qubit, similar to how the CV detection loss channel is a
generalization of single-photon attenuation dynamics De Palma  _et al._
(2016). This can be seen from the Eq. (18), where the action of the
decoherence process is the action of the ladder operator $J_{+}$, which
increases the $J_{z}$ projection, lowering the energy of the ensemble. The
steady state of the decoherence process is the uniform state in which each
spin is in the ground state, thus generalizing the single-qubit amplitude
damping channel.

Figure 4: Channel fidelity for stochastic relaxation. In (a) the channel
fidelity is given as a function of $\gamma$ for the noise model
$\mathcal{E}=\mathcal{N}_{\gamma}$ given in Eq. (19) for $N=64$ and cut off
$T=5$ for all codes except unigkp and oatgkp (uni) where we all optimize the
cut off value too. The inset in the figure gives the channel infidelity as a
function of $\gamma$ for the small values of $\gamma$ suggesting that the GKP
codes outperform the binomial code and rail code for a large range of $\gamma$
and thus showing its potential advantage compared to other codes for spin
systems for stochastic relaxation channel. In (b) the average fidelity for the
spingkp without any recovery for $20$ random states are given for reference to
show the fact the recovery is crucial.

The performance of the various code-words under the stochastic noise channel
is given in Fig. 4, one could see that the GKP codes outperform the binomial
code and rail code for a large range of $\gamma$. These results are very
similar to the results for the CV GKP codes in Matsuura  _et al._ (2020) for
the case of the photon loss case. In the inset, we show the small $\gamma$
behavior to further illustrate the behavior of these codes. This numerical
data indicate the usefulness of the GKP states based on TACT or displaced
coherent states for stochastic relaxation, similar to known results for the CV
GKP states with respect to amplitude damping.

Figure 5: Code states under stochastic relaxation. Husimi distribution of the
code states for the oatgkp and spingkp for the quantum channel considered in
Eq. (19) for $\gamma=0.2$. The plots are shown for $\delta=0.2,T=5$,
$\delta=0.04,T=5$, and $\delta=0.2,T=2$ respectively for the spingkp,oatgkp,
and tactgkp(uni). The states $\sum_{k}M_{k}|\hskip 1.0pt0\rangle\langle
0\hskip 1.0pt|M_{k}^{\dagger}$ and $\sum_{k}M_{k}|\hskip 1.0pt1\rangle\langle
1\hskip 1.0pt|M_{k}^{\dagger}$ are the states one obtains after the action of
the decoherence channel for the state $\ket{0}$ and $\ket{1}$ respectively.

However, the oatgkp performs worse than all other channels considered here and
we account for this fact that the error channel destroys the comb-like
structure and creates superposition between the combs which is the crucial
ingredient in recovering errors for the GKP kind of states. This also
underlies the significance of comb structure for the GKP states and the effect
of these codewords for $\gamma=0.2$ is given in Fig. 5. The effect of the
channel for these two codewords are significantly different and the
distinguishability of the combs in the oatgkp is lost whereas the spingkp
still preserves the comb-like structure even after the action of the quantum
channel. This manifests in the better recovery of the spingkp compared to the
oatgkp as given in Fig. 4. The results can also be interpreted by examining
the underlying physical interactions creating these states, which limits the
range of $\delta$ that define quasiorthogonal code states for the oatgkp (see
Fig. 3).

###  IV.2 Ballistic dephasing channel

Ballistic dephasing along a single spin axis is known to destroy the spin-
squeezing generated by one-axis twisting of an orthogonal spin direction
Baamara  _et al._ (2021). Such a one-axis ballistic dephasing channel can be
written Carnio  _et al._ (2015)

$\Phi_{\sigma}(\rho)\propto\int_{-\pi}^{\pi}d\theta\,e^{-{1\over
2\sigma}\theta^{2}}e^{-i\theta J_{z}}\rho e^{i\theta J_{z}}.$ (20)

where the $x$-axis is chosen just for illustration. More details about this
channel are given in App. (D).

Here we consider an isotropic ballistic dephasing channel, a generalized
version of the one-axis ballistic dephasing channel, defined as

$\Phi_{\sigma}(\rho)\propto\int_{-\pi}^{\pi}d\theta\int_{0}^{\pi/2}d\xi\int_{0}^{2\pi}d\phi\,\sin\xi\,e^{-{\theta^{2}\over
2\sigma}}e^{-i\theta\vec{n}\cdot\vec{J}}\rho e^{i\theta\vec{n}\cdot\vec{J}}$
(21)

where $\vec{n}=(\sin\xi\cos\phi,\sin\xi\sin\phi,\cos\xi)$ is a unit vector. We
focus on the isotropic case because the isotropic phase space structure of the
spin-$N/2$ GKP codes is an advantage in this case compared to codes that are
defined relative to a specific axis, such as cat. One can view this channel,
which delocalizes whatever information isotropically in all directions as can
be seen from Fig. (7)

Figure 6: Channel fidelity for isotropic ballistic dephasing. (a) The channel
fidelity as a function of $\sigma$ for the noise model given in Eq. (21) for
$N=64$. The inset figure is the same data given in a log scale for $\sigma$.
From the figure, one can infer that the spingkp and oatgkp outperforms all
other codes. In (b) the case of small values of $\sigma$ is given to further
illustrate the performance of different spin-$N/2$ GKP code states. Thus in
the very low error limit tactgkp outperforms other codes for small noise, and
oatgkp outperforms other codes for intermediate noise values

The noise strength regime $\sigma\gtrsim N^{-2}$ is where the optimal recovery
of different spin-$N/2$ codes can be usefully compared. The optimal recovery
results for the isotropic ballistic dephasing are shown in Fig. 6 for $N=64$
(see Appendix C for code parameters associated with the optimized data in the
figure). From the figure, one can see that the spingkp and oatgkp states
outperform all other codes for the ballistic dephasing channel. When the
dephasing is only along one axis, we find that cat outperforms all other codes
we considered (see Appendix D for detailed discussion). However, when the
noise is isotropic, cat is no longer the optimal code.

Figure 7: Code states under isotropic ballistic dephasing. Husimi distribution
of the code states for the oatgkp and spingkp for the quantum channel
considered in Eq. (19) for $\sigma=0.2$. The plots are shown for
$\delta=0.2,T=5$, $\delta=0.04,T=5$, respectively for the spingkp and oatgkp.
The effect of the channel is to isotropically displace the state as can be
seen clearly from the case of the coherent state. For the case of the
spin-$N/$ GKP codes oatgkp and spingkp, the comb-like structure is preserved
which helps in the recovery of the ideal code states. This in turn leads to
the better recovery fidelity (see Fig. 6) for the GKP based codes compared to
the other codes.

To further understand the nature of the isotropic ballistic dephasing channel,
the Husimi distribution of the code states for the oatgkp and spingkp are
shown in Fig. 7 for noise strength $\sigma=0.2$. The effect of the channel is
to isotropically widen the phase space support of the code states, as can be
seen clearly from the case of the coherent state and from the general equation

$\displaystyle\text{Var}_{\Phi_{\sigma}(\rho)}\vec{m}_{1}\cdot\vec{J}$
$\displaystyle=(1-{2\sigma\over 3})\text{Var}_{\rho}\vec{m}_{1}\cdot\vec{J}$
$\displaystyle{}+{\sigma\over
3}\langle(\vec{m}_{2}\cdot\vec{J})^{2}\rangle_{\rho}+{\sigma\over
3}\langle(\vec{m}_{3}\cdot\vec{J})^{2}\rangle_{\rho}+O(\sigma^{2})$ (22)

where $\vec{m}_{1,2,3}$ is an orthonormal basis of $\mathbb{R}^{3}$. This
behavior is reminiscent of the thermal noise bosonic Gaussian channel in the
CV setting. For the oatgkp and spingkp code states, the comb-like structure is
preserved which allows high optimal channel fidelity even for moderate noise
strengths.

