Квантовые вычисления в гетероядерных массивах ультрахолодных атомов тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Али Ахмед Мохамед Фарук Мохамед

• Али Ахмед Мохамед Фарук Мохамед
• кандидат наук
• 2025

Оглавление диссертации кандидат наук Али Ахмед Мохамед Фарук Мохамед

Contents

Abstract................................................................................................I

List of Figures..........................................................................................VI

List of Tables..........................................................................................IX

Declaration............................................................................................X

Acknowledgements....................................................................................XI

ABBREVIATIONS..........................................XII

List of Publications..........................................XIII

Synopsis................................................XV

1 Chapter 1: Rydberg Atoms and Atom-Light Interaction

1.1 State-of-the-art

1.2 Rydberg Atoms

1.3 Atom-light interactions

1.3.1 Two-level atom

1.3.2 Three-level A-type atom

1.3.3 The effective Hamiltonian of two-photon Rydberg excitation

1.4 Interactions of Rydberg atoms

1.5 The theory of Rydberg blockade

1.6 EIT

1.7 Quantum bit

1.8 Single-qubit gates

1.8.1 Pauli gates

1.8.2 Hadamard gate

1.9 Multi-qubit gates

1.9.1 Controlled-NOT (CNOT) gate

1.9.2 Toffoli (CCNOT) gate

2 Chapter 2: Parallel Implementation of CNOTn

2.1 Introduction

2.2 Scheme of Rydberg EIT

2.3 Interaction Energy

2.3.1 Heteronuclear Architecture

2.3.2 Homonuclear Architecture

2.4 Fidelity

2.4.1 Homonuclear Architecture

2.4.2 Heteronuclear Architecture

2.5 Scheme of C2NOT2

2.6 Gate Errors

2.7 Conclusion

3 Chapter 3: Scalable Heteronuclear Architecture

3.1 Introduction

3.2 Physical model

3.3 Fidelity of GHZ-state

3.4 Renyi entropy and Mutual information

3.5 Conclusion

4 Chapter 4: Quantum Phases of Matter and MWIS

4.1 Introduction

4.2 Quantum system

4.3 Quantum phase transitions (QPT)

4.4 Planar Graphs

4.4.1 Maximum Independent Set (MIS)

4.4.2 Maximum-weight independent set (MWIS)

4.5 Non-planar graph

4.6 Quantum Wire

4.7 Conclusion

5 Chapter 5: Results and Outlook

5.1 Summary

5.2 Outlook

Appendices

A Appendix A

A.1 Charge-charge interaction energy Vqq

A.2 Charge-dipole interaction Vqd

A.2.1 The Coulomb potential interact with dipole

A.2.2 Dipole field interact with point charge

A.3 Dipole-dipole interaction Vdd

B Appendix B

C Appendix C

D Appendix D

Bibliography

List of Figures

1 An info-graphic map for the funding on quantum research activity by each country .... XVI

1.1 The dependence of lifetime of 85Rb rubidium Rydberg state on the principal quantum number n

1.2 A schematic representation of a two-level atom

1.3 Probability to find the atom in the excited state for the case of resonant interaction of two-level atom with electromagnetic field

1.4 The scheme of three-level atom in A-configuration interacting with two laser fields, and

the corresponding probabilities

1.5 The atomic structure of a three-level atom in S-configuration

1.6 A scheme of atom-atom interaction where each atom is excited to Rydberg state

1.7 Constant electric field is applied to two alkali atoms in a spatially separated optical dipole traps

1.8 A scheme of energy levels of two atoms interacting and the evolution of blockade radius

Rb as a function of principal quantum number n

1.9 Schematic representation of three-level atom in A configuration interacting with probe

and coupling laser beams

1.10 Bloch sphere representation of a quantum state ^

1.11 The Bloch sphere representation of Pauli X gate

1.12 The Bloch sphere representation of Hadamard gate

2.1 Scheme of the generation of multi-atom entangled GHZ-state using a sequence of CNOT gates in homonuclear and heteronuclear architectures

2.2 The sequence of laser pulses for Rydberg EIT gate and the gate scheme in blocking and transfer of populations

2.3 Spatial configurations of control and target atoms for the CNOT gate

2.4 The dependence of the probability of blocking the population transfer |0)|A) ^ |0)|A)

on the ratio between Qc and Qp

2.5 The dependence of the interaction strength between control and target atoms on the angle between interatomic axis and quantization axis and the relation between blockade radius

and principal quantum number

2.6 Contour plot of the fidelity of entangled states in the homonuclear architecture for different spatial configurations of target atoms

2.7 Contour plot of the fidelity of entangled states in the heteronuclear architecture for different spatial configurations of target atoms

2.8 Scheme of spatial configurations of homonuclear/heteronuclear interactions for the implementation of C2NOT2 gate

2.9 Contour plot of fidelity FC-RbOT

2.10 Investigating the source of error resulting from exciting the target atom from ground state through the first of the second resonance level of intermediate state for different configurations

