# Quantum Computation Using Large Spin Qudits

Sivaprasad Thattupurackal Omanakuttan

B.Sc., Physics, Mahatma Gandhi University, Kerala, 2015
M.Sc., Physics, Indian Institute of Technology Madras, 2017

Ivan H. Deutsch Milad Marvian Tameem Albash

Michael J Martin

Doctor of Philosophy Ph.D. Physics 2024 Spring May Physics and Astronomy March
20th, 2024

## Introduction

The last decade of the $20$th century saw the marriage of two of its great
scientific pillars: quantum mechanics and information science and gave birth
to the field of quantum information science (QIS) [PRXQuantum.1.020101]. At
the foundational level of scientific understanding, quantum mechanics stands
as the most precise theory. It delineates the fundamental workings of the
world. The advent of information science ushered in a new era, bringing forth
computers, digital communication, and other transformative devices that have
reshaped our daily lives. Quantum Information Science (QIS) emerged from
inquisitive, curiosity-driven fundamental research, seeking to unravel the
implications of merging quantum mechanics with the principles of information
science.

QIS tries to harness the power of quantum systems for information processing
and as a field lies at the convergence of quantum optics, atomic molecular and
optical (AMO) physics, condensed matter physics, computer science, and several
other areas of science and engineering [caves2013quantum]. This has enabled
the broad application and integration of tools, techniques, and concepts
specific to quantum information science into various domains within
theoretical and experimental physics. QIS has since shown promise in diverse
applications, spanning quantum computation, quantum cryptography, quantum
sensing, quantum simulation, quantum networking, and more. Among these
applications, quantum computation stands out as one of the most anticipated,
holding significant advantages over classical computers[shor1994algorithms,
shor1999polynomial, farhi2001quantum, lloyd1996universal, biamonte2017quantum,
aspuru2005simulated].

Numerous intriguing problems remain beyond the reach of classical computers,
not due to inherent unsolvability, but rather because of the astronomical
resources needed to address practical instances of these challenges. The
spectacular promise of quantum computers is to use quantum superposition and
quantum entanglement to enable new quantum algorithms that tackle problems
that require exorbitant resources for their solution on a classical computer
[shor1994algorithms, shor1999polynomial, farhi2001quantum, lloyd1996universal,
biamonte2017quantum, aspuru2005simulated]. For example, one has a class of
algorithms based on quantum Fourier transform, and includes remarkable
algorithms for solving the factoring and discrete logarithm problems,
providing a striking exponential speedup over the best-known classical
algorithms [shor1994algorithms, shor1999polynomial, kitaev1995quantum].
Another example of algorithms is based on Grover’s algorithm for performing
quantum searching [grover1996fast, Bennett_1997]. The quantum searching
algorithm offers a notable quadratic speedup over the best classical
algorithms, presenting a significant advancement. Its importance stems from
the widespread utilization of search-based techniques in classical algorithms.
In many cases, a straightforward adaptation of the classical algorithm enables
the development of a faster quantum algorithm, making quantum search
particularly impactful. The exponentially expanding Hilbert space is important
for implementing quantum computation, facilitating the storage and processing
of information [blume2002climbing].

However, the scalability of quantum computation faces limitations imposed by
decoherence, which arises from the influence of the external environment on
the quantum system. Consequently, it becomes imperative to explore strategies
for scaling the system while mitigating the adverse impact of decoherence.
Unlocking the full power of quantum computation involves comprehending and
devising approaches to overcome the effects of decoherence
[knill1998resilient, aharonov1997fault, knill2005quantum,
raussendorf2007topological]. Recent years have witnessed significant
theoretical and experimental advancements toward realizing the full potential
of quantum computation, even in the presence of decoherence
[acharya2022suppressing, ryan2022implementing, krinner2022realizing,
Bluvstein_Lukin_2023_QEC_Logical].

In the standard paradigm of quantum information processing (QIP) one encodes
information in qubits, the quantum analog of classical bits, by isolating two
well-chosen energy levels of the system such that the computation space grows
as $2^{N}$ for $N$ qubits. In many platforms, one has access and control over
multiple levels per subsystem, which can enhance our ability to do QIP in a
variety of ways [wang2020qudits, Blok2021, Gross2021, puri2020bias,
Gottesman2001]. In particular, one can encode information in base-$d>2$ using
$d$-level qudits [wang2020qudits] such that the computational space goes as
$d^{N}$. With a larger state space per subsystem, qudits offer potential
advantages for quantum communication [PhysRevLett.90.167906], quantum
algorithms [luo2014geometry, luo2014universal, li2013geometry, lu2020quantum],
and topological quantum systems [cui2015universal, cui2015universal1,
bocharov2015improved]. Quantum computation with qudits can also reduce circuit
complexity and can be advantageous in a variety of noisy intermediate scalable
quantum(NISQ)-era applications [brylinski2002universal, lu2020quantum,
luo2014universal, li2013geometry, zobov2012implementation,
weggemans2022solving, PhysRevLett.129.160501]. Qudits may also provide
significant advantages in quantum error correction and fault-tolerant quantum
computation [campbell2014, PhysRevA.83.032310, gottesman1998fault,
campbell2012, Eliot2016].

This dissertation delves into the realm of quantum computation utilizing spin
qudits as its focal point. The primary emphasis is on addressing and
mitigating the adverse impacts of decoherence by leveraging access to qudits.
One avenue of exploration involves working towards universal quantum
computation, where the utilization of qudits allows for quantum computational
supremacy with fewer subsystems. Another facet of this research investigates
the feasibility of encoding a qubit within a qudit for the purpose of fault-
tolerant quantum computation. 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.

In the gate-based approach to quantum computation with qubits, a universal
gate set consists of single-qubit gates that generate the group SU$(2)$ and
one entangling two-qubit gate, such as CNOT [divincenzo1995two]. This
generalizes simply for qudits. The universal gate set consists of the
generators of single-qudit gates in SU($d$) and an entangling two-qudit gate
[muthukrishnan2000multivalued, zhou2003quantum, brennen2005criteria]. The
gates that are necessary for the implementation of the universal gate set have
been recently implemented for qudits in superconducting transmon [Blok2021,
goss2022high, fischer2022towards] as well as in trapped ions
[ringbauer2021universal, hrmo2022native] up to dimension $d=7$. In these
experiments, one implements qudit gates using constructive methods, e.g.,
through a prescribed set of Givens rotations [brennen2005criteria,
li2013decomposition].

While there has been substantial progress, much work remains to be done to
efficiently implement a high-fidelity universal qudit gate set. In this
dissertation I propose an alternative approach based on quantum optimal
control which was originally developed in NMR [vandersypen2005nmr] and for
coherent control of chemistry [rabitz2000whither, shapiro2012quantum], and has
been extensively used in quantum information processing [koch2022quantum].
This approach yields high-fidelity gates for qudits in the presence of
decoherence and can be made robust to experimental imperfections. As a
concrete example that demonstrates the power of the method, we present here an
optimal control scheme to implement universal gates in qudits encoded in the
nuclear spin of 87Sr atoms. The nuclear spin is a good memory for use in
quantum information processing given its weak coupling to the environment and
resilience to other background noise [barnes2021assembly,
PhysRevLett.101.170504, daley2011quantum].

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
[knill1998resilient, aharonov1997fault, knill2005quantum,
raussendorf2007topological]. 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
[aharonov1997fault, knill1998resilient, preskill1998reliable,
kitaev1997quantum, Aliferis2008fault]. 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
[acharya2022suppressing, ryan2022implementing, krinner2022realizing,
Bluvstein_Lukin_2023_QEC_Logical].

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 [knill2005quantum, raussendorf2007topological,
svore2006noise, spedalieri2008latency]. 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 [Aliferis2008fault,
zzPoulin]. One well-known case is when one noise channel dominates over all
other noises. For example, the cases in which dephasing noise dominates over
bit-flip noise for the qubit shows improved threshold as shown by Aliferis and
Preskill [Aliferis2008fault] and can be implemented in bosonic systems
[puri2020bias, guillaud2019repetition]. Another case is the Gottesman-
Preskill-Kitaev encoding of a qubit in an infinite dimensional oscillator
[Gottesman2001] which corrects for displacement errors in a bosonic mode.
Additionally, in scenarios where erasure errors dominate over Pauli errors,
tailored error-correcting codes have proven advantageous
[Grassl_erasure_1997_PRA, Wu_Puri_Thompson_2022_Nature_erasure,
Sahay_Puri_biased_erasure_PRX_2023]. By addressing the specific
characteristics of dominant noise sources, these tailored methods offer
promising avenues to enhance the performance of FTQC.

A similar but much less explored avenue is to encode a qubit in a spin $>1/2$
system (qudit). 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 [Gross2021,
omanakuttan2023multispin]. In this dissertation, I investigate how we can
achieve FTQC, specifically for a qubit encoded in a spin qudit. This approach
may be extended to a wide range of physical systems, including semiconductor
qubits [Gross2021, gross2021hardware], ion traps [ringbauer2021universal,
Low2020], atomic systems [omanakuttan2021quantum, Siva_Qudit_entangler_2023,
zache2023fermion], molecules [castro2021optimal, jain2023ae], and
superconducting systems [ozguler2022numerical, Blok2021], wherein spin qudits
offer the means to encode logical qubits.

### Outline

The remainder of this dissertation is organized as follows. Quantum Optimal
Control of Nuclear Spin Qudecimals in 87Sr is based on the publication
[omanakuttan2021quantum]. In this work we study the ability to implement
unitary maps on states of the $I=9/2$ nuclear spin in 87Sr, a $d=10$
dimensional (qudecimal) Hilbert space, using quantum optimal control. Qudit
entanglers using quantum optimal control is based on the publication
[Siva_Qudit_entangler_2023]. Here, we study the generation of two-qudit
entangling quantum logic gates using two techniques in quantum optimal
control. We take advantage of both continuous, Lie algebraic control and
digital, Lie group control. Fault-tolerant quantum computation using large
spin cat-codes is based on the publication [Omanakuttanerrorcorrection]. Here
I 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. The spin-cat codes we develop substantially
reduce the resource requirements for fault-tolerance in that a single atom can
encode the logical qubit, with only minimal repetition given the structure of
the noise. An important innovation is the development of a CNOT gate that
preserves the structure of the noise at the logical level. We do so in a way
that also is well-aligned with experimental capabilities. QND Cooling and
leakage detection in neutral atoms is based on the publication
[Omanakuttancooling]. Here I present ideas of converting leakage errors to
erasure errors when quantum information is encoded in the nuclear spin in the
electronic ground state. After doing so, erasure can be efficiently corrected
by standard error correction protocols. This protocol for erasure conversion
is compatible with a scheme to cool the atoms while preserving the coherence,
generalizing previous work on this problem [Reichenbach_cooling]. Lastly, I
summarize all of our work in Summary and Outlook and suggest potential avenues
of research for future work.

### List of Publications

Below is a chronological list of the papers that I coauthored during my PhD.
Not all works listed here appear as chapters in this dissertation

  * •

[omanakuttan2021quantum] S. Omanakuttan, A. Mitra, M. J. Martin, and I. H.
Deutsch. Quantum optimal control of ten-level nuclear spin qudits in 87Sr.
_Phys. Rev. A,_ 104, L060401 (2021).

  * •

[PhysRevD.104.123026] H. Duan, J. D. Martin and S. Omanakuttan. Flavor isospin
waves in one-dimensional axisymmetric neutrino gases._Phys. Rev. D,_ 104,
123026 (2021).

  * •

[omanakuttan2023scrambling] S. Omanakuttan, K. Chinni, P. D. Blocher, and P.
M. Poggi. Scrambling and quantum chaos indicators from long-time properties of
operator distributions. _Phys. Rev. A,_ 107, 032418 (2023).

  * •

[anupampra] A. Mitra, S. Omanakuttan, M. J. Martin, G. W. Biedermann, and I.
H. Deutsch. Neutral-atom entanglement using adiabatic rydberg dressing._Phys.
Rev. A,_ 107, 062609 (2023).

  * •

[buchemmavari2023entangling] V. Buchemmavari, S. Omanakuttan, Y.-Y. Jau, and
I. Deutsch. Entangling quantum logic gates in neutral atoms via the microwave-
driven spin-flip blockade. _Phys. Rev. A,_ 109, 012615 (2024)

  * •

[blocher2023probing]P. D. Blocher, K. Chinni, S. Omanakuttan, and P. M. Poggi.
Probing scrambling and operator size distributions using random mixed states
and local measurements. _Phys. Rev. Research,_ 6, 013309 (2024)

  * •

[omanakuttan2023multispin] S. Omanakuttan and J. A. Gross. Multispin Clifford
codes for angular momentum errors in spin systems. _Phys. Rev. A,_ 108, 022424
(2023).

