# Fault-tolerant quantum computation using large spin cat-codes

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  Vikas
Buchemmavari  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  Jonathan A. Gross  Google Quantum AI,
Venice, CA 90291, USA  Ivan H Deutsch<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  Milad Marvian<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  Department of
Electrical and Computer Engineering, University of New Mexico, Albuquerque,
New Mexico 87131, USA

(August 28, 2024)

###### Abstract

We construct a fault-tolerant quantum error-correcting protocol based on a
qubit encoded in a large spin qudit using a spin-cat code, analogous to the
continuous variable cat encoding. With this, we can correct the dominant error
sources, namely processes that can be expressed as error operators that are
linear or quadratic in the components of angular momentum. Such codes tailored
to dominant error sources can exhibit superior thresholds and lower resource
overheads when compared to those designed for unstructured noise models. A key
component is the CNOT gate that preserves the rank of spherical tensor
operators. Categorizing the dominant errors as phase and amplitude errors, we
demonstrate how phase errors, analogous to phase-flip errors for qubits, can
be effectively corrected. Furthermore, we propose a measurement-free error
correction scheme to address amplitude errors without relying on syndrome
measurements. Through an in-depth analysis of logical CNOT gate errors, we
establish that the fault-tolerant threshold for error correction in the spin-
cat encoding surpasses that of standard qubit-based encodings. We consider a
specific implementation based on neutral-atom quantum computing, with qudits
encoded in the nuclear spin of 87Sr, and show how to generate the universal
gate set, including the rank-preserving CNOT gate, using quantum control and
the Rydberg blockade. These findings pave the way for encoding a qubit in a
large spin with the potential to achieve fault tolerance, high threshold, and
reduced resource overhead in quantum information processing.

##  I Introduction

Quantum computers have the potential to provide a substantial advantage over
their classical counterparts Shor (1999); Farhi  _et al._ (2001); Lloyd
(1996); Biamonte  _et al._ (2017); Aspuru-Guzik  _et al._ (2005). However,
quantum computers are extremely susceptible to environmental noise and
imprecise control, which hinders achieving their full computational capacity.
Fault-tolerant quantum computation (FTQC), provides a solution to perform
reliable computation even in the presence of imperfect elementary components
Knill  _et al._ (1998); Aharonov and Ben-Or (1997); Knill (2005); Raussendorf
_et al._ (2007). The cornerstone of FTQC is the threshold theorem, which
states that if the error rate of individual components remains below a
constant threshold, then arbitrarily long quantum computation can be performed
Aharonov and Ben-Or (1997); Knill  _et al._ (1998); Preskill (1998); Kitaev
(1997); Aliferis and Preskill (2008). In addition to the value of noise
threshold, a critical aspect of FTQC is the resource overhead, quantifying the
number of physical systems required to encode logical information. Despite the
formidable challenges, there has been notable experimental progress in FTQC,
bringing us closer to harnessing the full potential of quantum computing ach
(2023); Ryan-Anderson  _et al._ (2022); Krinner  _et al._ (2022).

The conventional approaches for FTQC are mostly devoted to structureless and
uncorrelated noise. An instance of this is depolarizing noise, where all local
Pauli operators have an equal probability. However, such decoherence models
often entail stringent threshold requirements and result in significant
overheads for FTQC Knill (2005); Raussendorf  _et al._ (2007); Svore  _et al._
(2006); Spedalieri and Roychowdhury (2008). An alternative strategy involves
seeking error-correcting codes tailored to the prevalent noise sources of the
particular physical platform. When possible, these tailored approaches can
lead to improved thresholds and reduced resource overhead Aliferis and
Preskill (2008); Webster  _et al._ (2015). For instance, biased qubits in
bosonic systems can lead to exponentially suppressed bit flip errors compared
to phase flip error Cong  _et al._ (2022); Ofek  _et al._ (2016); Puri  _et
al._ (2020); Guillaud and Mirrahimi (2019). Additionally, in scenarios where
erasure errors dominate over Pauli errors, tailored error-correcting codes
have proven advantageous Grassl  _et al._ (1997); Wu  _et al._ (2022); Sahay
_et al._ (2023) . By addressing the specific characteristics of dominant noise
sources, these tailored methods offer promising avenues to enhance the
performance of FTQC.

Another avenue to develop more efficient error-corrected quantum processors is
to take advantage of the larger Hilbert spaces that can be controlled in
individual subsystems for a given physical platform. While many platforms
offer access to multiple levels, the focus is often on isolating two well-
defined levels for qubit-based computations. However, a more advantageous
approach emerges when we exploit these multiple levels to create qubits
naturally resilient to dominant noise channels Gottesman  _et al._ (2001);
Gross  _et al._ (2021); Omanakuttan and Gross (2023); Omanakuttan and Volkoff
(2023). The quintessential example is the Gottesman-Preskill-Kitaev encoding
of a qubit in an infinite dimensional oscillator Gottesman  _et al._ (2001).
In this work we will consider encoding a qubit in a spin-$J$ system,
corresponding to a qudit with $d=2J+1$ levels Omanakuttan  _et al._ (2021,
2023); Zache  _et al._ (2023). By harnessing the properties of this qudit with
multiple levels, we can establish logical qubits that possess inherent
resistance to the impact of dominant noise channels, paving the way for more
robust quantum computation.

Other works have previously explored the concept of encoding a qubit in a
large spin Gross (2021); Gross  _et al._ (2021); Omanakuttan and Gross (2023).
In this context, the angular momentum operators form the natural set of error
operators for such encodings, generalizing the Pauli operator basis for
qubits. Earlier studies identified quantum error-correcting encodings, but
these constructions were not fault-tolerant Gross (2021); Omanakuttan and
Gross (2023). Here, our main objective is to investigate how we can achieve
Fault-Tolerant Quantum Computation (FTQC), specifically for a qubit encoded in
a large spin. This approach may be extended to a wide range of physical
systems, including semiconductor systems Gross (2021); Gross  _et al._ (2021),
ion traps Ringbauer  _et al._ (2022); Low  _et al._ (2020), atomic systems
Omanakuttan  _et al._ (2021, 2023); Zache  _et al._ (2023), molecules Castro
_et al._ (2022); Jain  _et al._ (2023), and superconducting systems Özgüler
and Venturelli (2022); Blok  _et al._ (2021), wherein spin qudits offer the
means to encode logical qubits.

In this work, we direct our attention to a specific encoding we call the
“spin-cat encoding.” This choice is motivated by the cat encodings employed in
bosonic continuous variable systems Puri  _et al._ (2020); Guillaud and
Mirrahimi (2019), used to correct photon loss errors, the dominant errors for
the continuous variable systems. Similarly, spin-cat encoding can rectify the
dominant error operators in spin systems, namely, the linear and quadratic
angular momentum operators. Physically, these arise from uncontrolled Larmor
precession of the spins and optical pumping between magnetic sublevels. To
achieve fault tolerance with spin-cat encoding, we develop two key
ingredients. First, we show how to implement a universal gate set that
preserves the limited error space of interest. An essential element here is
the “rank-preserving CNOT” gate that ensures that one does not convert
correctable errors into uncorrectable ones. Second, aiming at a more easily
implemented scheme, we develop a measurement-free error correction gadget for
spin systems that require fresh ancilla spins and data-ancilla operations but
no measurements. As we will show, this scheme effectively utilizes the rank-
preserving CNOT gate in conjunction with standard phase flip error correction
to address and correct angular momentum errors.

A distinctive aspect of the spin-cat encoding, setting it apart from other
spin encodings Gross (2021); Omanakuttan and Gross (2023); Kubischta and
Teixeira (2023a, b), is its unique structural composition. In contrast to
these earlier methods, the error subspaces in the spin-cat encoding partition
the physical space into two-dimensional subspaces where logical operations act
identically. This gives the structure of a stabilizer code, a feature that
plays a pivotal role in enabling fault-tolerant schemes for error correction.

The remainder of this article is organized as follows. In Sec. II we define
the cat codes for spin systems and the natural basis for the dominant error
channels. In Sec. III, we discuss the requirements on gates in order to not
spread correctable errors. We describe the implementation of a rank-preserving
CNOT gate for the encoding of a qubit in the nuclear spin of 87Sr in Sec.
III.2 and the necessary measurement and state preparation steps to implement
the universal gate set in Sec. III.3. In Sec. IV, we explain the protocol for
syndrome extraction needed to correct the errors in spin-cat encoding, and the
measurement-free error correction native to the qubit encoded in the spin. In
Sec. V we obtain the threshold for FTQC based on the logical CNOT gate. We
conclude and explore possible future directions in Sec. VI.

##  II Generalization of cat code for Qudits/spin systems

(a)

(b)

Figure 1: Qubit encoded in a spin using spin-cat states. (a) The Bloch sphere
for the qubit encoded in a spin. The two spin-coherent states (stretched
states) are the computational basis states, lying on the $Z$-axis and the
spin-cat states then lie along the $X$-axis. The spin Wigner function of the
states is shown and its strong negativity indicates that spin-cats are highly
nonclassical. (b) The spin-cat encoding of a qubit in spin $J=9/2$,
$d=2J+1=10$ levels. The correctable errors divide the qudit into two
subspaces, $\bar{0}$ and $\bar{1}$, shown as blue and purple boxes,
respectively. One physical error channel is optical pumping, corresponding to
the absorption of photons (blur arrows) followed by spontaneous emission (wavy
red arrows), which can lead to amplitude damping.

In this section, we introduce our encoding, present the most prevalent types
of noises in spin systems, and look at how they affect an encoded qubit. We
consider quantum information encoded in large spins with angular momentum $J$,
a qudit of dimension $d=2J+1$. The space of local errors on a spin system is
spanned by the irreducible spherical tensor operators ${T}^{(k)}_{q}(J)$
Varshalovich  _et al._ (1988) which are polynomials in the spin angular
momentum components, $\\{{J}_{x},{J}_{y},{J}_{z}\\}$ of order $k$, with $2k+1$
components. The qudit operator space is spanned by the basis of tensors from
$k=1$ to $k=2J$. In most platforms, physical errors are associated with low
rank-$k$ tensors for $J\gg 1$. For example, erroneous Larmor precession caused
by noisy magnetic fields are generated by the SU(2) algebra, or rank-1
tensors. When controlled by laser light, as in atomic systems, optical pumping
arising from photon scattering can lead to rank-2 errors. Higher rank errors
are rare, as they involve multi-photon processes or higher rank tensor
perturbations. We thus design codes that can correct any errors in the space
spanned by the Kraus operators in the set of linear and quadratic spin
operators $\\{{T}^{(1)}_{q}(J),{T}^{(2)}_{q}(J)\\}$ Omanakuttan and Gross
(2023). For $J\gg 1$, this is a substantially reduced error space (dimension
8) compared to the total space of all possible errors (dimension
$(2J+1)^{2}-1$).

To design a spin-encoding that can efficiently correct this biased noise
structure, we consider the bosonic cat encoding of a qubit Puri  _et al._
(2020). In this encoding, the qubit states $\ket{0}$ and $\ket{1}$ are chosen
to be,

$\ket{C_{\alpha}^{\pm}}\propto\ket{\alpha}\pm\ket{-\alpha},$ (1)

where $\ket{\alpha}$ is a coherent state of a single bosonic mode, for e.g., a
mode of a microwave cavity as in superconducting systems. When the dominant
source of noise is photon loss, this encoding exhibits a biased noise channel
where increasing the amplitude $\alpha$, exponentially suppresses bit flip
errors when compared to phase flip errors. It has been shown that by using
simple codes such as a repetition code to correct phase flips, one can take
advantage of this bias in the noise to achieve significant improvement in the
threshold for FTQC Aliferis and Preskill (2008); Puri  _et al._ (2020) for cat
qubits.

In this work, we pursue a similar approach for finite-dimensional spin systems
and consider the spin-cat encoding with,

$\displaystyle\ket{\pm}$
$\displaystyle\equiv\frac{1}{\sqrt{2}}\left(\ket{J,-J}\pm\ket{J,+J}\right),$
(2)

where now $\ket{0}=\ket{J,-J}$ and $\ket{1}=\ket{J,J}$ are the spin coherent
states along the physical quantization ($z$) axis. We call this the spin-cat
encoding. Similar to previous works based on continuous variable bosonic cat
states Puri  _et al._ (2020); Guillaud and Mirrahimi (2019), the spin cat
states are defined along the $1$-axis of the qubit Bloch sphere; see Fig. 1a.
Note that, unlike the coherent states in the continuous variable setting, the
spin coherent states are perfectly orthogonal to each other.

Despite utilizing a similar encoding, there are significant differences
between the dominant sources of noise and the easy-to-implement operations in
the spin system compared to bosonic cats. Thus, this encoding requires the
development of new error-correction procedures that we address in this work.
Central to the continuous variable cat encoding, as explored in Puri  _et al._
(2020); Guillaud and Mirrahimi (2019), is the reduction in bit-flip errors.
The key to this bias is the presence of an energy gap between the excited
state manifold and the logical subspace, that scales with
$\lvert\alpha\rvert^{2}$. While this encoding offers significant advantages
compared to standard qubit-based encoding, the leakage to these excited states
can have detrimental effects on the energy-protected qubits. Dissipative
stabilization can be employed to overcome these errors Putterman  _et al._
(2022).

In contrast, in spin-cat encoding we use an alternative approach for fault
tolerance. We consider a primary layer of encoding where we correct for the
physically relevant errors and then use a second layer of concatenation to
achieve fault-tolerant quantum computation. We can achieve this because the
physically relevant errors are a small subset of all the possible errors for
the encoded qubit. For the spin-cat encoding, these physically relevant errors
are composed of spherical tensors of rank-1 and rank-2, as described above.
The key goal of the first layer of the encoding is to correct for these rank-1
and rank-2 errors. Our protocol is fault-tolerant because the universal gates
and error correction performed in the first layer of encoding do not convert
lower-rank spherical tensor operators to higher-rank operators. We call this
“rank-preserving” error correction. It is a generalization of the bias-
preserving error correction where the dominant error for the encoded qubit is
a single Pauli-error. In the second layer of encoding, the relevant errors are
Pauli errors on the logical qubit, which can be corrected by any standard
error correction protocol.

###  II.1 Error characterization

To categorize the relevant errors that can be corrected for the spin-cat
encoding, it is useful to define the generalized “kitten states” as,

$\displaystyle\ket{\pm}_{m}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{0}_{m}\pm\ket{1}_{m}\right).$ (3)

where,

$\displaystyle\ket{0}_{m}$ $\displaystyle=\ket{J,-J+m}\equiv\ket{-J+m},$ (4)
$\displaystyle\ket{1}_{m}$ $\displaystyle=\ket{J,J-m}\equiv\ket{J-m}.$

The case $m=0$ is the spin-cat state. The total Hilbert space of the spin-cat
encoding decomposes to $d/2$ qubit subspace where each of the qubit subspaces
is spanned by the kitten states $\ket{\pm}_{m}$. Thus we can write,

$\mathcal{H}_{d}=\bigoplus_{i=0}^{\frac{d}{2}}\mathcal{H}_{2}^{(i)},$ (5)

where each $\mathcal{H}_{2}$ is a kitten subspace and $\mathcal{H}_{d}$ is the
total Hilbert space of the qudit. These subspaces are preserved by rotations
about the spin quantization $z$-axis and by $\pi$ pulses around axes in the
equatorial plane that exchange $\ket{\pm J}$.

We also define the following projectors onto $\bar{0}$ and $\bar{1}$ subspaces
that define correctable errors,

$\displaystyle\Pi_{\overline{0}}$
$\displaystyle=\sum_{k=0}^{\lfloor\frac{2J-1}{2}\rfloor}\outerproduct{-J+k}{-J+k},$
(6) $\displaystyle\Pi_{\overline{1}}$
$\displaystyle=\sum_{k=0}^{\lfloor\frac{2J-1}{2}\rfloor}\outerproduct{J-k}{J-k}.$

See Fig. 1 for an illustration.

The relevant errors on the spin-cat encoding that we aim to correct are a
combination of amplitude and phase errors. The amplitude errors are defined by
the following transformation,

$\ket{\pm}_{m}\to\sum_{k=0}^{\lfloor\frac{2J-1}{2}\rfloor}c_{k}\ket{\pm}_{k},$
(7)

where $c_{k}$ is an arbitrary complex number. The phase error is given by the
transformation,

$\ket{+}_{k}\to\ket{-}_{k}.$ (8)

Physically, these occur as follows. First, consider spin rotations,

$\displaystyle U_{Z}$ $\displaystyle=\exp(-i\theta J_{z}),$ (9) $\displaystyle
U_{X}$ $\displaystyle=\exp(-i\theta J_{x}).$

For $\theta\ll 1$ their actions action on the spin-cat states is

$\displaystyle U_{Z}\ket{\pm}$ $\displaystyle\approx\left(\mathds{1}-i\theta
J_{z}\right)\ket{\pm}=\ket{\pm}-i\theta J\ket{\mp},$ (10) $\displaystyle
U_{X}\ket{\pm}$ $\displaystyle\approx\left(\mathds{1}-i\theta
J_{x}\right)\ket{\pm}$
$\displaystyle=\ket{\pm}-i\theta\frac{\sqrt{J}}{\sqrt{2}}\ket{+}_{1}.$

Thus, the effect of $U_{Z}$ is to introduce a phase error on the spin-cat
states whereas $U_{X}$ generates an amplitude error that takes a cat state to
a kitten state with $m=1$. The ratio of probabilities of amplitude errors to
phase errors due to random rotation errors goes as $1/J$, and hence approaches
zero for large values of $J$.