Although a specific recovery channel could be used to obtain a lower bound for
the channel fidelity for the noise channel (21), one can show that, for mixed
unitary channels, the correctable part of the QEC kernel is a rigorous lower
bound for the channel fidelity. Consider a quantum channel of the form
$\rho\mapsto\mathcal{E}(\rho):=\mathbb{E}_{U\sim p}U\rho U^{\dagger}$ where
the expectation is taken with respect to some probability density $p(U)$ on
the unitary group, and define the QEC kernel by

$\displaystyle\epsilon(U,U^{\prime})$
$\displaystyle=a_{U,U^{\prime}}P_{\text{code}}+b_{U,U^{\prime}}X_{\text{code}}$
$\displaystyle{}+c_{U,U^{\prime}}Y_{\text{code}}+d_{U,U^{\prime}}Z_{\text{code}}$
(23)

where the coefficient kernel $a(U,U^{\prime})={1\over
2}\text{tr}P_{\text{code}}U^{\dagger}U^{\prime}$ is designated as the
correctable part. In other words, the QEC kernel takes as input two unitary
operators and outputs a 2$\times$2 matrix. Note that channel (21) has the
mixed unitary form. We show that the expected square of the correctable part
of the QEC matrix provides a lower bound for the optimal channel fidelity. One
simply considers the recovery channel

$\mathcal{R}(\cdot)=R_{1}(\cdot)R_{1}^{\dagger}+R_{2}(\cdot)R_{2}^{\dagger}$
(24)

with $R_{1}=S^{\dagger}U^{\dagger}$ and $R_{2}$ is any other Kraus operator
that completes the channel and notes that

$\displaystyle\text{max}_{\mathcal{R}}\sum_{j}\mathbb{E}_{V\sim
p}|\text{tr}R_{j}VS\rho^{\prime}|^{2}$
$\displaystyle\geq\text{max}_{U}\mathbb{E}_{V\sim
p}|\text{tr}R_{1}VS\rho^{\prime}|^{2}$
$\displaystyle=\text{max}_{U}\mathbb{E}_{V\sim
p}|\text{tr}S^{\dagger}U^{\dagger}VS\rho^{\prime}|^{2}$
$\displaystyle\geq{1\over 4}\mathbb{E}_{(U,V)\sim p^{\times
2}}\big{|}\text{tr}U^{\dagger}VP_{\text{code}}\big{|}^{2}$
$\displaystyle\geq{1\over 4}\big{|}\mathbb{E}_{(U,V)\sim p^{\times
2}}\text{tr}U^{\dagger}VP_{\text{code}}\big{|}^{2}$ (25)

Figure 8: Channel fidelity lower bounds for isotropic ballistic dephasing.
Comparison of the channel fidelity obtained by the optimal recovery map for
the ballistic isotropic channel (solid lines) to the lower bounds on the
recovery map obtained in the second line of the Eq.(25) (color-corresponding
dashed lines).

where we refer to (53) for the notation $\rho^{\prime}$ and $S$. We also note
that the first two inequalities in (25) still hold when $\rho^{\prime}$ is
replaced by an arbitrary linear operator on $\mathbb{C}^{2}$. Since for small
noise values, the trivial recovery channel is optimal, we expect the right-
hand side of (25) to give an indication of $F_{\mathcal{E}}$ at small noise
strengths. Similarly, a rapidly vanishing expected non-correctable part with
respect to decreasing noise indicates a recovery rate rapidly approaching 1
for small noise. The benefit of analyzing (25) as a lower bound to the optimal
channel fidelity is that it does not require one to identify a cleverly chosen
recovery map. In Fig. 8, we compare the lower bound in the second line of the
(25) to the maximal channel fidelity obtainable by recovery map for the
ballistic isotropic channel.

###  IV.3 Noise Agnostic Recovery for Spin GKP

We have seen that the GKP codes for spin systems outperform other codes for
optimal recovery of specific error channels. However, neither the optimal
recovery channel solving (16) nor the optimal unitary recovery in (25) are
necessarily implementable in practice since they are obtained by optimizing
over large sets of channels acting on the symmetric subspace. Here we
introduce a recovery process that is defined irrespective of any specific
noise model (i.e., noise-agnostic), and is motivated by the comb-like
structure of the GKP codes. Similar to the continuous variable GKP recovery
process, the spin analog also follows from two steps as given in the following
circuits,

Figure 9: circuit for noise agnostic recovery

To understand how this recovery process works, we can focus on the rotation
error in the $y$ direction. First, note that the
$\ket{+}_{\mathrm{L}}\propto\ket{0}_{\mathrm{L}}+\ket{1}_{\mathrm{L}}$ has an
approximate comb structure with angular spacing $\sqrt{2\pi\over N}$ on the
circle defined at latitude $y$. An interaction between the spin ensembles
given by $e^{-2i{J_{x}\otimes J_{y}\over N}}$ can be used to approximately
imprint the rotation error onto an ancillary spin ensemble prepared in
$\ket{+}_{\mathrm{L}}$. Thus one could recover the state as long as the error
is small. Similarly, for large $N$, the state $\ket{0}_{\mathrm{L}}$ has an
approximate comb structure with angular spacing $\sqrt{2\pi\over N}$ on the
circle defined at latitude $x$ and again using the interaction
$e^{-2i{J_{x}\otimes J_{y}\over N}}$ can be used to approximately imprint the
rotation error onto an ancillary spin ensemble.

In the CV setting, one carries out a homodyne measurement to diagnose the
error. Here we introduce an analog of the homodyne measurement for the
spin-$N/2$ system defined by the positive operator-valued measurement
containing the rank one projections onto

$\\{\ket{\psi_{z}(a)}:=e^{-iaJ_{y}}e^{{z\over
2N}(J_{+}^{2}-J_{-}^{2})}\ket{N,0}\\}_{a=-\pi}^{\pi}$ (26)

(if one is considering correcting errors on tactgkp) allows to obtain a
syndrome $a$, from which a residue
$R_{\sqrt{2\pi/N}}(a)\in[-\sqrt{\pi/2N},\sqrt{\pi/2N}]$ is computed from
$R_{\sqrt{2\pi/N}}(a)=a-\lambda_{a}$, where $\lambda_{a}$ is the element of
$\\{-\lfloor\sqrt{N\pi/2}\rfloor,\ldots,-\sqrt{2\pi/N},0,\sqrt{2\pi/N},\ldots,\lfloor\sqrt{N\pi/2}\rfloor\\}$
nearest to $a$. Note that the measurement (26) is the spin-$N/2$ analog of the
position measurement for CV GKP correction of $q$-displacement errors, but
that for finite $N$, one must complete the measurement with the positive
operator $\mathbb{I}-\Pi_{z}$ where $\Pi_{z}$ is the projection onto the span
of the $\ket{\psi_{z}(a)}$ for all $a$.

To see the performance of noise agnostic recovery, consider the following
example where we have the initial state,

$\ket{\psi}_{\mathrm{initial}}=e^{{z\over 2N}(J_{+}^{2}-J_{-}^{2})}\ket{N,0}$
(27)

where $z$ is defined according to Eq. (6) with $\delta=0.2$ so that the state
is spin-squeezed. Now a unitary rotation error $e^{ibJ_{y}}$ acts on the state
and we want to see how the noise agnostic recovery performs in this simple
case. If the recovery process works, we will be able to obtain the ideal state
after the recovery process. The fidelity of the final state compared to the
initial state for both in the presence and absence of the noise agnostic
recovery is given in Fig. (10). The figure shows the fidelity of the recovered
state with (27) as a function of the spin $j$ ($j=N/2$) for the recovery
corresponding to the syndrome with the maximum probability for
$b=\sqrt{2\pi/N}/4$. From the figure, one can see the significant improvement
in fidelity when we do the recovery operation compared to the lack of
recovery, also as we go to a larger value of spin the recovered fidelity
increases, whereas in the recovery-less case, the fidelity decreases and
saturates to the value $e^{-(0.2)^{2}/4}$ predicted by the quantum central
limit theorem. This example indicates that the inter-ensemble interaction
defining the noise agnostic recoveries in (9) allows high-fidelity recovery.
Similarly, to implement the noise-agnostic recovery of a shift in $J_{y}$
expectation in the second figure of (9), one utilizes the rotated POVM
constructed from projections onto $e^{-i\pi J_{z}}\ket{\psi_{x}}$.

$\\{\ket{\phi_{z}(a)}:=e^{-i\pi J_{z}}e^{-iaJ_{y}}e^{{z\over
2N}(J_{+}^{2}-J_{-}^{2})}\ket{N,0}\\}_{a=-\pi}^{\pi}.$ (28)

Figure 10: Performance of the noise agnostic recovery. Comparison of the
fidelity of the recovered state with the state is given in Eq. (27) for a
rotation error $e^{ibJ_{y}}$ with $b=\sqrt{2\pi/N}/4$. The recovered fidelity
is calculated by identifying the syndrome with the maximum probability for the
first part of the recovery circuit given in Fig. (9). For larger values of
$j$, the recovery operation results in larger fidelity, indicative of the
success of our noise-agnostic recovery circuit (9).

More generally, the full recovery channel acting on the noisy state
$\mathcal{N}_{\gamma}(\rho)$ for the first part of the circuit in (9) is given
by,

$\displaystyle{}\mathcal{R}^{(q)}(\mathcal{N}_{\gamma}(\rho))=$
$\displaystyle{}\int_{-\pi}^{\pi}da\;e^{iR_{\sqrt{2\pi/N}}(a)J_{y}}\text{tr}_{2}\left[e^{-2i{J_{x}\otimes
J_{y}\over N}}\mathcal{N}_{\gamma}(\rho)\otimes\ket{+_{L}}\bra{+_{L}}\right.$
$\displaystyle{}\left.e^{2i{J_{x}\otimes J_{y}\over
N}}\left(\mathbb{I}\otimes\ket{\psi_{z}(a)}\bra{\psi_{z}(a)}\right)\right]e^{-iR_{\sqrt{2\pi/N}}(a)J_{y}}$
(29)

where $\mathcal{R}^{(q)}(\mathcal{N}_{\gamma}(\rho))$ becomes normalized as
$N\rightarrow\infty$ because the measurement (26) becomes complete in that
limit. Recovery channels correcting for $J_{x}$ rotation errors can be defined
act similar to (29), and concatenating the recovery channels would lead to
higher performance recovery protocols.