  * •

[Omanakuttan2023gkp] S. Omanakuttan and T. J. Volkoff. Spin-squeezed
gottesman-kitaev-preskill codes for quantum error correction in atomic
ensembles. _Phys. Rev. A,_ 108, 022428 (2023).

  * •

[Siva_Qudit_entangler_2023]S. Omanakuttan, A. Mitra, E. J. Meier, M. J.
Martin, and I. H. Deutsch. Qudit entanglers using quantum optimal control.
_PRX Quantum,_ 4:040333

  * •

[Omanakuttanerrorcorrection] S. Omanakuttan, V. Buchemmavari, J. Gross, I. H.
Deutsch, and M. Marvian. Fault-tolerant quantum computation using large spin
cat-codes. _arXiv:2401.04271,_ 2024.

  * •

[Omanakuttanmetrology] S. Omanakuttan, J. Gross, and T. J. Volkoff. Quantum
error correction inspired multiparameter quantum metrology. _in preparation,_
2024.

  * •

[Omanakuttancooling] S. Omanakuttan, V. Buchemmavari, M. J. Martin, and I. H.
Deutsch. Converting leakage errors to erasure errors and cooling atoms while
preserving coherence in neutral atoms for fault-tolerant quantum computation.
_in preparation,_ 2024

  * •

[Omanakuttanspingkp2] S. Omanakuttan, T. Thurtell, and B. Q. Baragiola.
Bridging the discrete and continuous variable quantum error correction _in
preparation,_ 2024.

## Quantum Optimal Control of Nuclear Spin Qudecimals in 87Sr

### Introduction

Ultracold ensembles of alkaline-earth atoms trapped in optical lattices or
arrays of optical tweezers are a powerful platform for quantum information
processing (QIP), including atomic clocks and sensors [Ludlow2015,
campbell2017fermi, norcia2019seconds, covey20192000, young2020half],
simulators of many-body physics [gorshkov2010two, daley2011quantum,
mukherjee2011many, banerjee2013atomic, isaev2016spin, kolkowitz2017spin], and
general purpose quantum computers [Madjarov_Endres_2020_Sr, daley2011quantum,
PhysRevLett.98.070501]. The ability to optically manipulate coherence in
single-atoms via ultranarrow optical resonances on the intercombination lines,
together with the ability to create high-fidelity entangling interactions
between atoms when they are excited to high-lying Rydberg states
[saffman2010quantum, Saffman_review_2016_Rydberg, Browaeys_2016] provides
tools that makes this system highly controllable for such applications. In
addition, fermionic species have nuclear spin. As the ground state is a closed
shell, there is no electron angular momentum, and the nuclear spin with its
weak magnetic moment is highly isolated from the environment. Such nuclear
spins in alkaline-earth atoms are thus natural carriers of quantum information
given their long coherence times and our ability to coherently control them
with magnetic and optical fields. Nuclear spins are also seen as excellent
carriers of quantum information in the solid state as demonstrated in
pioneering experiments including in NV-centers [morishita2020] and dopants in
silicon [soltamov2019, morello2018quantum, Godfrin2017, Leuenberger2003].

Using magneto-optical fields, [Lester2021] recently demonstrated the control
of qubits encoded in two nuclear-spin magnetic sublevels levels in 87Sr. The
nuclear spin in this atomic species, however, it is not a two-level system;
the spin is $I=9/2$ and there are $d=2I+1=10$ nuclear magnetic sublevels. Such
qudits, here “qudecimals,” have potential advantage for QIP. First and
foremost, one can encode a $D=d^{n_{d}}=2^{n_{2}}$ dimensional Hilbert space
associated with $n_{2}$ qubits in $n_{d}=n_{2}/\log_{2}d$ qudits. While only a
logarithmic saving, this is meaningful for the qudecimal ($\log_{2}d=3.32$),
especially when trapping and control of each atom is at a premium. This
savings extends to algorithmic efficiency, in that the number of elementary
two-qudit gates necessary to implement a general unitary map scales as
$O(n_{d}^{2}D^{2})=O\left(\frac{n_{2}^{2}D^{2}}{(\log_{2}d)^{2}}\right)$
[muthukrishnan2000multivalued]. Moreover, qudit architectures can show
increased resilience to noise [cozzolino2019high] and additional routes to
quantum error correction [gottesman1998fault]. For example, one can protect
against dephasing errors by encoding a qubit in a nuclear spin qudit [Li2017].
In addition, fault-tolerant operation of a quantum computer may be more
favorable based on qudit vs. qubit codes [PhysRevA.83.032310, campbell2014].

While QIP with qudits has great potential, there are substantial hurdles.
State preparation and readout are more challenging for systems with $d>2$.
Moreover, quantum logic with qudits is more complex. Universal quantum logic
with qubits can be achieved with a set of logic gates that include the
unitary-generators of SU(2) on each qubit, plus one entangling gate between
qubits pairwise. In the case of qudits, in addition to the entangling gate, we
require unitary-generators of SU($d$) for each subsystem
[muthukrishnan2000multivalued, zhou2003quantum, brennen2005criteria,
luo2014universal]. Unlike qubits, the Lie algebra of such gates are not
spanned by the native Hamiltonians, and thus implementation of this generating
set is not straightforward. Different approaches have been studied to
implement SU($d$) gates [Moreno2018, neeley2009emulation, Low2020, sawant2020,
moro2019]. One approach is to specify an arbitrary SU($d$) unitary matrix
through a sequence of so-called Givens rotations acting between pairs of
levels [O'Leary2006]. In a landmark experiment, the Innsbruck group employed
this construction to experimentally demonstrate universal quantum logic with
qudits in a trapped ions ion [ringbauer2021universal], with performance
similar to qubit quantum processors.

An alternative powerful approach to implementing universal quantum logic is to
employ the tools of quantum optimal control. In this paradigm, one numerically
searches for a time-dependent waveform that achieves the desired SU($d$)
unitary map when one has access to a Hamiltonian that makes the system
universally “controllable” [Merkel2009, jurdjevic1972control,
goerz2015optimizing, koch2016controlling, Frey2020, brockett1973lie,
schirmer2002degrees]. Optimal control is a powerful and flexible approach that
does not require specific pairwise Givens rotations, can be high-fidelity, and
can be made robust to imperfections such as inhomogeneities through the tools
of robust control [anderson2015accurate, goerz2015optimizing,
glaser2015training, koch2016controlling]. In seminal work, the Jessen group
used optimal control to demonstrate high-fidelity control of qudits encoded in
the hyperfine spin levels of ground-state cesium [Paul_experiment_Cs_2007,
Smith2013]. This flexible control has found potential application in studies
of quantum simulation [Poggi2020].

### Background

In this chapter we build on this approach to study implementation of SU(10)
gates on the nuclear spin of 87Sr-based on quantum optimal control. A nuclear-
spin encoding may have long-term advantages compared to hyperfine states that
couple electron and nuclear spins, in its strongly reduced sensitivity to to
background magnetic fields and resilience against decoherence driven by photon
scattering from optical tweezers or lattices [PhysRevLett.98.070501,
dorscher2018lattice]. Weak coupling to the environment, of course, comes with
increased challenges of weak coupling to control fields. We will show,
nonetheless, that with reasonable experimental parameters one can implement
high-fidelity qudecimal logic, with low decoherence.

We consider open loop-control in a Hilbert space with finite dimension $d$,
governed by a Hamiltonian
$H[\mathbf{c}(t)]=H_{0}+\sum_{\lambda}c_{\lambda}(t)H_{\lambda}$ where
$\mathbf{c}(t)=\\{c_{\lambda}(t)\\}$ is the set of time-dependent classical
control waveforms. The system is said to “controllable” if the set of
Hamiltonians, $\\{H_{0},H_{\lambda}\\}$, are generators of the Lie algebra
SU($d$). Then $\exists\hskip 2.84544pt\mathbf{c}(t)$ such that
$U[\mathbf{c},T]=\mathcal{T}\left[\exp\left(-i\int_{0}^{T}H[\mathbf{c}(t)]dt\right)\right]=U_{\mathrm{tar}}$
for any target unitary matrix $U_{\mathrm{tar}}=\mathrm{SU(d)}$ in this space.
The minimal time $T$ for which this is possible is known as the “quantum speed
limit” (QSL) [caneva2009optimal] . See Bandwidth limited Qudecimal Quantum
Optimal Control for additional details of the quantum control protocol used
here.

One can achieve quantum controllability of the nuclear spin qudecimal through
magneto-optical interactions. We combine magnetic spin resonance in the
presence of an off-resonant laser field as depicted in Fig. 1. The Hamiltonian
acting on the nuclear spin in the $5s^{2}$ ${}^{1}S_{0}$ ground state takes
the form $H=H_{\mathrm{mag}}+H_{\mathrm{LS}}$. Here
$H_{\mathrm{mag}}=-\bm{\mu}\cdot\mathbf{B}(t)$ is the magnetic spin-resonance
Hamiltonian, with $\bm{\mu}=g_{I}\mu_{N}\mathbf{I}$ the nuclear magnetic
dipole vector operator and
$\mathbf{B}(t)=B_{\parallel}\mathbf{e}_{z}+B_{T}\operatorname{Re}\left[(\mathbf{e}_{x}+i\mathbf{e}_{y})\mathbf{e}^{-i\left(\omega_{\text{rf}}t+\phi(t)\right)}\right]$
the magnetic field consisting of a strong bias defining the quantization axis
$\mathbf{e}_{z}$ and a transversely rotating rf-magnetic field with a time
dependent phase $\phi(t)$. Taken alone, the $H_{\mathrm{mag}}$ generates only
SU(2) rotations of nuclear spin. To achieve full SU($d$) control we add a
light-shift Hamiltonian due to the AC-Stark effect,
$H_{LS}=-\alpha_{zz}(\omega_{L})\left|E_{0}\right|^{2}/4$ where
$\alpha_{zz}(\omega_{L})$ is the $zz$-component of atomic AC-polarizability
tensor operator for a laser field at frequency $\omega_{L}$ linearly-polarized
along the quantization axis,
$\mathbf{E}_{L}(t)=\mathbf{e}_{z}\operatorname{Re}\left(E_{0}e^{-i\omega_{L}t}\right)$.
The form of $\alpha_{zz}$ depends on the atomic structure and the detuning of
the laser from atomic resonance. In particular, when the detuning is not large
compared to the hyperfine splitting in the excited state, the polarizability
has an irreducible rank-2 tensor component $\alpha_{zz}=\alpha^{(2)}I_{z}^{2}$
(there is also a trivial scalar term proportion to the identity)
[deutsch2010quantum], where $I_{x},I_{y},I_{z}$ are the nuclear spin operators
along the three Cartesian coordinates. This quadratic spin twist together with
the linear Larmor precession yields a set of control Hamiltonians
$\\{I_{x},I_{y},I_{z}^{2}\\}$ sufficient to generate the Lie algebra
SU($2I+1$) for an arbitrary spin $I$ [Giorda2003]. Such control was first
demonstrated in the alkali atom cesium, for the hyperfine spin $F=3$ in the
electronic ground state, in order to generate nonclassical spin states in the
$d=7$ dimensional Hilbert space [Paul_experiment_Cs_2007].

Importantly, the size of tensor polarizability $\alpha^{(2)}$ depends on the
ratio of the excited state hyperfine splitting to the laser detuning
[deutsch2010quantum] , achieving its maximum when these are of the same order.
Thus, to achieve high-fidelity control, one must tune sufficiently close to
resonance, while avoiding photon scattering that leads to decoherence.
Critically, in alkaline-earth atoms, the first excited ${}^{3}P_{1}$ states
have long lifetimes and large hyperfine splittings. This leads to a very
favorable figure of merit for optimal control, as measured by the ratio of the
characteristic tensor light shift to the photon scattering rate $\gamma_{s}$,
$\kappa\equiv\alpha^{(2)}\left|E_{0}\right|^{2}/4\gamma_{s}$. For example, in
87Sr, the hyperfine splitting between the $F=7/2$ and $F=9/2$ levels in the
singly-excited $5s5p\;^{3}P_{1}$ state is $\omega_{\mathrm{HF}}/2\pi=1130$
MHz, while the spontaneous emission linewidth is $\Gamma/2\pi=7.5$ kHz. For a
scattering rate averaged over all magnetic sublevels [deutsch2010quantum], we
find that when we detune about halfway between these resonances, we obtain the
maximum figure of merit $\kappa=6.8\times 10^{3}$ (see Fig. 1). In contrast,
$\kappa=18.6$ for $F=3$ hyperfine spin in the cesium ground state when the
laser is tuned halfway between the $F=3$ and $F=4$ hyperfine levels in the
excited $6P_{1/2}$ D1-resonance. This small figure of merit limited the
fidelity to around $0.85$ for the arbitrary state preparation. A factor of
$364$ increase in the figure of merit for alkaline earths shows the potential
power of this approach to yield high-fidelity quantum optimal control of the
nuclear spin qudit.