Next, we consider errors resulting from optical pumping associated with photon
scattering. For example, given a laser photon linearly polarized along the
quantization axis, followed by the emission of $q=0,\pm 1$ helicity photon,
the Lindblad jump (Kraus) operators $W_{q}$ are given by Deutsch and Brennen
(2000),

$\displaystyle W_{0}$ $\displaystyle=\beta T^{(2)}_{0},$ (11) $\displaystyle
W_{+1}$ $\displaystyle=i\alpha
T^{(1)}_{-1}-\beta\sqrt{\frac{3}{4}}T^{(2)}_{-1},$ $\displaystyle W_{-1}$
$\displaystyle=i\alpha T^{(1)}_{1}+\beta\sqrt{\frac{3}{4}}T^{(2)}_{1}.$

where $\alpha,\beta$ are real numbers that depend on the atomic structure and
the states being excited by a near resonance laser. (See App. B details.)
Optical pumping can include rank-2 tensors as it involves two photons. The
effect of optical pumping introduces both amplitude errors that change the
kitten subspace Eq. 7, and phase errors as given in Eq. 8. In contrast to
errors that result from rank-1 SU(2) rotation, in optical pumping, it is
equally important to correct both amplitude damping and phase errors and
ultimately, we must do so fault-tolerantly.

Amplitude errors up to rank $K=\lfloor 2J-1/2\rfloor$ can be corrected by
identifying whether the system is in a specific kitten state with a given $m$
value. To correct for the phase errors, we concatenate the spin-cat code in a
repetition code with logical states,

$\displaystyle\ket{+_{\mathrm{L}}}$ $\displaystyle=\ket{+}\ket{+}\ket{+},$
(12) $\displaystyle\ket{-_{\mathrm{L}}}$
$\displaystyle=\ket{-}\ket{-}\ket{-}.$

While we consider a three-qubit repetition code here and throughout Sec. IV
for simplicity, in Sec. V we will look at repetition codes with more than
three qubits in order to calculate the threshold for fault-tolerance. One can
then perform the corresponding error correction steps similar to the approach
taken in the continuous variable encoding Puri  _et al._ (2020); Guillaud and
Mirrahimi (2019). We call this the “logical-level encoding” to differentiate
it from the physical-level encoding in Eq. 4.

More formally, in App. C we show that the logical-level encodings in Eq. 12
can correct any single spin angular momentum errors of the form,

$\small\mathcal{E}_{K}=\left\\{J_{x}^{l}J_{y}^{m}J_{z}^{n};0\leq l+m+n\leq
K=\lfloor\frac{2J-1}{2}\rfloor\right\\}.$ (13)

In practice we can restrict our attention to quadratic polynomials.

###  II.2 The irreducible spherical tensor basis

The irreducible spherical tensor basis provides a natural basis to
characterize the action of the error operators. In the basis of the magnetic
sublevels, the normalized tensors are Varshalovich  _et al._ (1988)

$T^{(k)}_{q}(J)=\sqrt{\frac{2k+1}{2J+1}}\sum\limits_{m,m^{\prime}=-J}^{J}C_{Jm^{\prime};kq}^{Jm}\outerproduct{J,m}{J,m^{\prime}},\
$ (14)

where $C_{J,m^{\prime};kq}^{J,m}=\innerproduct{J,m}{J,m^{\prime};k,q}$ are the
Clebsch-Gordan coefficients. The spherical tensor operators of rank-$k$ are
the solid harmonics consisting of polynomials on the angular momentum
operators of order $k$. To track how errors occur, it is convenient to
introduce the following linear combination of the spherical tensor operators,

$\displaystyle S^{(k)}_{q}(J)$
$\displaystyle=\frac{1}{\sqrt{2}}\left[T^{(k)}_{q}(J)+(-1)^{k}T^{(k)}_{-q}(J)\right],$
(15) $\displaystyle A^{(k)}_{q}(J)$
$\displaystyle=\frac{1}{\sqrt{2}}\left[T^{(k)}_{q}(J)-(-1)^{k}T^{(k)}_{-q}(J)\right],$
$\displaystyle S^{(k)}_{0}(J)$ $\displaystyle=T^{(k)}_{0}(J).$

for $0\leq k\leq 2J+1$ and $q>0$. It is straightforward to check that these
operators form another orthonormal basis for a spin-$J$ system, i.e.,

$\displaystyle\Tr{\left(S^{(k)}_{q}\right)^{\dagger}S^{(k^{\prime})}_{q^{\prime}}}$
$\displaystyle=\Tr{\left(A^{(k)}_{q}\right)^{\dagger}{A}^{(k^{\prime})}_{q^{\prime}}}=\delta_{k,k^{\prime}}\delta_{q,q^{\prime}},$
(16)
$\displaystyle\Tr{\left(S^{(k)}_{q}\right)^{\dagger}A^{(k^{\prime})}_{q^{\prime}}}$
$\displaystyle=0,$

for $0\leq k,k^{\prime}\leq 2J+1$, $0\leq q\leq k$, and $0\leq q^{\prime}\leq
k^{\prime}$. The action of the operators on the cat and kitten states are
given (for $q>0$) as,

$\displaystyle S^{(k)}_{q}\ket{\pm}_{l}$
$\displaystyle=\sqrt{\frac{2k+1}{2(2J+1)}}\left[(-1)^{k}C^{J,-J+l-q}_{J,-J+l;k,-q}\ket{\pm}_{l-q}+C^{J,-J+l+q}_{J,-J+l;k,q}\ket{\pm}_{l+q}\right],$
(17) $\displaystyle A^{(k)}_{q}\ket{\pm}_{l}$
$\displaystyle=\sqrt{\frac{2k+1}{2(2J+1)}}\left[(-1)^{k}C^{J,-J+l-q}_{J,-J+l;k,-q}\ket{\mp}_{l-q}-C^{J,-J+l+q}_{J,-J+l;k,q}\ket{\mp}_{l+q}\right],$
$\displaystyle S^{(k)}_{0}\ket{\pm}_{l}$
$\displaystyle=\begin{cases}\sqrt{\frac{2k+1}{2J+1}}C^{J,-J+l}_{J,-J+l;k,0}\ket{\pm}_{l},&\text{if
}\ k\text{ mod }2=0\\\
\sqrt{\frac{2k+1}{2J+1}}C^{J,-J+l}_{J,-J+l;k,0}\ket{\mp}_{l},&\text{otherwise.}\end{cases}$

Note that the states on the righthand side of the equations are not
normalized, as the operators $S^{(k)}_{q},A^{(k)}_{q}$ are not unitary. They
are the Kraus operators corresponding to the relevant errors.

The action of the Kraus operator $S^{(k)}_{q}$ is the amplitude error given in
Eq. 7. The Kraus operator $S^{(k)}_{0}$ flips the kitten states for
$k\;\mathrm{mod}2$=1 which corresponds to the phase error in Eq. 8; the Kraus
operators $A^{(k)}_{q}$ change the value of the kitten state and also flip
their sign. This corresponds to the action of both amplitude and phase error.
This basis of the Kraus operators tracks whether the error is amplitude,
phase, or the product of two. The single spin errors that a spin-cat code can
correct can be written in terms of the new basis as,

$\mathcal{E}_{K}=\left\\{S_{q}^{(k)},{A}_{q}^{(k)}\;|\;0\leq k\leq K,-k\leq
q\leq k\right\\},$ (18)

where $K=\lfloor\frac{2J-1}{2}\rfloor$.

The logical encoding defined in Eq. 12 introduces a biased logical qubit so
that the rate of bit flip errors is exponentially suppressed compared to the
phase flip errors as a function of the total value of spin $J$. Any
uncorrectable amplitude error at the physical level of the spin-cat encoding
is transformed into a bit-flip error on the logical qubit. In Fig. 12 we
compare the ratio of uncorrectable amplitude error to phase error for rotation
error. It is evident that even for modest values of $J=5/2,7/2,\text{ and
}9/2$, the bit-flip error rate for the logical qubit is significantly
suppressed compared to phase-flip errors.

The proposed encoding can be considered a generalized version of the Shor
code,

$\displaystyle\ket{0}$
$\displaystyle=\frac{1}{\sqrt{8}}\left(\ket{\uparrow}^{\otimes
2J+1}+\ket{\downarrow}^{\otimes 2J+1}\right)^{\otimes 3}$ (19)
$\displaystyle\ket{1}$
$\displaystyle=\frac{1}{\sqrt{8}}\left(\ket{\uparrow}^{\otimes
2J+1}-\ket{\downarrow}^{\otimes 2J+1}\right)^{\otimes 3}$

For the Shor code Shor (1995), the inner encoding protects against bit-flip
errors and the outer encoding protects against phase-flip errors. In our case,
the inner layer protection originates from the encoding of the qubit in the
spin-$J$ qudit, $\ket{\uparrow}^{\otimes 2J+1}=\ket{J,J}$,
$\ket{\downarrow}^{\otimes 2J+1}=\ket{J,-J}$.

##  III Universal gate set and Rank-Preserving CNOT gate

In this section, we establish a set of universal fault-tolerant operations for
spin-cat qubits. As discussed above, similar to Aliferis and Preskill (2008);
Puri  _et al._ (2020), our strategy is to first correct for the dominant
errors by encoding the biased qubit in a repetition code $\mathcal{C}_{1}$.
After performing error correction corresponding to code $\mathcal{C}_{1}$, we
obtain a logical qubit with reduced (but less biased) effective errors. We can
then achieve FTQC by employing another level of concatenation using a generic
CSS code $\mathcal{C}_{2}$, as long as the effective noise strength is below
the threshold of the code $\mathcal{C}_{2}$.

To construct the universal gate sets, we target the following physical level
gates,

$\\{\mathcal{P}_{\ket{0}},\mathcal{P}_{\ket{+}},\mathcal{M}_{X},\mathcal{M}_{Z},\mathrm{CNOT},ZZ(\theta),X,Y,Z\\}.$
(20)

We require these spin-cat qubit operations to be “rank-preserving” so that
they do not convert correctable errors into uncorrectable ones. Using this
gate set, one can construct the following logical universal gate set for
$\mathcal{C}_{1}$,

$\\{\mathcal{P}_{\ket{0}_{\mathrm{L}}},\mathcal{P}_{\ket{+}_{\mathrm{L}}},\mathcal{M}_{X_{\mathrm{L}}},\mathcal{M}_{Z_{\mathrm{L}}},\mathrm{CNOT}_{\mathrm{L}}\\}\cup\\{\mathcal{P}_{\ket{i}_{\mathrm{L}}},\mathcal{P}_{\ket{T}_{\mathrm{L}}}\\}.$
(21)

Here $\mathcal{P}$ denotes state preparation, and $\mathcal{M}$ represents the
measurement operators. To prepare the magic states
$\mathcal{P}_{\ket{i}_{\mathrm{L}}},\mathcal{P}_{\ket{T}_{\mathrm{L}}}$, we
can utilize rank-preserving $ZZ(\theta)$ at the physical level, similar to the
bias-preserving case of qubits Webster  _et al._ (2015) and cat codes Puri
_et al._ (2020).

###  III.1 Single qubit gates

To ensure fault tolerance, a gate $U$ must not turn correctable errors into
uncorrectable errors in a specific level of encoding, i.e., we require that

$U\mathcal{E}_{K}U^{\dagger}\in\mathcal{E}_{K},$ (22)

where $\mathcal{E}_{K}$ represents the set of correctable errors for the spin-
cat encoding as defined in Eq. 18. Further, to prevent the propagation of
correctable errors into uncorrectable ones during subsequent computations, the
gates $U$ should act on states for which an error has occurred in the same
manner as they act on states within the logical subspace. Specifically, these
gates must exhibit identical behavior whether the states are in the cat
subspace or the kitten subspace with $m>1$, the subspace corresponding to
amplitude damping errors.

By building the gates $U$ in the universal gate set using operations solely
from the spin-$J$ representations of $\mathrm{SU}(2)$, we can guarantee the
condition in Eq. 22. To see this, recall the definition of spherical tensor
operators:

$UT_{q}^{(k)}U^{\dagger}=\sum_{-k\leq q^{\prime}\leq
k}D_{q,q^{\prime}}T_{q^{\prime}}^{(k)},$ (23)

where $U=e^{-i\theta\hat{\mathbf{n}}.\mathbf{J}}$ is a spin-$J$
$\mathrm{SU}(2)$ rotation operator and

$D_{q,q^{\prime}}=\bra{k,M=q^{\prime}}\exp(-i\theta\hat{n}.\mathbf{J})\ket{k,M=q},$
(24)

are the elements of Wigner $D$-matrices Sakurai and Napolitano (2014). As a
result, $\mathrm{SU(2)}$ operators do not change the rank of spherical tensor
operators. Using the above relationships for the basis of errors introduced in
Eq. 15, we get,

$\displaystyle US^{(k)}_{q}U^{\dagger}$
$\displaystyle=\sum_{q^{\prime}}\left(g_{q,q^{\prime}}S^{(k)}_{q^{\prime}}+\widetilde{g}_{q,q^{\prime}}A^{(k)}_{q^{\prime}}\right)$
(25) $\displaystyle UA^{(k)}_{q}U^{\dagger}$
$\displaystyle=\sum_{q^{\prime}}\left(h_{q,q^{\prime}}S^{(k)}_{q^{\prime}}+\widetilde{h}_{q,q^{\prime}}A^{(k)}_{q^{\prime}}\right)$

where the coefficients
$\\{g_{q,q^{\prime}},\widetilde{g}_{q,q^{\prime}},h_{q,q^{\prime}},\widetilde{h}_{q,q^{\prime}}\\}$
are given in App. D. Therefore, the SU(2) rotations do not change the rank of
the error operators and obey the condition given in Eq. 22.

For the remainder of this article, we consider the case of $J$ half-integer
(even $d$). These schemes can be easily refashioned for odd $d$ with minor
modifications. The single-qubit Pauli gates for the qubit encoded in the spin-
qudit can be implemented using the following general $\mathrm{SU}(2)$
operations,

$\displaystyle X$ $\displaystyle=\exp(-i\pi J_{x}),$ (26) $\displaystyle Y$
$\displaystyle=\exp(-i\pi J_{y}),$ $\displaystyle Z$ $\displaystyle=\exp(-i\pi
J_{z}).$

These are easily implemented by Larmor precession of the spin.

In contrast, and critically, the Hadamard gate $H$ for the spin-cat encoding,
defined by

$\displaystyle H\ket{0}$ $\displaystyle=\ket{+}.$ (27) $\displaystyle
H\ket{1}$ $\displaystyle=\ket{-},$

cannot be achieved by $\mathrm{SU}(2)$ operations alone. To see this, note
that an SU(2) rotation preserves the projection of the spin onto a rotated
axis. As $\ket{0}$ and $\ket{1}$ are spin coherent states (so-called
“stretched states”), an SU(2) rotation cannot be used to prepare a cat state,
which is a superposition of spin coherent states. Therefore,

$H\mathcal{E}_{K}H^{\dagger}\not\in\mathcal{E}_{K}.$ (28)

The essential feature of our protocol is to circumvent this restriction by
using ancilla qubits and rank-preserving CNOT gates to effectively apply a
Hadamard gate that preserves the set of correctable errors.

###  III.2 Rank-preserving CNOT gate

In this section, we develop a rank-preserving CNOT gate, the key ingredient to
realize the universal gate set, using only $\mathrm{SU}(2)$ operations. For
concreteness, we provide a detailed protocol based on the platform of neutral-
atom quantum computing Brennen  _et al._ (1999); Deutsch and Brennen (2000);
Jaksch  _et al._ (2000); Saffman (2016); Henriet  _et al._ (2020), which has
shown increasing promise for scalable FTQC Bluvstein  _et al._ (2022); Cong
_et al._ (2022); Singh  _et al._ (2022); Graham  _et al._ (2022); Ebadi  _et
al._ (2022). In particular, we consider 87Sr atoms, with a spin-qudit encoded
in the nuclear spin $I=9/2$, providing a qudit with $d=10$ levels Omanakuttan
_et al._ (2021, 2023). When in the ground electronic state, the weak coupling
to the environment and resilience to other background noise makes the nuclear
spin an ideal candidate for quantum information processing Barnes  _et al._
(2022a); Daley  _et al._ (2008); Daley (2011); Hayes  _et al._ (2007).

Note, when considering the physical spins of atoms, in standard notation $I$
is the nuclear spin, $J$ is the total angular momentum of the electrons, and
$F$ is the total electronic angular momentum plus nuclear spin. Our qudit is
encoded in spin $I$ in the electronic ground state with $J=0$ for 87Sr, so
that $F=I=9/2$. In this section, the spin angular momentum in which we encode
the qudit is $\mathbf{F}$. In the other sections of this article, we use
$\mathbf{J}$ to denote a generic spin, without reference to its physical
encoding.

We target a CNOT gate for the spin-cat encoding that operates the same for all
kitten states. As discussed above (see Eq. 6), we divide the qudit into “left”
and “right” subspaces, with projectors onto them $\Pi_{\overline{0}}$ and
$\Pi_{\overline{1}}$ respectively. The gate is formally given as,

$\mathrm{CNOT}=\Pi_{\overline{0}}\otimes\mathds{1}+\Pi_{\overline{1}}\otimes
X,$ (29)

where $X=\exp(-i\pi F_{x})$. That is, we apply a $\pi$-rotation (NOT) to every
kitten subspace of the target atom if the control atom is in the
$\overline{1}$-subspace (the amplitude damped states of $\ket{1}$ we can
correct), and the identity, if the control atom is in the
$\overline{0}$-subspace (the amplitude damped states of $\ket{0}$ we can
correct). Clearly, if the amplitude damping takes an atom from the
$\overline{0}$ to $\overline{1}$ space, or vice versa, the error cannot be
corrected.