2.11 The fidelity for different spatial arrangement of qubits in CNOTn (C2NOT2) gate

3.1 A scheme of an array of dual-species atoms with parallel implementation of single CNOT gates

3.2 Scalable architecture of neutral heteronuclear atoms to perform CNOT gate using EIT protocol by coherently transporting the control atom (ancilla qubit) between four target atoms (data qubits)

3.3 The populations of the logical computational states for only one target atom as a function

of the number of cycles n

3.4 Bell-state fidelity and parity oscillation of the logical computational states for only one target atom as a function of the number of cycles n

3.5 Truth tables of the generation of GHZ-state after implementing CNOTn gates of each of target atoms

3.6 Truth tables of the generation of the other case of GHZ-state after implementing CNOTn gates of each of target atoms

3.7 The population transfer from |1)|AAAA) to |1)|BBBB)

3.8 The evolution of fidelity, and Renyi entropy as a function of average speed of transporting

the ancilla qubit over a distance d

3.9 The calculated mutual information 1AB as a function of the subset A

4.1 Arbitrary undirected planar graphs being represented by Rydberg atoms

4.2 Scheme of the atomic energy levels and laser excitation scheme for 87Rb atoms, the time-profiles of the detuning of Rydberg state and Rabi frequency, the strength of atoms interaction, and the probability amplitudes

4.3 A scheme of quantum phases of matter and the probability distributions of commensurate

Z2, Z3- ordered states and incommensurate states

4.4 The probability of obtaining the ordered state Z2 and the susceptibility for different values

of the sweeping rates

4.5 Phase diagram of P4 graph

4.6 Probability distributions indicating the MIS solutions of the considered arbitrary graphs

4.7 Probability distributions indicating the MWIS solutions of the considered arbitrary graphs

4.8 The scheme of Johnson solid J14 graph in 2D representation and the probabilities of MWIS for J14

4.9 The graph scheme of 3-Pan graph with vertices 1 and 2 are connected using two different spatial configurations of quantum wires

4.10 The pair-state energy (GHz) for the heteronuclear and homonuclear interactions, the probabilities indicating the MWIS of 5-Pan graph (3-Pan+W1), and the phase diagram of mediating the vertices v1, and v2 by the quantum wire

4.11 The probability P of the incommensurate state |rggr)G|gggr)w2

5.1 (a) The spatial configuration of Steane code. (b) The quantum circuit for implementing

the QEC on 7-qubit Steane code

A.1 Charge-charge interaction Vqiqj

A.2 Schemes for (a) charge-dipole interaction, and (b) dipole-charge interaction

A.3 A scheme for dipole-dipole interaction

C.1 Hyperfine structure of the collective atomic energy levels with control 133Cs atom and

target 87Rb atom

C.2 The contour plots of fidelity for implementing CNOTn gate using multi-Rydberg model

D.1 A scheme of energy levels of 87Rb target atom, the dependence of the fidelity of the transfer gate on two-photon Raman detuning

List of Tables

4.1 Sites of atoms demonstrating the considered graphs as a function of graph spacing constant

A

C.1 The calculated dipole-dipole coefficient C3a) (GHz-^m3), the energy defect 6Ea (GHz), the

van derWaals C<f) (GHz •^mb ) interaction coefficients

Введение диссертации (часть автореферата) на тему «Квантовые вычисления в гетероядерных массивах ультрахолодных атомов»

Synopsis

Since the beginning of the industrial revolution in 18th century people are continuously inspired to build powerful machines required for further development of communities. The burst of the technological revolution in the 20th century started with inventing the transistor, integrated circuits and microchips, which allowed rapid creation of the computing machines. Such computing machines changed from the mechanical prototypes of the beginning of 20th century and even earlier time to the room-size computers developed by IBM in 1964, and now to highly-efficient supercomputers. The commonly used computers, which are based on microchips, can be denoted as classical computers. For data processing, they use the binary system consisting of zeros and ones. The computing devices are able to store, access and manipulate all types of information using a central processing unit and memory. In classical computing the circuits/transistors can be switched between two states (on and off), and there is no third option. Regardless the increased size and capabilities of modern computers, there is a number of problems which are challenging for them. The examples of challenging optimization problems are simulation of the interaction of molecules, route planning [1,2], or Netflix's recommendation [3].

As optimization problems get harder, even powerful computers require months and maybe years to solve them. These challenges resulted in the necessity to develop a new concept of quantum computers based on features of quantum physics which allow overcoming these problems. Quantum computer consists of quantum bits (qubits) which are a two-state (or two-level) quantum-mechanical systems. Qubits are among the simplest quantum systems displaying the spookiness of quantum mechanics. Qubits can be initialized in two logical quantum states or in their quantum superposition. The ability to prepare a qubit in the superposition state allows performing large-scale parallel computations. However, this requires building large-scale quantum registers consisting of large numbers of qubits. Similar to a classical register, processing of information in the quantum register requires designing of sequences of logical gates which can be classical or quantum gates. With the development of large-scale universal quantum computer, the ability for eavesdropping modern secured communications will be inevitable since a commonly used 2,048-bit digital key (has 617 decimal digits), which is protected by the well-known

Rivest-Shamir-Adleman (RSA) encryption algorithm, can be decrypted by a quantum computer composed of 20 million qubits (physical qubits) [4-6] in almost 8 hours compared with estimated 300 trillion years for a classical computer, which is far beyond the age of the universe. Even for a non uttermost problem, quantum algorithms promise less number of steps to complete a task. As an example, Grover's search algorithm [7], requires only y/N iterations instead of N/2 iterations of classical algorithms.