Figure 1:  Schematic for magneto-optical control. The qudecimal is encoded in
the ten magnetic sublevels of the nuclear spin,
$\ket{-9/2}\rightarrow\ket{9/2}$, in the $5s^{2}$ ${}^{1}S_{0}$ ground state.
Their levels are shifted by a linear Zeeman effect due to a bias magnetic
field and a quadratic tensor AC-Stark effect induced by an off-resonant laser
beam, polarized along the quantization axis, and detuned $\Delta$ between the
hyperfine levels of the $5s5p$ ${}^{3}P_{1}$ intercombination line. Control of
the qudecimal is then achieved with a phase modulated radio-frequency magnetic
field, co-rotating at the bare Larmor precession frequency, whose amplitude
causes Rabi rotations at frequency $\Omega_{\mathrm{rf}}$. The figure of merit
for the control is the ratio of the AC-Stark shift to the photon scattering,
$\kappa$ , shown in the inset (see text).

We consider control of the nuclear spin qudecimal with on-resonance rf fields
on resonance with the Zeeman splitting, $\Delta
E_{0}=|g_{I}|\mu_{N}B_{\parallel}$, where $g_{I}\mu_{N}/h=-184$ Hz/Gauss in
87Sr [olschewski1972messung]. In the rotating frame, the control Hamiltonian
is

$H(t)=\Omega_{\mathrm{rf}}\left(\cos[c(t)\pi]I_{x}+\sin[c(t)\pi]I_{y}\right)+\beta
I_{z}^{2},$ (1)

where $\Omega_{\text{rf}}=-g_{I}\mu_{N}B_{T}$ is the rf-Rabi frequency and
$\beta=\alpha^{(2)}\left|E_{0}\right|^{2}/4$ is the strength of the tensor
light shift (here and to follow $\hbar=1$). Note, for a rotating rf-field,
there is no rotating wave approximation, and this Hamiltonian is valid even
when $\Omega_{\text{rf}}\geq\omega_{\text{rf}}$. Here the control waveform is
solely the rf-phase $c(t)\equiv\phi(t)/\pi$. It was proven in [Merkel2008]
that varying $c(t)$ is sufficient to achieve universal control the system.

### Numerical Results

We consider two classes of quantum control tasks, preparation of a target pure
state $\ket{\psi_{\text{tar}}}$ and implementation of a unitary map
$U_{\text{tar}}$. Optimal control follows by maximizing the relevant fidelity,

$\displaystyle\mathcal{F}_{\psi}[\bm{c},T]$ $\displaystyle=$
$\displaystyle\left|\bra{\psi_{\text{tar}}}U[\bm{c},T]\ket{\psi_{0}}\right|^{2},$
(2) $\displaystyle\mathcal{F}_{U}[\bm{c},T]$ $\displaystyle=$
$\displaystyle\left|\Tr\left(U^{\dagger}_{\text{tar}}U[\bm{c},T]\right)\right|^{2}/d^{2}.$
(3)

This is achieved by discretizing the control waveform and then numerically
maximizing the fidelity with gradient ascent. In a series of works, the Rabitz
group showed that the fidelity landscape is favorable for this purpose
[rabitz2004quantum, Hsieh2008]. We choose here a piecewise constant
parameterization (as in [Merkel2008]) and write the control function as a
vector $\mathbf{c}=\\{c(t_{j})|j=1,\dots,n\\}$ where $t=j\Delta t$ and
$n=T/\Delta t$, parameterizing waveforms that are constant over the duration
$\Delta t$. A minimal choice of $n$ depends on the number of parameters
necessary for the control task; for state-maps $n_{\text{min}}=2d-2$ and for
arbitrary SU($d$) maps $n_{\text{min}}=d^{2}-1$. In practice, we choose $n$ to
be a larger than $n_{\text{min}}$ which improves the fidelity landscape when
$T$ is close the the QSL. To numerically optimize $\mathcal{F}$ we use a
variation of the well-known GRAPE algorithm [khaneja2005optimal]. See
Bandwidth limited Qudecimal Quantum Optimal Control for further details on the
choice of parameterization and optimization.

For a fixed value of $\Omega_{\text{rf}}$, the optimal choice of $\beta$ and
total time $T$ are found empirically. Figures 2a(b) show the infidelity,
$1-\mathcal{F}$, for state preparation (unitary maps), when averaged over 20
Haar random target vectors (10 random unitary maps). As expected, when
$T\rightarrow\infty$ the infidelity is essentially zero, when the number of
steps $n>n_{\mathrm{min}}$. The QSL is highly dependent on the value of
$\beta$. As expected, the optimal choice is $\beta\approx\Omega_{\text{rf}}$
as this provides the optimal mixing between Larmor precession and one-axis
twisting. The characteristics of state preparation and unitary maps are
similar in nature. The major difference between these two cases is that
unitary mapping requires more time for the simple reason that unitary mapping
has $d^{2}-1$ parameters compared to the $2d-2$ for the state preparation. The
quantum speed limit at $\beta=\Omega_{\mathrm{rf}}$ is $T_{*}\approx
1.5\pi/\Omega_{\mathrm{rf}}$ for state preparation and $T_{*}\approx
8\pi/\Omega_{\mathrm{rf}}$ for SU(10) unitary maps.

In principle, one can achieve arbitrarily high fidelity with increasing $T$.
In practice $T$ is limited by the coherence time of the system. Here, the
coherence time is fundamentally limited by decoherence arising from photon
scattering and optical pumping due to the off-resonant light-shift laser. We
model the effects of decoherence in the state preparation protocols using the
Lindblad Master equation [deutsch2010quantum],

$\displaystyle\frac{d\rho[\bm{c},t]}{dt}$ $\displaystyle=$
$\displaystyle-i\commutator{H_{\text{eff}}[\bm{c}]}{\rho[\bm{c},t]}+\Gamma\sum_{i}W_{q}\rho[\bm{c},t]W_{q}^{\dagger}$
(4) $\displaystyle\equiv$
$\displaystyle\mathcal{L}[\bm{c}]\left[\rho[\bm{c},t]\right].$

where the jump operators for optical pumping between magnetic sublevels
describing absorption followed by emission of a $q$-polarized photon are
$W_{q}$,

$W_{q}=\sum_{F^{\prime}}\frac{\Omega/2}{\Delta_{FF^{\prime}}+i\Gamma/2}(\bm{e}_{q}^{*}.\bm{D}_{FF^{\prime}})(\vec{\epsilon}_{L}.\bm{D}_{FF^{\prime}}^{\dagger}).$
(5)

Here $\bm{D}_{FF^{\prime}}^{\dagger}$ are the dimensionless dipole raising
operators from ground state manifold $F=I$ to the excited state manifold
$F^{\prime}$, as defined in [deutsch2010quantum].
$H_{\text{eff}}[\mathbf{c}]=H[\mathbf{c}]-i\Gamma\sum_{q}W_{q}^{\dagger}W_{q}$/2
is the non-Hermitian control Hamiltonian, Eq. (4), now including absorption of
the laser light.

For gates, we define a $d^{2}\times d^{2}$ superoperator matrix acting on the
density matrix. For the open quantum system, the superoperator describing the
evolution of an arbitrary input state is the Completely Positive (CP)-map,
$\mathcal{E}[\bm{c},T]=\mathcal{T}\left(\exp\\{\int_{0}^{T}\mathcal{L}[\bm{c}(t^{\prime})]\\}dt^{\prime}\right)$,
where $\mathcal{L}$ is the Lindbladian superoperator of the master equation,
defined implicitly in Eq. (4).

We compared the output in the open quantum system dynamics given the ideal
control solution $\mathbf{c}$ found in closed-system optimization. The
fidelities for state preparation and full SU(10) maps are, respectively,

$\displaystyle\mathcal{F}_{\psi}[\bm{c},T]$ $\displaystyle=$
$\displaystyle\Tr{\rho_{\psi_{\text{tar}}}\rho[\bm{c},T]},$ (6)
$\displaystyle\mathcal{F}_{U}[\bm{c},T]$ $\displaystyle=$
$\displaystyle\left|\Tr{\mathcal{E}_{U_{\text{tar}}}^{\dagger}\mathcal{E}[\bm{c},T]}\right|/d^{2}.$
(7)

Here $\rho_{\psi_{\text{tar}}}=\ket{\psi_{\text{tar}}}\bra{\psi_{\text{tar}}}$
is the target state and $\rho[\bm{c},T]$ is the solution to the master
equation. $\mathcal{E}_{U_{\text{tar}}}=U_{\text{tar}}^{*}\otimes
U_{\text{tar}}$ is the CP-map corresponding to the target unitary gate and
$\mathcal{E}[\bm{c},T]$ is the CP-map with decoherence. Eq. (7) is the
“process fidelity,” a key quantity of interest in determining the thresholds
for fault-tolerant quantum computation [schulte2011].

Numerical results are given in Fig. 2 for both state preparation and unitary
mapping. In contrast to closed-system control, Fig. 2c and Fig. 2d show that
there is an island where the infidelity is smallest. This reflects the trade
off between coherent control and decoherence. There is an optimal total time
of evolution $T$ than larger than the QSL but not too large when compared to
the optical pumping time. In addition, the optimal choice of $\beta$ is now
smaller than we found for the closed quantum system, as increased tensor-light
shift is accompanied by increased photon scattering. Including decoherence,
for the case of state preparation, averaged over $20$ random states, we find
the fidelity $\langle\mathcal{F}_{\psi}\rangle\approx 0.9997$. Here the island
of high fidelity is large, occurring for $\beta<1.2$. For the case of unitary
mapping the island of lowest infidelity occurs for $\beta<1.2$ where the
fidelity $\langle\mathcal{F}_{U}\rangle\approx 0.9970$ which is averaged over
$10$ Haar random unitaries. We emphasize that these qudecimal maps act on a
10-dimension Hilbert space. Thus a fair comparison of the effective fidelity
acting on qubits is
$\langle\mathcal{F}\rangle_{\text{qubit}}=\langle\mathcal{F}\rangle_{\text{qudecimal}}^{0.3}$,
since, in principle, one can encode more than 3 qubits in a qudecimal

Figure 2: Fidelity of objectives found by optimal control as a function of the
strength of AC-stark shift, $\beta$, and the total time $T$, in units of the
rf-Rabi frequency $\Omega_{\text{rf}}$. Predictions based on closed-unitary
evolution for state-maps (a) and SU(10) unitary-maps (b) averaged over $120$
Haar-random target states and $10$ Haar-random target SU(10) matrices,
respectively. The control waveforms are piecewise constant, over times $\delta
t=T/n$. For state maps we choose $n=120$ time steps; the unitary maps we take
$n=500$. The bottom layer gives the similar figures in the presence of
decoherence using the master equation, Eq. (5): state fidelity(c), Eq. (6):
and process fidelity (d).

Coherence is also limited when there are inhomogenieties arising from
uncertainties in the Hamiltonian parameters such as the laser intensity and
detuning. When the decoherence time is longer than than the inhomogeneous
dephasing time, one can mitigate this with the numerical tools of robust
control [PhysRevLett.82.2417, PhysRevA.58.2733, anderson2015accurate]. We
consider here an uncertainty in the tensor light shift arising from the
thermal velocity of the atoms. To perform robust control, we replace the
control Hamiltonian by $H[\bm{c}]\rightarrow
H^{\prime}[\bm{c},\epsilon]=H[\bm{c}]+\epsilon I_{z}^{2}$, where $\epsilon$ is
the variation in $\beta$ around the fiducial value, and define a new objective
function as the average fidelity,
$\langle\mathcal{F}[\mathbf{c},T]\rangle=\int
d\epsilon\;p(\epsilon)\mathcal{F}[\mathbf{c},T,\epsilon]$. While in principle
one can design inhomogeneous control with detailed knowledge of the
probability distribution $p(\epsilon)$, in practice, when the standard
deviation of the distribution $\delta$ is sufficiently narrow, it is
sufficient to simultaneously optimize at two points[anderson2015accurate], and
choose the objective function as

$\langle\mathcal{F}[\mathbf{c},T]\rangle=(\mathcal{F}[\mathbf{c},T,\epsilon=+\delta]+\mathcal{F}[\mathbf{c},T,\epsilon=-\delta])/2.$
(8)

The numerical results of robust control are shown in Fig. 3 for
$\beta=0.4\Omega_{\mathrm{rf}}$ and an error of $\delta=.005\beta$. We see
that robust control outperforms the bare waveforms, even in presence of
decoherence, but one does not reach the fidelity without any inhomogeneity due
to optical pumping occurring over the extended time of the control pulses. For
the parameters chosen here, we find that for state preparation one could
achieve a fidelity of $\langle\mathcal{F}_{\psi}\rangle\approx 0.9992$ in a
time $T={4.5}\pi/\Omega_{\text{rf}}$, and for unitary mapping one achieved a
fidelity $\langle\mathcal{F}_{U}\rangle\approx 0.9923$ in a time
$T={24}\pi/\Omega_{\text{rf}}$. Other practical considerations such as the
bandwidth needed for rapidly varying waveform may limit the speed of operation
(see Bandwidth limited Qudecimal Quantum Optimal Control).