The noise agnostic recovery introduced in this section works in the limit of
small $\delta$ and large $N$ because in those limits it becomes equivalent to
the CV error correction circuit proposed in the original GKP paper Gottesman
_et al._ (2001). Note that when finite energy versions of the CV GKP codes are
utilized in the recent experimental implementations, it was also necessary to
adapt the original recovery method Sivak  _et al._ (2022); Royer  _et al._
(2020); de Neeve  _et al._ (2022). One can view the finite $N$ and non-
infinitesimal $\delta$ cases of the spin-$N/2$ GKP code as analogous to
finite-energy versions of the CV GKP code, and in future work, it may prove
advantageous to port noise-agnostic error correction schemes for finite-energy
CV GKP codes to the atomic ensemble setting.

##  V Preparation of GKP states using Linear combination of unitaries

Even though CV GKP states are widely considered as optimal states for error
correction of various optical decoherence models, high-fidelity preparation
protocols for these states remain a major roadblock for their use in
applications. There have been multiple approaches for preparing the CV GKP
states Terhal and Weigand (2016); Shi  _et al._ (2019). In this section we
discuss a general approach for preparing spin-$N/2$ GKP states using the well-
known technique of linear combination of unitaries (LCU), which is widely
deployed in quantum algorithms Berry  _et al._ (2015); Low and Chuang (2019);
Holmes  _et al._ (2022). The goal of the LCU is to implement a linear
combination of unitaries $X=\sum_{j}\alpha_{j}U_{j}$, with $\alpha_{j}>0$,
using an ancillary qubits and entangling interaction. The following circuit
implements the LCU,

where the action of the preparation unitary is
$U_{\mathrm{prep}}\ket{0}={1\over\sum_{j}\alpha_{j}^{2}}\sum_{j}\alpha_{j}\ket{j}$
and the select unitary defined by

$U_{\mathrm{select}}=\sum_{j}|\hskip 1.0ptj\rangle\langle j\hskip
1.0pt|\otimes U_{j}$ (30)

implements $U_{j}$ controlled on the $\ket{j}$ state of the ancilla register,
which we assume is a quantum system with discrete addressable levels. The
state $\ket{\psi}$ in the above figure is the parent state for the spin-$N/2$
GKP code and is given by a pure coherent state or spin-squeezed coherent state
for the code states discussed in this work. Thus the final state is given as,

$\ket{\Phi}=\frac{1}{\sum_{j}\alpha_{j}^{2}}\ket{0}X\ket{\psi}+\ket{\phi}$
(31)

where the state $\ket{\phi}$ has only support on the space orthogonal to the
ancilla state $\ket{0}$. Further, the oblivious amplitude amplification
techniques can be used to increase the probability of heralding the desired
state $X\ket{\psi}$ Berry  _et al._ (2015).

LCU approach is ideal for the preparation of GKP state as in general a GKP
state can be written as

$\ket{\mu}=\sum_{t=-T}^{T}\alpha_{t}U(t)R_{r}(\mu)\ket{\text{VAC}}$ (32)

for an appropriate notion of vacuum state $\ket{\text{VAC}}$ and structure
unitary $R_{r}(\mu)$ which defines the phase space squeezing structure and
logical index of the code state. The coefficients $\alpha_{t}$ define the
weights of the superposed nonorthogonal states according to the specific code
at hand. For example consider code states such as those for tactgkp and oatgkp
which have the general form

$\displaystyle\ket{\mu}$
$\displaystyle=\sum_{t=-T}^{T}\alpha_{t}e^{2itJ_{y}\sqrt{2\pi\over
N}}R_{r}(\mu)\ket{N,0}$ (33)

where the generic vacuum state has been defined as $\ket{N,0}$, the unitary
$U(t)$ is now an indexed rotation around $J_{y}$, and $R_{r}(\mu)\ket{N,0}$ is
the spin-squeezed state produced using TACT or OAT, respectively ($r$ is a
generic parameter characterizing the squeezing).

To apply the LCU method, one can write the preparation unitary
$U_{\mathrm{prep}}$ on the ancilla mode as

$U_{\mathrm{prep}}\ket{0}=\sum_{t=-T}^{T}\alpha_{t}\ket{t}$ (34)

and

$\displaystyle U_{\mathrm{select}}$
$\displaystyle=\exp\left(\frac{2i\sqrt{2\pi}}{\sqrt{N}}\mathcal{T}\otimes
J_{y}\right)$ (35) $\displaystyle=\sum_{t=-T}^{T}\ket{t}\bra{t}\otimes
e^{2it\sqrt{2\pi\over N}J_{y}}.$

The coupling to the ancilla in (35) is given here by an operator with integer
spectrum $\mathcal{T}=\sum_{t=-T}^{T}t|\hskip 1.0ptt\rangle\langle t\hskip
1.0pt|$, but could be replaced by an operator with positive integer spectrum
such as the photon number operator if the ancilla is a CV mode. This would
require a rotation of the system at the end of the LCU circuit if a certain
orientation of the code state is desired. Application of the LCU circuit,
therefore, yields

$\ket{\Phi}=\frac{1}{\sum_{t=-T}^{T}{\alpha_{t}}^{2}}\ket{0}\ket{\mu}+\ket{\phi}.$
(36)

The $U_{\mathrm{select}}$ in (35) has been implemented for the light-matter
interaction for spin systems Davis  _et al._ (2016); Li  _et al._ (2022) and
for Bose-Einstein condensates Ilo-Okeke and Byrnes (2014). The main limitation
of the above approach is the number of accessible levels in the ancilla. To
obtain a large number of combs in the phase space distribution for large atom
number $N$ requires increasing the $T$ value, and the decoherence rate scales
with the number of accessible levels in the ancilla Davis  _et al._ (2016).
Similar techniques can be implemented for the CV GKP case using the cross-Kerr
interaction as the $U_{\mathrm{select}}$ interaction Fukui  _et al._ (2022).
In this case, the correct GKP state is heralded on a vacuum (no-photon)
measurement result on the ancilla.

As an illustrative example consider the case of tactgkp(uni)  given in Eq.(5),
for this case one can find that the $U_{\mathrm{prep}}$ is proportional to a
complex Hadamard matrix, viz.,

$U_{\mathrm{prep}}=\frac{1}{\sqrt{2T+1}}\sum_{i,j=-T}^{T}\omega^{ij}|\hskip
1.0pti\rangle\langle j\hskip 1.0pt|$ (37)

where $\omega=\exp(2i\pi/(2T+1))$.

Now as a possible source of error, one can consider coherent errors acting on
the ancilla system register prior to application of $U_{\mathrm{select}}$.
This model of errors is chosen because the entangling interaction is the
hardest step of the LCU method to implement in practice. Specifically, the
select unitary is replaced by the noisy select unitary

$U_{\mathrm{select}}\to U_{\mathrm{select}}\exp(-i\epsilon
H_{\mathrm{rand}})\otimes\mathds{1}$ (38)

where $H_{\mathrm{rand}}$ is a random Hermitian operator acting on the ancilla
and we restrict the operator norm of the error-generating random Hermitian
matrices to be $T$. Fig. 11 shows the state preparation fidelity as a function
of the error parameter $\epsilon$ for the case of $T=5$, $\delta=0.1$, and
$N=64$. Numerically it is found that the state fidelity for the preparation of
$\ket{0_{L}}$ for tactgkp(uni) is large even for a significant error
$\epsilon$ and thus our approach is robust to imperfections. A similar
analysis can be done for other GKP states as well showing that the state
preparation is robust to imperfections.

Figure 11: Performance of the LCU approach for GKP state preparation. The
fidelity of a noisily-prepared tactgkp(uni) to an ideal tactgkp(uni) as a
function of the error parameter for $T=5,\delta=0.1$ and $N=64$. The fidelity
remains large even for a significant error.