For the past few decades, governments and mega-size corporations invested billions of USD$ to fund academic research activities of quantum technologies, develop new computer programming languages. In industry we clearly observe the race in building a scalable quantum computer. Figure 1 is an info-graphic map of investment of each country to the field of quantum initiatives in 2023 [8].

Figure 1: An info-graphic map for the funding on quantum research activity by each country. Coprights are reserved for QURECA.

The results of this thesis are obtained as a part of research project titled "Multi-qubit entangled states in scalable heteronuclear arrays of single atoms" funded by the Russian Science Foundation [9]. This project aims for developing of methods for precision control of the quantum states of ultracold neutral atoms in order to obtain multi-qubit entangled states with high fidelity by using the properties of het-eronuclear interactions of Rydberg atoms.

Quantum information processing

The ultimate goal of research in quantum technologies is obtaining the ability to prepare a set of quantum objects (atoms, photons, ions, etc.) in a well-defined quantum state and using different methods exploiting properties of quantum systems to surpass the limitations of classical devices. This can be achieved by mimicking a complex quantum system by a simpler physical system which Hamiltonian is as close as possible to the Hamiltonian of the quantum system under study. This approach is known as analog quantum computing. The model quantum systems, known as quantum simulators [10], can mimic the behavior and properties of molecules or materials if the parameters of model quantum system are chosen properly. The alternative is digital quantum computing, where a quantum state of the target system is encoded into a quantum state of a quantum register, and a set of programmable quantum gates is applied to to simulate the time evolution of the system under study.

In 2000, David P. DiVincenzo [11] outlined five (plus two) requirements for the physical implementation of quantum computer, which are:

1. Scalability of the physical system,

2. The ability of the physical system to be initialized in a desired state,

3. Coherence time should be much longer than the gate operation time,

4. The ability to perform a universal quantum gate,

5. The capability of measuring the state of each qubit individually,

+ The ability to interconvert stationary and flying qubits,

+ The ability to transmit flying qubits between specific locations.

The last extra two requirements are basic requirements for quantum communications. A flying qubit is a stationary qubit able to reliably store quantum information. It is used to propagate the quantum information at macroscopic distances (> 50 Km). Stationary qubits are used to store quantum information and perform information processing.

All existing approaches to build a quantum computer satisfy the Di-Vincenzo criteria in different manner and with different limitations. Examples of these approaches are superconducting qubits [12,13],

trapped ions [14,15], single photons [16,17], neutral atoms [18,19], etc. Superconducting qubits

Superconductors are among the most popular approaches for building qubits. Superconducting qubits are synthesized of superconducting materials. They are capable of performing gates quickly, but are very susceptible to quantum noise and short coherence time. Corporations like Google, IBM, etc. are focused in application of this approach to quantum computers. Sycamore, Google's superconducting processor with 53 qubits, has achieved quantum supremacy over classical computers in 2019 [20].

Neutral atom qubits

We are interested in the use of neutral atoms for quantum computing. This approach is based on trapping atoms inside the vacuum cell in arrays of focused laser beams which are used as optical tweezers. This approach results in longer coherence times (but slower gate operation time) and potenital for the scalability from thousands to tens of millions of interacting qubits [21]. The quantum information is stored in the long-lived hyperfine sublevels of the ground state of single atoms or superatoms, and processed using laser excitation to Rydberg states [22,23]. Mediating the interaction of atoms (alkali [24] or alkalineEarth [25] atoms) is the main technique for implementing entangling gates. Microscopic arrays of dipole traps in optical tweezers are used to trap the single atoms which are loaded stochastically and arranged randomly. Using traps generated by moving acousto-optic deflector (AOD) or spatial light modulator (SLM) allowed developing heuristic algorithms for trapping and sorting atoms to obtain the desired spatial configurations [26-28]. In chapter 1, the Rydberg atoms and the properties of their interactions will be discussed in details.

Large number of startup companies are developing the hardware and the technologies for realizing quantum computer based on ultracold neutral atoms. Among them we can list Pasqal, Infleqtion (previously was known as ColdQuanta), and QuEra. In 2023, QuEra reported the realization of a programmable quantum processor based on logical qubits operating with up to 280 physical qubits of Rb atoms [29]. QuEra adopted a three-year roadmap for a model with 100 logical qubits and over 10,000 physical qubits of a third-generation quantum error-corrected system [30]. Infleqtion revealed a qubit-array of 1600 qubits and unveiled a five-year quantum computing roadmap for creating 100 error-corrected logical qubits tailored for commercial applications capable of executing circuits of depth above 1 million [31].

Pasqal issued a roadmap for delivering systems with 10,000 physical qubits in 2026 and more than 128 logical qubits with fault-tolerant operations in 2028 [32]. Achievements and roadmaps for different corporations and startups can be found in the reference [33].