Figure 3:  Comparison of infidelity with and without decoherence and robust
control to counteract dephasing due to inhomogeneities at the level of $.5\%$
of $\beta$ and $\beta=0.4\Omega_{\mathrm{rf}}$. (a) state preparation
(averaged over $20$ Haar-random target states), (b) SU(10) mapping (averaged
over $10$ Haar-random unitary matrices). Robust control can largely remove
dephasing and achieve almost same the infidelity seen due solely to
decoherence.

### Conclusion and Summary

In this chapter, we have shown that in the presence of fundamental decoherence
and small inhomogeneities, quantum optimal control allows for the realization
of high-fidelity arbitrary state maps and SU(10) qudecimal gates acting on
nuclear spin in the ground state of 87Sr. While we proposed one protocol that
leverages the strong tensor light shift induced by a laser tuned near the
${}^{3}P_{1}$ hyperfine manifold, the richness of magneto-optical controls
in87Sr provides multiple possible approaches, e.g., by employing the tensor
light shift when tuned near the ${}^{3}P_{0}$ clock state. Quantum optimal
control of nuclear spins should find a variety of applications in QIP,
including metrological enhancement with qudits [Noris2012], quantum simulation
[Poggi2020, Blok2021], and universal quantum computation [daley2011quantum].
For the latter additional components are necessary. One must enable readout of
all 10 magnetic sublevels though appropriate shelving and fluorescence
protocols [boyd2007nuclear]. Most importantly, we must study the
implementation of entangling gates consistent with qudit logic. Advances in
Rydberg-state control for alkaline earth atoms show great promise in this
direction [Madjarov_Endres_2020_Sr]. Finally, while we have studied here two
extremes of the control tasks, state preparation and SU(10) maps, optimal
control allows for arbitrary partial isometries to encode a $d^{\prime}<10$
qudit in the qudecimal. For example one can encode a qubit in the logical
states $\ket{0}=\ket{M_{I}=9/2}$, $\ket{1}=\ket{M_{I}=-9/2}$ and potentially
protect it from dephasing noise, analogous to a cat-code [Li2017] or other
encodings of a qubit in a large spin that leverages the available interactions
and dominant error channels [Gross2021]. The flexibility of arbitrary control
provides avenues to explore the best approach to encoding and error
mitigation.

## Qudit entanglers using quantum optimal control

### Introduction

In the gate-based approach to quantum computation with qubits, a universal
gate set consists of single-qubit gates that generate the group SU$(2)$ and
one entangling two-qubit gate, such as CNOT [divincenzo1995two]. This
generalizes simply for qudits. The universal gate-set consists of the
generators of single-qudit gates in SU($d$) and an entangling two-qudit gate
[muthukrishnan2000multivalued, zhou2003quantum, brennen2005criteria]. Unlike
qubits, where native Hamiltonians can be used to naturally implement the
desired gate set, qudits require more complex protocols. The gates that are
necessary for the implementation of the universal gate set have been recently
implemented for qudits in superconducting transmon [Blok2021, goss2022high,
fischer2022towards] as well as in trapped ions [ringbauer2021universal,
hrmo2022native] up to dimension $d=7$. In these experiments, one implements
qudit gates using constructive methods through a prescribed set of Givens
rotations [brennen2005criteria, li2013decomposition].

Following the ideas from the previous chapter, here we study an alternative
approach based on quantum optimal control for the implementation of entangling
gates between two qudits. We study qudit entangling gates for any $k\leq d$
within the $d$-dimensional Hilbert space of each subsystem. As a concrete
example that demonstrates the power of the method, we present here an optimal
control scheme to implement entangling gates in qudits encoded in the nuclear
spin of 87Sr atoms. The ground state of the 87Sr is also studied in a recent
paper as a possible candidate for qudit encoding with entangling interaction
enabled by the Rydberg blockade [zache2023fermion]. Also, the recent
significant achievements of quantum information processing using the Rydberg
blockade [Levine_Pichler_gate, bluvstein2022quantum, Saffman_Nature_2022] make
this an ideal platform for exploring quantum computation. Using a combination
of a tunable radio-frequency magnetic field and interactions that arise when
atoms are excited to high-lying Rydberg states, the atomic qudit is fully
controllable. We find that one can use quantum optimal control to implement
high-fidelity entangling qudit gates even in the presence of decoherence
arising from the finite Rydberg-state lifetime.

### Controllability

A complete universal gate set for qudits requires one entangling gate. A
standard choice is the CPhase gate, which is the generalization of CZ gate for
qubits, defined

$\mathrm{CPhase}\ket{j}\ket{k}=\omega^{jk}\ket{j}\ket{k},$ (9)

where $\omega=\exp(2\pi i/d)$ , the $d$-th primitive root of identity for a
subsystem of dimension $d$ and $0\leq j,k\leq d-1$. We can see that for $d=2$
we recover the CZ gate. This gate is locally equivalent to the qudit-analog of
the CNOT gate, known as CSUM gate,

$C_{\mathrm{SUM}}\ket{i}\ket{j}=\ket{i}\ket{i\oplus j(\text{mod }d)}$ (10)

by the Hadamard gate for qudits,
$H_{d}\ket{j}=\frac{1}{\sqrt{d}}\sum_{i}\omega^{ij}\ket{i}$. Previous works
have studied how to implement these gates through a well-defined sequence of
maps generated by one-qudit and two-qudit Hamiltonians [brennen2005criteria,
muthukrishnan2000multivalued, vlasov2002noncommutative,
brylinski2002mathematics]. We study here the use of numerical optimization and
the theory of optimal control.

#### Lie algebraic approach

Figure 4:  Comparison of Lie algebra versus Lie group approach for quantum
control. (a) Schematic of the continuous-time Lie algebraic approach for
quantum control. The physical systems are governed by the time-dependent
Hamiltonian, $H[\mathbf{c}(t)]=H_{\mathrm{ent}}+\sum_{j=1}^{k}c_{j}(t)H_{j}$,
here with a time-dependent entangling Hamiltonian, $H_{\mathrm{ent}}$. The
time-dependent waveforms $\\{c_{j}(t)\\}$ are found through numerical
optimization, and this defines the target unitary map of interest through the
solution to the time-dependent Schrödinger equation. (b) Schematic for a
digital, Lie group approach to quantum control of entangling two-qudit gates.
The target unitary is achieved through a discrete series of layers consisting
of unitary maps from a given family. One layer of the scheme consists of
single-qudit gates on each subsystem and an entangling interaction between
them, applied for a given time $t_{j}$. Through numerical optimization, one
finds the parameters of the local $SU($d$)$-gates and the entangling time
$t_{j}$ in each layer.

In the Lie algebraic approach to quantum control which we also studied in the
last chapter, we consider a Hamiltonian of the form
$H[\mathbf{c}(t)]=H_{0}+\sum_{j=1}^{k}c_{j}(t)H_{j}$, where
$\mathbf{c}(t)=\\{c_{j}(t)\\}$ is the set of time-dependent classical control
waveforms, and $H_{0}$ is called the drift Hamiltonian. The system is said to
be “controllable” if the set of Hamiltonians,
$\\{H_{0},H_{1},H_{2},\ldots,H_{k}\\}$, are generators of the desired Lie
algebra, e.g., $\mathfrak{su}{(d)}$. Then $\exists\hskip
2.84544pt\mathbf{c}(t)$ such that
$U[\mathbf{c},T]=\mathcal{T}\left[\exp\left(-i\int_{0}^{T}H[\mathbf{c}(t)]dt\right)\right]=U_{\mathrm{tar}}$
for any target unitary in desired Lie Group, e.g.,
$U_{\mathrm{tar}}\in\mathrm{SU}(d)$. In addition, we require $T\geq T_{*}$,
where $T_{*}$ is known as the “quantum speed limit time,” which sets the
minimal time needed for the system to be fully controllable.

We consider here open-loop control determined by a well-defined Hamiltonian of
the general form,

$H(t)=H^{(1)}(t)+H^{(2)}(t)+H_{\mathrm{ent}},$ (11)

where $H^{(i)}(t)$ are time-dependent Hamiltonians acting on the individual
subsystems, and $H_{\mathrm{ent}}$ is the interaction that entangles them.
Here we include the time dependence in the Hamiltonian that acts on the
individual system as these will be generally easier to implement
experimentally. In this formulation, $H_{\mathrm{ent}}=H_{0}$, is the drift
Hamiltonian. However, one could in principle include time dependence in the
entangling Hamiltonian as well and this may achieve faster gates.

#### Lie group approach

In the digital, Lie group approach to quantum control, we consider a family of
unitary maps in the desired group that are easily implementable,
$U(\lambda_{j})$, where $\\{\lambda_{j}\\}$ are the parameters that specify
the unitary matrices at our disposal. The relevant Lie group of interest here
is $\mathrm{SU}(d^{2})$, the group of two-qudit unitary matrices in $d^{2}$
dimensions, where the overall phase is removed. The system is controllable if
$\forall U_{\mathrm{tar}}\in\mathrm{SU}(d^{2})$, $\exists\\{\lambda_{i}\\}$
such that $\prod_{j=1}^{k}U(\lambda_{j})=U_{\mathrm{tar}}$. Similar to the Lie
algebraic quantum control approach, the goal is to find $\\{\lambda_{j}\\}$
through numeric optimization, e.g., via gradient-based methods.

For the case of two-qudit gates, a controllable Lie group structure is given
as,

$U_{\lambda_{j}}=U_{\mathrm{ent}}*(U_{1}\otimes U_{2}),$ (12)

where $U_{1,2}\in\mathrm{SU}(d)$ and
$U_{\mathrm{ent}}=\exp(-iH_{\mathrm{ent}}t)\notin\mathrm{SU}(d)\otimes\mathrm{SU}(d)$.
Thus, we can achieve the target gate to the desired fidelity by intertwining a
sequence of local $\mathrm{SU}(d)$ gates and the available entangling
interaction in alternating layers of single qudit gates and entangling gates,
as shown in Fig. 4(b). This approach is similar to the construction based on
Givens rotation [ringbauer2021universal]. Here, the possibility of accessing
arbitrary local $\mathrm{SU}(d)$ gates makes this protocol very powerful. A
schematic comparison of both these approaches is shown in Fig. 4.

#### Physical Platform: Rydberg atoms

To make these ideas concrete, we consider the implementation of entangling
gates in neutral atoms using the strong van der Waals interactions between
atoms in high-lying Rydberg states. We use the Rydberg dressing paradigm in
which one adiabatically superposes the Rydberg state into the ground states to
introduce interactions between dressed ground states [johnson2010interactions,
keating2015robust, jau2016entangling, zeiher2016many, zeiher2017coherent,
borish2020transverse]. Rydberg dressing has been studied with multiple
applications including the dynamics of interacting spin models
[zeiher2016many, zeiher2017coherent, borish2020transverse] as well as to
prepare metrologically-useful states [Kaubruegger_Zoller_metrology].
Entanglement between neutral atoms via Rydberg dressing has been theoretically
proposed for creating qubit entangling gates [keating2015robust,
mitra_martin_gate, anupampra] and experimentally implemented
[jau2016entangling, martin2021molmer, schine2022long]. The dressing approach
has a potential advantage in that it exhibits reduced sensitivity to some
noise sources [keating2015robust, mitra_martin_gate, schine2022long]. For the
specific protocol based on optimal control, the utilization of Rydberg
dressing confines our operations to the qudit subspace, as one can work with
the dressed basis [anupampra, mitra_martin_gate]. This restriction effectively
reduces the dimension of the Hilbert space for optimization from $(2d)^{2}$ to
$d^{2}$ for a $d$ dimensional qudit. This dimension reduction significantly
accelerates the numerical optimization of the pulses required for quantum
control.