The protocol for implementing this gate is presented in Fig. 2b. We note that
this protocol requires individual addressing of the atoms. In step I of the
protocol, the population from the ground state memory is coherently
transferred to an auxiliary state where it is more easily controlled. In 87Sr,
we utilize the auxiliary hyperfine state,
$\ket{5\mathrm{s}5\mathrm{p};\;^{3}{P}_{2};\;F=9/2,M_{F}}$ with hyperfine
quantum numbers $F=9/2,M_{F}$. This manifold possesses a large magnetic dipole
moment and a long lifetime. For the control atom, only the population of
$\overline{1}$-subspace is transferred to the auxiliary manifold, whereas for
the target atom, the population from both $\overline{1}$-subspace and
$\overline{0}$-subspace is transferred. Both of these are facilitated by an
effective $\pi$-pulse between the ground and the auxiliary states, which one
can implement using quantum optimal control, as discussed below.

In step II, an effective $\pi$-pulse is applied on the control atom between
the auxiliary and the Rydberg state. In step III, we apply the same
$\pi$-pulse on the target atom. Due to the Rydberg blockade, this population
exchange only occurs when the control atom is in $\overline{0}$-subspace. If
the state of the control atom is in $\overline{1}$-subspace, the population
from the auxiliary state of the target atom is blockaded from transferring to
the Rydberg state.

Subsequently in step IV, using a global interaction and quantum optimal
control, we simultaneously implement a $X=\exp(-i\pi F_{x})$ rotation in the
auxiliary manifold and an identity operator in the Rydberg manifold of the
target atom. The net effect is that if the control atom is in
$\overline{1}$-subspace an $X$ gate has been applied to the target atom and if
the control atom is in $\overline{0}$-subspace the identity operator has been
applied on the target. We transfer all the states back to the ground state by
applying steps III-I in reverse order. The whole procedure implements the
desired rank-preserving CNOT gate for the spin-cat encoding in Eq. 29.

(a)

(b)

Figure 2:  Protocol for implementing a rank-preserving CNOT-gate in neutral
atomic 87Sr based of optimal control and the Rydberg blockade. The spin-cat
qubit is encoded in the nuclear spin, ${I}=F=9/2$, in the electric ground
state, $5\mathrm{s}^{2}\;{}^{1}{S}_{0}$. (a) Detailed level diagram and
protocol; (b) High-level schematic. When a gate is to be performed, the qudit
is excited from the ground-state memory to the long-lived auxiliary metastable
state, $5\mathrm{s}5\mathrm{p}\;^{3}{P}_{2}$, ${F}=9/2$. Entangling
interactions occur through excitation from the auxiliary state to the Rydberg
state, $5\mathrm{s}60\mathrm{s}\;^{3}S_{1}$, ${F}=11/2$. The error-correctable
subspaces, $\overline{0}$ and $\overline{1}$, are represented by blue and
purple colored boxes respectively, in the ground (g), auxiliary (a), and
Rydberg (r) manifolds. The gate is performed in four steps. Step I: Using
quantum optimal control the population from the ground state is transferred to
the auxiliary state while preserving coherence between magnetic sublevels.
Each two-level resonance,
$\ket{\mathrm{a},M_{F}}\rightarrow\ket{\mathrm{r},M_{F}}$, has a detuning
$\Delta_{\mathrm{a},M_{F}}$ and Rabi frequency $\Omega_{\mathrm{a},M_{F}}$.
For the control atom, we only promote the population from the
$\bar{1}$-subspace, whereas for the case of the target atom, we promote the
population from both the $\overline{0}$ and $\overline{1}$ subspaces to the
auxiliary state (see main text for details). Step II: Using $\pi$-polarized
light, local addressing, and quantum control, transfer the population from the
auxiliary to Rydberg states only for the control atom. Step III: Apply the
same pulse to the target atom. Due to the Rydberg blockade, this population
transfer only occurs when the control atom is in $\overline{0}$-subspace; for
the $\overline{1}$-subspace the population is otherwise blockaded. Step IV:
Using global rf-phase-modulated optimal control, we perform the SU(2) rotation
$X=\exp(-i\pi F_{x})$ in the auxiliary manifold and simultaneously the
identity operator in the Rydberg manifold. The result is a CNOT gate – if the
control atom is in $\overline{1}$-subspace we apply an $X$ gate to the target
atom if the control atom is in $\overline{0}$-subspace we implement an
identity operator $\mathds{1}$. Finally, we will transfer all the states back
to the ground state by reversing steps III-I, thus implementing a rank-
preserving CNOT gate for the spin-cat encoding.

(a)

(b)

(c)

Figure 3: Examples of control waveforms that achieve the transfer of
populations between spin manifolds while preserving the coherence between
magnetic sublevels. Based on Hamiltonian Eq. 30, we modulate the lasers’
amplitude, detuning, and phase, as piecewise constant functions of time. Using
the GRAPE optimal control we find the target isometries. (a) The waveform that
implements $V_{\mathrm{tar}}^{(\mathrm{C})}$, which transfer population from
$\overline{1}_{\mathrm{g}}$-subspace to $\bar{1}_{\mathrm{a}}$-subspace while
the population in the $\bar{0}_{\mathrm{g}}$-subspace is unchanged. (b) The
waveform that implements $V_{\mathrm{tar}}^{(\mathrm{T})}$, which transfer
population from $\overline{1}_{\mathrm{g}}$-subspace to
$\bar{1}_{\mathrm{a}}$-subspace and $\overline{0}_{g}$-subspace to
$\bar{0}_{\mathrm{a}}$-subspace . (c) The waveform that implements
$V_{\mathrm{tar}}^{(\mathrm{Ryd})}$ that transfers the population from the
auxiliary states to the Rydberg states. For all these three cases we divide
the time into $12$ equal time steps.

In steps I and II of the rank-preserving CNOT gate, one needs to implement the
transfer of population from the ground to the auxiliary manifold and from the
auxiliary manifold to the Rydberg manifold, respectively. This can be achieved
by an effective $\pi$-pulse between these respective states and using quantum
optimal control. In both these cases we use the control Rabi Hamiltonian

$\displaystyle H_{e}(t)=$
$\displaystyle\sum_{M=-\frac{9}{2}}^{\frac{9}{2}}-\Delta_{e,M}(t)\outerproduct{e,M}{e,M}$
(30)
$\displaystyle+\Omega_{e,M}(t)\left[e^{i\phi_{e}(t)}\sigma^{+}_{e,M}+\mathrm{h.c}\right].$

To simplify the notation we have denoted the two excited metastable manifolds
by $e$, where $e=\mathrm{a}$ (auxiliary states) and $e=\mathrm{r}$ (Rydberg
states). Together with the ground-state manifold,

$\displaystyle\ket{r,M}$
$\displaystyle\equiv\ket{5\text{s}60\text{s};\;^{3}S_{1};\hskip
2.84544ptF=\frac{11}{2},M_{F}=M},$ (31) $\displaystyle\ket{a,M}$
$\displaystyle\equiv\ket{5\text{s}5\text{p};\;^{3}P_{2};\hskip
2.84544ptF=\frac{9}{2},M_{F}=M},$ $\displaystyle\ket{g,M}$
$\displaystyle\equiv\ket{5\text{s}^{2};\;^{1}S_{0},;\hskip
2.84544ptF=\frac{9}{2},M_{F}=M},$

and

$\sigma^{+}_{e,M}\equiv\outerproduct{e,M}{e^{\prime},M}.$ (32)

where $e^{\prime}=\mathrm{g}$ (for the interaction between the ground and
auxiliary states) and $e^{\prime}=\mathrm{a}$ (for the interaction between
auxiliary and Rydberg states). The control task is achieved by modulation of
the amplitude, detuning, and phase of the exciting lasers. The time-dependent
Rabi frequency and detuning are,

$\displaystyle\Omega_{e,M}(t)$ $\displaystyle=\mathcal{C}_{e,M}\Omega_{e}(t),$
(33) $\displaystyle\Delta_{e,M}(t)$
$\displaystyle=\Delta_{e}(t)+\delta_{e,M},$

where $\mathcal{C}_{e,M}$ is the ratio of Clebsch-Gordan coefficients,

$\mathcal{C}_{e,M}=\frac{\bra{F,M}\ket{1,0;F,M}}{\bra{F,\frac{9}{2}}\ket{1,0;F,\frac{9}{2}}}.$
(34)

$\Delta_{e}$ is the detuning, and $\delta_{e,M}$ is the additional detuning
due to the relative Zeeman shift. To implement the particular target unitary
map interest ($U_{\mathrm{tar}}$) we consider modulation of the amplitude,
detuning, and phase of the two lasers that drive the
$\ket{g}\rightarrow\ket{a}$ transitions and the $\ket{a}\rightarrow\ket{r}$
transitions. The GRAPE algorithm searches for the optimal control parameters
$\Phi=\\{\Omega_{e}(t),\Delta_{e}(t),\phi_{e}(t)\\}$ that maximizes the
fidelity with the target map $U_{\mathrm{tar}}$

$\mathcal{F}[\Phi]=\frac{1}{{d}^{2}}\left|\tr\left\\{U_{\mathrm{tar}}^{\dagger}U[\Phi,T]\right\\}\right|^{2},$
(35)

where ${d}$ is the dimension of the qudit and
$U[\Phi,T]=\mathcal{T}\left[\exp\left(-i\int_{0}^{T}H[\Phi(t)]dt\right)\right]$
is the solution to the time-dependent Schrödinger equation.

We consider partial isometries for our target maps. These have fewer
constraints than unitary transformations and hence require fewer resources
(time, bandwidth etc.). For a system of dimension $d$, one can define a
partial isometry as,

$V_{\mathrm{tar}}=\sum_{i=1}^{k}\outerproduct{f_{i}}{e_{i}}$ (36)

where $1\leq k\leq d$ is the dimension of the partial isometry of interest and
$\\{\ket{e_{i}}\\},\\{\ket{f_{i}}\\}$ are two orthonormal bases. The unitary
maps of interest then take the form,

$U_{\mathrm{tar}}=V_{\mathrm{tar}}+V_{\perp},$ (37)

where $V_{\perp}$ acts on the orthogonal subspace, with dimension $d-k$. To
find the control waveform that generates the partial isometry, one then
optimizes the fidelity between the target isometry and the isometry generated
using quantum control Pedersen  _et al._ (2007)

$\mathcal{F}_{V}[\Phi]=\left|\Tr\left(V^{\dagger}_{\text{tar}}V[\Phi,T]\right)\right|^{2}/k^{2}.$
(38)

For the case of the rank-preserving CNOT gate, one needs to implement three
target isometries. Firstly, on the control atom (C) we need to transfer the
population from the $\overline{1}$-subspace of the ground manifold to that of
the auxiliary manifold while keeping the population in the
$\overline{0}$-subspace unchanged. The isometry we need to implement is,

$\displaystyle V_{\mathrm{tar}}^{(\mathrm{C})}$
$\displaystyle=\sum_{M=-\frac{9}{2}}^{-\frac{1}{2}}\outerproduct{\mathrm{a},M}{\mathrm{a},M}+\sum_{M=\frac{1}{2}}^{\frac{9}{2}}\outerproduct{\mathrm{a},M}{\mathrm{g},M}.$
(39)

Secondly, we seek to transfer the entire population from the ground manifold
to the auxiliary manifold on the target atom (T). The isometry is

$\displaystyle V_{\mathrm{tar}}^{(\mathrm{T})}$
$\displaystyle=\sum_{M=-\frac{9}{2}}^{\frac{9}{2}}\outerproduct{\mathrm{a},M}{\mathrm{g},M}.$
(40)

Finally, we need to implement an isometry that transfers the population from
the auxiliary manifold to the Rydberg manifold,

$\displaystyle V_{\mathrm{tar}}^{(\mathrm{Ryd})}$
$\displaystyle=\sum_{M=-\frac{9}{2}}^{\frac{9}{2}}\outerproduct{\mathrm{r},M}{\mathrm{a},M}.$
(41)

All three can be implemented using the Rabi Hamiltonian.

As a proof of principle, we numerically optimize a piece-wise constant
waveform based on the well-known GRAPE algorithm for quantum optimal control
Merkel  _et al._ (2009); Merkel (2009); Jurdjevic and Sussmann (1972); Goerz
(2015). Example waveforms that implement the target isometries are given in
Fig. 3. The total time required is $4\pi/\Omega_{\mathrm{rf}}$, where
$\Omega_{\mathrm{rf}}$ is the rf-Larmor precession rate, chosen to be resonant
with the Zeeman splitting in the auxiliary auxiliary manifold. To achieve high
fidelity control, we have divided the time into $12$ equal time steps. In
practice, other parameterizations could be used to yield smoother waveforms if
bandwidth is limited.

Figure 4: Evolution of the spin vector $\langle\mathbf{F}\rangle$ for the
auxiliary (a) and Rydberg (r) manifolds resulting from rf-driven Larmor
precession with time-varying phases. Optimal control is based on Hamiltonian
Eq. 43 for the piece-wise constant phases and total time
$T_{\mathrm{tot}}=\sqrt{2}\pi/\Omega_{\mathrm{rf}}$. The blue and black dots
correspond to the first and second steps respectively (see text). An
$X=\exp(-i\pi F_{x})$ gate acts on the auxiliary manifold and transfers the
population from $\bar{1}_{\mathrm{a}}$ to $\bar{0}_{\mathrm{a}}$ and vice-
versa.However, for the Rydberg manifold, the pulse sequence acts as an
identity operator, and the population in the $\bar{0}_{\mathrm{r}}$ and
$\bar{1}_{\mathrm{r}}$ subspaces remain unaffected.

Another important ingredient for the rank-preserving CNOT gate in Fig. 2 is
that we need to apply an rf-pulse that rotates the auxiliary ${}^{3}{P}_{2}$
state and the Rydberg ${}^{3}{S}_{1}$ state differently. For the case of the
rank-preserving CNOT gate, one needs to implement an $X$ gate in the auxiliary
manifold and identity in the Rydberg manifold. This can be achieved because of
the different magnetic $g$-factors of the two spin manifolds. For our specific
choice of Rydberg manifold and auxiliary manifold
$g_{\mathrm{r}}/g_{\mathrm{a}}\approx 2$ Urech (2023). The Hamiltonian
describing Larmor precession in each of the excited manifolds, driven by an
rf-magnetic field oscillating at frequency $\omega$, in the presence of a
basis magnetic field is then

$\displaystyle H_{a}$ $\displaystyle=\Omega_{\mathrm{rf}}\left[\cos(\omega
t+\phi)F_{x}+\sin(\omega t+\phi)F_{y}\right]+\omega_{0}F_{z},$ (42)
$\displaystyle H_{r}$ $\displaystyle=2\Omega_{\mathrm{rf}}\left[\cos(\omega
t+\phi)F_{x}+\sin(\omega t+\phi)F_{y}\right]+2\omega_{0}F_{z}.$

Here $\Omega_{\mathrm{rf}}$ is the Larmor precession frequency and
$\omega_{0}$ is the Zeeman shift induced by the bias B-field in the
${}^{3}P_{2}$ auxiliary manifold. The spin angular moment operators act in the
respective manifolds. Going to the rotation frame of the rf-oscillation, using
the unitary operator $U=\exp(-i\omega tF_{z})$, and choosing the rf-frequency
to be off-resonant with $\omega=4/3\omega_{0}$, gives

$\displaystyle H_{a}^{\mathrm{rot}}$
$\displaystyle=\Omega_{\mathrm{rf}}\left[\cos(\phi)F_{x}+\sin(\phi)F_{y}\right]-\frac{1}{3}\omega_{0}F_{z}.$
(43) $\displaystyle H_{r}^{\mathrm{rot}}$
$\displaystyle=2\Omega_{\mathrm{rf}}\left[\cos(\phi)F_{x}+\sin(\phi)F_{y}\right]+\frac{2}{3}\omega_{0}F_{z}.$

Because of the finite detuning, the total Larmor precession frequency in the
auxiliary and Rydberg manifold is then

$\displaystyle\Omega_{\mathrm{a}}$
$\displaystyle=\sqrt{\Omega_{\mathrm{rf}}^{2}+\frac{\omega_{0}^{2}}{9}},$ (44)
$\displaystyle\Omega_{\mathrm{r}}$
$\displaystyle=\sqrt{4\Omega_{\mathrm{rf}}^{2}+\frac{4\omega_{0}^{2}}{9}}=2\Omega_{\mathrm{a}}.$

Since the total Larmor frequency of the auxiliary auxiliary and Rydberg
manifolds are different, one can use optimization techniques such composite
pulses Levitt (1986) or quantum optimal control Merkel  _et al._ (2009); Goerz
(2015) to implement separate unitaries in the auxiliary and Rydberg manifold.

For example, when $\Omega_{\mathrm{rf}}=\omega_{0}/3$ using optimal control
one achieves an $X$ gate in the auxiliary manifold and the identity in the
Rydberg manifold by taking the phase to be a piece-wise constant function
time, corresponding to a series of rf-pulses, and a total time,
$T_{\mathrm{tot}}=\frac{\sqrt{2}\pi}{\Omega_{\mathrm{rf}}}$. The resultant
dynamics for the auxiliary and Rydberg manifold are given in Fig. 4. Since the
optimization is purely geometric in nature the same pulse schemes work for any
value of the spin as long as the $g$-factors have this ratio. For further
details on the optimization see App. E.

The protocol described above can be generalized for other entangling gates.
One can optimize rf-phases in Eq. 43 to implement the identity operator in the
Rydberg manifold and $R(\theta)=\exp(-i\theta\hat{\bm{n}}.\mathbf{F})$, an
SU(2) operator, in the auxiliary manifold. Thus one can implement the gate
$ZZ(\theta)=\exp{-i\theta Z\otimes Z}$ with any angle $\theta$, up to local
$Z$ rotations, for the spin-cat qubits.