A modification of the previous analysis allows one to use LCU to generate the
codes spingkp and unigkp, which have the generic form

$\ket{\nu}=\sum_{t_{1},t_{2}}\alpha_{t_{1},t_{2}}U(t_{1})V(t_{2})\ket{N,0},$
(39)

for rotations $U$ and $V$. Unlike the previous case, two ancillae are
sufficient and the LCU circuit is given by

where we have

$U_{\mathrm{prep}}\ket{0}\ket{0}={1\over\sum_{t_{1},t_{2}}\alpha_{t_{1},t_{2}}^{2}}\sum_{t_{1}}\alpha_{t_{1},t_{2}}\ket{t_{1}}\ket{t_{2}}.$
(40)

The select unitary now has two control registers

$\displaystyle U_{\mathrm{select}}$
$\displaystyle=\sum_{t_{1},t_{2}}\ket{t_{1}}\bra{t_{1}}\otimes\ket{t_{2}}\bra{t_{2}}\otimes
U_{t_{1}}V_{t_{2}}$ (41)

and the final state is given as,

$\ket{\Phi}=\frac{1}{\sum_{t_{1},t_{2}}\alpha_{t_{1},t_{2}}^{2}}\ket{0,0}X\ket{N,0}+\ket{\phi}$
(42)

where $X=\sum_{t_{1},t_{2}}\alpha_{t_{1},t_{2}}U_{t_{1}}V_{t_{2}}$.

##  VI Universal fault-tolerant quantum computation

The fact that the Clifford gates for the single-mode CV GKP code are given by
symplectic transformations of the single-mode CV phase space $\mathbb{R}^{2}$
allows one to utilize (1) to define approximate Clifford gates for the
spin-$N/2$ GKP codes. Specifically, because every symplectic transformation
corresponds to a CV unitary generated by a self-adjoint operator quadratic in
$a$ and $a^{\dagger}$, we can write the analogue of the fault-tolerant CV GKP
Clifford group generators for a spin-$N/2$ GKP code such as tactgkp:

$\displaystyle\mathrm{SUM}$ $\displaystyle=\exp\left(-2i{J_{x}\otimes
J_{y}\over N}\right)$ (43) $\displaystyle F$
$\displaystyle=\exp\left(-i\frac{\pi}{2}J_{z}\right),$ $\displaystyle P$
$\displaystyle=\exp\left(i{J_{x}^{2}\over N}\right).$

The approximate stabilizer generators are

$\displaystyle\exp\left(-2i\sqrt{\frac{2\pi}{N}}J_{x}\right)\text{ and
}\exp\left(2i\sqrt{\frac{2\pi}{N}}J_{y}\right)$ (44)

where the choice of rotation generators can be inferred from the phase space
distributions of the code states shown in Fig. 2. The $\mathrm{SUM}$ gate for
two spin-$N/2$ systems requires access to an all-to-all coupling between the
individual atoms of the two subsystems. For example, in ion traps the
availability of all-to-all couplings provides a path to implement the SUM gate
using the Mølmer-Sorenson interaction Sørensen and Mølmer (1999); Mølmer and
Sørensen (1999). Also, in a Rydberg atom system in which coherent transport of
entangled atom arrays has recently been demonstrated, this technique could be
used to implement the $\mathrm{SUM}$ gate Bluvstein  _et al._ (2022). It may
also be possible to extend methods for generating spin squeezing in a single
atomic ensemble Sørensen and Mølmer (2002); Davis  _et al._ (2016); Li  _et
al._ (2022), to a scheme in which total spin operators of two atomic ensembles
are coupled by off-resonant interaction with a single laser. Such an approach
would not require single-atom addressability, but may not be able to enter a
strongly-interacting regime.

A non-Clifford resource is required to make a universal gate set for quantum
computing. One approach is to utilize a magic state and, analogous to the
method proposed in Baragiola  _et al._ (2019), one can create an H-type magic
state for the spin-$N/2$ GKP code by initializing a spin-$N/2$ coherent state
(analog of the CV vacuum state) and using the GKP error correction protocol
using the $\mathrm{SUM}$ gate a state of the distillable H-type magic state
Bravyi and Kitaev (2005); Reichardt (2005).

The main difference between the fault-tolerant gates for the continuous
variable case to the spin case we defined here is that all the gates for the
continuous variable case are generated by symplectic interactions whereas, for
the case of the spin case, we need $\mathrm{SU(2)}$ plus one non-linear
interaction for the $P$ gate.

###  VI.1 Propagation of errors

To show the feasibility of the fault-tolerant gates for computation with
spin-$N/2$ GKP states, one needs to determine whether the gates amplify errors
present in a specific implementationGrimsmo  _et al._ (2020). For simplicity,
we focus on coherent rotation errors,

$\displaystyle M_{\theta}$ $\displaystyle=\exp(-i\theta J_{x}),$ (45)
$\displaystyle N_{\phi}$ $\displaystyle=\exp(-i\phi J_{y}).$

Now consider the $F$ gate, we get that,

$\displaystyle FM_{\theta}F^{\dagger}$
$\displaystyle=\exp\left(-i\frac{\pi}{2}J_{z}\right)M_{\theta}\exp\left(i\frac{\pi}{2}J_{z}\right)$
$\displaystyle=N_{-\theta},$

and similarly

$FN_{\phi}F^{\dagger}=\exp(-i\phi J_{x})=M_{\phi}.$ (46)

Concatenating the noise-agnostic recovery channels $\mathcal{R}^{(q)}$,
$\mathcal{R}^{(p)}$ of Section IV.3 allows to correct for small $J_{x}$ and
$J_{y}$ errors, so the error propagation is controllable.

Next consider the gate$P$, it commutes with $M_{\theta}$ and the effect of $P$
on $N_{\phi}$ is given as,

$\displaystyle PN_{\phi}P^{\dagger}$ $\displaystyle=\exp\left(i{J_{x}^{2}\over
N}\right)N_{\phi}\exp\left(-i{J_{x}^{2}\over N}\right).$ (47)

Using the canonical commutation relationship between the continuous variables
$q$ and $p$, and the isometry $V$ from (1) we get for large $N$,

$\displaystyle Ve^{i{J_{x}^{2}\over N}}e^{-i\phi J_{y}}e^{-i{J_{x}^{2}\over
N}}V^{\dagger}$ $\displaystyle\sim e^{iq^{2}/2}e^{-i\phi\sqrt{N\over
2}p}e^{-iq^{2}/2}$ $\displaystyle VN_{\phi}M_{-\phi}V^{\dagger}$
$\displaystyle\sim e^{-i\phi\sqrt{N\over 2}p}e^{i\phi\sqrt{N\over 2}q}.$ (48)

But the operators on the right-hand sides of these equations are equal up to
an overall phase. We conclude that for large $N$, the $N_{\phi}$ error
propagates through the $P$ gate by introducing $N_{\phi}M_{-\phi}$ errors,
which are again correctable by concatenation of the recovery channels
$\mathcal{R}^{(q)}$, $\mathcal{R}^{(p)}$.

Next, first noting that certain local $M_{\theta}$ and $N_{\phi}$ errors on
the SUM gate are benign due to

$\displaystyle\mathrm{SUM}\left(M_{\theta}\otimes\mathds{1}\right)\mathrm{SUM}^{\dagger}$
$\displaystyle=0,$ (49) $\displaystyle\mathrm{SUM}\left(\mathds{1}\otimes
N_{\phi}\right)\mathrm{SUM}^{\dagger}$ $\displaystyle=0,$

we find that

$\displaystyle
V\mathrm{SUM}\left(N_{\phi}\otimes\mathds{1}\right)\mathrm{SUM}^{\dagger}V^{\dagger}\sim
e^{-i\sqrt{N\over 2}(p\otimes\mathbb{I}+\mathbb{I}\otimes p)}$ $\displaystyle
VN_{\phi}^{\otimes 2}V^{\dagger}\sim e^{-i\phi\sqrt{N\over 2}p}\otimes
e^{-i\phi\sqrt{N\over 2}p}$ (50)

so the effect of the $N_{\phi}$ error on the SUM gate is to fan out the error
to another atom ensemble register. If the other register is also a spin-$N/2$
GKP qubit, the local errors are individually correctable. An analogous
conclusion holds for the $M_{\phi}$ error.

The final ingredient in the fault-tolerant quantum computation scheme is magic
state preparation, which involves initial preparation of a spin coherent state
followed by an error correction scheme composed of SUM gates. Therefore, the
possibility of controlling error propagation in the SUM gate indicates the
possibility of fault-tolerant magic state preparation, at least for coherent
rotation errors. Thus all the gadgets in the fault-tolerant scheme for the
spin-$N/2$ GKP code propagate these errors in a controllable manner.