We study here encoding a qudit in the spin of 87Sr. To implement entangling
two-qudit control, we will make use of the excitation to the $5sns\hskip
2.84544pt^{3}S_{1}$ Rydberg series from one of the metastable $5s5p\hskip
2.84544pt^{3}P_{J}$ first excited states in the triplet series. For optimal
control based on the combination of rf-driven Larmor precession and Rydberg
dressing one can compare different choices of metastable states. One natural
choice is the ${{}^{3}P_{0}}$ clock state, whose spin is essentially solely
nuclear, and thus robust in the presence of magnetic field noise. By contrast,
the ${{}^{3}P_{2}}$ state involves electronic angular momentum with a large
magnetic dipole moment and commensurate sensitivity to noise, including
possible tensor light shifts induced by the trapping laser. However, within
the specific approach addressed in this study, access to a large magnetic
dipole moment enables faster gate operations compared to the Rydberg lifetime.
For the ${{}^{3}P_{2}}$ states, the strength of the rf-Larmor precession
frequency is closer to that of the available Rydberg dressing interaction. In
this regime, the quantum speed limit (i.e., the minimum time required to
implement gates) is more favorable compared to the situation that the rf-
interaction is much weaker than the Rydberg interaction, as would be the case
for the ${{}^{3}P_{0}}$ states. This regime, characterized by similar
strengths of competing Hamiltonians, is known to be optimal for achieving the
quantum speed limit [omanakuttan2021quantum, buchemmavari2023entangling]. We
consider here coherently transferring qudits from the ${}^{1}{S}_{0}$ ground
state to the $F=9/2$ state hyperfine states of the ${}^{3}{P}_{2}$ manifold,
which provides for faster and more flexible control [trautmann20221], putting
technical noise aside.

To achieve the entangling interaction, we consider Rydberg dressing,
generalizing the mechanism discussed in [jau2016entangling, keating2015robust,
mitra_martin_gate]. The AC Stark shift (light shift) associated with a dressed
state when a laser is tuned near a Rydberg resonance is modified for two atoms
because of the Rydberg blockade. The deficit between the two-atom light shift
and twice the one-atom light shift determines the entangling energy
[keating2015robust]. For the case of qudits, the same physics holds, but now
with a multilevel structure and a spectrum of entangling energies. When the
spectrum is nonlinear, the system is controllable.

Figure 5: Schematic for designing two-qudit entangling interactions in 87Sr
neutral atoms. (a) A $k\leq d$-dimensional qudit is encoded in memory in the
nuclear spin with $d=10$ magnetic sublevels in the electronic ground state
$(5s^{2})\;^{1}S_{0}$. When the gate is to be performed, the $k$ levels (here
$k=3$) are transferred coherently to the metastable clock states
$(5s5p)\;^{3}P_{2},F=11/2$ in the presence of a bias magnetic field. The
system becomes controllable by adiabatically dressing the ${}^{3}P_{2}$ with
Rydberg character through the application of a near-resonant laser with Rabi
frequency $\Omega_{\mathrm{L}}$ and detuning $\Delta_{\mathrm{L}}$ with
respect to the hyperfine manifold $(5sns)\;^{3}S_{1},F^{\prime}=9/2$ in the
Rydberg series. Control is then achieved through the application of a phase-
modulated rf-field with Rabi rate $\Omega_{\mathrm{rf}}$ which acts on the
dressed states to generate a nonlinear Larmor precession. The entanglement
arises due to the Rydberg blockade. The coupling of the state of two qudits
for a perfect blockade as depicted in (b), where $i$ is a state from the first
qudit and $j$ is from the second qudit, excited by two Rabi frequencies and
detunings determined by the Clebsch-Gordan coefficients and Zeeman shifts. The
state $\ket{ij}\to\ket{\widetilde{ij}}$ is the dressed state given in Eq.
(14). The spectrum of eigenvalues of the entangling Hamiltonian Eq.(13) is
given in (c) as a function of $i$ and $j$ where the function chosen is
$f(i,j)=10i+j;0\leq i,j<10$. The spectrum indicates 10 parabolas, where each
parabola corresponds to the effect of a single state in the first atom sees
due to all the states in the second atom. This nonlinear spectrum arises
through a combination of the tensor AC Stark shift and the Rydberg blockade,
making the system controllable, allowing us to implement any symmetric two
qudit gate in this system of interest.

Fig. 5 depicts the basic scheme. Those levels of the qudit that we chose to
participate in the gate are excited from the ground ${{}^{1}S_{0}}$ to the
first excited ${{}^{3}P_{2}}$ state. The Rydberg states in 87Sr have well-
resolved hyperfine splitting. We consider UV dressing laser near the resonance
between the ${{}^{3}P_{2}}$, $F=9/2$ hyperfine manifold and the
${{}^{3}S_{1}}$, $F^{\prime}=11/2$ Rydberg hyperfine states. In the presence
of a bias magnetic field, due to the difference in the g-factors, the two
manifolds will be differently Zeeman shifted. The different magnetic sublevels
that define the qudit will thus be differently detuned to the Rydberg magnetic
sublevels. Due to this and the Clebsch-Gordan coefficients associated with the
different transitions, each sublevel will be differently dressed
(equivalently, there is a tensor light shift). When two atoms are dressed, the
effect of the Rydberg blockade modifies the spectrum as discussed above.

An example of two sublevels (one from each atom) is shown in Fig. 5(b).
Diagonializing this atom-laser Hamiltonian under the approximation of a
perfect Rydberg blockade yields the representation

$H_{\mathrm{ent}}=\sum_{ij}E^{ij}\ket{\widetilde{ij}}\bra{\widetilde{ij}},$
(13)

where the tilde indicates dressed states,

$\ket{\widetilde{ij}}=C_{ij}\ket{ij}+C_{r_{i}j}\ket{r_{i}j}+C_{ir_{j}}\ket{ir_{j}},$
(14)

and $E^{ij}$ are the light shifts originating from these interactions. The
spectrum of the entangling Hamiltonian shown in Fig. 5(c) gives us insight
into the controllability of the system. In the chosen order, the spectrum
reveals the structure of $10$ quadratic potentials arising from a combination
of the tensor light shift and Rydberg blockade. This inturn creates a
Hamiltonian that has support on spherical tensor operators with rank $K\leq 2$
and makes the Hamiltonian controllable; further details are discussed in
Appendix (Controllability).

The time-dependent Hamiltonian necessary for the Lie algebraic control can be
chosen as phase-modulated Larmor precession,
$H_{\mathrm{mag}}=-\bm{\mu}\cdot\mathbf{B}(t)$, with
$\bm{\mu}=g_{F}\mu_{B}\mathbf{F}$ the magnetic dipole vector operator, and
where
$\mathbf{B}(t)=B_{\parallel}\mathbf{e}_{z}+B_{T}\operatorname{Re}\left[(\mathbf{e}_{x}+i\mathbf{e}_{y})\mathbf{e}^{-i\left(\omega_{\text{rf}}t+\phi(t)\right)}\right]$.
Defining the auxilary subspace, $a$, for the levels in hyperfine manifold
$\\{5s5p\hskip 2.84544pt^{3}P_{2},\hskip 2.84544ptF=9/2\\}$ and the subspace,
$r$, for the levels $\\{5sns\hskip 2.84544pt^{3}S_{1},\hskip
2.84544ptF^{\prime}=11/2\\}$ in the Rydberg hyperfine manifold, we have
$g_{F}(r)/g_{F}(a)\approx 2$. Thus defining the Zeeman shift
$\omega_{0}=g_{F}(a)B_{\parallel}$, the Larmor precession frequency
$\Omega_{\mathrm{rf}}=g_{F}(a)B_{T}$, and choosing rf drive on resonance in
the $a$-manifold, $\omega_{\mathrm{rf}}=\omega_{0}$, in the co-rotating frame
at $\omega_{0}$, the Hamiltonian is

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

where $F_{i}^{a},F_{i}^{r}$ are the spin angular momentum operators in the
respective subspaces along axis $i\in\\{x,y,z\\}$.

As the $H_{\mathrm{mag}}$ acts on the laser-dressed states defined in Eq.
(14), which are superpositions of $a$ and $r$ states that have different
$g$-factors, one needs to find the action of the magnetic interaction in the
dressed basis. Due to the nonlinearity, the action of the rf-magnetic driving
on the dressed states is no longer simple Larmor precession. Considering a
global rf-magnetic interaction, the $H_{\mathrm{mag}}$ acts on both qudits as

$\displaystyle\left(H_{\mathrm{mag}}(t)\otimes\mathds{1}+\mathds{1}\otimes
H_{\mathrm{mag}}(t)]\right)\ket{\widetilde{ij}}$
$\displaystyle=C_{ij}\left[H_{\mathrm{mag}}^{(a)}(t)\otimes
H_{\mathrm{mag}}^{(a)}(t)\right]\ket{ij}$ (16)
$\displaystyle+C_{r_{i}j}\left[H_{\mathrm{mag}}^{(r)}(t)\otimes
H_{\mathrm{mag}}^{(a)}(t)\right]\ket{r_{i}j}$
$\displaystyle+C_{ir_{j}}\left[H_{\mathrm{mag}}^{(a)}(t)\otimes
H_{\mathrm{mag}}^{(r)}(t)\right]\ket{ir_{j}}.$

Thus in the dressed basis, the Hamiltonian is
$H(t)=\widetilde{H}\left[\phi(t)\right]+H_{\mathrm{ent}}$, where the action of
the magnetic field in the dressed basis is given by the Hamiltonian,

$\displaystyle\widetilde{H}\left[\phi(t)\right]=\sum_{i,j,k,l}\bra{\widetilde{ij}}H_{\mathrm{mag}}(t)\otimes\mathds{1}+\mathds{1}\otimes
H_{\mathrm{mag}}(t)]\ket{\widetilde{kl}}\outerproduct{\widetilde{ij}}{\widetilde{kl}}.$
(17)

By modulating the phase $\phi(t)$ one can generate any target unitary gate.

### Numerical Methods

We consider encoding a $k$-dimensional qudit in the $d=10$ dimensional Hilbert
space associated with $10$ magnetic sublevels of the nuclear spin of 87Sr. To
implement gates based on optimal control for $k<10$, we use techniques based
on the structure of partial isometries. A partial isometry of dimension $k\leq
d$ in a physical system of dimension $d$ is defined as,

$V_{\mathrm{{tar}}}=\sum_{i=1}^{k}\ket{f_{i}}\bra{e_{i}}$ (18)

where $\\{\ket{e_{i}}\\},\\{\ket{f_{i}}\\}$ are two orthonormal bases for the
qudit. The unitary of maps of interest then has the form,

$\displaystyle U_{\mathrm{tar}}=V_{\mathrm{tar}}+V_{\perp},$ (19)

where $V_{\perp}$ acts on the orthogonal subspace, with dimension $d-k$. To
find the control waveform, one then optimizes the fidelity between the target
isometry and the isometry generated using quantum control
[pedersen2007fidelity]

$\mathcal{F}_{V}[{c},T]=\left|\Tr\left(V^{\dagger}_{\text{tar}}V[{c},T]\right)\right|^{2}/k^{2}.$
(20)

#### Numerical results for Lie algebraic approach

As discussed in Sec. Controllabilityc, one can implement an arbitrary
entangling gate through a combination of Rydberg dressing and phase-modulated
Larmor precession driven by rf-fields. Because our control Hamiltonian is
symmetric with respect to the exchange of the qudits, we consider here
symmetric gates, with global control. We seek, through numerical optimization,
the time-dependent rf-phase, $\phi(t)$. To achieve this we employ the well-
known GRAPE algorithm [khaneja2005optimal]. To implement GRAPE, we discretize
the control waveform, $\phi(t)$, and numerically maximize the fidelity by
gradient ascent. We choose here a piecewise constant parameterization (as in
[anderson2013unitary]) and write the control waveform as a vector
$\mathbf{c}=\\{\phi(t_{j})/\pi\;|\;j=1,\dots,n\\}$ where $t=j\Delta t$ and
$n=T/\Delta t$. The waveform is thus a series of square rf-pulses with
constant amplitude and phase over the duration $\Delta t$.