###  III.3 State preparation and Measurement

To complete the universal gate set, one needs to implement the state
preparation and measurement at the physical level given in Eq. 20.
$\mathcal{P}_{\ket{0}}$, which is the preparation of the spin coherent state
can be achieved with high fidelity using optical pumping Chow  _et al._
(2023). Also, $\mathcal{M}_{Z}$, which is the measurement in the
$\ket{F,M_{F}}$ basis can be achieved with high fidelity in principle Barnes
_et al._ (2022b); Ringbauer  _et al._ (2022). However, $\mathcal{P}_{\ket{+}}$
and $\mathcal{M}_{X}$ are not straightforward to implement without an SU(2)
Hadamard gate. We describe here new approaches unique to spin-cat encoding and
the rank-preserving CNOT gate.

####  III.3.1 Preparation of the spin-cat state

We can generate the spin-cat state $\ket{+}$ using multiple approaches. For
example, one can use quantum optimal control by considering the controllable
Hamiltonian

$H(t)=\Omega_{\mathrm{rf}}\left(\cos\phi(t)F_{x}+\sin\phi(t)F_{y}\right)+\beta
F_{z}^{2}.$ (45)

This can be implemented in atomic systems using a combination of tensor light
shifts and rf rotations Chaudhury  _et al._ (2007). For the specific case of
87Sr, we have previously studied how this can be implemented with high
fidelity through the tensor light shift imparted on the ground-electronic
state nuclear spin Omanakuttan  _et al._ (2021). Using quantum optimal control
protocols one can generate the state $\ket{+}$ from an initial state
$\ket{F,M_{F}=F}$.

The light-shift will also be accompanied by decoherence to photon scattering
and optical pumping. We study this in App. B to calculate the fidelity for the
state preparation,

$\mathcal{F}_{\mathrm{state}}=\bra{+}\rho\ket{+}.$ (46)

For the particular choice of 87Sr, we find the fidelity for quantum optimal
control is $\mathcal{F}_{\mathrm{state}}=0.9998$.

####  III.3.2 Measurement of $X$

Figure 5: Circuit diagram implementing $\mathcal{M}_{X}$. Consider an initial
state $\alpha\ket{+}_{k}+\beta\ket{-}_{k}$, where $0\leq
k\leq\lfloor\frac{2J-1}{2}\rfloor$, The action of the CNOT gate for an ancilla
state $\ket{+}_{0}\equiv\ket{+}$ gives us the state,
$\alpha\ket{+}_{k}\ket{+}+\beta\ket{-}_{k}\ket{-}$, thus to identify whether
the state is in $\ket{+}_{k}$ or $\ket{-}_{k}$, we need to measure whether the
ancilla is in $\ket{+}_{0}$ or $\ket{-}_{0}$. One can achieve this using a
destructive measurement, for more details (see Eq. 47).

To measure the $X$ operator ($\mathcal{M}_{X}$), we need to identify whether
the state is in $\ket{+}_{k}$ or $\ket{-}_{k}$ for $0\leq
k\leq\lfloor\frac{2J-1}{2}\rfloor$. We cannot implement the $X$ measurement
fault-tolerantly by applying a Hadamard followed by measuring in the
computational basis since Hadamard is not an SU(2) rank-preserving gate. To
surmount these challenges, similar to Puri  _et al._ (2019), we use an
ancilla-assisted measurement protocol, where measurement errors will lead to
syndrome errors without disturbing the encoded data. Hence, we implement the
$X$-measurement by adding an ancilla qubit in the spin-cat state
$\ket{+}_{0}$, applying a CNOT gate, and then destructively measuring the
ancilla. Since the ancilla is measured destructively and discarded, we do not
need to implement the $X$-measurement using rank-preserving operators.

The circuit diagram which implements the measurement is shown in Fig. 5. After
the application of the CNOT gate, the joint state of the system is
$\alpha\ket{+}_{k}\ket{+}_{0}+\beta\ket{-}_{k}\ket{-}_{0}$. Measuring whether
the ancilla is in $\ket{+}_{0}$ or $\ket{-}_{0}$ gives the value of $X$ on the
data qubit. To measure the ancilla in the $\ket{\pm}_{0}$ basis, we use
quantum optimal control techniques to implement the required transformation to
the $\mathcal{M}_{z}$ basis using $\mathrm{SU}(d)$ optimal control. We employ
the control Hamiltonian in Eq. 45 to implement the isometry Omanakuttan  _et
al._ (2023),

$V_{\mathrm{targ}}=\ket{F,M_{F}=F}\bra{+}+\ket{F,M_{F}=-F}\bra{-}.$ (47)

In practice, this operation will be accompanied by decoherence, and the actual
map we implement may be written as

$V=e^{-\int\mathcal{L}(t)dt}V(0),$ (48)

where

$V(0)=\ket{+}\bra{+}+\ket{-}\bra{-}.$ (49)

and $\mathcal{L}(t)$ is the Lindbladian. Thus the fidelity for the
implementation of the isometry is defined as

$\mathcal{F}_{\mathrm{iso}}=\frac{1}{4}\lvert\Tr(V_{\mathrm{targ}}V^{\dagger})\rvert^{2}.$
(50)

As an example, we consider the effect of photon scattering and optical pumping
that accompanies the tensor light shift. In our simulation, we achieve
fidelity of $\mathcal{F}_{\mathrm{iso}}=0.999$ for 87Sr in the presence of
optical pumping described above.

We have now constructed all the required operations at the level of the qubit
encoded in the spin, as given in Eq. 20. We can use these operations to
implement a universal gate set on the spin-cat qubits and to construct the
error correction and logical operations of the $\mathcal{C}_{1}$ code Aliferis
and Preskill (2008). (See App. G for the implementation of logical operations
in $\mathcal{C}_{1}$.)

Generalizations of rank-preserving gate sets at the physical level can reduce
the circuit size for specific applications Guillaud and Mirrahimi (2019); Katz
_et al._ (2023). For example, we can easily generalize our construction of the
CNOT gate in Sec. III.2 to implement a Toffoli gate in spin systems as
discussed in App. H. The scheme is similar to the $\mathrm{CCZ}$ gate
implemented in Levine  _et al._ (2019). This utilizes the capability to move
neutral atoms in tweezer arrays, arranging the nearest neighbors to interact
via the Rydberg blockade, while leaving the next-to-nearest neighbors
unaffected. With access to such a gate, similar to the recent development in
the bosonic system Guillaud and Mirrahimi (2019), we can implement the
following operations,

$\\{P_{\ket{\pm}},\mathcal{M}_{X},\mathrm{CNOT},\mathrm{Toffoli}\\}.$ (51)

Such a gate set can be used to construct more efficient fault-tolerant
logical-level operations.

##  IV Syndrome Measurement and error recovery

The design of error correction gadgets plays a major role in determining the
threshold of tolerable noise and also the required overhead of fault-tolerant
schemes mainly due to the fact that current fault-tolerant designs require
many rounds of error correction to control the spread of errors. The standard
method to perform an error recovery is to measure the syndromes to identify
the errors and then correct the errors by applying an appropriate unitary
operator. This is the approach we take to correct the phase errors. We use a
repetition code of size $n$, capable of correcting up to
$\lfloor(n-1)/2\rfloor$ phase errors. In this case, the $(n-1)$ syndrome
measurements for phase error correction are

$\mathcal{S}_{\text{phase}}=\\{X_{1}X_{2},X_{2}X_{3},\ldots,X_{n-1}X_{n}\\}.$
(52)

These syndrome measurements can be implemented according to the standard
circuits in Fig. 6 (for $n=3$) using the universal operations described in
Sec. III.

When the probability of phase errors is larger than amplitude errors in each
spin, increasing the size of the repetition code $n$ can reduce the
probability of logical phase errors. However, increasing $n$ will increase the
probability of logical amplitude errors due to the increase in the number of
the required CNOT gates for the syndrome circuits. Therefore we can choose the
optimal $n$ that brings the two types of errors to the same level, determined
by the noise threshold required by the outer CSS code $\mathcal{C}_{2}$.

Figure 6: Circuit for error correction of a phase error for a qubit encoded in
$3$ spins. The error correction is achieved by measuring the syndromes
$\\{X_{1}X_{2},X_{2}X_{3}\\}$ followed by $Z=\exp(-i\pi J_{z})$ gate(s)
according to the syndrome outcomes.

For the case of amplitude damping, one approach to diagnose the syndrome is to
perform nondestructive measurements to identify the amplitude errors, for
example, by measuring $J_{z}^{2}$. In practice this can be difficult to
implement experimentally. (In this section and below we return to denote a
generic spin $J$, without reference to a specific platform.) Instead, we take
advantage of the cat encoding and the unique properties of our proposed CNOT
gate to coherently apply the recovery map using fresh ancilla without
performing any measurement. Our construction is a new example of measurement-
free quantum error correction (MFQEC) Crow  _et al._ (2016); Ercan  _et al._
(2018); Li  _et al._ (2011); Premakumar  _et al._ (2020); Perlin  _et al._
(2023) motivated by the experimental constraints of spin systems.

To describe our proposed error recovery, we first observe that we can “swap”
the state of two qubits encoded in the kitten states. Let

$\displaystyle\ket{\psi}_{k}$
$\displaystyle=\alpha\ket{+}_{k}+\beta\ket{-}_{k},$ (53)
$\displaystyle\ket{\phi}_{l}$
$\displaystyle=\gamma\ket{+}_{l}+\delta\ket{-}_{l},$

where $\alpha,\beta,\gamma,\text{ and }\delta$ are arbitrary complex
amplitudes. Three applications of our proposed CNOT gate, as shown in Fig. 7a,
implement the following transformation (see App. J for a proof):

$\displaystyle\ket{\psi}_{k}$
$\displaystyle\otimes\ket{\phi}_{l}\rightarrow\ket{\phi}_{k}\otimes\ket{\psi}_{l}$
(54)

We expect this construction which implements the swap of kitten states, to
find applications beyond error correction, in particular in algorithmic
subroutines native to qudits platforms, but in this work, we focus on its
application in amplitude correction. We denote this gate by $V_{s}$.

If we replace one of the input states with a fixed cat state, $\ket{+}_{0}$,
then the recovery circuit can be simplified to the circuit in Fig. 7b.
Therefore amplitude errors can be corrected by consuming fresh ancilla qudits
in the cat state, $\ket{+}_{0}$, and applying two CNOT gates. The operation
coherently transfers the qubit that is damped at level $k$ back to level $0$,
which is our encoded qubit. In App. J, we show that the action of this quantum
channel, after tracing the extra subsystem, is exactly equivalent to a
recovery channel implemented by measuring $J_{z}^{2}$ and then applying a
unitary correction to transfer the state into the $k=0$ subspace.

(a)

(b)

Figure 7:  (a) General circuit for swapping the state of the two qubits in two
different kitten subspaces. (b) The circuit that swaps the information between
the data and ancilla, when the initial state of the ancilla state is
$\ket{+}_{0}$.

###  IV.1 Error correction for Optical Pumping

To see how phase and amplitude error correction combines to correct any local
errors, it is illuminating to describe the procedure for correcting a dominant
noise channel in atomic systems, optical pumping. (The details of optical
pumping are discussed in App. B). In particular, consider the example of
absorption of a linear $\pi$-polarized laser photon, followed by the
spontaneous emission of a circularly polarized $\sigma_{+}$ photon. This
process results in mapping $\ket{J,J}$ to $\ket{J,J-1}$ and also annihilating
any amplitude in the state $\ket{J,-J}$. On the cat states, this
transformation can be re-written as,

$\displaystyle\ket{+}$
$\displaystyle\to\ket{J,J-1}=\frac{\ket{+}_{1}-\ket{-}_{1}}{\sqrt{2}},$ (55)
$\displaystyle\ket{-}$
$\displaystyle\to-\ket{J,J-1}=-\frac{\ket{+}_{1}-\ket{-}_{1}}{\sqrt{2}}.$

Consider an arbitrary logical state
$\ket{\psi}=\alpha\ket{+}_{\mathrm{L}}+\beta\ket{-}_{\mathrm{L}}$. The action
of the optical pumping on the first qudit gives

$\ket{\psi}\to\frac{\ket{+}_{1}-\ket{-}_{1}}{\sqrt{2}}\otimes\left(\alpha\ket{+}_{0}\ket{+}_{0}-\beta\ket{-}_{0}\ket{-}_{0}\right)\equiv\ket{\phi}.$
(56)

Now we can consider the states after the phase and amplitude error correction
steps. (As these error correction steps commute with each other, the order in
which we perform them is irrelevant.) The phase error correction is specified
by the syndromes $X_{1}X_{2}$ and $X_{2}X_{3}$. If we measure both the
syndromes as $+1$, the state $\ket{\phi}$ collapses to,

$\ket{\phi}\to\alpha\ket{+}_{1}\ket{+}_{0}\ket{+}_{0}+\beta\ket{-}_{1}\ket{-}_{0}\ket{-}_{0}.$
(57)

If the syndrome measurement gives outcome $-1$ and $1$ for the syndrome
$X_{1}X_{2}$ and $X_{2}X_{3}$ the state becomes,

$\ket{\phi}\to\alpha\ket{-}_{1}\ket{+}_{0}\ket{+}_{0}+\beta\ket{+}_{1}\ket{-}_{0}\ket{-}_{0}.$
(58)

Applying the correction unitary $Z_{1}$ corresponding to this syndrome yields

$\alpha\ket{+}_{1}\ket{+}_{0}\ket{+}_{0}+\beta\ket{-}_{1}\ket{-}_{0}\ket{-}_{0}\equiv\ket{\phi}_{\mathrm{ph}}.$
(59)

The same state is achieved after performing the correction for the other two
possible syndromes. Thus the state after the phase error correction collapses
to the state Eq. 59.

Next, we can apply measurement-free amplitude error correction by consuming
three ancilla states $\ket{+}_{0}$, which gives,

$\displaystyle
V_{s}^{\otimes^{3}}\ket{\phi}_{\mathrm{ph}}\ket{+}_{0}\ket{+}_{0}\ket{+}_{0}$
$\displaystyle=\ket{+}_{1}\ket{+}_{0}\ket{+}_{0}\otimes\ket{\psi}$ (60)

Tracing out the first three subsystems yields the initial state $\ket{\psi}$
in the three ancilla subsystems. The error correction scheme developed here
thus corrects the optical pumping errors.

This quantum error correction gadget is especially well suited to the neutral
atom platform due to the ability to move atoms mid-circuit. Firstly, the swap
gates are easy to implement as we can move individual ancillas and data atoms
into a pairwise configuration to apply the CNOT gates parallelly. Secondly, at
the end of the protocol, the ancilla atoms can be used as the new data atoms
by simply moving them into the right positions.

##  V logical CNOT gate and Fault-tolerant threshold

Figure 8: The error corrected logical CNOT gadget. The logical $\mathrm{CNOT}$
gate is implemented by applying a physical CNOT gate between each qubit
(encoded in the spin) of the control and target blocks transversely. Error
correction steps are performed before and after the logical CNOT. We apply a
total of $r_{1}$ rounds of phase error correction and $r_{2}$ rounds of
amplitude error correction.

In this section, we provide lower bounds on the noise level that can be
tolerated in our proposed spin-cat code, while still achieving fault-tolerant
quantum computation. As discussed in Sec. III, to achieve fault-tolerance, we
need to guarantee that the effective noise strength in our implementation of
the logical gadgets of the inner code $\mathcal{C}_{1}$, as specified by Eq.
21, is below the noise threshold needed for the outer code $\mathcal{C}_{2}$
used in concatenation.

In this concatenated scheme, the main source of error is the logical CNOT gate
of $\mathcal{C}_{1}$, and hence, an upper bound on its failure probability
will provide an estimate for the threshold of all $\mathcal{C}_{2}$ gadgets
Puri  _et al._ (2020); Aliferis and Preskill (2008). The logical CNOT gadget
for the code $\mathcal{C}_{1}$ can be realized using transversal physical CNOT
gates between two code blocks, accompanied by error correction procedures to
correct phase and amplitude errors, which is illustrated in Fig. 8. For the
sake of generality, we consider each logical CNOT gadget to consist of $r_{1}$
applications of phase error correction and $r_{2}$ applications of amplitude
error correction. We define $r=r_{1}+r_{2}$ and denote the number of the data
qudits in each code block by $n$. The recovery operation for phase error
correction is determined by majority voting of the $r_{1}$ rounds of syndrome
measurement.

We start by estimating the probability of dephasing errors. In this case, the
analysis is similar to the analysis of biased cat qubits in bosonic systems
Puri  _et al._ (2020). Suppose each physical CNOT gate causes (independent)
dephasing errors on the target and control qubits with probability $\epsilon$.
During the application of each phase correction or amplitude correction
procedures, every qudit is acted upon by at most two physical CNOTs. Hence,
after $r_{1}$ repetition of phase corrections and $r_{2}$ repetition of
amplitude corrections the probability of dephasing error on each qudit, in
both the control and target block, will be at most $2r\epsilon$. After the
implementation of error correction steps, the next step is to implement the
transversal CNOTs between the control and target blocks of data qudits. This
operation can propagate phase errors from the target block to the control
block. Therefore, after the action of the transversal CNOT gates, the
probability of dephasing error on each qubit of the target and control blocks
is at most $2r\epsilon+\epsilon$ and $4r\epsilon+\epsilon$ respectively.