Recently, a research team in CalTech managed to trap 6100 highly coherent neutral atomic qubits in around 12,000 sites by optical tweezers with trapping lifetimes close to 23 minutes in a room-temperature apparatus [34].

Thesis' goal, tasks and provisions

The aim of this thesis is to develop and optimize methods of quantum computing in large-scale arrays of ultracold neutral atoms using heteronuclear Rydberg interactions. This includes the search for new promising methods for implementing quantum algorithms in ordered ensembles of ultracold neutral atoms interacting with laser radiation, including methods for improving the accuracy of multi-qubit gates in heteronuclear dual-species arrays, methods for parallel execution of two-qubit gates using coherent transport.

Tasks

To fulfill this goal, it is necessary to solve the following tasks:

1. Development and improvement of the schemes of high-fidelity multi-qubit CkNOTN gates using electromagnetically-induced transparency and heteronuclear Rydberg interactions.

2. Design of the scheme of parallel implementation of CNOTN quantum gates in large-scale atomic arrays using coherent transport and heteronuclear Rydberg blockade.

3. Design of the algorithm for solving maximum-weight independent set problem for unit-disk graphs represented by the spatially arranged atomic array and investigation of quantum phases of matter represented by the quantum state of the atomic array.

Provisions

The provisions presented at the defence are the following:

1. The schemes of multiqubit CkNOTN gates for atomic qubits based on heteronuclear Rydberg interactions and electromagnetically induced transparency allow achievement of fidelities above 99% both for C2NOT2 and CNOT4 gates which are important for quantum error correction.

2. Parallel implementation of sequence of CNOT gates in large-scale heteronuclear atomic arrays using coherent transport results in CNOT4 fidelities reaching 94.95% which is potentially useful for quantum error correction and non-destructive measurement in large-scale atomic arrays.

3. The NP-hard problem of finding maximum-weight independent set of planar and non-planar unit-disk graphs is efficiently solved using Rydberg excitation of the spatially arranged arrays of Rydberg atoms using specially designed quasiadiabatic time-dependent profile of the detuning from the resonance at Rydberg excitation and facilitation of interaction by a heteronuclear quantum wire.

Novelty and practical significance of the results

The scientific novelty of the obtained results results from the development of new methods for implementation of high-fidelity multi-qubit quantum gates for quantum computing with large-scale atomic quantum processors and design of the algorithms for solving NP-hard optimization problems using arrays of interacting Rydberg atoms. The schemes designed in this thesis are peculiar for the use of heteronu-clear dual-species architecture of atomic ensembles, which outstandingly expands the possibilities of controlling the energy of interaction of ultracold neutral atoms. The proposed schemes are confirmed by numerical simulation of the interaction of atoms with laser radiation and with each other for various spatial configurations. It is proposed to use quasi-adiabatic laser excitation of an ordered ensemble of atoms into Rydberg states to solve the problems of MIS and MWIS. The analysis of commensurate and incommensurable phases in ordered atomic ensembles, which are representations for planar and non-planar graphs, is carried out.

The practical significance of the results obtained is determined by their applicability to the development of quantum computation with ultracold neutral atoms, including the possibility of modeling the quantum dynamics of many-body quantum systems, increasing the fidelity of quantum gates, optimizing the implementation of quantum algorithms due to the parallel execution of individual quantum gates, the possibility of implementing the proposed quantum optimization algorithms on a quantum platform.

Our results on the implementation of CNOT gates using single-species homonuclear and dual-species heteronuclear architectures where obtained in parallel with the foremost experimental demonstration of using the electromagnetic induced transparency for implementing CNOT gate with homonuclear architecture [35,36] and later with the first experimental realization of dual-species arrays of atoms for implementing the Controlled-Z gate [37].

Also, our results on finding the solutions of MWIS was obtained in parallel with the experimental realization for weighted graphs using local light-shifts [38].

The main results of the work on the topic of the dissertation are published in 3 articles in peer-reviewed scientific journals, including the publication in Physical Review A which is a Q1 scientific journal.

Thesis Outline

This thesis is divided to the following parts:

Chapter Rydberg Atoms and Atom-Light Interaction is devoted to introducing the state-of-the-art of quantum computing with Rydberg atoms and includes a brief theoretical description of atom-light, atom-atom interactions, and basic principles of quantum computing including the action of single- and multi-qubit gates.

Chapter Parallel Implementation of CNOTN and C2NOT2 Gates via Homonuclear and Heteronuclear Forster Interactions of Rydberg Atoms details our results on parallel implementation of CNOT gates with homonuclear and heteronuclear architectures and generalization of gate schemes to multi-control and multi-target gates using Rydberg blockade and electromagnetically induced transparency.

Chapter Scalable Heteronuclear Architecture of Neutral Atoms Based on EIT describes the sequential implementation of CNOT gate which can be performed by coherently transporting the control atom among several target atoms in a parallel manner for a large atomic array. This expands the scalability during quantum information processing with neutral atoms.