##  VII Discussion

CV GKP codes are considered as primary candidates for implementing fault-
tolerant, scalable quantum computation because of the natural robustness of
these codes to errors in CV bosonic systems. In this paper, we utilized spin
squeezing interactions and the quantum central limit theorem to develop a
notion of GKP codes for atomic ensembles, which are candidates for large scale
quantum information processing with ultracold matter. In particular, our code
states exhibit a well-defined comb-like phase space structure similar to the
CV GKP states, and the free parameters defining the comb-like structure can be
optimized to obtain high optimal recovery rates under the stochastic
relaxation noise and ballistic dephasing. The stochastic relaxation channel is
analogous to the CV amplitude damping process and the isotropic ballistic
dephasing noise is analogous to CV Gaussian thermal noise. For the stochastic
relaxation channel, the spinGKP code is the spin-$N/2$ GKP code allowing the
best optical recovery, and it further outperforms the standard bosonic codes
even for intermediate noise strengths. For isotropic ballistic dephasing,
oatgkp, which presently has no well-studied CV GKP analogue, outperforms other
bosonic codes for intermediate noise strength values. We studied how to use
techniques from quantum algorithms to create the spin-$N/2$ GKP states, as
there is no fundamental atomic interaction that generates the GKP state. We
used the linear combination of unitaries (LCU) to prepare the spin-$N/2$ GKP
code states with the only controlled being rotations. This approach is in
principle adaptable to CV GKP and other non-trivial state preparations which
are superpositions of Gaussian states or approximately Gaussian states with
fixed interaction strength. Finally, we again used the quantum central limit
theorem to outline the correspondence between a gate set allowing universal,
fault-tolerant computation with CV GKP-encoded qubits and a gate set with
generators given by total spin operators of a spin-$N/2$ system. For
sufficiently large $N$, these gates provide a route to universal, fault-
tolerant quantum computation using a spin-$N/2$ GKP-encoded qubits.

In this work, we have focused on atomic ensembles which can be prepared in
$\mathrm{SU}(2)$ coherent states, which are relevant when the atoms are
distributed among two orthogonal modes (e.g., internal or motional). Similar
codes can be defined for $\mathrm{SU}(d)$ coherent states, which are relevant
in the case of cold atoms with a larger number of accessible internal states
or distributed in an optical trapping latticeSmith  _et al._ (2013);
Omanakuttan  _et al._ (2022, 2021); Zache  _et al._ (2023). Analyses of such
GKP codes for extended ultracold matter are an important direction for future
studies. We also expect that concatenation of spin-$N/2$ GKP codes with
standard surface code schemes potentially provides a route to fault-tolerant
quantum computation with ultracold atoms, analogous to proposals for
incorporating CV GKP codes in larger QEC schemes. Detailed threshold analyses
and comparison to state-of-the-art atomic gates are necessary to substantiate
this claim as a realistic proposal.

Lastly, we note that spin-$N/2$ GKP codes are not the only quantum error
correction codes for atomic ensembles in the symmetric subspace. Such
symmetric logical qubits have previously been considered as possible
candidates to achieve fault-tolerant quantum computation Ouyang (2014); Ouyang
and Fitzsimons (2016); Omanakuttan and Gross (2023). In Ouyang (2014), the
ground state space of Heisenberg ferromagnets was used as physical motivation
for introducing permutation-invariant codes for ensembles of $N$ atoms. In
contrast, the present work was motivated by the relationship between
quadrature squeezing in infinite-dimensional CV systems and spin squeezing in
the symmetric subspace of $N$ two-level atoms. This relationship is made
rigorous by the QCLT, which we further utilized in translating noise-agnostic
error correction protocols and fault-tolerant gates from the CV GKP setting.
Because the relevant spin-squeezing interactions have already been implemented
experimentally, and because several methods for utilizing CV GKP codes for
fault-tolerant measurement-based CV quantum computation have now been proposed
Bourassa  _et al._ (2021); Noh and Chamberland (2020); Noh  _et al._ (2022);
Larsen  _et al._ (2021); Fukui  _et al._ (2018); Fukui (2023), it is
interesting to consider the possibility of implementing spin-$N/2$ GKP codes
in schemes for fault-tolerant quantum computation in atomic ensembles.

###### Acknowledgements.

The authors thank Victor Albert for discussions about bosonic codes and
optimal recovery methods as well as other useful discussions. S.O. would like
to acknowledge fruitful discussions with Tyler G Thurtell, Austin K Daniel,
and Karthik Chinni during various stages of work. T.J.V. acknowledges support
from the Laboratory Directed Research and Development (LDRD) Program at Los
Alamos National Laboratory (LANL). S.O. acknowledges the support by the
Laboratory Directed Research and Development program of Los Alamos National
Laboratory under Project No. 20200015ER and by the U.S. Department of Energy,
Office of Science, National Quantum Information Science Research Centers,
Quantum Systems Accelerator. LANL is operated by Triad National Security, LLC,
for the National Nuclear Security Administration of U.S. Department of Energy
(Contract No. 89233218CNA000001).

##  Appendix A Diversity combining method for optimal recovery

Here we provide an overview of our implementation of the diversity combining
method for finding the recovery channel that maximizes the channel fidelity
Fletcher  _et al._ (2007). Let the logical code states be elements of a
$d$-dimensional Hilbert space $\mathcal{H}$. The diversity combining method
offers a quadratic advantage in the size of the semidefinite program (from
$d^{4}$ to $4d^{2}$ optimized parameters) compared to a standard optimization
of the entanglement fidelity because the logical states are replaced by qubit
states. Consider the ensemble of code states: $\rho={1\over
2}\ket{0_{L}}\bra{0_{L}}+{1\over 2}\ket{1_{L}}\bra{1_{L}}$, where we assume
$\langle 0_{L}|1_{L}\rangle=0$ (if the code states are not orthogonal, use the
Gram-Schmidt procedure to define orthonormal code states $\ket{0_{L}}$ and
$\ket{1_{L}}$). The encoding isometry is defined as
$S=\ket{0_{L}}\bra{0}+\ket{1_{L}}\bra{1}$. We consider the following error
channel:

$\mathcal{N}_{\gamma}\circ\mathcal{S}\circ\mathcal{S}^{\dagger}(\rho)=\mathcal{N}_{\gamma}(SS^{\dagger}\rho
SS^{\dagger}).$ (51)

Since $SS^{\dagger}=P_{\text{code}}$, the above error map is equal to
$\mathcal{N}_{\gamma}(\rho)$ (note that $P_{\text{code}}\rho
P_{\text{code}}=\rho$). Note that

$\rho^{\prime}:=\mathcal{S}^{\dagger}(\rho)=S^{\dagger}\rho S={1\over
2}\ket{0}\bra{0}+{1\over 2}\ket{1}\bra{1}$ (52)

is a 2$\times$2 matrix. The map $\mathcal{N}_{\gamma}\circ\mathcal{S}$ has
Kraus operators $\\{E_{\ell}S\\}_{\ell}$, where each Kraus operator is a
$d\times 2$ matrix, and one can see that the map in (51) is
$\mathcal{N}_{\gamma}\circ\mathcal{S}(\rho^{\prime})$. We implement recovery
for $\mathcal{N}_{\gamma}\circ\mathcal{S}$ by a recovery channel $\mathcal{R}$
with Kraus operators $R_{j}$ which are $2\times d$ matrices (so $\mathcal{R}$
takes the noisy encoded state back to the qubit space). Then any purification
$\ket{\psi_{\rho^{\prime}}}$ of $\rho^{\prime}$ satisfies

$\langle\psi_{\rho^{\prime}}|(\mathcal{R}\circ\mathcal{N}_{\gamma}\circ\mathcal{S})\otimes\mathbb{I}\left(\ket{\psi_{\rho^{\prime}}}\bra{\psi_{\rho^{\prime}}}\right)|\psi_{\rho^{\prime}}\rangle=\sum_{j,\ell}|\text{tr}R_{j}E_{\ell}S\rho^{\prime}|^{2}.$
(53)

The latter expression is expressed as an inner product of Choi matrices by
using the vec map from $\mathcal{H}\otimes\mathcal{H}^{\text{dual}}$ to
$\mathcal{H}^{\otimes 2}$ given by
$\textbf{vec}E_{i,j}=|E_{i,j}\rangle\rangle:=\ket{i}\ket{j}$, where
$E_{i,j}:=\ket{i}\bra{j}$. Write (53) as

$\displaystyle\sum_{j,\ell}|\text{tr}R_{j}E_{\ell}S\rho^{\prime}|^{2}$
$\displaystyle=\sum_{j,\ell}\langle\langle\rho^{\prime}|R_{j}E_{\ell}S\rangle\rangle\langle\langle
R_{j}E_{\ell}S|\rho^{\prime}\rangle\rangle$
$\displaystyle{}=\sum_{j,\ell}\langle\langle\rho^{\prime}|(\mathbb{I}\otimes
S^{T}E_{\ell}^{T})|R_{j}\rangle\rangle\langle\langle R_{j}|(\mathbb{I}\otimes
E_{\ell}^{*}S^{*})|\rho^{\prime}\rangle\rangle$
$\displaystyle=\sum_{\ell}\langle\langle\rho^{\prime}S^{\dagger}E_{\ell}^{\dagger}|J_{\mathcal{R}}|\rho^{\prime}S^{\dagger}E_{\ell}^{\dagger}\rangle\rangle$
$\displaystyle=\text{tr}C_{\rho^{\prime},\mathcal{N}_{\gamma}}J_{\mathcal{R}}$
(54)

with
$C_{\rho^{\prime},\mathcal{N}_{\gamma}}:=\sum_{\ell}|\rho^{\prime}S^{\dagger}E_{\ell}^{\dagger}\rangle\rangle\langle\langle\rho^{\prime}S^{\dagger}E_{\ell}^{\dagger}|$.
Note that $J_{\mathcal{R}}$ is a $2d\times 2d$ Hermitian matrix and that
$\text{tr}C_{\rho^{\prime},\mathcal{N}_{\gamma}}=1/2$.