A logical error would occur if more than $(n+1)/2$ qubits are faulty in either
the target or the control code blocks. Thus the upper bound on the logical
phase error probability in the control and the target blocks can be given as
(keeping only the dominant term),

$\displaystyle\mathcal{\epsilon}_{\mathrm{target}}^{\mathrm{phase}}$
$\displaystyle\leq\binom{n}{\frac{n+1}{2}}(2r\epsilon+\epsilon)^{(n+1)/2},$
(61) $\displaystyle\mathcal{\epsilon}_{\mathrm{control}}^{\mathrm{phase}}$
$\displaystyle\leq\binom{n}{\frac{n+1}{2}}(4r\epsilon+\epsilon)^{(n+1)/2}.$

To account for the possible errors in the syndrome measurements in the phase
error correction step, we repeat measurements of $(n-1)$ syndromes in the
control and the target blocks $r_{1}$ times and take the majority vote to
apply error correction. A logical error happens if the syndrome is incorrect
for at least $(r_{1}+1)/2$ rounds of this procedure. Each syndrome measurement
requires two physical CNOT gates and we also need to account for state
preparation and measurement errors used in each syndrome measurement, both of
which can be performed with much higher accuracy compared to the rank-
preserving CNOT gate. Also one needs to account for the dephasing error
induced by the amplitude error correction following the phase error correction
which has two physical CNOT gates. Therefore the upper bound on the
probability of a dephasing error in each syndrome bit is at most $6\epsilon$.
As a result, the upper bound on the logical error for the syndrome measurement
is given by (only keeping the dominant term):

$\mathcal{\epsilon}_{\mathrm{ec}}^{\mathrm{phase}}\leq
2(n-1)\binom{r_{1}}{\frac{r_{1}+1}{2}}(6\epsilon)^{\frac{r_{1}+1}{2}}.$ (62)

Next, we establish an upper bound on the probability of logical errors
resulting from amplitude errors on the control and target, just before the
amplitude error correction step. An amplitude error on an individual qudit
occurs when a minimum of $k_{\mathrm{max}}=(2J-1)/2$ jumps has taken place.
This can be determined by summing the probabilities of $k_{\mathrm{max}}$
jumps, given a total of $s$ CNOT gates and is expressed as
$q(s,k_{\mathrm{max}})$ as given in Eq. 120 ($s=2r$). Following the error
correction steps, the subsequent phase involves implementing transversal CNOT
gates between the control and target blocks of data qudits. This operation,
however, has the potential to propagate amplitude errors from the control
block to the target block. Consequently, after the application of transversal
CNOT gates, the probability of amplitude errors on each qubit in the target
and control blocks is bounded by

$\displaystyle\mathcal{\epsilon}_{\mathrm{target}}^{\mathrm{amp}}$
$\displaystyle\leq 2nq(s=2r,k_{\mathrm{max}})+nq(s=1,k_{\mathrm{max}}),$ (63)
$\displaystyle\mathcal{\epsilon}_{\mathrm{control}}^{\mathrm{amp}}$
$\displaystyle\leq nq(s=2r,k_{\mathrm{max}})+nq(s=1,k_{\mathrm{max}}).$

Next, we provide upper bounds on the probability of logical error in the
amplitude error correction procedure. An ideal implementation of the swap
protocol described in Sec. IV would correct the amplitude errors by putting
back the state into the cat manifold, defined as the support of the projector
$\Pi_{0}$, where

$\Pi_{l}=\ket{+}_{l}\bra{+}_{l}+\ket{-}_{l}\bra{-}_{l}.$ (64)

Imperfect amplitude error correction may arise due to factors such as small
random rotations during the swapping process intended for error correction,
errors caused by optical pumping, or imperfections in ancilla preparation. For
the case of small random rotation errors and optical pumping, the error
operators involve at most two amplitude jumps as discussed in Sec. II.1.
Similarly, as discussed in Sec. III.3.1, optical pumping and random rotation
errors can create at most two amplitude jumps during the preparation of the
ancilla state. Thus the imperfect amplitude error correction can cause at most
four amplitude jumps. This phenomenon is conceptualized in Fig. 9, where the
population in the cat manifold can leak to $\Pi_{i}$ for $i=\\{1,2,3,4\\}$
manifolds with probabilities $p_{i}$.

Errors in the preparation of the ancilla can in principle result in a
superposition of $\ket{+}_{k}$ states with $k\leq 4$ instead of $\ket{+}_{0}$.
However, the amplitude error correction destroys any coherence between the cat
and kitten subspaces, resulting in a mixed state in the cat manifold (see App.
J). Hence, to find an upper bound on the success probability of amplitude
correction, we only need to consider the probability of error in preparing
$\ket{+}_{k}$ states with $k\leq 4$, rather than an arbitrary state in that
subspace.

Figure 9: Imperfect amplitude error correction gadget. There are two sources
of imperfection one can associate with the amplitude error correction. The
first one is a rotation error or optical pumping error occurring during the
swapping approach to correct amplitude errors. The second one is due to
imperfect preparation of the ancilla state, where ideally
$\rho_{A}=\ket{+}_{0}$, however, in a non-ideal setting the ancilla can be in
a mixture of $\ket{\pm}_{i}$ states where $i=\\{0,1,2,3,4\\}$, due to optical
pumping or rotation error during the state preparation. For an ideal amplitude
error correction, the final state lives in the
$\Pi_{0}=\ket{+}\bra{+}_{0}+\ket{-}_{0}\bra{-}_{0}$, whereas for a non-ideal
setting, there is a small probability to be in other manifold $\Pi_{l}$. The
figure shows when the final state is in the $\Pi_{i}$ where
$i=\\{0,1,2,3,4\\}$.

We denote the failure probability of the amplitude error correction given that
the ancilla states is in $\ket{+}_{k}$ by $q(s,k_{\mathrm{max}}|k)$ where $s$
is the total number of CNOT gates before the application of error correction,
and $k_{\mathrm{max}}$ is the minimum rank of the amplitude errors which
create a logical error, i.e. $k_{\mathrm{max}}=\lfloor(2J+1)/2\rfloor$ in our
construction. This probability can be calculated by adding the probabilities
of cascades of single and two jumps that push the population from level $k$ to
at least $k_{\mathrm{max}}$ level. Assuming the population only leaks to
$\Pi_{i}$ for $i=\\{1,2,3,4\\}$ the logical error probability after $r_{2}$
rounds of amplitude error correction can be bounded by

$\displaystyle\epsilon^{\mathrm{amp}}\leq
r_{2}\left(\sum_{k=0}^{4}q(s,k_{\mathrm{max}}|k)p_{k}\right),$ (65)

where $p_{0}=1-\sum_{i=1}^{4}p_{i}$. (For a detailed calculation see Sec.
J.1.) As we have $2n$ total qudits, the logical error probability of the
amplitude error correction blocks for the logical CNOT gate can be bounded by

$\displaystyle\mathcal{\epsilon}_{\mathrm{ec}}^{\mathrm{amp}}\leq
2n\epsilon^{\mathrm{amp}}.$ (66)

Note that unlike phase error correction where the measurement is repeated
$r_{1}$ many times and the correction is applied based on a majority vote of
syndrome results, amplitude error correction does not involve direct
measurement. Therefore repeated applications of amplitude error correction
without a phase correction step in between do not provide extra error
correction power.

Finally adding up all the probabilities of failures for the various components
of the logical CNOT gate, yields an upper bound on its total logical error
probability,

$\displaystyle\epsilon_{\mathrm{logical}}$
$\displaystyle\leq\mathcal{\epsilon}_{\mathrm{ec}}^{\mathrm{phase}}+\mathcal{\epsilon}_{\mathrm{control}}^{\mathrm{phase}}+\mathcal{\epsilon}_{\mathrm{target}}^{\mathrm{phase}}$
(67)
$\displaystyle+\mathcal{\epsilon}^{\mathrm{amp}}_{\mathrm{ec}}+\mathcal{\epsilon}_{\mathrm{control}}^{\mathrm{amp}}+\mathcal{\epsilon}_{\mathrm{target}}^{\mathrm{amp}}.$

To assess the improvement provided by our construction, we provide estimates
of $\epsilon$ for various noise parameters that guarantee a logical error
$\epsilon_{\mathrm{logical}}$ below the threshold demanded by the CSS code
$\mathcal{C}_{2}$. For the CSS code $\mathcal{C}_{2}$ we use the fault-
tolerant construction of Aliferis and Preskill (2009), with a provable
threshold of $\mathcal{\epsilon}_{\mathrm{CSS}}=0.67\times 10^{-3}$.

(a)

(b)

Figure 10: Logical error as a function of the physical level error (for
details of the relation between phase error and amplitude error, see App. A)
for the random rotation error for different value of $n$. Also, the threshold
one needs to achieve CSS encoding in the second layer of concatenation is
given for reference. Figure (a) is for the case of $p_{i}=0$ for $i\neq 0$ and
figure (b) is for an imperfect ancilla state preparation with $p_{i}=10^{-4}$
for $i\neq 0$. We can see whether the swapping error ideal or non-ideal does
not affect much except for very low noise and this in turn is because the
contribution of the amplitude error is very low for the random rotation error.
The black circle shows the intersection of the logical error with $y=x$ line
for the optimal case shown here and the gray circle shows the intersection of
the $\mathcal{\epsilon}_{\mathrm{CSS}}$ with the logical error for the optimal
case. The simulation is shown for $r_{1}=7$ and $r_{2}=1$.

In Fig. 10 we present the case of the small rotation error for encoding a
qubit in a qudit $J=9/2$ with $r_{1}=7$, $r_{2}=1$, and for different choices
of $n$. The figure on the left assumes no leakage error in the ancilla state
preparation, i.e. $p_{i}=0$ for $i\neq 0$, and the figure on the right is for
a leakage error of $p_{i}=10^{-4}$ for $i\neq 0$. As is evident in the figure,
the logical error rates for scenarios with and without leakage error exhibit
similar characteristics except for very low noise. This is expected since for
small rotation errors, the probability of amplitude error is exponentially
suppressed as a function of $J$ compared to the phase errors, see Fig. 12 for
more details. In particular, we find that for $n=21$, $r_{1}=7$, and
$r_{2}=1$, the physical error $\mathcal{\epsilon}$ needed to achieve the
targeted CSS threshold is less than $0.0054$.

(a)

(b)

Figure 11: Logical error as a function of the physical level error (for
details to the relation between phase error and amplitude error, see App. B)
for the optical pumping error for different value of $n$. The targeted
threshold for the CSS code in the second layer of concatenation is given for
reference. Figure (a) is for the case of $p_{i}=0$ for $i\neq 0$ and figure
(b) is for an imperfect ancilla state preparation with $p_{i}=10^{-4}$ for
$i\neq 0$. We can see a significant change in the behavior depending on
whether the amplitude error correction is ideal or not specifically in the low
noise regime. This in turn is due to the fact that for the case of optical
pumping, as seen in App. B, there is a significant contribution to the logical
error from the amplitude errors. The black circle shows the threshold value
for the optimal value of $n$ and the gray circle shows the intersection of the
$\mathcal{\epsilon}_{\mathrm{CSS}}$ with the logical error for the optimal
value of $n$. The simulation are shown for $r_{1}=7$ and $r_{2}=1$.

Next, in Fig. 11 we explore the impact of stronger photon scattering and
optical pumping on the encoding of a qubit in a qudit with $J=9/2$ with
$r_{1}=7$, $r_{2}=1$, considering various choices of $n$. (For the case of
$J=9/2$, we get $\alpha=0.0137$ and $\beta=0.2$ in Eq. 11 for stronger photon
scattering and optical pumping. Details of the noise model and parameters can
be found in App. B.) The left panel is the case with no leakage error
$p_{i}=0$ for $i\neq 0$, while the right panel incorporates a leakage error
with $p_{i}=10^{-4}$ for $i\neq 0$.

As can be seen in the figure, for the ideal amplitude error correction the
behavior of both the rotation error and case when photon scattering and
optical pumping are stronger are very similar in nature. However, when photon
scattering and optical pumping are stronger, the imperfect ancilla preparation
during amplitude error correction plays a more severe role in the overall
logical error of the low noise regime. The competition between the error
correction power of the gadget and the extra error due to the increased number
of qudits needed to encode a logical qubit leads to identifying a “sweet spot”
that determines the optimal number of qudits needed to encode a logical qubit.
In particular, we find that for $n=21$, $r_{1}=7$, and $r_{2}=1$, the physical
error needed to achieve the targeted CSS threshold is $\mathcal{\epsilon}\leq
0.0053$.

As discussed in detail in App. A, the primary error source for the considered
spin systems is the first-order angular momentum operators, stemming from
potential unwanted magnetic fields. Additionally, there are second-order terms
in the angular momentum operators due to optical pumping Deutsch and Jessen
(2010); Omanakuttan  _et al._ (2021). Despite this, the presence of extra
levels in the qudit results in a logical error contribution from amplitude
errors that is notably lower than that from phase errors. Thus the threshold
behavior for both these error models only impacts the low noise regimes.

##  VI Summary and Outlook

To achieve the full power of quantum computing, one needs to execute quantum
algorithms on error-corrected logical qubits. However, meeting the demanding
requirements for physical qubits and achieving low error rates, essential for
error-corrected logical qubits, remains a significant challenge in current
quantum implementations Knill (2005); Raussendorf  _et al._ (2007); Svore  _et
al._ (2006); Spedalieri and Roychowdhury (2008). Recent advancements in noise-
tailored error correction provide a promising avenue for achieving this by
substantially alleviating the stringent demands of error-corrected logical
qubits Aliferis and Preskill (2008); Webster  _et al._ (2015); Grassl  _et
al._ (1997); Wu  _et al._ (2022); Sahay  _et al._ (2023).

In this article, we follow this direction and introduce a fault-tolerant
quantum computation protocol by encoding a qubit into a spin system, with a
spin larger than $J=1/2$. The general scheme that we introduce in this work is
applicable to a wide range of physical spins, including in semiconductors
Gross (2021); Gross  _et al._ (2021), atomic ions Ringbauer  _et al._ (2022);
Low  _et al._ (2020), neutral atoms Omanakuttan  _et al._ (2021, 2023); Zache
_et al._ (2023), molecules Castro  _et al._ (2022), and superconducting
systems Özgüler and Venturelli (2022); Blok  _et al._ (2021), where we have
spin qudits that can be coherently controlled and entangled.

The specific encoding we consider in this article is the spin-cat encoding
which draws inspiration from the cat-code encoding for continuous variable
bosonic systems Puri  _et al._ (2020); Aliferis and Preskill (2008). For this
implementation we develop techniques to perform reliable computation in the
presence of dominant noise in spin systems, taking advantage of natively
available interactions. One key factor that distinguishes the spin-cat
encoding from the other encodings of a qubit in a qudit is that the total
Hilbert space of the spin-cat encoding decomposes into a direct sum of qubit
subspace. This induces the structure of a stabilizer code, a feature that
plays a pivotal role in enabling fault-tolerant schemes for error correction.

Spherical SU(2) tensor operators provide a basis in which to characterize the
error channels and identify the set of correctable errors. The dominant error
sources for encoding a qubit in a spin are the rank-1 SU(2) rotations and the
rank-2 tensors which can arise, e.g., from optical pumping between magnetic
sublevels. Our codes are constructed with these physical errors in mind. We
use the concatenation scheme of Aliferis and Preskill (2008) to perform fault-
tolerant computation. In addition to using an inner repetition code that
corrects phase errors, we correct for amplitude-damping errors by consuming
fresh ancilla spins and performing measurement-free error correction natively
for spin systems.

As a concrete application of our proposed scheme, we focus on the encoding of
a qubit in the nuclear spin of 87Sr, characterized by a spin of $9/2$. In this
scenario, we systematically build a universal gate set for fault-tolerant
quantum computing, leveraging the available interaction mechanisms. A pivotal
element in the formulation of the physical-level gate is the rank-preserving
CNOT gate. We elaborate on the implementation details of this gate, by taking
advantage of the metastable states available in 87Sr and the well-known
Rydberg blockade. In addition to the swap gadget that helps us correct
amplitude errors, this CNOT gate is used in the construction of a universal
gate set.

We also studied the threshold for fault-tolerant error correction and found
that it is much higher than found in standard protocols of error correction
with physical qubits, and it is similar to the threshold observed in bosonic
cat-codes Puri  _et al._ (2020). As a result, our approach demonstrates a
significant reduction in the required overhead and exhibits higher fault
tolerance thresholds compared to conventional qubit-based techniques.

Our work represents another example of designing resource-efficient fault-
tolerant schemes by taking advantage of the native noise characteristics of a
given hardware. In contrast to the earliest work in quantum error correction
where models were constructed for hypothetical qubits and generic noise
models, efforts are being made to develop error correcting codes that are
symbiotic with the control methods and noise structures of physical quantum
systems Gross (2021); Omanakuttan and Gross (2023); Puri  _et al._ (2020,
2019); Cong  _et al._ (2022); Gross  _et al._ (2021). A related direction of
research is to engineer qubit encodings with favorable noise properties Puri
_et al._ (2017); Wu  _et al._ (2022). This has been made possible because of
the substantial experimental advances in quantum computing ach (2023); Ryan-
Anderson  _et al._ (2022); Krinner  _et al._ (2022).

In a similar vein, the structure of our protocol works well with spin systems
and their control methods, regardless of the platform in which they are
implemented. It is particularly well-suited for the neutral atom platform,
where significant experimental advances have been achieved recently Bluvstein
_et al._ (2023, 2022); Graham  _et al._ (2022). We have previously explored
the use of quantum optimal control of spin-$9/2$ nuclei in 87Sr atoms for
arbitray single qudit gates Omanakuttan  _et al._ (2021) and two-qudit
entangling gates Omanakuttan  _et al._ (2023), where this protocol would be a
natural fit. The unique capabilities of neutral atom platforms, such as
reconfigurable connectivity and the ability to implement hundreds of parallel
entangling gates Bluvstein  _et al._ (2023) would assist in the implementation
of the fault-tolerant protocol we proposed here.