The minimum number of elements in the control vector $\mathbf{c}$ is
determined by the number of parameters needed to specify the target isometry.
A $K$-dimensional partial isometry is defined by the $K$ columns in a $D\times
D$-dimensional unitary matrix. Hence, to find the number of free parameters
for a $K$-dimensional isometry one can count the number of parameters needed
to specify $K$ orthonormal vectors uniquely in a $D$-dimensional vector space.
This is given by

$\displaystyle n_{\mathrm{min}}(K,D)=$
$\displaystyle\sum_{j=1}^{K}2(D-j)-1+K-1$ (21) $\displaystyle=$ $\displaystyle
2\left[KD-\frac{K(K+1)}{2}\right]+K-1$ $\displaystyle=$ $\displaystyle
2KD-K^{2}-1,$

where in the first line, we subtracted one from the parameter count in since
the overall phase of the isometry is neglected. Eq. (21) recovers well-known
limits. When $K=1$ and $D=d$, $n_{\mathrm{min}}=2d-2$, which is the number of
free parameters needed to specify a pure state in a $d$-dimensional Hilbert
space. When $K=D=d$, $n_{\mathrm{min}}=d^{2}-1$, which is the number of free
parameters needed to specify a special unitary map in $d$-dimensions.

In the Lie algebraic protocol for designing entangling gates, the control
Hamiltonian, as well as the target unitary matrices, are symmetric under the
exchange of qudits. In this case, one can work in the symmetric subspace for
two qudits. Using the hook length formula [frame1954hook], the dimension of
the symmetric subspace of the total vector space and isometry is,

$D=\frac{d(d+1)}{2},K=\frac{k(k+1)}{2}.$ (22)

Thus, using Eq. (21), we find the number of free parameters required for the
two-qubit entangling unitary given in Table 1.

$k$ | $n_{\mathrm{min}}(K,D)$
---|---
2 | 320
3 | 623
5 | 1424
7 | 2295
Table 1: The minimum number of parameters required for encoding a partial
isometry of dimension $k$ in the $d=10$ dimensional Hilbert space according to
Eq. (21) for the prime dimensions $k\leq 10$ with $K$ and $D$ given by Eq.
(22) Figure 6: Waveforms of the CPhase gate. Quantum control is achieved by
modulating the phase of an rf-field as a function of time, $\phi(t)$. We
parameterize this by a piecewise constant waveform. The figure shows proof-of-
principle examples of $\phi(t)$ that generate the CPhase gate, optimized using
the GRAPE algorithm for different qudit dimensions. (a) The case of the $d=3$
for a total time of $T=50\pi/\Omega_{\mathrm{rf}}$ with $700$ piecewise
constant steps. (b) The case of the $d=5$ for a total time of
$T=240\pi/\Omega_{\mathrm{rf}}$ with $1600$ piecewise constant steps. (c) The
case of the $d=7$ for a total time of $T=400\pi/\Omega_{\mathrm{rf}}$ with
$2500$ piecewise constant steps. For all of these calculations, the rf-field
is on resonance with the Zeeman splitting $\omega_{\mathrm{rf}}=\omega_{0}$
and we choose the rf-Larmor frequency
$\Omega_{\mathrm{rf}}=\omega_{\mathrm{rf}}$. Control is achieved by Rydberg
dressing with laser Rabi frequency
$\Omega_{\mathrm{L}}=6\Omega_{\mathrm{rf}}$.

Proof-of-principle numerical examples of waveforms that generate the CPhase
gate are given in Fig. 6. The figure gives the $\phi(t)$ as a piecewise
constant function of time, obtained using the GRAPE algorithm. We consider
prime-dimensional qudits, the cases of most interest in quantum algorithms.
Fig. 6(a) shows the case of the $k=3$, a qutrit encoded in $d=10$. The total
time is $T=50\pi/\Omega_{\mathrm{rf}}$, which is divided into $700$ intervals
for the quantum control. Fig. 6(b) shows an example waveform for the case of
$k=5$. Here, the total time is $T=240\pi/\Omega_{\mathrm{rf}}$, divided into
$1600$ intervals. Similarly, Fig. 6(c) shows the case of $k=7$ in our $d=10$
level system. The total time is $T=400\pi/\Omega_{\mathrm{rf}}$, divided into
2500 time intervals. This controllable Hamiltonian can also be used to
generate other two-qudit gates. The qudit generalization of the
M$\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}$lmer-S$\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}$rensen
gate, as is given in the Appendix Creating other symmetric qudit entanglers
for the Lie algebraic approach.

The waveforms found here are a proof-of-principle set of square pulses and are
not intended to be taken as the best choice for experimental implementation.
In practice, one can design and optimize for much smoother waveforms using
well-known techniques by imposing additional constraints on bandwidth and slew
rate. Alternatively, one can optimize in the Fourier domain or in any other
complete basis of functions using the techniques of gradient optimization of
analytic controls (GOAT) [machnes2015gradient].

#### Numerical results for Lie group approach

In the Lie group control protocol discussed in Sec.Controllabilityc we
parameterize the target unitary map as

$\displaystyle U_{\mathrm{tar}}$ $\displaystyle=\prod_{j}U_{\lambda_{j}},$
(23)
$\displaystyle=\prod_{j}e^{-iH_{\mathrm{ent}}t_{j}}\,U_{1}(\vec{\alpha}^{(j)})\otimes
U_{2}(\vec{\beta}^{(j)}).$

The control parameters $\\{\lambda_{i}\\}$ consist of the set of times
$\\{t_{i}\\}$ and the $2(d^{2}-1)$ parameters $\vec{\alpha}^{(j)}$,
$\vec{\beta}^{(j)}$, which specify each of the local $\mathrm{SU}(d)$ unitary
maps. We can parameterize these according to

$\displaystyle
U_{i}(\vec{\alpha}^{(j)})=\exp(-i\sum_{i=1}^{d^{2}-1}\alpha_{i}^{(j)}\Lambda_{i}),$
(24)

where $\Lambda$ is the generalized Gell-Mann matrices that span the Lie
algebra $\mathfrak{su}(d)$. The matrices can be categorized as,

symmetric:
$\displaystyle\Lambda_{jk}^{x}=\outerproduct{j}{k}+\outerproduct{k}{j},$ (25)
anti-symmetric:
$\displaystyle\Lambda_{jk}^{y}=-i\outerproduct{j}{k}+i\outerproduct{k}{j},$
diagonal:
$\displaystyle\Lambda_{l}^{z}=\sum_{j=1}^{l}\outerproduct{j}{j}-l\outerproduct{l+1}{l+1}.$

The task of the numerical optimization, thus, is to find the set of times of
the entangling interaction $\\{t_{j}\\}$, and the expansion coefficients of
the Gell-Mann matrices $\\{\alpha_{i}^{(j)}\\}$ and $\\{\beta_{i}^{(j)}\\}$.
We denote this whole set of parameter as
$\\{\lambda_{j}\\}=\\{t_{j},\vec{\alpha}^{(j)},\vec{\beta}^{(j)}\\}$.

We define one layer of the control as consisting of a pair of local SU$(d)$
gates followed by the entangling Hamiltonian for a time $t_{j}$. The total
number of free parameters for a CPhase gate is $d^{2}(d^{2}+1)/2$, as follows
from Eq. (22) for a symmetric gate in $\mathrm{SU}(d^{2})$. Thus, the minimum
number of layers required to obtain the CPhase gate is given by

$\displaystyle N_{\text{min}}\left(2(d^{2}-1)+1\right)=$
$\displaystyle\frac{d^{2}(d^{2}+1)}{2}$ (26) $\displaystyle N_{\text{min}}=$
$\displaystyle\frac{d^{2}(d^{2}+1)}{2(2d^{2}-1)}.$

The numerical results for the minimum number of layers needed in the system
are given in Table 2 for the cases of $d=3,5,$ and $d=7$. In practice, we find
that one needs more than this minimum number of layers to implement the target
unitary gate with high fidelity. This improves the optimization landscape for
gradient ascent [larocca2018quantum].

For our case under study, we choose the same entangling Hamiltonian as we used
in the Lie algebraic approach given in Eq. (13). However, unlike that
approach, we interleave the entangling interaction with local single-qudit
SU($d$) gates. Implementation of this requires another layer of optimization.
As we do not have access to native Hamiltonians proportional to the Gell-Mann
matrices, to implement local qudit gates we can employ local SU($d$) optimal
control [omanakuttan2021quantum]. From a practical perspective, this might be
implemented directly in the ${}^{3}P_{2}$ manifold, either through a
combination of tensor-light shift and rf-driven Larmor precession similar to
[omanakuttan2021quantum], or alternatively through a combination of microwave-
driven Rabi oscillations between different hyperfine levels in ${}^{3}P_{2}$
and rf-driven Larmor procession as in [anderson2013unitary]. In either case,
optimal control can be used to find the relevant experimental waveform that
generates the desired local $\mathrm{SU}(d)$ gates.

$d$ | $N_{\mathrm{min}}$ | $N_{\text{local}}$ | $N_{\text{global}}$
---|---|---|---
3 | 3 | 6 | 7
5 | 7 | 10 | 12
7 | 13 | 14 | 15
Table 2: The number of layers of primitive gates in the Lie group approach
required to achieve the CPhase gate. The theoretical minimum is $N_{\rm{min}}$
according to Eq. (26). If we allow locally addressable single qudit gates, the
number of layers required is $N_{\rm{local}}$. If we have only global control
but allow for a sign change in the entangling Hamiltonian, the number of
layers required is $N_{\rm{global}}$

In this analysis, we included locally addressable control on each qudit.
Though the CPhase gate is symmetric under exchange, we find that this symmetry
breaking is necessary for effective optimization of this parameterization,
similar to that seen in [PhysRevResearch.3.023092]. An alternative protocol is
to employ symmetric global control of the local unitaries,
$\vec{\alpha}^{(j)}=\vec{\beta}^{(j)}$, but to reverse the sign of the
entangling Hamiltonian $H_{\mathrm{ent}}\to-H_{\mathrm{ent}}$ in alternating
layers. This allows for effective optimization, and the corresponding result
is given in Table (2).

#### Decoherence

In a closed quantum system, quantum optimal control employing either the Lie
algebraic or the Lie group approaches can be used in principle to implement
any qudit entangling gate to any desired fidelity. In our numerical
optimization, we took the target infidelity to be $10^{-3}$. In the absence of
decoherence, we could achieve that target in a reasonable time for $d\leq 5$.
For $d=7$, more time is required. However, the fundamentally achievable
fidelity is limited by decoherence associated with the particular physical
platform. For the system at hand, decoherence occurs due to the finite
lifetime of the Rydberg states, which predominantly leads to leakage and loss
outside the computational basis. In that case, we can model the gate as
generated by a non-Hermitian effective Hamiltonian, $H_{\mathrm{eff}}[c(t)]$,
where the Hermitian part is the control Hamiltonian and the anti-Hermitian
represents decay out of the Rydberg states. The fidelity of interest is given
by

$\mathcal{F}_{V}[\bm{c},T]=\left|\Tr\left(V^{\dagger}_{\text{tar}}V_{\mathrm{eff}}[\bm{c},T]\right)\right|^{2}/d^{2},$
(27)

where
$V_{\mathrm{eff}}[\mathbf{c},T]=\mathcal{T}\left[\exp\left(-i\int_{0}^{T}H_{\mathrm{eff}}[\mathbf{c}(t)]dt\right)\right]$.
Here the decay amplitude from a dressed state is
$\gamma_{\mathrm{decay}}^{ij}=|C_{r_{i}j}|^{2}\Gamma_{r_{i}}+|C_{ir_{j}}|^{2}\Gamma_{r_{j}}$,
which in turn gives the effective Hamiltonian as

$H_{\mathrm{ent}}^{\mathrm{eff}}=\sum_{ij}\left(E_{\mathrm{LS}2}^{ij}-i\gamma_{\mathrm{decay}}^{ij}/2\right)\ket{\widetilde{ij}}\bra{\widetilde{ij}}.$
(28)

With this model for decoherence in hand, the numerical results for the Lie
algebraic approach are given in Fig. 7, which shows the infidelity as a
function of time for a CPhase gate for different dimension isometries. We
focus here on the case of the prime dimensional qudits. In contrast to closed-
system control, in the presence of decoherence, infidelity decreases at first
and then increases. This is due to the fact there is an optimal time of
evolution, larger than the quantum speed limit, but not too large when
compared to the coherence time of the system. As expected, one needs more time
as the qudit dimension increases, which in turn results in an increase in the
minimum infidelity one could achieve in each of these cases as shown in Fig.
7. We obtain a maximum fidelity of 0.9985, 0.9980, 0.9942, and 0.9800 for
$d=2$, $d=3$, $d=5$, and $d=7$ respectively for the CPhase gate. Note, the
values of fidelity for different dimensional qudits should be considered in
the context of a particular application. For example, the threshold for fault
tolerance for qudits, in general, is larger for larger $d$ [anwar2014fast,
watson2015fast]. For the particular scheme considered in [anwar2014fast], the
threshold for $d=2$, $d=3$, $d=5$, and $d=7$ are close to $0.008$, $0.012$,
$0.0135$, and $0.015$ respectively. Hence, the proof-of-principle fidelity
obtained here is promising and can be further optimized.