Chapter Generation of quantum phases of matter and finding maximum weight independent set of unit-disk graphs using Rydberg atom is devoted for investigating the quantum phases of matter and finding the solutions of optimization problems using variational quantum adiabatic algorithm. The maximum independent sets and the maximum-weight independent sets of undirected unit-disk graphs are obtained by quasi-adiabatic sweeping of the Rydberg detuning during laser excitation of spatially ordered

arrays of atoms representing the graphs.

Список литературы диссертационного исследования кандидат наук Али Ахмед Мохамед Фарук Мохамед, 2025 год

Bibliography

[1] Peter E. Hart, Nils J. Nilsson, and Bertram Raphael. A Formal Basis for the Heuristic Determination of Minimum Cost Paths. IEEE Transactions on Systems Science and Cybernetics, 4(2):100-107, 1968, doi: 10.1109/TSSC.1968.300136. [Cited on page XV.]

[2] Wei Zeng and Richard L Church. Finding shortest paths on real road networks: the case for A*. International journal of geographical information science, 23(4):531-543, 2009, doi: 10.1080/13658810801949850. [Cited on page XV.]

[3] Robert M Bell, Yehuda Koren, and Chris Volinsky. The bellkor solution to the netflix prize. KorBell Team's Report to Netflix, 2007. [Cited on page XV.]

[4] Peter W. Shor. Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM Journal on Computing, 26(5):1484-1509, 1997, doi: 10.1137/S0097539795293172. [Cited on page XVI.]

[5] Peter W Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings 35th annual symposium on foundations of computer science, pages 124-134. Ieee, 1994. [Cited on page XVI.]

[6] Craig Gidney and Martin Ekera. How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits. Quantum, 5:433, 2021, doi: 10.22331/q-2021-04-15-433. [Cited on page XVI.]

[7] Lov K. Grover. Quantum Mechanics Helps in Searching for a Needle in a Haystack. Phys. Rev. Lett., 79:325-328, Jul 1997, doi: 10.1103/PhysRevLett.79.325. [Cited on page XVI.]

[8] QURECA on July 19, 2023, Overview of Quantum Initiatives Worldwide 2023. [Cited on page XVI.]

[9] Russian Science Foundation - Grant No. 23-42-00031, Multi-qubit entangled states in scalable heteronuclear arrays of single atoms. [Cited on page XVI.]

[10] Iulia Buluta and Franco Nori. Quantum simulators. Science, 326(5949):108-111, 2009, doi: 10.1126/science.1177838. [Cited on page XVII.]

[11] David P DiVincenzo. The physical implementation of quantum computation. Fortschritte der Physik: Progress of Physics, 48(9-11):771-783, 2000, doi: 10.48550/arXiv.quant-ph/0002077. [Cited on page XVII.]

[12] Morten Kjaergaard, Mollie E Schwartz, Jochen Braumüller, Philip Krantz, Joel I-J Wang, Simon Gustavsson, and William D Oliver. Superconducting qubits: Current state of play. Annual Review of Condensed Matter Physics, 11:369-395, 2020, doi: 10.1146/annurev-conmatphys-031119-050605. [Cited on page XVII.]

[13] Irfan Siddiqi. Engineering high-coherence superconducting qubits. Nature Reviews Materials, 6(10):875-891, 2021, doi: 10.1038/s41578-021-00370-4. [Cited on page XVII.]

[14] CJ Ballance, TP Harty, NM Linke, MA Sepiol, and DM Lucas. High-fidelity quantum logic gates using trapped-ion hyperfine qubits. Physical review letters, 117(6):060504, 2016, doi: 10.1103/PhysRevLett.117.060504. [Cited on page XVIII.]

[15] Stephen D Erickson, Jenny J Wu, Pan-Yu Hou, Daniel C Cole, Shawn Geller, Alex Kwiatkowski, Scott Glancy, Emanuel Knill, Daniel H Slichter, Andrew C Wilson, et al. High-fidelity indirect readout of trapped-ion hyperfine qubits. Physical Review Letters, 128(16):160503, 2022, doi: 10.1103/PhysRevLett.128.160503. [Cited on page XVIII.]

[16] Fulvio Flamini, Nicolo Spagnolo, and Fabio Sciarrino. Photonic quantum information processing: a review. Reports on Progress in Physics, 82(1):016001, 2018, doi: 10.1088/1361-6633/aad5b2. [Cited on page XVIII.]

[17] Jianwei Wang, Fabio Sciarrino, Anthony Laing, and Mark G Thompson. Integrated photonic quantum technologies. Nature Photonics, 14(5):273-284, 2020, doi: 10.1038/s41566-019-0532-1. [Cited on page XVIII.]

[18] H-J Briegel, Tommaso Calarco, Dieter Jaksch, Juan Ignacio Cirac, and Peter Zoller. Quantum computing with neutral atoms. Journal of modern optics, 47(2-3):415-451, 2000, doi: 10.1080/09500340008244052. [Cited on page XVIII.]

[19] M Morgado and S Whitlock. Quantum simulation and computing with Rydberg-interacting qubits. AVS Quantum Science, 3(2):023501, 2021, doi: 10.1116/5.0036562. [Cited on pages XVIII and 1.]