This work opens many directions for future research. One can extend the
current protocol for the rank-preserving CNOT gate in neutral atoms to other,
more experimental-friendly protocols. Specifically, one can explore using the
geometric phase approach Levine  _et al._ (2019) or Rydberg dressing-based
approaches Mitra  _et al._ (2020); Schine  _et al._ (2022); Martin  _et al._
(2021); Mitra  _et al._ (2023), typically used for entangling gates in qubits,
to realize the rank-preserving CNOT gate. In addition, similar to continuous-
variable cat encoding Puri  _et al._ (2020), the proposed gate set enables the
use of other codes, including topological codes. Another direct extension is
to develop gate sets to perform computation by encoding a qudit, rather than a
qubit, into the large spin.

Lastly, while we focus on errors caused by random rotations and optical
pumping in this paper. Another very important source of errors we didn’t
consider is leakage out of computational subspace, especially in the form of
atom loss in neutral atom platforms. The conventional approach to circumvent
these errors is to use leakage reduction units Suchara  _et al._ (2015). We
plan to address this by converting these errors into erasure errors, which are
easier to deal with Wu  _et al._ (2022), in future work.

##  VII Acknowledgments

This material is based upon work supported by NSF CAREER award No.
CCF-2237356. Additional support by the Quantum Leap Challenge Institutes
program (Grant No. 2016244) is acknowledged. The authors acknowledge fruitful
discussions with Anupam Mitra, Tyler Thurtel, Austin Daniel, and Karthik
Chinni.

###### Contents

  1. I Introduction
  2. II Generalization of cat code for Qudits/spin systems
    1. II.1 Error characterization
    2. II.2 The irreducible spherical tensor basis
  3. III Universal gate set and Rank-Preserving CNOT gate
    1. III.1 Single qubit gates
    2. III.2 Rank-preserving CNOT gate
    3. III.3 State preparation and Measurement
      1. III.3.1 Preparation of the spin-cat state
      2. III.3.2 Measurement of $X$
  4. IV Syndrome Measurement and error recovery
    1. IV.1 Error correction for Optical Pumping
  5. V logical CNOT gate and Fault-tolerant threshold
  6. VI Summary and Outlook
  7. VII Acknowledgments
  8. A Small Rotation errors
  9. B Photon scattering and optical pumping
  10. C Correctable set of errors
  11. D Action of the SU(2) operators
  12. E Rotating the ground and excited manifold differently using optimal control
  13. F Implementing Hadamard gate from the Physical level gates
  14. G Implementing the Logical operator
  15. H Toffoli gate
  16. I Alternate approaches for cat-state preparation and measurement of $X$
  17. J Error correction without measurement
    1. J.1 Upper bounds on the probability of the logical error in the amplitude error correction
  18. K Commutativity of the Error correction steps

##  Appendix A Small Rotation errors

(a)

(b)

Figure 12: Logical amplitude error probabilities due to rotation errors. (a)
The ratio of logical amplitude error to phase error is given as a function of
phase error. The probability of logical errors falls as the overall error rate
decreases. A logical error occurs when we have $\lfloor(2J-1)/2\rfloor$
amplitude errors and thus as spin $J$, increases, the ratio decreases
exponentially. However, for $J=3/2$, a single amplitude jump creates a logical
error and thus the ratio of logical error to phase error is a constant equal
to $1/2J$. (b) The ratio of logical error probability due to amplitude errors
to phase error for rotation error as a function of spin $J$. We can see that
this ratio exhibits an exponential trend, and the logical error becomes
negligible for sufficiently large values of $J$. Consequently, there is no
need for amplitude error correction in such cases.

A main source of decoherence for a qubit encoded in a spin is small random
rotation errors Gross  _et al._ (2021); Gross (2021). As given in Eq. 10, for
the spin-cat encoding the ratio of phase error to amplitude error decreases
with spin $J$ as $1/J$. However, for the spin-cat encoding, we need
$\lfloor(2J-1)/2\rfloor$ amplitude errors/jumps for a logical error (logical
amplitude error) to occur, such that these errors are not correctable by the
amplitude error correction (a logical bit flip error for the encoding in Eq.
12). As such, we look at the probability of such logical amplitude errors in
Fig. 12. In Fig. 12a we show that for a spin $J$, the logical amplitude error
decreases with phase error probability, and the decrease shows an exponential
behavior with spin $J$.

To further illustrate the exponential suppression of the logical error arising
from amplitude errors as a function of spin due to random rotation errors, in
Fig. 12b, the ratio of logical amplitude error probability to phase error for
rotation error is given as a function of spin $J$ for different value of phase
error. Notably, this ratio exhibits an exponential trend, and for sufficiently
large values of $J$, the logical amplitude error becomes negligible.
Consequently, there is no need for amplitude error correction in such cases.

##  Appendix B Photon scattering and optical pumping

Figure 13: The error process corresponding to the photon scattering and
optical pumping for encoding a qudit in an atomic spin $\mathbf{F}$. The
information is stored in the ground state and is controlled by laser light
with Rabi frequency $\Omega_{\mathrm{L}}$ and detuning $\Delta_{\mathrm{L}}$
from an excited state manifold, with spin $\mathbf{F}^{\prime}$. Absorption of
a laser photon (here $\pi$-polarized) is followed by a spontaneous emission
given by wavy lines. The process causes amplitude errors and can collapse a
cat-state to a single magnetic sublevel.

Another major source of decoherence for the qubit encoded in a spin is the
optical pumping arising from photon scattering when the spin are manipulated
by laser light. We consider here optical pumping arising from laser excitation
with Rabi frequency $\Omega_{\mathrm{L}}$ and detuning $\Delta_{\mathrm{L}}$
from a dominant resonance. Absorption of a laser with polarization
$\vec{\epsilon}_{L}$ is followed by a spontaneous emission of photon
$\mathbf{e}_{q}$. A schematic of the error process corresponding to the photon
scattering and optical pumping for atomic spins is shown in Fig. 13 for the
case $\vec{\epsilon}_{L}=\mathbf{e}_{0}$.

In this section, the spin angular momentum in which we encode the qudit is
$\mathbf{F}$, and $\mathbf{J}$ is the total angular momentum of the electrons.
The jump operators for the optical pumping followed by photon scattering are,
Deutsch and Jessen (2010):

$W_{q}=\sum_{F^{\prime}}\frac{\Omega_{\mathrm{L}}/2}{\Delta_{FF^{\prime}}+i\Gamma/2}(\bm{e}_{q}^{*}.\bm{D}_{FF^{\prime}})(\vec{\epsilon}_{L}.\bm{D}_{FF^{\prime}}^{\dagger}),$
(68)

where $\Omega_{\mathrm{L}}$ is the Rabi frequency and $\Delta_{FF^{\prime}}$
is the detuning between the ground state and excited with total spin $F$ and
$F^{\prime}$ respectively. $\Gamma$ is the characteristic linewidth of the
excited state, $\vec{\epsilon}_{L}$ is the polarization of the laser, and
$q={-1,0,1}$ represent the polarization of the scattered light.
$\bm{D}_{FF^{\prime}}$ are the dimensionless raising operators from a ground
state with total spin $F$ to an excited state with spin $F^{\prime}$ and see
Deutsch and Jessen (2010) for a detailed analysis of these operators.

By decomposing the dyadic into irreducible tensors, one can derive a basis
independent representation for the jump operators Deutsch and Jessen (2010),

$\displaystyle=\bm{e}_{q}^{*}.(\bm{D}_{FF^{\prime}}\bm{D}_{FF^{\prime}}^{\dagger}).\vec{\epsilon}_{L}$
(69)
$\displaystyle=C_{J^{\prime}FF^{\prime}}^{0}\bm{e}_{q}^{*}.\vec{\epsilon}_{L}+iC^{1}_{J^{\prime}FF^{\prime}}(\bm{e}_{q}^{*}\crossproduct\vec{\epsilon}_{L}).\bm{F}$
$\displaystyle+C_{J^{\prime}FF^{\prime}}^{2}\left[\frac{(\bm{e}_{q}^{*}.\bm{F})(\vec{\epsilon}_{L}.\bm{F})+(\vec{\epsilon}_{L}.\bm{F})(\bm{e}_{q}^{*}.\bm{F})}{2}-\frac{1}{3}|\bm{e}_{q}^{*}.\vec{\epsilon}_{L}|\bm{F}^{2}\right]$

where $J$ is the electron angular momentum. The above expression involves only
angular momentum operators of the form $\bm{F}$ (rank-1) and $\bm{F}^{2}$
(rank-2), and thus for photon scattering and optical pumping the error
operators are linear and quadratic powers of angular momentum operators. Then
the Lindblad master equation gives us:

$\displaystyle\frac{d\rho(t)}{dt}$
$\displaystyle=-i\left(H_{\mathrm{eff}}\rho(t)-\rho(t)H_{\mathrm{eff}}^{\dagger}\right)+\Gamma\sum_{i}W_{q}\rho(t)W_{q}^{\dagger}$
$\displaystyle\equiv\mathcal{L}\rho(t).$

where $\mathcal{L}$ is the Lindbladian and
$H_{\text{eff}}=H-i\sum_{q}W_{q}^{\dagger}W_{q}/2$.

From the jump operators, one can find the probability of phase errors and
amplitude errors by finding the overlap of the jump operators with the basis
operators as given in Eq. 15.

##  Appendix C Correctable set of errors

In this section, we find the set of correctable errors for the logical level
encoding $\mathcal{C}_{1}$ in Eq. 12. To find the correctable set of errors
$\\{E_{a}\\}$, one can use the Knill-Laflamme conditions Knill and Laflamme
(1997):

$\bra{\psi_{i}}E_{a}^{\dagger}E_{b}\ket{\psi_{j}}=C_{ab}\delta_{ij},$ (70)

where $i,j=\\{0,1\\}$ represents the codespace of interest.

The local angular momentum errors of interest here are of the form
$J_{x}^{l}J_{y}^{m}J_{z}^{n}$. From the locality assumption of the errors, one
can find that for the spin-cat encoding in Eq. 12,

$\bra{\psi_{i}}E_{a}^{\dagger}E_{b}\ket{\psi_{j}}=0\hskip 5.69046pt\forall
i\neq j.$ (71)

The next condition we need to satisfy for the spin-cat encoding is,

$\bra{+_{\mathrm{L}}}E_{a}^{\dagger}E_{b}\ket{+_{\mathrm{L}}}=\bra{-_{\mathrm{L}}}E_{a}^{\dagger}E_{b}\ket{-_{\mathrm{L}}}.$
(72)

From the locality assumption of the noise, this condition translates into two
cases. In the first case the error operators $E_{a}$ and $E_{b}$ act on the
same physical system, thus for the angular momentum errors the error
correction condition in Eq. 72 becomes,

$\displaystyle\bra{+}J_{x}^{l}J_{y}^{m}J_{z}^{n}J_{x}^{l^{\prime}}J_{y}^{m^{\prime}}J_{z}^{n^{\prime}}\ket{+}=\bra{-}J_{x}^{l}J_{y}^{m}J_{z}^{n}J_{x}^{l^{\prime}}J_{y}^{m^{\prime}}J_{z}^{n^{\prime}}\ket{-}.$
(73)

Using an alternate definition of the spin-cat codes,

$\displaystyle\ket{\pm}=\frac{\mathds{1}\pm\exp(i\pi
J_{y})}{\sqrt{2}}\ket{J,-J},$ (74)

Eq. 73 transforms into a compact expression:

$\displaystyle\bra{J,J}J_{x}^{l}J_{y}^{m}J_{z}^{n}J_{x}^{l^{\prime}}J_{y}^{m^{\prime}}J_{z}^{n^{\prime}}\ket{J,-J}$
(75)
$\displaystyle=\bra{J,-J}J_{x}^{l}J_{y}^{m}J_{z}^{n}J_{x}^{l^{\prime}}J_{y}^{m^{\prime}}J_{z}^{n^{\prime}}\ket{J,J}=0.$

Plugging the ladder operators,

$\displaystyle J_{+}$ $\displaystyle=J_{x}+iJ_{y}$ (76) $\displaystyle J_{-}$
$\displaystyle=J_{x}-iJ_{y}$

into Eq. 75, and using the condition that one needs at least $2J-1$ operations
of $J_{+}$ or $J_{-}$ to make the overlap between the states $\ket{J,J}$ and
$\ket{J,-J}$ non-zero, the error correction condition in Eq. 73 simplifies to,

$l+m+n+l^{\prime}+m^{\prime}+n^{\prime}\leq 2J-1.$ (77)

Thus we can correct the errors of the form $J_{x}^{l}J_{y}^{m}J_{z}^{n}$ if

$l+m+n\leq\lfloor\frac{2J-1}{2}\rfloor.$ (78)

The second case for Eq. 72 is when the two error operators $E_{a}$ and $E_{b}$
act on different physical systems. For the angular momentum errors this
simplifies to,

$\displaystyle\bra{+}J_{x}^{l}J_{y}^{m}J_{z}^{n}\ket{+}\bra{+}J_{x}^{l^{\prime}}J_{y}^{m^{\prime}}J_{z}^{n^{\prime}}\ket{+}$
(79)
$\displaystyle=\bra{-}J_{x}^{l}J_{y}^{m}J_{z}^{n}\ket{-}\bra{-}J_{x}^{l^{\prime}}J_{y}^{m^{\prime}}J_{z}^{n^{\prime}}\ket{-}.$

Again using the Eq. 74 and Eq. 76, the error correction condition is given as:

$\displaystyle l+m+n$ $\displaystyle\leq 2J-1,$ (80) $\displaystyle
l^{\prime}+m^{\prime}+n^{\prime}$ $\displaystyle\leq 2J-1.$

Hence the spin-cat encoding can correct all the errors of the form,

$\displaystyle\mathcal{E}_{K}=\left\\{J_{x}^{l}J_{y}^{m}J_{z}^{n};0\leq
l+m+n\leq K=\lfloor\frac{2J-1}{2}\rfloor\right\\}.$ (81)

##  Appendix D Action of the SU(2) operators

The Euler angle representation of an $\mathrm{SU}(2)$ operator
$V=\exp(-i\theta\hat{n}.\mathbf{J})$ is,

$V(\alpha,\beta,\gamma)=\exp(-i\theta\hat{n}.\mathbf{J})=e^{-i\alpha
J_{z}}e^{-i\beta J_{y}}e^{-i\gamma J_{z}}.$ (82)

The Wigner $D$ matrix defined in Eq. 24 can be expressed in terms of Euler
angles as,

$\displaystyle D_{q,q^{\prime}}(\alpha,\beta,\gamma)$
$\displaystyle=\bra{k,J_{z}=q^{\prime}}\exp(-i\theta\hat{n}.\mathbf{J})\ket{k,J_{z}=q}$
(83)
$\displaystyle=e^{-iq^{\prime}\alpha}d_{q,q^{\prime}}(\beta)e^{-iq\gamma}.$

Hence, deriving from the definitions of the spherical tensor operators in Eq.
23, the operators in Eq. 15, and the inherent properties of the Wigner $d$
matrices,

$d_{q,q^{\prime}}=(-1)^{q-q^{\prime}}d_{-q,-q^{\prime}},$ (84)

we find the action of an SU(2) rotation acting on the error operator, Eq. (15)
is

$\displaystyle VS^{(k)}_{q}V^{\dagger}$ (85)
$\displaystyle=\sum_{q^{\prime}}\frac{f_{q,q^{\prime}}(\vec{\theta})}{\sqrt{2}}\left(T^{(k)}_{q^{\prime}}+(-1)^{q-q^{\prime}+k}e^{-2i\left(q\alpha+q^{\prime}\beta\right)}T^{(k)}_{-q^{\prime}}\right),$
$\displaystyle=\sum_{q^{\prime}}f_{q,q^{\prime}}S^{(k)}_{q}+\frac{\widetilde{f}_{q,q}}{2}\left({F}^{(k)}_{q}-A^{(k)}_{q}\right).$

where to lighten the notation we defined,

$\widetilde{f}_{q,q}=(-1)^{k}\left[1-(-1)^{q-q^{\prime}}e^{-2i\left(q\alpha+q^{\prime}\beta\right)}\right]f_{q,q^{\prime}}.$
(86)

Thus,

$\displaystyle VS^{(k)}_{q}V^{\dagger}$
$\displaystyle=\sum_{q^{\prime}}g_{q,q^{\prime}}S^{(k)}_{q}+\widetilde{g}_{q,q}A^{(k)}_{q}$
(87)

where we have defined,

$\displaystyle g_{q,q^{\prime}}$
$\displaystyle=f_{q,q^{\prime}}+\frac{\widetilde{f}_{q,q^{\prime}}}{2},$ (88)
$\displaystyle\widetilde{g}_{q,q^{\prime}}$
$\displaystyle=-\frac{\widetilde{f}_{q,q^{\prime}}}{2}.$

Similarly,

$\displaystyle VA^{(k)}_{q}V^{\dagger}$
$\displaystyle=\sum_{q^{\prime}}h_{q,q^{\prime}}S^{(k)}_{q}+\widetilde{h}_{q,q}A^{(k)}_{q}$
(89)

where again for simplification of notation,

$\displaystyle h_{q,q^{\prime}}$
$\displaystyle=\frac{(-1)^{k}\left[1+(-1)^{q-q^{\prime}}e^{-2i\left(q\alpha+q^{\prime}\beta\right)}\right]f_{q,q^{\prime}}}{2},$
(90) $\displaystyle\widetilde{h}_{q,q^{\prime}}$
$\displaystyle=f_{q,q^{\prime}}-h_{q,q^{\prime}}.$

Thus the action of the $\mathrm{SU}(2)$ does not change the rank of the error
operators, $A^{(k)}_{q},{F}^{(k)}_{q}$ and obey the condition given in Eq. 22.