Figure 7: Infidelity as a function of time. Simulated infidelity with and
without decoherence as a function of control time divided by the dimension $d$
for CPhase gate with different prime dimensions with $d\leq 10$, as found
using Lie algebraic quantum control and the GRAPE algorithm. Decoherence due
to Rydberg decay outside the computational basis is included through an
imaginary part of the Hamiltonian. We take the Rydberg lifetime to be
$140\mu$s and choose the rf-Larmor frequency to be
$\Omega_{\mathrm{rf}}/2\pi=10$ MHz. In the absence of decoherence (dashed
lines), for a time greater than the “quantum speed limit” (the time required
to obtain ideal fidelity) we achieve a minimal error (infidelity) of $10^{-3}$
due to our threshold in the numerics for $d\leq 5$. This speed-limit time
increases as we increase the qudit dimension, which in turn results in an
increased decay in maximum fidelity. For the CPhase gate, we obtain a fidelity
of $0.9985$, $0.9980$, $0.9942$, and $0.9800$ for $d=2,d=3,d=5,\text{ and
}d=7$ respectively. For all of these calculations, we have taken the dressing
laser Rabi frequency to be $\Omega_{\mathrm{L}}=6\Omega_{\mathrm{rf}}$ and the
lifetime of the Rydberg states to be $140\mu$s.

In the Lie group approach, we can use the effective Hamiltonian to describe
the evolution when the Rydberg dressing is employed. In this case, we have,

$\displaystyle U_{\mathrm{tar}}^{\mathrm{eff}}$
$\displaystyle=\prod_{j}U_{\lambda_{j}},$ (29)
$\displaystyle=\prod_{j}e^{-iH_{\mathrm{ent}}^{\mathrm{eff}}t_{j}}U_{1}(\vec{\alpha}^{(j)})\otimes
U_{2}(\vec{\beta}^{(j)}).$

We neglect here any decoherence associated with the local SU($d$) gates. Thus
the fidelity including the decoherence effects is given as,

$\mathcal{F}_{\mathrm{eff}}=\left|\Tr\left(U^{\dagger}_{\text{tar}}U_{\mathrm{tar}}^{\mathrm{eff}}\right)\right|^{2}/d^{2},$
(30)

Figure 8: A comparison of the optimized fidelity, $\mathcal{F}$ of the CPhase
gate achieved for the Lie algebraic and Lie group approaches (including both
local single-qudit control and only global control) is plotted as a function
of the total Hilbert space dimension $d^{2}$, for the qudits of dimension
$d=3,5,$ and $d=7$. For all of these simulations, we have taken the parameters
given in Fig. 7.

A comparison of the fidelities achieved based on the Lie algebraic and Lie
group approaches is given in Fig. 8 for $d=3,5,$ and $d=7$. The results
suggest that the Lie algebraic protocol slightly outperforms the Lie group
protocol in the presence of decoherence. This difference in the performance
can be attributed to the time spent in the Rydberg state for these two
approaches, as shown in Fig. 9. Fundamentally, we can understand this from the
fact that the Lie algebraic approach has more control parameters as compared
to the Lie group protocol. Thus, based on the Magnus expansion
[merkel2009quantum, jurdjevic1972control, brockett1973lie], the nested
commutators which are at the heart of controllability become easier to
achieve. Both approaches yield high fidelities in large dimensional qudits.
Nevertheless, the Lie group approach may be preferable when considering the
complexity necessary for experimental control. The difference in the behavior
of Lie-group(local) to Lie-group(global) is due to the fact that for the
global approach we allow $H_{\mathrm{ent}}\to-H_{\mathrm{ent}}$ in alternating
layers.

In general, a key experimental consideration for the successful implementation
of open-loop quantum control is the effect of uncertainties in Hamiltonian
parameters. These can be mitigated to some degree using the tools of robust
quantum control [anderson2015accurate, goerz2015optimizing,
glaser2015training, koch2016controlling]. Such techniques are generalizations
of spin-echo type composite pulses which can be useful when there is
sufficient coherence time. With a detailed understanding of the dominant
inhomogeneities, robust optimal control can be used to implement suitable
composite waveforms for qudit entanglers on any platform.

The specific experimental foundation of this proposal is well-motivated by
existing literature, particularly the work of the Jessen group
[anderson2013unitary]. One particular issue discussed above is the trap-
induced differential light shifts between the ground state and excited state
${}^{3}\mathrm{P}_{2}$ manifold [PhysRevResearch.5.013219]. It will be
necessary to mitigate motional dephasing arising from vector- and tensor-
shifts, which induce an $m_{F}$-dependence on polarizability, thus inducing
possible motional dephasing between $m_{F}$ levels. The easiest way around
this problem is to operate with a linearly-polarized optical trap, with
polarization vector aligned at the “magic angle” [PhysRevX.8.041054] and
corresponding magic wavelength [doi:10.1126/science.1148259] for the
${}^{1}\mathrm{S}_{0}\rightarrow{}^{3}\mathrm{P}_{2}$ transition. This allows
intra-state coherence within the ${}^{3}\mathrm{P}_{2}$ $F=9/2$ (and other
$F$-levels) manifold, and inter-state (i.e., optical qubit) coherence between
the ${}^{1}\mathrm{S}_{0}$ and ${}^{3}\mathrm{P}_{2}$ $F=9/2$. We can also
mitigate motional effects via high-fidelity ground-state cooling
[kaufman2012cooling, thompson2013coherence, lester2014raman].

Figure 9: A comparison of the minimum time spent in the Rydberg state to
implement the CPhase gate achieved for the Lie algebraic and Lie group
approaches (including both local single-qudit control and only global control)
is plotted as a function of the total Hilbert space dimension $d^{2}$, for the
qudits of dimension $d=3,5,$ and $d=7$. For all of these simulations, we have
taken the parameters given in Fig. 7. Thus the time required for the Lie
algebraic control is smaller than the Lie group control which in turn
contributes to the fidelity.

### Conclusion and Outlook

Quantum computation with qudits has potential advantages when compared with
architectures employing qubits. Implementing gates for qudit-based quantum
computation is fundamentally more challenging, as the generators for these
gates are not native Hamiltonians on physical platforms. One way to overcome
this challenge is to use the tools of quantum optimal control, whereby we
combine native Hamiltonians with time-dependent waveforms that drive the
system in order to implement a universal gate set with high fidelity.

In this chapter, we introduced two classes of numerical methods of quantum
optimal control for implementing the qudit entangling gates, an essential
component of the universal gate set. The first approach is based on
continuous-time driving given a controllable Hamiltonian with tunable
parameters and uses the Lie algebraic structure of the control problem. The
second approach is more “digital,” using the Lie group structure to design a
family of unitary maps that can be applied in sequence to achieve any
nontrivial entangling gate of interest.

As a specific example, we studied encoding a qudit in the nuclear spin of
87Sr, a species of atoms that is particularly important in quantum information
processing. The nuclear spin can accommodate a qudit of dimension $d\leq 10$.
We have previously studied protocols for implementing single-qudit gates in
SU($d$). To implement entangling gates we studied how we make two atoms
interact using the well-known Rydberg blockade mechanism, and in particular,
we studied Rydberg dressing schemes. Using this we are able to generate any
two-qudit entangling gate, both using the Lie algebraic and Lie group based
approaches.

We also studied how the fundamental effects of decoherence introduced by the
finite lifetime of the Rydberg states reduce the gate fidelity. To model this
we used a nonHermitian Hamiltonian and found that even when including
decoherence, one could achieve high fidelity for these qudit entanglers. Given
the flexibility of arbitrary control, we can seek the best approach to
encoding qudits and mitigating errors.

Finally, while we have studied a particular case study in the context of
neutral-atom quantum computing, the general methods we have developed here can
be applied in other platforms, including trap ions transmon qudits, and
nanomagnets [petiziol2021counteracting, chiesa2021embedded], which also have
natural encoding and control Hamiltonians.

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

### Introduction

In this chapter, we develop more efficient error-corrected quantum processors
by taking 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 [Gottesman2001,
gross2021hardware, omanakuttan2023multispin, Omanakuttan2023gkp]. In this
chapter we will consider encoding a qubit in a spin-$J$ system, corresponding
to a qudit with $d=2J+1$ levels [omanakuttan2021quantum,
Siva_Qudit_entangler_2023, zache2023fermion]. 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 in this direction have previously explored the concept of encoding
a qubit in a large spin [Gross2021, gross2021hardware,
omanakuttan2023multispin]. 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
[Gross2021, omanakuttan2023multispin]. 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 qubits
[Gross2021, gross2021hardware], ion traps [ringbauer2021universal, Low2020],
atomic systems [omanakuttan2021quantum, Siva_Qudit_entangler_2023,
zache2023fermion], molecules [castro2021optimal, jain2023ae], and
superconducting systems [ozguler2022numerical, Blok2021], wherein spin qudits
offer the means to encode logical qubits.

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 [puri2020bias, guillaud2019repetition], 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 [Gross2021, omanakuttan2023multispin, kubischta2023family,
kubischta2023not], 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.

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

(a)

(b)

Figure 10: 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 ${T}^{(k)}_{q}(J)$
[sakurai2014modern, klimov2008generalized, varshalovich1988quantum] 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
$\\{{T}^{(1)}_{q}(J),{T}^{(2)}_{q}(J)\\}$ [omanakuttan2023multispin]. 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 [puri2020bias]. In
this encoding, the qubit states $\ket{0}$ and $\ket{1}$ are chosen to be,

$\ket{C_{\alpha}^{\pm}}\propto\ket{\alpha}\pm\ket{-\alpha},$ (31)

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 [Aliferis2008fault, puri2020bias] for cat qubits.

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

$\displaystyle\ket{\pm}$
$\displaystyle\equiv\frac{1}{\sqrt{2}}\left(\ket{J,-J}\pm\ket{J,+J}\right),$
(32)

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 [puri2020bias, guillaud2019repetition], the spin cat states are defined
along the $1$-axis of the qubit Bloch sphere; see Fig. 10a. 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 [puri2020bias,
guillaud2019repetition], 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 [PhysRevLett.128.110502].

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.

#### 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,

$\displaystyle\ket{\pm}_{m}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{0}_{m}\pm\ket{1}_{m}\right).$ (33)

where,

$\displaystyle\ket{0}_{m}$ $\displaystyle=\ket{J,-J+m}\equiv\ket{-J+m}$ (34)
$\displaystyle\ket{1}_{m}$ $\displaystyle=\ket{J,J-m}\equiv\ket{J-m}.$

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,

$\mathcal{H}_{d}=\bigoplus_{i=0}^{\frac{d}{2}}\mathcal{H}_{2}^{(i)},$ (35)

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,

$\displaystyle\Pi_{\overline{0}}$ $\displaystyle=\sum_{k=0}^{\lfloor
J-1/2\rfloor}\outerproduct{-J+k}{-J+k},$ (36)
$\displaystyle\Pi_{\overline{1}}$ $\displaystyle=\sum_{k=0}^{\lfloor
J-1/2\rfloor}\outerproduct{J-k}{J-k}.$

See Fig. 10 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,

$\ket{\pm}_{m}\to\sum_{k=0}^{\lfloor\frac{2J-1}{2}\rfloor}c_{k}\ket{\pm}_{k},$
(37)

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

$\ket{+}_{k}\to\ket{-}_{k}.$ (38)

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

$\displaystyle U_{Z}$ $\displaystyle=\exp(-i\theta J_{z}),$ (39)
$\displaystyle U_{X}$ $\displaystyle=\exp(-i\theta J_{x}).$

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

$\displaystyle U_{Z}\ket{\pm}$ $\displaystyle\approx\left(\mathds{1}-i\theta
J_{z}\right)\ket{\pm}=\ket{\pm}-i\theta J\ket{\mp},$ (40) $\displaystyle
U_{X}\ket{\pm}$ $\displaystyle\approx\left(\mathds{1}-i\theta
J_{x}\right)\ket{\pm}$
$\displaystyle=\ket{\pm}-i\theta\frac{\sqrt{J}}{\sqrt{2}}\ket{+}_{1}.$

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 [Deutsch2000],

$\displaystyle W_{0}$ $\displaystyle=\beta T^{(2)}_{0},$ (41) $\displaystyle
W_{+1}$ $\displaystyle=i\alpha
T^{(1)}_{-1}-\beta\sqrt{\frac{3}{4}}T^{(2)}_{-1},$ $\displaystyle W_{-1}$
$\displaystyle=i\alpha T^{(1)}_{1}+\beta\sqrt{\frac{3}{4}}T^{(2)}_{1}.$

where $\alpha,\beta$ are real numbers that depend on the atomic structure and
the states being excited by a near resonance laser. (See Photon scattering and
optical pumping 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. 37, and phase errors as given in
Eq. 38. 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,

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

While we consider a three-qubit repetition code here and throughout Syndrome
Measurement and error recovery for simplicity, in logical CNOT gate and Fault-
tolerant threshold 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 [puri2020bias, guillaud2019repetition]. We
call this the “logical-level encoding” to differentiate it from the physical-
level encoding in Eq. 34.