[20] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Ru-pak Biswas, Sergio Boixo, Fernando GSL Brandao, David A Buell, et al. Quantum supremacy using a programmable superconducting processor. Nature, 574(7779):505-510, 2019, doi: 10.1038/s41586-019-1666-5. [Cited on pages XVIII and 1.]

[21] Ehud Altman, Kenneth R Brown, Giuseppe Carleo, Lincoln D Carr, Eugene Demler, Cheng Chin, Brian DeMarco, Sophia E Economou, Mark A Eriksson, Kai-Mei C Fu, et al. Quantum simulators: Architectures and opportunities. PRX quantum, 2(1):017003, 2021, doi: 10.1103/PRXQuan-tum.2.017003. [Cited on page XVIII.]

[22] D Jaksch, JI Cirac, P Zoller, SL Rolston, R Côté, and MD Lukin. Fast quantum gates for neutral atoms. Physical Review Letters, 85(10):2208, 2000, doi: 10.1103/PhysRevLett.85.2208. [Cited on pages XVIII and 2.]

[23] Mikhail D Lukin, Michael Fleischhauer, Robin Cote, LuMing Duan, Dieter Jaksch, J Ignacio Cirac, and Peter Zoller. Dipole blockade and quantum information processing in meso-scopic atomic ensembles. Physical Review Letters, 87(3):037901, 2001, doi: 10.1103/Phys-RevLett.87.037901. [Cited on pages XVIII and 2.]

[24] Harry Levine, Alexander Keesling, Giulia Semeghini, Ahmed Omran, Tout T Wang, Sepehr Ebadi, Hannes Bernien, Markus Greiner, Vladan Vuletic, Hannes Pichler, et al. Parallel implementation of high-fidelity multiqubit gates with neutral atoms. Physical Review Letters, 123(17):170503, 2019, doi: 10.1103/PhysRevLett.123.170503. [Cited on pages XVIII, 1, 2, 25,30,31, and 38.]

[25] Ivaylo S Madjarov, Jacob P Covey, Adam L Shaw, Joonhee Choi, Anant Kale, Alexandre Cooper, Hannes Pichler, Vladimir Schkolnik, Jason R Williams, and Manuel Endres. High-fidelity entanglement and detection of alkaline-earth Rydberg atoms. Nature Physics, 16(8):857-861, 2020, doi: 10.1038/s41567-020-0903-z. [Cited on page XVIII.]

[26] Yevhen Miroshnychenko, Wolfgang Alt, Igor Dotsenko, Leonid Förster, Mkrtych Khudaverdyan, Dieter Meschede, Dominik Schrader, and Arno Rauschenbeutel. An atom-sorting machine. Nature, 442(7099):151-151, 2006, doi: 10.1038/442151a. [Cited on page XVIII.]

[27] Daniel Barredo, Sylvain De Leseleuc, Vincent Lienhard, Thierry Lahaye, and Antoine Browaeys. An atom-by-atom assembler of defect-free arbitrary two-dimensional atomic arrays. Science, 354(6315):1021-1023, 2016, doi: 10.1126/science.aah3778. [Cited on page XVIII.]

[28] Cheng Sheng, Jiayi Hou, Xiaodong He, Kunpeng Wang, Ruijun Guo, Jun Zhuang, Bahti-yar Mamat, Peng Xu, Min Liu, Jin Wang, et al. Defect-free arbitrary-geometry assembly of mixed-species atom arrays. Physical Review Letters, 128(8):083202, 2022, doi: 10.1103/Phys-RevLett.128.083202. [Cited on pages XVIII, 3, and 4.]

[29] Dolev Bluvstein, Simon J Evered, Alexandra A Geim, Sophie H Li, Hengyun Zhou, Tom Manovitz, Sepehr Ebadi, Madelyn Cain, Marcin Kalinowski, Dominik Hangleiter, et al. Logical quantum processor based on reconfigurable atom arrays. Nature, 626(7997):58-65, 2024, doi: 10.1038/s41586-023-06927-3. [Cited on page XVIII.]

[30] QuEra on January 9, 2024, QuEra Computing Releases a Groundbreaking Roadmap for Advanced Error-Corrected Quantum Computers, Pioneering the Next Frontier in Quantum Innovation. [Cited on page XVIII.]

[31] Infleqtion on February 8, 2024, Infleqtion Unveils 5-year Quantum Computing Roadmap, Advancing Plans to Commercialize Quantum at Scale. [Cited on page XVIII.]

[32] Pasqal on March 12, 2024, PASQAL Announces New Roadmap Focused on Business Utility and Scaling Beyond 1,000 Qubits Towards Fault Tolerance Era. [Cited on page XIX.]

[33] Ian Hellström, Quantum Computer Roadmaps. [Cited on page XIX.]

[34] Hannah J Manetsch, Gyohei Nomura, Elie Bataille, Kon H Leung, Xudong Lv, and Manuel Endres. A tweezer array with 6100 highly coherent atomic qubits. arXivpreprint arXiv:2403.12021, 2024, doi: 10.48550/arXiv.2403.12021. [Cited on page XIX.]