##  Appendix E Rotating the ground and excited manifold differently using
optimal control

To implement the rank-preserving CNOT gate in Fig. 2, one needs to implement
$X=\exp(-i\pi J_{x})$ gate on the auxiliary manifold while applying the
identity operator on the Rydberg manifold. For the specific choice of
auxiliary and Rydberg states considered, we have the Hamiltonian in the
rotating field as given by Eq. 43. As we are dealing with $\mathrm{SU}(2)$
representations of the spin $J$, the problem is isomorphic to the simultaneous
control of two two-level systems/two qubits with different Rabi frequencies
and different detuning. The objective would be to apply a Pauli $X$ operation
on the first qubit and identity on the second system. This problem has a
Quantum-Speed-Limit(QSL) of $\pi/\Omega_{\mathrm{rf}}$Hegerfeldt (2013).

Since $\Omega_{\mathrm{r}}=2\Omega_{\mathrm{a}}$, a pulse of length
$\pi/\Omega_{\mathrm{a}}$ would cause a full Rabi rotation in the Rydberg
manifold and only a half rotation in the auxiliary manifold. By choosing the
phases of N such pulses, $\vec{\phi}$ in Eq. 43, one can use quantum optimal
control algorithms to implement the desired transformation. The minimum number
of pulses $N$ required depends on the ratio $\omega_{0}/\Omega_{\mathrm{rf}}$.
While a solution with $N=2$ only exists when
$\omega_{0}/\Omega_{\mathrm{rf}}=3$, a solution with $N=3$ is possible if
$\omega_{0}/\Omega_{\mathrm{rf}}<3\sqrt{3}$ for example the case of
$\omega_{0}=5\Omega_{\mathrm{rf}}$ and $T=3\pi/\Omega_{\mathrm{rf}}$ is given
in Fig. 14. The overall trend is that with an increasing ratio
$\omega_{0}/\Omega_{\mathrm{a}}$, we need a larger $N$. This protocol is
similar to Levine  _et al._ (2019), and takes $N\pi/\Omega_{\mathrm{a}}$,
which is longer than the QSL. We can use waveforms with a large number of
steps to implement a gate in the minimum time $\pi/\Omega_{\mathrm{rf}}$, as
shown in the example below.

Figure 14:  Evolutions of the spin vector $\langle\vec{F}\rangle$ for the
auxiliary (a) and Rydberg (r) manifolds resulting from rf-driven Larmor
precession with time-varying phases in Eq. 43 for piecewise constant function
with $3$ time steps with a total time
$T_{\mathrm{tot}}=3\pi/\Omega_{\mathrm{rf}}$ and
$\omega_{0}=5\Omega_{\mathrm{rf}}$. For the specific choice of parameters, an
$X$ gate acts on the auxiliary manifold and transfers the population from
$\overline{0}_{\mathrm{a}}$-subspace to $\overline{1}_{\mathrm{a}}$-subspace
and vice-versa. However, for the Rydberg manifold, the pulse sequence acts as
an identity operator, and the population in the $\overline{0}_{\mathrm{r}}$
and $\overline{1}_{\mathrm{r}}$ subspaces remain unaffected.

Using the Hamiltonians in Eq. 43, one can also optimize the phase $\phi$ to
implement a gate $R(\theta)=\exp(-i\theta\hat{\bm{n}}.\mathbf{J})$ in the
auxiliary manifold and identity on the Rydberg manifold. For example, the
pulse scheme for the $R=\exp(i\pi J_{z})$ for the auxiliary manifold, which
can be used to implement the rank-preserving $\mathrm{CZ}$ gate is given in
Fig. 15. The total time is $\Omega_{\mathrm{rf}}T=\pi$ and total time is
divided into $10$ equal time steps with $\omega_{0}=3\Omega_{\mathrm{rf}}$.

Finally, for $\omega_{0}\gg\Omega_{\mathrm{rf}}$, a field that is resonant for
the auxiliary spin will be far off-resonant for the Rydberg manifold. So we
can implement any desired transformation $\mathrm{SU}(2)$ operation in the
auxiliary subspace without disturbing the Rydberg manifold populations.

Figure 15:  The phase $\phi(t)$ which generates an $R=\exp(i\pi J_{z})$ for
the auxiliary manifold and an identity in the Rydberg manifold, which can be
used to implement the rank-preserving $\mathrm{CZ}$ gate. The total time is
$\Omega_{\mathrm{rf}}T=\pi$, which is divided into $10$ equal time steps with
$\omega_{0}=3\Omega_{\mathrm{rf}}$ and pulse sequence is found using the
quantum optimal control algorithm GRAPE.

##  Appendix F Implementing Hadamard gate from the Physical level gates

Figure 16: Circuit implementing a fault-tolerant Hadamard gate using the
physical level gates for the spin-cat encoding. This differs from the standard
implementation as we use both CNOT and CZ gate to implement the action of the
target unitary of interest.

The physical level gates for the spin-cat encoding are given as,

$\\{\mathcal{M}_{z},\mathcal{M}_{X},\mathcal{P}_{\ket{+}},\mathcal{P}_{\ket{0}},\mathrm{CNOT},X,Y,Z,ZZ(\theta)\\}.$
(91)

The Hadamard gate is not in the universal gate set as it does not preserve the
rank. Here we show the implementation of the Hadamard gate using the rank-
preserving physical level gates and an ancilla qubit. The circuit diagram
corresponding to a teleportation-based scheme for the Hadamard gate is given
in the Fig. 16. Consider an initial arbitrary state,

$\ket{\psi}=\alpha\ket{0}_{k}+\beta\ket{1}_{k},$ (92)

and ancilla state,

$\ket{+}_{0}=\frac{1}{\sqrt{2}}\left(\ket{0}+\ket{1}\right).$ (93)

Define $\ket{\phi}=\ket{\psi}\otimes\ket{+}_{0}$, then

$\displaystyle\mathrm{CNOT}\ket{\phi}=\mathrm{CNOT}\ket{\psi}\otimes\ket{+}_{0}$
(94)
$\displaystyle=\frac{1}{\sqrt{2}}\left(\alpha\ket{0}_{k}\ket{0}+\alpha\ket{1}_{k}\ket{1}+\beta\ket{1}_{k}\ket{0}+\beta\ket{0}_{k}\ket{1}\right),$

and

$\displaystyle\mathrm{CZ}\text{ }\mathrm{CNOT}\ket{\phi}$
$\displaystyle=\left(\alpha\ket{-}_{k}+\beta\ket{+}_{k}\right)\ket{+}$ (95)
$\displaystyle+\left(\alpha\ket{+}_{k}-\beta\ket{-}_{k}\right)\ket{-}.$

Thus one can act $Z$ or $X$ gate depending on the measurement of the $X$
operator in the ancilla to get the state,

$H\ket{\psi}=\alpha\ket{+}_{k}+\beta\ket{+}_{k},$ (96)

and implement the action of the Hadamard gate.

##  Appendix G Implementing the Logical operator

(a)

(b)

(c)

(d)

Figure 17: Circuits implementing logical level gates in $\mathcal{C}_{1}$
using the physical level gates. (a) Logical $\mathrm{CNOT}$
$\mathrm{CNOT}_{\mathrm{L}}$. To implement $\mathrm{CNOT}_{\mathrm{L}}$, we
apply physical $\mathrm{CNOT}$ gates transversally on all qubit pairs. (b)
Preparation of $\ket{0}_{\mathrm{L}}$ $\mathcal{P}_{\ket{0}_{\mathrm{L}}}$.
$\ket{0}_{\mathrm{L}}$ is prepared by initializing the system with the state
$\mathcal{P}_{\ket{+}_{\mathrm{L}}}$ and measuring the parity. To measure the
parity we use an ancilla initialized with $\mathcal{P}_{\ket{0}}$ and use
physical CNOT gates followed by measuring the $\mathcal{M}_{Z}$, the final
state is $\ket{0}_{\mathrm{L}}$ or $\ket{1}_{\mathrm{L}}$ for the measurement
outcomes $1$ and $-1$ respectively. (c) The Logical $Z$ measurement
$\mathcal{M}_{Z_{\mathrm{L}}}$. An ancilla state is prepared in $\ket{+}$ and
physical CZ gates with the data qubits are applied followed by measuring the
ancilla in the $X$ basis. (d) Logical $X$ measurement
$\mathcal{M}_{X_{\mathrm{L}}}$. The logical $X$ is measured by applying the
physical CNOT gates and then measurement along $X$.

In this section, we demonstrate the universal gate set at the logical level
with the physical level gates for the spin-cat encoding. The rank-preserving
physical level gates for the spin-cat encoding are,

$\\{\mathcal{M}_{Z},\mathcal{M}_{X},\mathcal{P}_{\ket{+}},\mathcal{P}_{\ket{0}},\mathrm{CNOT},X,Y,Z\\}.$
(97)

Consider a universal gate set,

$\\{\mathcal{M}_{Z_{\mathrm{L}}},\mathcal{M}_{X_{\mathrm{L}}},\mathcal{P}_{\ket{+}_{\mathrm{L}}},\mathcal{P}_{\ket{0}_{\mathrm{L}}},\mathrm{CNOT}_{\mathrm{L}}\\},$
(98)

where $\mathcal{P}$ refers to preparation and $\mathcal{M}$ denotes
measurement. The logical preparation of the
$\mathcal{P}_{\ket{+}_{\mathrm{L}}}$ can be done transversally by preparing
the $\mathcal{P}_{\ket{+}}$ in the individual systems. For example in the case
of three physical systems, the logical level state preparation is,

$\displaystyle\mathcal{P}_{\ket{+}_{\text{L}}}=\ket{+}_{\text{L}}$
$\displaystyle=\ket{+++}.$ (99)

In a similar fashion, the construction of additional logical-level gates
follows the approach detailed in Aliferis and Preskill (2008); Puri  _et al._
(2020). Comprehensive details for the implementation of all other logical
gates are provided in Fig. 17. In (a), the $\mathrm{CNOT}_{\mathrm{L}}$ is
implemented using the physical CNOT gates. One can implement the
$\mathrm{CNOT}_{\mathrm{L}}$ by transversal application of the CNOT gates. In
(b), the $\mathcal{P}_{\ket{0}_{\mathrm{L}}}$ is prepared by initializing the
system with the state $\mathcal{P}_{\ket{+}_{\mathrm{L}}}$ and measuring the
parity. To measure the parity we use an ancilla initialized with
$\mathcal{P}_{\ket{0}}$ and use physical CNOT gates followed by measuring the
$\mathcal{M}_{Z}$, the final state is $\mathcal{P}_{\ket{0}_{\mathrm{L}}}$ and
$\mathcal{P}_{\ket{1}_{\mathrm{L}}}$ for the measurement outcomes $1$ and $-1$
respectively. (c) implements the logical measurement of $Z$ with an ancilla
state prepared in $\ket{+}$ and physical CZ gates followed by measuring the
ancilla in the $X$ basis. Finally (d) implements the logical measurement by
applying the physical CNOT gates and measurement of $X$.

##  Appendix H Toffoli gate

Figure 18:  Protocol for a rank-preserving Toffoli gate for spin-cat encoding
using $\mathrm{SU}(2)$ operations. Similar to the rank-preserving CNOT gate
Fig. 2, we implement the Toffoli gate in the ground state of 87Sr and the
physical setting is the same as given in Fig. 2a. We consider a geometry of
atoms such that the nearest neighbors are constrained by the Rydberg blockade,
but the next-nearest neighbors are not constrained. In step I the population
is promoted to the auxiliary manifold in the atoms. In the control atoms we
only promote the population of the $\overline{0}$-subspace whereas for the
target atom, the population from both the $\overline{0}$ and $\overline{1}$
subspaces are promoted to the auxiliary state. In step II, we transfer the
population between the auxiliary and the Rydberg manifolds of the control
atoms. In step III, we transfer the population from the auxiliary to the
Rydberg manifold of the target atom. However, due to the Rydberg blockade,
this population transfer only happens when both the control atoms are in
$\overline{0}$-subspaces. If even one of the control atoms is in
$\overline{1}$-subspace this transition is blockaded. Then similar to the
rank-preserving CNOT gate, in step IV we implement aa $X=\exp(-i\pi J_{x})$
gate in the auxiliary manifold and an identity operator in the Rydberg
manifold. Finally, we will transfer all the states back to the ground state by
acting steps III-I in reverse, thus implementing a rank-preserving Toffoli
gate for the spin-cat encoding.

One can generalize the rank-preserving CNOT gate in Fig. 2 to construct a
Toffoli gate, also known as a controlled-controlled NOT gate. Fig. 18 gives
the protocol for creating the rank-preserving Toffoli gate for the spin-cat
encoding using only $\mathrm{SU(2)}$ interactions. Again, similar to the rank-
preserving CNOT gate, the Toffoli gate is implemented in the ground state of
87Sr. The key to the scheme is the availability of special geometries for the
neutral atoms Levine  _et al._ (2019); Bluvstein  _et al._ (2022). Here we use
a geometry such that for three linearly arranged atoms, the nearest neighbors
are constrained by the Rydberg blockade, but the next-nearest neighbors are
not constrained by it. The central atom acts as the target atom while its two
neighbors are the control atoms.

In step I of the Toffoli gate, the population is promoted to the auxiliary
state. For the case of the control atoms we only promote the population of the
$\overline{0}$-subspace whereas for the target atom, the population from both
the $\overline{0}$ and $\overline{1}$ subspaces are promoted to the auxiliary
state. In step II, we use a pulse sequence similar to the Fig. 3c to transfer
the population between the auxiliary and the Rydberg state of the control
atoms using $\pi$ polarized light. In step III we apply the same pulse
sequence as in step II to the target atom however, due to the Rydberg
blockade, the population transfer between the auxiliary and Rydberg state only
happens when both the control atoms are in $\overline{1}$-subspace. Then
similar to the case of the rank-preserving CNOT gate in step IV, we implement
a $X=\exp(-i\pi J_{x})$ gate in the auxiliary manifold and an identity
operator in the Rydberg manifold. Finally, we will transfer all the states
back to the ground state by acting steps III-I in reverse, thus implementing a
rank-preserving Toffoli gate for the spin-cat encoding up to local rotations.

Thus when one of the control atoms is in the $\overline{0}$-subspace, $X$ gate
is applied target atom, and when both the control atoms are in the
$\overline{1}$-subspace, the target atoms remain unchanged. This is the
Toffoli gate up to a local $X=\exp(-i\pi J_{x})$ rotation on the target atom.

##  Appendix I Alternate approaches for cat-state preparation and measurement
of $X$

One can use alternative approaches than quantum optimal control for cat-state
preparation and measurement of $X$. In this section, we detail some of those
approaches. For example, one can use an adiabatic approach and one-axis
twisting to create a spin-cat state.

I) Adiabatic approach. Starting with an initial state $\ket{J,J_{z}=J}$ and
evolving the Hamiltonian

$H(s)=(1-s)J_{x}-\frac{s}{2J}J_{z}^{2},$ (100)

adiabatically from $s=0$ to $s=1$ guarantees the final state to be close to a
cat state $\ket{+}$ Puri  _et al._ (2017). This can be implemented in atomic
systems using a combination of tensor light shifts and rf rotation Chaudhury
_et al._ (2007); Omanakuttan  _et al._ (2021).

II) One-axis twisting: Using a time-independent Hamiltonian, $H=\beta
J_{z}^{2}$, for a certain time $T=\pi/(2\beta)$, one can evolve a spin
coherent state along $J_{x}$ to prepare a high-fidelity cat state.

$\ket{+}=\exp(-i\pi J_{x})\exp(-i\frac{\pi}{2}J_{z}^{2})\ket{J,J_{x}=J}.$
(101)

Including the effect of decoherence due to photon scattering and optical
pumping for 87Sr, we find the fidelity for one-axis twisting is $0.9998$
whereas for the adiabatic preparation, one can achieve a fidelity of $0.9889$.

Similarly one can use an alternative approach to measure $X$, in particular,
to know if the ancilla state is in $\ket{+}$ or $\ket{-}$. We can
adiabatically rotate the states using the Hamiltonian

$H(s)=-(1-s)J_{z}^{2}/(J)+sJ_{x},$ (102)

which implements the following transformations:

$\displaystyle\ket{+}_{0}$ $\displaystyle\to\ket{J,J_{x}=J},$ (103)
$\displaystyle\ket{-}_{0}$ $\displaystyle\to\ket{J,J_{x}=J-1},$

and then then measuring $J_{x}$.

To evaluate the accuracy of $X$ measurement, we define the target isometry as:

$V_{\mathrm{targ}}=\ket{J,J_{x}=J}\bra{+}+\ket{J,J_{x}=J-1}\bra{-}.$ (104)

The implemented isometry using the adiabatic approach is given as,

$V=e^{-\int\mathcal{L}(s)ds}V(0)$ (105)

where $\mathcal{L}(s)$ is the Lindbladian including the effects of decoherence
and

$V(0)=\ket{+}\bra{+}+\ket{-}\bra{-}.$ (106)

Thus the fidelity for the implementation of the isometry is defined as:

$\mathcal{F}_{\mathrm{iso}}=\frac{1}{4}\lvert\Tr(V_{\mathrm{targ}}V^{\dagger})\rvert^{2}.$
(107)

This approach is similar to the approach taken in bosonic cat qubits Puri  _et
al._ (2019). To measure $J_{x}$, we first implement the unitary transformation
$U=\exp(-i\pi/2J_{y})$ to rotate the basis to $\ket{J,J_{z}}$ and then perform
the readily accessible measurement $\mathcal{M}_{Z}$ which we can in principle
achieve with a fidelity larger than $99\%$ Barnes  _et al._ (2022b). Including
the effects of optical pumping as discussed in App. B, one can implement this
transformation with a fidelity of $\mathcal{F}_{\mathrm{iso}}=0.98$ for the
87Sr nuclear spin qudit.