More formally, in Correctable set of errors we show that the logical-level
encodings in Eq. 42 can correct any single spin angular momentum errors of the
form,

$\small\mathcal{E}_{K}=\left\\{J_{x}^{l}J_{y}^{m}J_{z}^{n};0\leq l+m+n\leq
K=\lfloor\frac{2J-1}{2}\rfloor\right\\}.$ (43)

In practice we can restrict our attention to quadratic polynomials.

#### 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 [varshalovich1988quantum]

$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}},\
$ (44)

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,

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

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.,

$\displaystyle\Tr{\left(S^{(k)}_{q}\right)^{\dagger}S^{(k^{\prime})}_{q^{\prime}}}$
$\displaystyle=\Tr{\left(A^{(k)}_{q}\right)^{\dagger}{A}^{(k^{\prime})}_{q^{\prime}}}=\delta_{k,k^{\prime}}\delta_{q,q^{\prime}},$
(46)
$\displaystyle\Tr{\left(S^{(k)}_{q}\right)^{\dagger}A^{(k^{\prime})}_{q^{\prime}}}$
$\displaystyle=0,$

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,

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

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. 37. The Kraus operator $S^{(k)}_{0}$ flips the kitten states for
$k\;\mathrm{mod}2$=1 which corresponds to the phase error in Eq. 38; 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 correctable single spin errors can be
written in terms of the new basis as,

$\mathcal{E}_{K}=\left\\{S_{q}^{(k)},{A}_{q}^{(k)}\;|\;0\leq k\leq K,-k\leq
q\leq k\right\\},$ (48)

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

The logical encoding defined in Eq. 42 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. 37 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,

$\displaystyle\ket{0}$
$\displaystyle=\frac{1}{\sqrt{8}}\left(\ket{\uparrow}^{\otimes
2J+1}+\ket{\downarrow}^{\otimes 2J+1}\right)^{\otimes 3}$ (49)
$\displaystyle\ket{1}$
$\displaystyle=\frac{1}{\sqrt{8}}\left(\ket{\uparrow}^{\otimes
2J+1}-\ket{\downarrow}^{\otimes 2J+1}\right)^{\otimes 3}$

For the Shor code [shorcode], 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, $\ket{\uparrow}^{\otimes 2J+1}=\ket{J,J}$,
$\ket{\downarrow}^{\otimes 2J+1}=\ket{J,-J}$.

### 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 [Aliferis2008fault,
puri2020bias], 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,

$\\{\mathcal{P}_{\ket{0}},\mathcal{P}_{\ket{+}},\mathcal{M}_{X},\mathcal{M}_{Z},\mathrm{CNOT},ZZ(\theta),X,Y,Z\\}.$
(50)

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
$\mathcal{C}_{1}$,

$\\{\mathcal{P}_{\ket{0}_{\mathrm{L}}},\mathcal{P}_{\ket{+}_{\mathrm{L}}},\mathcal{M}_{X_{\mathrm{L}}},\mathcal{M}_{Z_{\mathrm{L}}},\mathrm{CNOT}_{\mathrm{L}}\\}\cup\\{\mathcal{P}_{\ket{i}_{\mathrm{L}}},\mathcal{P}_{\ket{T}_{\mathrm{L}}}\\}.$
(51)

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 [zzPoulin] and cat codes [puri2020bias].

#### 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

$U\mathcal{E}_{K}U^{\dagger}\in\mathcal{E}_{K},$ (52)

where $\mathcal{E}_{K}$ represents the set of correctable errors for the spin-
cat encoding as defined in Eq. 48. 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. 52. To see this, recall the definition of spherical tensor
operators [klimov2008generalized, sakurai2014modern]:

$UT_{q}^{(k)}U^{\dagger}=\sum_{-k\leq q^{\prime}\leq
k}D_{q,q^{\prime}}T_{q^{\prime}}^{(k)},$ (53)

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

$D_{q,q^{\prime}}=\bra{k,M=q^{\prime}}\exp(-i\theta\hat{n}.\mathbf{J})\ket{k,M=q},$
(54)

are the elements of Wigner $D$-matrices [sakurai2014modern]. 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. 45, we get,

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

where the coefficients
$\\{g_{q,q^{\prime}},\widetilde{g}_{q,q^{\prime}},h_{q,q^{\prime}},\widetilde{h}_{q,q^{\prime}}\\}$
are given in Action of the SU(2) operators. Therefore, the SU(2) rotations do
not change the rank of the error operators and obey the condition given in Eq.
52.

For the remainder of this chapter, 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,

$\displaystyle X$ $\displaystyle=\exp(-i\pi J_{x}),$ (56) $\displaystyle Y$
$\displaystyle=\exp(-i\pi J_{y}),$ $\displaystyle Z$ $\displaystyle=\exp(-i\pi
J_{z}).$

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

$\displaystyle H\ket{0}$ $\displaystyle=\ket{+}.$ (57) $\displaystyle
H\ket{1}$ $\displaystyle=\ket{-},$

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,

$H\mathcal{E}_{K}H^{\dagger}\not\in\mathcal{E}_{K}.$ (58)

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.

#### 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 [Brennen1999, Deutsch2000, Jaksch_gate_2000,
Saffman_review_2016_Rydberg, Browaeys_review_2020_Quantum], which has shown
increasing promise for scalable FTQC [bluvstein2022quantum,
Cong_Lukin_QEC_Rydberg_PRX_2022, Singh_Bernien_2022_dual_array,
Saffman_Nature_2022, Ebadi_Lukin_2022_MIS_optimization]. In particular, we
consider 87Sr atoms which we studied in detail in the previous two chapters,
with a spin-qudit encoded in the nuclear spin $I=9/2$, providing a qudit with
$d=10$ levels [omanakuttan2021quantum, Siva_Qudit_entangler_2023].

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 chapter, 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. 36), 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,

$\mathrm{CNOT}=\Pi_{\overline{0}}\otimes\mathds{1}+\Pi_{\overline{1}}\otimes
X,$ (59)

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. 11b. 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. 59.

(a)

(b)

Figure 11:  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 12: Examples of control waveforms that achieve the transfer of
populations between spin manifolds while preserving the coherence between
magnetic sublevels. Based on Hamiltonian Eq. 60, 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

$\displaystyle H_{e}(t)=$
$\displaystyle\sum_{M=-\frac{9}{2}}^{\frac{9}{2}}-\Delta_{e,M}(t)\outerproduct{e,M}{e,M}$
(60)
$\displaystyle+\Omega_{e,M}(t)\left[e^{i\phi_{e}(t)}\sigma^{+}_{e,M}+\mathrm{h.c}\right].$

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,

$\displaystyle\ket{r,M}$
$\displaystyle\equiv\ket{5\text{s}60\text{s};\;^{3}S_{1};\hskip
2.84544ptF=\frac{11}{2},M_{F}=M},$ (61) $\displaystyle\ket{a,M}$
$\displaystyle\equiv\ket{5\text{s}5\text{p};\;^{3}P_{2};\hskip
2.84544ptF=\frac{9}{2},M_{F}=M},$ $\displaystyle\ket{g,M}$
$\displaystyle\equiv\ket{5\text{s}^{2};\;^{1}S_{0},;\hskip
2.84544ptF=\frac{9}{2},M_{F}=M},$

and

$\sigma^{+}_{e,M}\equiv\outerproduct{e,M}{e^{\prime},M}.$ (62)

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,

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

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

$\mathcal{C}_{e,M}=\frac{\bra{F,M}\ket{1,0;F,M}}{\bra{F,\frac{9}{2}}\ket{1,0;F,\frac{9}{2}}}.$
(64)

$\Delta_{e}$ 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. As given in detail in Lie group approach one can use the GRAPE
algorithm to find the optimal control parameters
$\Phi=\\{\Omega_{e}(t),\Delta_{e}(t),\phi_{e}(t)\\}$ that maximizes the
fidelity with the target map $U_{\mathrm{tar}}$

$\mathcal{F}[\Phi]=\frac{1}{{d}^{2}}\left|\tr\left\\{U_{\mathrm{tar}}^{\dagger}U[\Phi,T]\right\\}\right|^{2},$
(65)

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 as given in detail in
Numerical Methods. These have fewer constraints than unitary transformations
and hence require fewer resources (time, bandwidth etc.). 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,

$\displaystyle V_{\mathrm{tar}}^{(\mathrm{C})}$
$\displaystyle=\sum_{M=-\frac{9}{2}}^{-\frac{1}{2}}\outerproduct{\mathrm{a},M}{\mathrm{a},M}+\sum_{M=\frac{1}{2}}^{\frac{9}{2}}\outerproduct{\mathrm{a},M}{\mathrm{g},M}.$
(66)

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

$\displaystyle V_{\mathrm{tar}}^{(\mathrm{T})}$
$\displaystyle=\sum_{M=-\frac{9}{2}}^{\frac{9}{2}}\outerproduct{\mathrm{a},M}{\mathrm{g},M}.$
(67)

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

$\displaystyle V_{\mathrm{tar}}^{(\mathrm{Ryd})}$
$\displaystyle=\sum_{M=-\frac{9}{2}}^{\frac{9}{2}}\outerproduct{\mathrm{r},M}{\mathrm{a},M}.$
(68)

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
[Merkel2009, merkel2009quantum, jurdjevic1972control, goerz2015optimizing].
Example waveforms that implement the target isometries are given in Fig. 12.
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 13: 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. 70 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. 11 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
$g_{\mathrm{r}}/g_{\mathrm{a}}\approx 2$ [urech2023single]. 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

$\displaystyle H_{a}$ $\displaystyle=\Omega_{\mathrm{rf}}\left[\cos(\omega
t+\phi)F_{x}+\sin(\omega t+\phi)F_{y}\right]+\omega_{0}F_{z},$ (69)
$\displaystyle H_{r}$ $\displaystyle=2\Omega_{\mathrm{rf}}\left[\cos(\omega
t+\phi)F_{x}+\sin(\omega t+\phi)F_{y}\right]+2\omega_{0}F_{z}.$

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

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

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

$\displaystyle\Omega_{\mathrm{a}}$
$\displaystyle=\sqrt{\Omega_{\mathrm{rf}}^{2}+\frac{\omega_{0}^{2}}{9}},$ (71)
$\displaystyle\Omega_{\mathrm{r}}$
$\displaystyle=\sqrt{4\Omega_{\mathrm{rf}}^{2}+\frac{4\omega_{0}^{2}}{9}}=2\Omega_{\mathrm{a}}.$

Since the total Larmor frequency of the auxiliary auxiliary and Rydberg
manifolds are different, one can use optimization techniques such composite
pulses [levitt1986composite] or quantum optimal control [Merkel2009,
goerz2015optimizing] 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. 13. 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 Rotating the ground and excited manifold
differently using optimal control.

The protocol described above can be generalized for other entangling gates.
One can optimize rf-phases in Eq. 70 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.

#### 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. 50.
$\mathcal{P}_{\ket{0}}$, which is the preparation of the spin coherent state
can be achieved with high fidelity using optical pumping [chow2022high]. Also,
$\mathcal{M}_{Z}$, which is the measurement in the $\ket{F,M_{F}}$ basis can
be achieved with high fidelity in principle [barnes2021assembly,
ringbauer2021universal]. 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.

##### 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

$H(t)=\Omega_{\mathrm{rf}}\left(\cos\phi(t)F_{x}+\sin\phi(t)F_{y}\right)+\beta
F_{z}^{2}.$ (72)

This can be implemented in atomic systems using a combination of tensor light
shifts and rf rotations [Paul_experiment_Cs_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 [omanakuttan2021quantum]. 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 Photon scattering and optical pumping to
calculate the fidelity for the state preparation,

$\mathcal{F}_{\mathrm{state}}=\bra{+}\rho\ket{+}.$ (73)

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

##### Measurement of $X$

Figure 14: Circuit diagram implementing $\mathcal{M}_{X}$. Consider an initial
state $\alpha\ket{+}_{k}+\beta\ket{-}_{k}$, where $0\leq