[35] Katie McDonnell. Demonstration of a CNOT gate using electromagnetically induced transparency. PhD thesis, University of Strathclyde, 2021. [Cited on pages XXI, 8, and 26.]

[36] K. McDonnell, L. F. Keary, and J. D. Pritchard. Demonstration of a Quantum Gate Using Elec-tromagnetically Induced Transparency. Physcial Review Letters, 129:200501, Nov 2022, doi: 10.1103/PhysRevLett.129.200501. [Cited on pages XXI, 26, 27,30,31, 48, and 98.]

[37] Shraddha Anand, Conor E Bradley, Ryan White, Vikram Ramesh, Kevin Singh, and Hannes Bernien. A dual-species Rydberg array. Nature Physics, pages 1-7, 2024. [Cited on pages XXI and 78.]

[38] AG de Oliveira, E Diamond-Hitchcock, DM Walker, MT Wells-Pestell, G Pelegri, CJ Picken, GPA Malcolm, AJ Daley, J Bass, and JD Pritchard. Demonstration of weighted graph optimization on a Rydberg atom array using local light-shifts. arXiv preprint arXiv:2404.02658, 2024, doi: 10.48550/arXiv.2404.02658. [Cited on pages XXI and 61.]

[39] Yulin Wu, Wan-Su Bao, Sirui Cao, Fusheng Chen, Ming-Cheng Chen, Xiawei Chen, Tung-Hsun Chung, Hui Deng, Yajie Du, Daojin Fan, et al. Strong quantum computational advantage using a superconducting quantum processor. Physical Review Letters, 127(18):180501, 2021, doi: 10.1103/PhysRevLett.127.180501. [Cited on page 1.]

[40] Han-Sen Zhong, Yu-Hao Deng, Jian Qin, Hui Wang, Ming-Cheng Chen, Li-Chao Peng, Yi-Han Luo, Dian Wu, Si-Qiu Gong, Hao Su, et al. Phase-programmable Gaussian boson sampling using stimulated squeezed light. Physical Review Letters, 127(18):180502, 2021, doi: 10.1103/Phys-RevLett.127.180502. [Cited on page 1.]

[41] TM Graham, Y Song, J Scott, C Poole, L Phuttitarn, K Jooya, P Eichler, X Jiang, A Marra, B Grinkemeyer, et al. Multi-qubit entanglement and algorithms on a neutral-atom quantum computer. Nature, 604:457, 2022, doi: 10.1038/s41586-022-04603-6. [Cited on pages 1 and 38.]

[42] Yong Zeng, Peng Xu, Xiaodong He, Yangyang Liu, Min Liu, Jin Wang, DJ Papoular, GV Shlyapnikov, and Mingsheng Zhan. Entangling two individual atoms of different isotopes via Rydberg blockade. Physical Review Letters, 119(16):160502, 2017, doi: 10.1103/Phys-RevLett.119.160502. [Cited on pages 1 and 3.]

[43] TM Graham, M Kwon, B Grinkemeyer, Z Marra, X Jiang, MT Lichtman, Y Sun, M Ebert, and M Saffman. Rydberg-mediated entanglement in a two-dimensional neutral atom qubit array. Physical Review Letters, 123(23):230501, 2019, doi: 10.1103/PhysRevLett.123.230501. [Cited on pages 1 and 30.]

[44] Simon J Evered, Dolev Bluvstein, Marcin Kalinowski, Sepehr Ebadi, Tom Manovitz, Hengyun Zhou, Sophie H Li, Alexandra A Geim, Tout T Wang, Nishad Maskara, et al. High-fidelity parallel entangling gates on a neutral-atom quantum computer. Nature, 622(7982):268-272, 2023, doi: 10.1038/s41586-023-06481-y. [Cited on page 1.]

[45] Pascal Scholl, Michael Schuler, Hannah J Williams, Alexander A Eberharter, Daniel Barredo, Kai-Niklas Schymik, Vincent Lienhard, Louis-Paul Henry, Thomas C Lang, Thierry Lahaye, et al. Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms. Nature, 595(7866):233-238, 2021, doi: 10.1038/s41586-021-03585-1. [Cited on pages 1 and 63.]

[46] Sepehr Ebadi, Tout T Wang, Harry Levine, Alexander Keesling, Giulia Semeghini, Ahmed Om-ran, Dolev Bluvstein, Rhine Samajdar, Hannes Pichler, Wen Wei Ho, et al. Quantum phases of matter on a 256-atom programmable quantum simulator. Nature, 595(7866):227-232, 2021, doi: 10.1038/s41586-021-03582-4. [Cited on pages 1,4, 63, 69, and 85.]

[47] TF Gallagher. Rydberg atoms. Reports on Progress in Physics, 51(2):143, 1988, doi: 10.1088/0034-4885/51/2/001. [Cited on pages 1 and 5.]

[48] Kilian Singer, Jovica Stanojevic, Matthias Weidemüller, and Robin Cote. Long-range interactions between alkali Rydberg atom pairs correlated to the ns-ns, np-np and nd-nd asymptotes. Journal of Physics B: Atomic, Molecular and Optical Physics, 38(2):S295, 2005, doi: 10.1088/09534075/38/2/021. [Cited on page 1.]