##  Appendix J Error correction without measurement

An alternative to syndrome-based quantum error correction is measurement-free
quantum error correction (MFQEC) Crow  _et al._ (2016); Ercan  _et al._
(2018); Li  _et al._ (2011); Premakumar  _et al._ (2020). The standard
syndrome-based error correction is given by recovery operation:

$R(\rho)=\sum_{i}U_{i}M_{i}\rho M_{i}^{\dagger}U_{i}^{\dagger},$ (108)

where for a general state $\rho$, $M_{i}$ is the syndrome measurement and
$U_{i}$ is the correction unitary according to the outcome of the syndrome
measurement.

MFQEC is based on the unitary operator $V$, which couples the data and ancilla
qubits. The action of which is given as,

$V\ket{\psi}\ket{0}=\sum_{i}\left(U_{i}M_{i}\otimes\mathds{1}\right)\ket{\psi}\ket{i}.$
(109)

Defining $\rho=\sum_{kl}\alpha_{kl}\ket{\psi}_{k}\bra{\psi}_{l}$, we can find
that,

$V\rho\otimes\outerproduct{0}{0}V^{\dagger}=\sum_{k,l,i,j}\alpha_{kl}U_{i}M_{i}\ket{\psi}_{k}\bra{\psi}_{l}M_{j}^{\dagger}U_{j}^{\dagger}\otimes\outerproduct{i}{j}$
(110)

Partial tracing of the ancilla gives,

$\rho_{\mathrm{rec}}=\sum_{i}U_{i}M_{i}\rho M_{i}^{\dagger}U_{i}^{\dagger}.$
(111)

Thus the MFQEC is equivalent to syndrome-based error correction and the key
for MFQEC is a specific unitary gate between the ancilla and the data.

One can consider a fault-tolerant MFQEC scheme for the amplitude errors. The
syndrome for the amplitude errors is the eigenvalue of $J_{z}^{2}$, which can
be extracted by the projective measurement,

$M_{k}=\ket{+}_{k}\bra{+}_{k}+\ket{-}_{k}\bra{-}_{k},$ (112)

where $0\leq k\leq(2J-1)/2$. Recovery unitaries corresponding to the
projective measurement outcomes are

$\displaystyle U_{k}$
$\displaystyle=\ket{+}_{0}\bra{+}_{k}+\ket{+}_{k}\bra{+}_{0}+\ket{-}_{0}\bra{-}_{k}+\ket{-}_{k}\bra{-}_{0}$
(113) $\displaystyle+\sum_{j\neq k,j\neq
0}\ket{+}_{j}\bra{+}_{j}+\ket{-}_{j}\bra{-}_{j},$

which takes the state from the subspace,

$\\{\ket{+}_{k},\ket{-}_{k}\\}\to\\{\ket{+}_{0},\ket{-}_{0}\\}.$ (114)

Consider the following unitary operator, using the definitions from Eqs. 6, 26
and 29 the product of three alternating CNOT gates can be written as:

$\displaystyle V_{s}$
$\displaystyle=\Pi_{\overline{0}}\otimes\Pi_{\overline{0}}+\Pi_{\overline{1}}\otimes\Pi_{\overline{1}}$
(115) $\displaystyle+X\Pi_{\overline{0}}\otimes
X\Pi_{\overline{1}}+X\Pi_{\overline{1}}\otimes X\Pi_{\overline{0}},$

Consider the following states,

$\displaystyle\ket{\psi}_{k}$
$\displaystyle=\alpha\ket{+}_{k}+\beta\ket{-}_{k},$ (116)
$\displaystyle\ket{\phi}_{l}$
$\displaystyle=\gamma\ket{+}_{l}+\delta\ket{-}_{l},$

where $\alpha,\beta,\gamma,\text{ and }\delta$ are arbitrary complex
amplitudes. The action of the $V_{s}$ on the state,
$\ket{\xi}=\ket{\psi}_{k}\otimes\ket{\phi}_{l}$ gives,

$V_{s}\ket{\xi}=\ket{\phi}_{k}\otimes\ket{\psi}_{l}.$ (117)

Thus $V_{s}$ gate swaps the information between two kitten or cat states. The
circuit diagram for the $V_{s}$ gate for a qubit encoded in the qudit is given
in Fig. 7a.

When the second qudit is prepared in $\ket{+}_{0}$ state, as shown in Fig. 7b,
the application of the $V_{s}$ gate gives

$\ket{\phi}=V_{s}\ket{\psi}\ket{+}_{0}=\ket{+}_{k}\otimes\left(\alpha\ket{+}_{0}+\beta\ket{-}_{0}\right).$
(118)

The above state can also be written as,

$\ket{\phi}=\sum_{k}U_{k}M_{k}\ket{\psi}\ket{+}_{k},$ (119)

where the notion of data and ancilla qubits are swapped for convenience. Thus
the unitary operator $V_{s}$ followed by partial tracing implements the
desired recovery operation. Thus one can correct the amplitude error fault
tolerantly using a combination of two rank-preserving CNOT gates and fresh
$\ket{+}_{0}$ state.

For fault-tolerant gadgets, one needs to repeat the phase and amplitude error
correction multiple times and one needs to ensure that these two error
correction steps commute with each other. The phase error correction Fig. 6
commutes with measurement-free error correction of the amplitude error and the
details of the calculation are given in App. K.

###  J.1 Upper bounds on the probability of the logical error in the amplitude
error correction

In this section, we provide a detailed analysis to find an upper bound on the
probability of a logical error in the amplitude error correction used in the
error-corrected logical CNOT gadget in Fig. 8.

First, consider the case where ancilla is prepared perfectly, i.e., we have
$\rho_{A}=\ket{+}_{0}$ and $p_{i}=0$ for $i\neq 0$ in Fig. 9. In this case, a
logical amplitude error occurs after $s$ faulty CNOT gates if they create at
least $k_{\mathrm{max}}=\lfloor(2J+1)/2\rfloor$ many jumps, the probability of
which we denote by $q(s,k_{\mathrm{max}})$. The number of CNOT gates $s$ is
determined by the number of phase error corrections that appear before an
amplitude correction, in addition to the two CNOT acting in the amplitude
error correction itself. To find the probability $q(s,k_{\mathrm{max}})$, we
note that each physical CNOT gate can create one or two jumps with
probabilities $p_{1}$ and $p_{2}$ respectively, and therefore we need to add
the probabilities of cascades of one and two jumps that can create more than
$k_{\mathrm{max}}$ jumps. Therefore $q(s,k_{\mathrm{max}})$ can be written as

$q(s,k_{\mathrm{max}})=\sum_{i}\lambda_{i}(s,k_{\mathrm{max}}),$ (120)

where $\lambda_{i}$ represents the probability of one path such that we have
at least $k_{\mathrm{max}}$ jumps. For example, consider the case of $s=4$ and
$J=9/2$, then $\lambda_{i}$ represents all the possible combinations of one
and two jumps, such that the total sum of these jumps is at least $5$. One
such possibility is a combination of $(1,1,1,2)$ where we have one jump
occurring at the first three CNOTs and two jumps occurring at the last CNOT.

When the ancilla is imperfect, for example, if it is prepared in $\ket{+}_{k}$
state rather than $\ket{+}_{0}$, one needs to find the paths that create
$k_{\mathrm{max}}-k$ many jumps. Thus we get,

$q(s,k_{\mathrm{max}}|k)=\sum_{i}\lambda_{i}(s,k_{\mathrm{max}}|k),$ (121)

where $\lambda_{i}(s,k_{\mathrm{max}}\lvert k)$ is the probability of a path
where we have at least $k_{\mathrm{max}}$ jumps given that we already had $k$
jumps to start with.

We repeat the amplitude error correction $r_{2}$ many times in one error-
corrected logical CNOT gate. Thus the upper bound of the logical amplitude
error probability after $r_{2}$ rounds of error correction in Fig. 8 is,

$\epsilon^{\mathrm{amp}}=r_{2}\left(\sum_{k=0}^{4}q(s,k_{\mathrm{max}}|k)p_{k}\right).$
(122)

where $p_{k}$ is the probability of ancilla starting at $\ket{+}_{k}$.

##  Appendix K Commutativity of the Error correction steps

The error correction for the spin-cat encoding follows two steps. The first
step is the phase error correction in Fig. 6 and the second step is the
measurement-free error correction for correcting amplitude errors given in
Fig. 7. For fault-tolerant gadgets, one needs to repeat these steps multiple
times and we need to ensure that these two error correction steps commute with
each other such that the errors do not proliferate uncontrollably. For this,
we need to satisfy,

$\mathcal{R}_{\mathrm{amp}}\mathcal{R}_{\mathrm{ph}}\left(\mathcal{E}\left(\rho\right)\right)=\mathcal{R}_{\mathrm{ph}}\mathcal{R}_{\mathrm{amp}}\left(\mathcal{E}\left(\rho\right)\right),$
(123)

where $\mathcal{R}_{\mathrm{amp}},\mathcal{R}_{\mathrm{ph}}$ are the recovery
maps corresponding to the amplitude and phase error correction respectively.
The recovery map for the amplitude error can be expressed in terms of the
Kraus operators as,

$\mathcal{R}_{\mathrm{amp}}(\rho)=\sum_{j,i}M_{j,i}^{\mathrm{amp}}\rho\left(M_{j,i}^{\mathrm{amp}}\right)^{\dagger},$
(124)

where,

$\displaystyle M^{\mathrm{amp}}_{j,1}$
$\displaystyle=\left(\bra{j}^{(1)}V_{s}\ket{+}_{0}^{(2)}\right)\otimes\mathds{1}\otimes\mathds{1},$
(125) $\displaystyle M^{\mathrm{amp}}_{j,2}$
$\displaystyle=\mathds{1}\otimes\left(\bra{j}^{(1)}V_{s}\ket{+}_{0}^{(2)}\right)\otimes\mathds{1},$
$\displaystyle M^{\mathrm{amp}}_{j,3}$
$\displaystyle=\mathds{1}\otimes\mathds{1}\otimes\left(\bra{j}^{(1)}V_{s}\ket{+}_{0}^{(2)}\right),$

and $V_{s}$ is the unitary operator given in Eq. 115. The Kraus operator
representation of the phase error correction for spin-cat encoding is,

$\mathcal{R}_{\mathrm{ph}}(\rho)=\sum_{i,j}M_{i,j}^{\mathrm{ph}}\rho\left(M_{i,j}^{\mathrm{ph}}\right)^{\dagger}$
(126)

where,

$\displaystyle M_{00}^{\mathrm{ph}}$
$\displaystyle=\sum_{i,j,k}\ket{+}_{i}\ket{+}_{j}\ket{+}_{k}\bra{+}_{i}\bra{+}_{j}\bra{+}_{k}+\ket{-}_{i}\ket{-}_{j}\ket{-}_{k}\bra{-}_{i}\bra{-}_{j}\bra{-}_{k},$
(127) $\displaystyle M_{01}^{\mathrm{ph}}$
$\displaystyle=Z_{3}\sum_{i,j,k}\ket{+}_{i}\ket{+}_{j}\ket{-}_{k}\bra{+}_{i}\bra{+}_{j}\bra{-}_{k}+\ket{-}_{i}\ket{-}_{j}\ket{+}_{k}\bra{-}_{i}\bra{-}_{j}\bra{+}_{k},$
$\displaystyle M_{10}^{\mathrm{ph}}$
$\displaystyle=Z_{1}\sum_{i,j,k}\ket{-}_{i}\ket{+}_{j}\ket{+}_{k}\bra{-}_{i}\bra{+}_{j}\bra{+}_{k}+\ket{+}_{i}\ket{-}_{j}\ket{-}_{k}\bra{+}_{i}\bra{-}_{j}\bra{-}_{k},$
$\displaystyle M_{11}^{\mathrm{ph}}$
$\displaystyle=Z_{2}\sum_{i,j,k}\ket{+}_{i}\ket{-}_{j}\ket{+}_{k}\bra{+}_{i}\bra{-}_{j}\bra{+}_{k}+\ket{-}_{i}\ket{+}_{j}\ket{-}_{k}\bra{-}_{i}\bra{+}_{j}\bra{-}_{k}.$

To prove the commutativity of the two error correction steps first consider
the Kraus operators $M_{00}^{\mathrm{ph}}$ and $M^{\mathrm{amp}}_{j,1}$, we
get

$\displaystyle
M_{j,1}^{\mathrm{amp}}M_{00}^{\mathrm{ph}}=\sum_{klm}\bra{j}\ket{+}_{k}\left(\ket{+}_{0}^{(b)}\ket{+}_{l}^{(a)}\ket{+}_{m}^{(a)}\bra{+}_{k}^{(a)}\bra{+}_{l}^{(a)}\bra{+}_{m}^{(a)}+\ket{-}_{0}^{(b)}\ket{-}_{l}^{(a)}\ket{-}_{m}^{(a)}\bra{-}_{k}^{(a)}\bra{-}_{l}^{(a)}\bra{-}_{m}^{(a)}\right),$
(128)

and,

$\displaystyle M_{00}^{\mathrm{ph}}M_{j,1}^{\mathrm{amp}}$
$\displaystyle=\sum_{k,l,m}\ket{+}_{k}^{(b)}\ket{+}_{l}^{(a)}\ket{+}_{m}^{(a)}\bra{+}_{k}^{(b)}\bra{+}_{l}^{(a)}\bra{+}_{m}^{(a)}\bra{j}^{(a)}\otimes\mathds{1}^{(b)}V_{s}^{(ab)}\mathds{1}^{(a)}\otimes\ket{+}_{0}^{(b)}$
(129)
$\displaystyle+\sum_{k,l,m}\ket{-}_{k}^{(b)}\ket{-}_{l}^{(a)}\ket{-}_{m}^{(a)}\bra{-}_{k}^{(b)}\bra{-}_{l}^{(a)}\bra{-}_{m}^{(a)}\bra{j}^{(a)}\otimes\mathds{1}^{(b)}V_{s}^{(ab)}\mathds{1}^{(a)}\otimes\ket{+}_{0}^{(b)}.$

Using the resolution of the identity
$\mathds{1}=\sum_{p}\ket{+}_{p}\bra{+}_{p}+\ket{-}_{p}\bra{-}_{p}$ in the
above equation yields,

$\displaystyle
M_{00}^{\mathrm{ph}}M_{j,1}^{\mathrm{amp}}=M_{j,1}^{\mathrm{amp}}M_{00}^{\mathrm{ph}}.$
(130)

Thus these two Kraus operators commute with each other. Similarly, one can
find that,

$\displaystyle\commutator{M_{j,2}^{\mathrm{amp}}}{M_{00}^{\mathrm{ph}}}$
$\displaystyle=0,\commutator{M_{j,3}^{\mathrm{amp}}}{M_{00}^{\mathrm{ph}}}$
$\displaystyle=0.$ (131)

Similar calculations also give,

$\displaystyle\commutator{M_{j,2}^{\mathrm{amp}}}{M_{10}^{\mathrm{ph}}}$
$\displaystyle=0,\commutator{M_{j,2}^{\mathrm{amp}}}{M_{01}^{\mathrm{ph}}}$
$\displaystyle=0,$ (132)
$\displaystyle\commutator{M_{j,3}^{\mathrm{amp}}}{M_{01}^{\mathrm{ph}}}$
$\displaystyle=0,\commutator{M_{j,3}^{\mathrm{amp}}}{M_{11}^{\mathrm{ph}}}$
$\displaystyle=0,$
$\displaystyle\commutator{M_{j,1}^{\mathrm{amp}}}{M_{10}^{\mathrm{ph}}}$
$\displaystyle=0,\commutator{M_{j,1}^{\mathrm{amp}}}{M_{10}^{\mathrm{ph}}}$
$\displaystyle=0.$

Next, consider the Kraus operators, $M_{10}^{\mathrm{ph}}$ and
$M^{\mathrm{amp}}_{j,1}$ we get,

$\displaystyle M_{j,1}^{\mathrm{amp}}M_{10}^{\mathrm{ph}}$
$\displaystyle=\sum_{klm}(-1)^{k}\bra{j}\ket{+}_{k}\left(\ket{+}_{0}^{(b)}\ket{+}_{l}^{(a)}\ket{+}_{m}^{(a)}\bra{-}_{k}^{(a)}\bra{+}_{l}^{(a)}\bra{+}_{m}^{(a)}+\ket{-}_{0}^{(b)}\ket{-}_{l}^{(a)}\ket{-}_{m}^{(a)}\bra{+}_{k}^{(a)}\bra{-}_{l}^{(a)}\bra{-}_{m}^{(a)}\right),$
(133) $\displaystyle M_{10}^{\mathrm{ph}}M_{j,1}^{\mathrm{amp}}$
$\displaystyle=\sum_{k,l,m}\bra{j}\ket{+}_{k}\left(\ket{-}_{0}^{(b)}\ket{+}_{l}^{(a)}\ket{+}_{m}^{(a)}\bra{-}_{k}^{(a)}\bra{+}_{l}^{(a)}\bra{+}_{m}^{(a)}+\ket{-}_{0}^{(b)}\ket{-}_{l}^{(a)}\ket{-}_{m}^{(a)}\bra{+}_{k}^{(a)}\bra{-}_{l}^{(a)}\bra{-}_{m}^{(a)}\right),$
$\displaystyle\neq M_{j,1}^{\mathrm{amp}}M_{10}^{\mathrm{ph}}.$

Thus these two Kraus operators do not commute with each other, however looking
at the full recovery operation,

$\displaystyle\sum_{j}M_{10}^{\mathrm{ph}}M_{j,1}^{\mathrm{amp}}\rho\left(M_{j,1}^{\mathrm{amp}}\right)^{\dagger}\left(M_{10}^{\mathrm{ph}}\right)^{\dagger}$
$\displaystyle=\sum_{j}\sum_{k,l,m,k^{\prime},l^{\prime},m^{\prime}}\bra{+}_{k^{\prime}}\ket{j}\bra{j}\ket{+}_{k}A_{k,l,m}\rho