Therefore, the maximal channel fidelity

$\displaystyle
F_{\mathcal{N}_{\gamma}}=\max_{\mathcal{R}}\langle\psi_{\rho^{\prime}}|(\mathcal{R}\circ\mathcal{N}_{\gamma}\circ\mathcal{S})\otimes\mathbb{I}\left(\ket{\psi_{\rho^{\prime}}}\bra{\psi_{\rho^{\prime}}}\right)|\psi_{\rho^{\prime}}\rangle$

is given by the solution to the semidefinite program

$\displaystyle\underset{X}{\text{maximize}}$
$\displaystyle\text{tr}(C_{\rho^{\prime},N_{\gamma}}X)$ (55) subject to
$\displaystyle X>0$ $\displaystyle\text{tr}_{1}X=\mathbb{I}_{d}$
$\displaystyle\underset{Y}{\text{minimize}}$ $\displaystyle\text{tr}(Y)$ (56)
subject to $\displaystyle-
C_{\rho^{\prime},\mathcal{N}_{\gamma}}+\mathbb{I}_{2}\otimes Y>0$

over $d\times d$, Hermitian $Y$.

Note that if one considers the Kraus operators
$R_{s,j}:=\\{{1\over\sqrt{2}}\ket{s}\bra{j}\\}_{s=1,2;j=1,\ldots,d}$ for the
recovery channel $\mathcal{R}$, one has
$\sum_{s,j}R_{s,j}^{\dagger}R_{s,j}=\sum_{s,j}\text{tr}_{1}|R_{s,j}\rangle\rangle\langle\langle
R_{s,j}|=\mathbb{I}_{d}$, as required. Since
$\sum_{s,j}\text{tr}_{1}|R_{s,j}\rangle\rangle\langle\langle R_{s,j}|={1\over
2}\mathbb{I}_{2}\otimes\mathbb{I}_{d}$, it follows that the maximum value of
the above semidefinite program is at least $1/4$. Physically, one can see that
this lower bound is attained by considering the identity noise channel and the
recovery that randomly sends each of the orthogonal code states to $\ket{0}$,
$\ket{1}$ with equal probability.

##  Appendix B Properties of stochastic relaxation channel

We have the stochastic relaxation channel given as in (19):

$T_{t}(\rho)=\sum_{\ell=0}^{N}{(1-e^{-\lambda
t})^{\ell}\over\ell!\lambda^{\ell}}e^{-{\lambda t\over
4}(N-2J_{z})}(\mathcal{S}^{(1)})^{\ell}(\rho)e^{-{\lambda t\over
4}(N-2J_{z})},$ (57)

where we have defined,

$\displaystyle{}\mathcal{S}^{(1)}(\ket{N-m,m}\bra{N-m^{\prime},m^{\prime}})$
$\displaystyle{}=\lambda
J_{+}{\ket{N-m,m}\bra{N-m^{\prime},m^{\prime}}\over\sqrt{(N-m+1)(N-m^{\prime}+1)}}J_{-}.$
(58)

To show that this map is trace-preserving, it suffices to verify that
$\text{tr}\left[T_{t}(\ket{N-m,m}\bra{N-m^{\prime},m^{\prime}})\right]=\delta_{m,m^{\prime}}$.
For $m\neq m^{\prime}$, it is clear that
$\text{tr}\left[T_{t}(\ket{N-m,m}\bra{N-m^{\prime},m^{\prime}})\right]=0$
because

$\displaystyle{}J_{+}^{\ell}\ket{N-m,m}\bra{N-m^{\prime},m^{\prime}}J_{-}^{\ell}\propto$
$\displaystyle{}\ket{N-m+\ell,m-\ell}\bra{N-m^{\prime}+\ell,m^{\prime}-\ell}.$
(59)

For $m=m^{\prime}$, one has

$\displaystyle\text{tr}\left[T_{t}(\ket{N-m,m}\bra{N-m,m})\right]$
$\displaystyle=\text{tr}\left[\sum_{\ell=0}^{N}{(1-e^{-\lambda
t})^{\ell}\over\ell!(N-m+1)\cdots(N-m+\ell)}e^{-{\lambda t\over
4}(N-2J_{z})}J_{+}^{\ell}\ket{N-m,m}\bra{N-m,m}J_{-}^{\ell}e^{-{\lambda t\over
4}(N-2J_{z})}\right]$
$\displaystyle=\text{tr}\left[\sum_{\ell=0}^{N}{(1-e^{-\lambda
t})^{\ell}\over\ell!}{m!\over m-\ell!}e^{-{\lambda t\over
4}(N-2J_{z})}\ket{N-m+\ell,m-\ell}\bra{N-m+\ell,m-\ell}e^{-{\lambda t\over
4}(N-2J_{z})}\right]$
$\displaystyle=\text{tr}\left[\sum_{\ell=0}^{N}(1-e^{-\lambda
t})^{\ell}{m\choose\ell}e^{-\lambda
t(m-\ell)}\ket{N-m+\ell,m-\ell}\bra{N-m+\ell,m-\ell}\right]$
$\displaystyle=\sum_{\ell=0}^{m}(1-e^{-\lambda
t})^{\ell}{m\choose\ell}e^{-\lambda t(m-\ell)}$ $\displaystyle=e^{-\lambda
tm}\sum_{\ell=0}^{m}{m\choose\ell}(e^{\lambda t}-1)^{\ell}$ $\displaystyle=1.$
(60)

We now consider the $\ell=0$ and $\ell=1$ contributions to the channel in the
limit of small $\gamma:=t\lambda$ and find that up to linear order in
$\gamma$, one has $\sum_{\ell=0}^{1}E_{\ell}^{\dagger}E_{\ell}=\mathbb{I}$,
where

$\displaystyle E_{0}$ $\displaystyle=\begin{pmatrix}1&\\\ &1-{\gamma\over
2}&\\\ &&1-{2\gamma\over 2}&\\\ &&&\ddots&\\\ &&&&1-{N\gamma\over
2}\end{pmatrix},E_{1}$ $\displaystyle=\sqrt{\gamma}\begin{pmatrix}0&1&\\\
&0&\sqrt{2}&\\\ &&\ddots&\ddots&\\\ &&&0&\sqrt{N}\\\ &&&&0\end{pmatrix}.$

are matrices on the $N+1$ dimensional symmetric subspace. Embedding the
symmetric subspace into two CV oscillator modes according to
$\ket{N-m,m}\mapsto\ket{N-m}\otimes\ket{m}$, one can write

$\displaystyle E_{0}$
$\displaystyle=\mathbb{I}-\frac{\gamma}{2}\mathbb{I}_{1}\otimes
a_{2}^{\dagger}a_{2}$ $\displaystyle E_{1}$
$\displaystyle=\sqrt{\gamma}R\otimes a_{2}.$ (61)

where $a_{1},a_{2}$ are the respective annihilation operators of the CV modes
and $R$ is the unilateral shift $R\ket{n}=\ket{n+1}$. Therefore, for small
$\gamma$, one can think of the channel $T_{t}$ as an attenuation of the second
mode which still conserves the global total particle number.

##  Appendix C Numerical Benchmarking details

$\gamma$ |  spingkp ($\delta$) |  oatgkp ($\delta$) |  tactgkp($\delta$) |  unigkp ($T$)
---|---|---|---|---
0.0100 | 0.2926 | 0.0777 | 0.2715 | 3
0.0303 | 0.3055 | 0.0785 | 0.2746 | 3
0.0505 | 0.3110 | 0.0790 | 0.2746 | 3
0.0708 | 0.3096 | 0.0794 | 0.2732 | 3
0.0910 | 0.3067 | 0.0797 | 0.2717 | 3
0.1113 | 0.3035 | 0.0799 | 0.2698 | 3
0.1315 | 0.3002 | 0.0801 | 0.2678 | 3
0.1518 | 0.2970 | 0.0803 | 0.2657 | 2
0.1721 | 0.2938 | 0.0804 | 0.2657 | 2
0.1923 | 0.2909 | 0.0805 | 0.2628 | 2
0.2126 | 0.2882 | 0.0806 | 0.2618 | 2
0.2328 | 0.2858 | 0.0807 | 0.2613 | 2
0.2531 | 0.2838 | 0.0807 | 0.2616 | 2
0.2733 | 0.2824 | 0.0807 | 0.2627 | 2
0.2936 | 0.2818 | 0.0807 | 0.2650 | 2
0.3138 | 0.2821 | 0.0807 | 0.2687 | 2
0.3341 | 0.2835 | 0.0807 | 0.2739 | 2
0.3544 | 0.2862 | 0.0807 | 0.2815 | 2
0.3746 | 0.2904 | 0.0808 | 0.2922 | 2
0.3949 | 0.2963 | 0.0808 | 0.3084 | 2
0.4151 | 0.3045 | 0.0808 | 0.3349 | 2
0.4354 | 0.3153 | 0.0808 | 0.3725 | 2
0.4556 | 0.3301 | 0.0808 | 0.3991 | 1
0.4759 | 0.3509 | 0.0808 | 0.3999 | 1
0.4962 | 0.3816 | 0.0809 | 0.3999 | 1
0.5164 | 0.4000 | 0.0809 | 0.3999 | 1
0.5367 | 0.4000 | 0.0809 | 0.4000 | 1
0.5569 | 0.4000 | 0.0810 | 0.4000 | 1
0.5772 | 0.4000 | 0.0811 | 0.4000 | 1
0.5974 | 0.4000 | 0.0814 | 0.4000 | 1
0.6177 | 0.4000 | 0.0817 | 0.4000 | 1
0.6379 | 0.4000 | 0.0821 | 0.4000 | 1
0.6582 | 0.4000 | 0.0825 | 0.4000 | 1
0.6785 | 0.4000 | 0.0829 | 0.4000 | 1
0.6987 | 0.4000 | 0.0834 | 0.4000 | 1
0.7190 | 0.4000 | 0.0839 | 0.4000 | 1
0.7392 | 0.4000 | 0.0843 | 0.4000 | 1
0.7595 | 0.4000 | 0.0848 | 0.4000 | 1
0.7797 | 0.4000 | 0.0852 | 0.4000 | 1
0.8000 | 0.4000 | 0.0855 | 0.4000 | 1
Table 1: The optimized code parameters for the stochastic relaxation channel
using diversity combining approaches for the data Fig. 4 for $N=64$. For the
case of the unigkp we optimize over various choices of the $1\leqslant
T\leqslant 5$ and the best possible $T$ is chosen. For the codes
$\\{\texttt{spingkp},\texttt{oatgkp},\texttt{tactgkp}\\}$ we choose $T=5$, and
for the case of the spingkp and tactgkp we restrict the value of $\delta$ to
be with $0\leqslant\delta\leqslant 1$ whereas for the case of the oatgkp the
value of $\delta$ is restricted to $0\leqslant\delta\leqslant 1/\sqrt{N}$. The
restrictions are made such that the code states retain the comb-like structure
essential for the GKP decoding process and the optimal value occurs
approximately when the two code words become orthogonal as shown in Fig. (3).

The value of the parameters that optimize the channel fidelity
$F_{\mathcal{E}}$ for the stochastic relaxation channel data in Fig. 4 is
given in Table 1. The codewords we are interested here are are the
$\\{\texttt{spingkp},\texttt{oatgkp},\texttt{tactgkp},\texttt{unigkp}\\}$. For
the case of the unigkp we optimize over various choices of the $1\leqslant
T\leqslant 5$ and the best possible $T$ is chosen. For the codes
$\\{\texttt{spingkp},\texttt{oatgkp},\texttt{tactgkp}\\}$ we choose $T=5$, and
for the case of the spingkp and tactgkp we restrict the value of $\delta$ to
be with $0\leqslant\delta\leqslant 1$ whereas for the case of the oatgkp the
value of $\delta$ is restricted to $0\leqslant\delta\leqslant 1/\sqrt{N}$. The
parameter domain restrictions are made such that the comb-like structure of
the phase space support of the code states persists, which is essential for
the GKP decoding process.

$\sigma$ |  spingkp($\delta$) |  oatgkp($\delta$) |  tactgkp($\delta$) |  unigkp($T$)
---|---|---|---|---
0.0010 | 0.2200 | 0.0682 | 0.2200 | 3
0.0012 | 0.2200 | 0.0682 | 0.2200 | 3
0.0014 | 0.2200 | 0.0682 | 0.2200 | 3
0.0017 | 0.2200 | 0.0682 | 0.2200 | 3
0.0020 | 0.2200 | 0.0682 | 0.2200 | 3
0.0024 | 0.2200 | 0.0682 | 0.2200 | 3
0.0029 | 0.2200 | 0.0682 | 0.2200 | 3
0.0035 | 0.2200 | 0.0682 | 0.2200 | 3
0.0041 | 0.2200 | 0.0682 | 0.2200 | 3
0.0049 | 0.2200 | 0.0682 | 0.2200 | 3
0.0059 | 0.2200 | 0.0682 | 0.2200 | 3
0.0070 | 0.2200 | 0.0682 | 0.2200 | 3
0.0084 | 0.2200 | 0.0682 | 0.2200 | 3
0.0100 | 0.2196 | 0.0682 | 0.2200 | 3
0.0119 | 0.2183 | 0.0682 | 0.2196 | 3
0.0143 | 0.1926 | 0.0681 | 0.1926 | 3
0.0170 | 0.2200 | 0.0682 | 0.2049 | 3
0.0203 | 0.2201 | 0.0668 | 0.2178 | 3
0.0242 | 0.1925 | 0.0629 | 0.2023 | 3
0.0289 | 0.2116 | 0.0682 | 0.2062 | 3
0.0346 | 0.2201 | 0.0597 | 0.2113 | 3
0.0412 | 0.2187 | 0.0673 | 0.2171 | 3
0.0492 | 0.2199 | 0.0665 | 0.2199 | 3
0.0588 | 0.2178 | 0.0511 | 0.2178 | 3
0.0702 | 0.2199 | 0.0544 | 0.2199 | 3
0.0838 | 0.2193 | 0.0683 | 0.2308 | 3
0.1000 | 0.2199 | 0.0685 | 0.2299 | 3
0.1194 | 0.2525 | 0.0683 | 0.2194 | 3
0.1425 | 0.2157 | 0.0683 | 0.2199 | 3
0.1701 | 0.2393 | 0.0683 | 0.2289 | 2
0.2031 | 0.2428 | 0.0630 | 0.2202 | 3
0.2424 | 0.2135 | 0.0709 | 0.2173 | 2
0.2894 | 0.2362 | 0.0702 | 0.2141 | 1
0.3455 | 0.2221 | 0.0513 | 0.1980 | 1
0.4125 | 0.1799 | 0.0707 | 0.1652 | 1
0.4924 | 0.1909 | 0.0751 | 0.1752 | 1
0.5878 | 0.1925 | 0.0731 | 0.1652 | 1
0.7017 | 0.1925 | 0.0751 | 0.1864 | 1
0.8377 | 0.1652 | 0.0676 | 0.1787 | 1
1.000 | 0.2205 | 0.0682 | 0.2013 | 1
Table 2:  The optimized parameters for the isotropic ballistic dephasing
channel using diversity combining approaches given in Fig. 6 for $N=64$. For
the case of the spingkp we restrict the value of $\delta$ to be with
$0\leqslant\delta\leqslant 1$ whereas for the case of the oatgkp the value of
$\delta$ is restricted to $0\leqslant\delta\leqslant 1/\sqrt{N}$. For both
cases, we studied the case of $T=5$. The restrictions are made such that the
code states retain the comb-like structure essential for the GKP decoding
process and the optimal value occurs approximately when the two code words
become orthogonal as shown in Fig. (3).

The value of the parameters that optimizes the $F_{\mathcal{E}}$ for the
isotropic ballistic dephasing channel in Fig. 6 is given in Table 2. The
codewords we are interested here are are the
$\\{\texttt{spingkp},\texttt{oatgkp},\texttt{tactgkp}\\}$. For the case of the
spingkp we restrict the value of $\delta$ to be with
$0\leqslant\delta\leqslant 1$ whereas for the case of the oatgkp the value of
$\delta$ is restricted to $0\leqslant\delta\leqslant 1/\sqrt{N}$. For both
cases, we studied the case of $T=5$. The restrictions are made such that we
are in a regime where we still have the comb-like structure essential for the
GKP decoding process

##  Appendix D Ballistic dephasing along one axis

A one-axis ballistic dephasing, which describes a Hamiltonian with fluctuating
energy levels and studied in detail in Baamara  _et al._ (2021) is defined by
the following mixed-unitary channel in Eq. (20). For $\sigma\ll 1$, the effect
of this channel can be seen by considering, e.g., the observable $J_{x}$, and
noting that

$\displaystyle\text{Var}_{\Phi_{\sigma}(\rho)}J_{x}=(1-\sigma)\text{Var}_{\rho}J_{x}+\sigma\langle
J_{y}^{2}\rangle_{\rho}+O(\sigma^{2}).$ (62)

Consider the case when the dephasing occurs about the $z$-axis and write the
Kraus operators as,

$E_{\theta}=e^{-\frac{\theta^{2}}{4\sigma}}e^{-i\theta J_{z}}.$ (63)

When $\sigma$ is taken to be small, one can focus on the $\theta\approx 0$
limit. In this limit, the quantum error correction condition depends on only
the first and second moments of the total spin operator $J_{z}$. To identify
codes that have large value of the correctable part of the QEC kernel (23) in
the small $\sigma$ limit, we expand about $\theta,\theta^{\prime}=0$:

$\displaystyle
P_{\mathrm{code}}E_{\theta}^{\dagger}E_{\theta^{\prime}}P_{\mathrm{code}}$
$\displaystyle=e^{-\frac{\theta^{2}+\theta^{\prime
2}}{4\sigma}}P_{\mathrm{code}}$ (64)
$\displaystyle-\frac{1}{2}e^{-\frac{\theta^{2}+\theta^{\prime
2}}{4\sigma}}(\theta-\theta^{\prime})^{2}P_{\mathrm{code}}J_{z}^{2}P_{\mathrm{code}}$
$\displaystyle+\mathcal{O}(\theta^{3})+\mathcal{O}(\theta^{\prime 3}),$

where $P_{\mathrm{code}}=|\hskip 1.0pt0_{L}\rangle\langle 0_{L}\hskip
1.0pt|+|\hskip 1.0pt1_{L}\rangle\langle 1_{L}\hskip 1.0pt|$ and we have
assumed that the code under consideration has no first $J_{z}$ moment, i.e.,
$P_{\mathrm{code}}J_{z}P_{\mathrm{code}}=0$. Codes for which the code states
are not connected by pair tunneling between single-particle modes will retain
a diagonal matrix in the second line of Eq.(64), indicating greater
correctability. Examples include the coherent state codes
$\ket{J,\vec{n}\cdot\vec{J}=\frac{N}{2}}$ where $\vec{n}$ is a unit vector in
the $xy$-plane, or cat codes along direction $\vec{n}$.

We now examine the optimal recovery properties of spin-$N/2$ GKP codes under
one-axis ballistic dephasing. In Fig. 12, the effect of this channel for the
case of dephasing along $z$ direction for $\gamma=0.2$ is shown. The effect of
the noise channel is similar for the spingkp and oatgkp, in particular, the
code states have more delocalized support along an axis in phase space.
However, the comb-like structure is maintained, leading to high optimal
recovery for both these codes. The spreading of the comb structure causes the
fidelity to decay faster compared to the stochastic relaxation case.

Figure 12: Husimi distribution for ballistic dephasing. Husimi distribution of
the codewords for the oatgkp and spingkp for the quantum channel considered in
Eq. (20) for $\gamma=0.2$. The plots are shown for $\delta=0.2$ and
$\delta=0.04$ respectively for the spingkp and oatgkp. The effect of the
channel for these two codewords are similar in nature for the spingkp and
oatgkp. We can see the effect of this quantum channel for these codewords is
to make the codewords more delocalized in phase space; while still maintaining
the comb-like structure. The persistence of these comb-like structures makes
the recovery accurate for both these codes. But the spreading of the comb
structure makes the fidelity decay faster compared to the stochastic
relaxation case.

Fig. 13 shows the optimal recovery performance of various spin-$N/2$ codes
with respect to the one-axis ballistic dephasing channel for $N=64$. A cutoff
$T=5$ is used for all codes except unigkp codes where we all optimize for all
values of $T$. The figure shows that the spin-$N/2$ codes outperform binomial
and rail for a large range of $\sigma$, however, cat has the highest optimal
recovery performance as predicted from calculation in (64).

Figure 13: Channel fidelity for ballistic dephasing. The channel fidelity as a
function of $\sigma$ for the noise model $\mathcal{E}=\Phi_{\sigma}$ given in
Eq. (20). The ballistic dephasing along $z$ direction is given for different
code words with $N=64$ and cut off, $T=5$ for all codes except unigkp codes
where we all optimize for all values of $T$ . The figure shows that the GKP
codes outperform the binomial code and rail code for large range of $\sigma$,
however the cat state is the best as predicted from the QEC matrix calculation
in Eq.(64).

##  Appendix E  Optimal recovery for CV GKP codes

Figure 14:  The position space wavefunctions for the code states for all four
codes are given. From the figure, one can see the well-separated comb-like
structure for all these four codewords. For the equal energy codewords, the
position space peaks for the logical $\ket{0}_{\mathrm{L}}$ and
$\ket{1}_{\mathrm{L}}$ are in one-to-one correspondence. The square grid equal
energy code is taken to have energy $5$ and $T=8$; the vertical lines are
shown at ${\sqrt{\pi}\over 2}$ and ${\sqrt{\pi}\over 2}+2\sqrt{\pi}$ for
reference.

There is not a unique finite-energy regularization scheme for the CV GKP code,
and a specific convention can be chosen with regard to accessible experimental
state preparation schemes Matsuura  _et al._ (2020). However, at a fixed
energy the finite-energy GKP codes can have different optimal recovery
properties. We define four different kinds of finite energy CV GKP codes.
First, we consider Gottesman’s original finite energy code CVGKP,

$\displaystyle|\mu_{\lambda,d}\rangle$
$\displaystyle\propto\sum_{t\in\mathbb{Z}}e^{-{\pi\over
2}\lambda^{2}(2t+\mu)^{2}}$ (65) $\displaystyle{}D\left(\sqrt{\pi\over
2}(2t+\mu)\right)S\left({1\over 2}\log d\right)|0\rangle$

$\mu\in\\{0,1\\}$. Often one takes $d={1\over\lambda^{2}}$, so that $\lambda$
is the only parameter. However, one can also choose $d$ based on fidelity
minimization or best optimal recovery. In practice, the infinite sum over $t$
is truncated to the domain $[-T,T]$ for integer $T$.

Another finite energy CV GKP code is defined by a superposition of CV coherent
states on a square lattice in $\mathbb{R}^{2}$. We refer to this code as
SgGKP,

$\displaystyle|\mu_{\lambda}\rangle\propto\sum_{t\in\mathbb{Z}^{\times 2}}$
$\displaystyle e^{-{\pi\over 2}\lambda^{2}((2t_{1}+\mu)^{2}+t_{2}^{2})}$ (66)
$\displaystyle D\left(\sqrt{\pi\over
2}(2t_{1}+\mu)\right)D\left(i\sqrt{\pi\over 2}t_{2}\right)|0\rangle,$

where $\mu\in\\{0,1\\}$.

We can also define CV GKP codes for which the code states have exactly equal
energy in expectation. Analogous to CVGKP, one defines the code ECVGKP as,

$\displaystyle|\nu_{\lambda,d}\rangle\propto\sum_{t\in\mathbb{Z}}$
$\displaystyle e^{-2\pi\lambda^{2}t^{2}}$ (67) $\displaystyle
D\left(\sqrt{\pi\over 2}(2t+\nu)\right)S\left({1\over 2}\log
d\right)|0\rangle$

$\nu\in\\{-{1\over 2},{1\over 2}\\}$. There is also an equal expected energy
version of SgGKP which we call ESgGKP. It is given by

$\displaystyle|\nu_{\lambda}\rangle\propto\sum_{t\in\mathbb{Z}^{\times 2}}$
$\displaystyle e^{-{\pi\over 2}\lambda^{2}((2t_{1}+2\nu)^{2}+t_{2}^{2})}$ (68)
$\displaystyle D\left(\sqrt{\pi\over
2}(2t_{1}+\nu)\right)D\left(i\sqrt{\pi\over 2}t_{2}\right)|0\rangle$

$\nu\in\\{-{1\over 2},{1\over 2}\\}$.

In Fig. 14, the position space wavefunctions for each of these four codes are
shown. From the figure, one can identify a well-defined comb-like structure
for all codewords. Also for the codes SgGKP and call ESgGKP the position space
peaks have the same amplitudes for the logical $\ket{{1\over 2}_{L}}$ and
$\ket{-{1\over 2}_{L}}$.

Figure 15: The optimal recovery performance of finite-energy CV GKP codes
under the quantum channel given in Eq. (17). In (a) the case of $T=2$ is shown
whereas in (b) the case of $T=5$ is shown.

Similar to the case of the spin-$N/2$ GKP states, one can use the diversity
combining method and optimization of the code parameters to compare the
optimal recovery of the four finite-energy CV GKP codes above (Fig. 15). The
noise channel is given by (17). In (a) the case of $T=2$ is shown whereas in
(b) the case of $T=5$ is shown. From these curves, it is clear that the equal
expected energy version of the square grid CV GKP code has lower optimal
achievable fidelity. This suggests that given a specific noise channel, it is
important to consider an appropriate finite-energy CV GKP code to maximize
recovery.

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