[49] E Urban, Todd A Johnson, T Henage, L Isenhower, DD Yavuz, TG Walker, and M Saffman. Observation of Rydberg blockade between two atoms. Nature Physics, 5(2):110-114, 2009, doi: 10.1038/nphys1178. [Cited on page 2.]

[50] Alpha Gaetan, Yevhen Miroshnychenko, Tatjana Wilk, Amodsen Chotia, Matthieu Viteau, Daniel Comparat, Pierre Pillet, Antoine Browaeys, and Philippe Grangier. Observation of collective excitation of two individual atoms in the Rydberg blockade regime. Nature Physics, 5(2):115-118, 2009, doi: 10.1038/nphys1183. [Cited on page 2.]

[51] L Isenhower, E Urban, XL Zhang, AT Gill, T Henage, Todd A Johnson, TG Walker, and M Saffman. Demonstration of a neutral atom controlled-NOT quantum gate. Physical Review Letters, 104(1):010503, 2010, doi: 10.1103/PhysRevLett.104.010503. [Cited on page 2.]

[52] Shi-Lei Su and Weibin Li. Dipole-dipole interaction driven antiblockade of two Rydberg atoms. Physical Review A, 104(3):033716, 2021, doi: 10.1103/PhysRevA.104.033716. [Cited on page 2.]

[53] Jin-Lei Wu, Yan Wang, Jin-Xuan Han, Yu-Kun Feng, Shi-Lei Su, Yan Xia, Yongyuan Jiang, and Jie Song. One-step implementation of Rydberg-antiblockade SWAP and controlled-SWAP gates with modified robustness. Photonics Research, 9(5):814-821, 2021, doi: 10.1364/PRJ.415795. [Cited on page 2.]

[54] Jin-Lei Wu, Yan Wang, Jin-Xuan Han, Shi-Lei Su, Yan Xia, Yongyuan Jiang, and Jie Song. Un-selective ground-state blockade of Rydberg atoms for implementing quantum gates. Frontiers of Physics, 17(2):1-13, 2022, doi: 10.1007/s11467-021-1104-7. [Cited on page 2.]

[55] Mohammadsadegh Khazali and Klaus M0lmer. Fast multiqubit gates by adiabatic evolution in interacting excited-state manifolds of Rydberg atoms and superconducting circuits. Physical Review X, 10(2):021054, 2020, doi: 10.1103/PhysRevX.10.021054. [Cited on pages 2 and 94.]

[56] Jeremy T Young, Przemyslaw Bienias, Ron Belyansky, Adam M Kaufman, and Alexey V Gor-shkov. Asymmetric blockade and multiqubit gates via dipole-dipole interactions. Physical Review Letters, 127(12):120501, 2021, doi: 10.1103/PhysRevLett.127.120501. [Cited on pages 2 and 41.]

[57] Meng Li, Jing-Yang Li, Fu-Qiang Guo, Xiao-Yu Zhu, Erjun Liang, Shou Zhang, Lei-Lei Yan, Mang Feng, and Shi-Lei Su. Multiple-Qubit CkUm Logic Gates of Rydberg Atoms via Optimized Geometric Quantum Operations. Annalen der Physik, page 2100506, 2022, doi: 10.1002/andp.202100506. [Cited on pages 2 and 41.]

[58] Li-Na Sun, L-L Yan, Shi-Lei Su, and Y Jia. One-Step Implementation of Time-Optimal-Control Three-Qubit Nonadiabatic Holonomic Controlled Gates in Rydberg Atoms. Physical Review Applied, 16(6):064040, 2021, doi: 10.1103/PhysRevApplied.16.064040. [Cited on page 2.]

[59] Zhuo Fu, Peng Xu, Yuan Sun, Yang-Yang Liu, Xiao-Dong He, Xiao Li, Min Liu, Run-Bing Li, Jin Wang, Liang Liu, et al. High-fidelity entanglement of neutral atoms via a Rydberg-mediated single-modulated-pulse controlled-phase gate. Physical Review A, 105(4):042430, 2022, doi: 10.1103/PhysRevA.105.042430. [Cited on page 2.]

[60] James M Auger, Silvia Bergamini, and Dan E Browne. Blueprint for fault-tolerant quantum computation with Rydberg atoms. Physical Review A, 96(5):052320, doi: 10.1103/Phys-RevA.96.052320. [Cited on page 2.]

[61] Dolev Bluvstein, Harry Levine, Giulia Semeghini, Tout T Wang, Sepehr Ebadi, Marcin Kali-nowski, Alexander Keesling, Nishad Maskara, Hannes Pichler, Markus Greiner, et al. A quantum processor based on coherent transport of entangled atom arrays. Nature, 604(7906):451-456, 2022, doi: 10.1038/s41586-022-04592-6. [Cited on page 2.]

[62] M Müller, I Lesanovsky, H Weimer, HP Büchler, and P Zoller. Mesoscopic Rydberg gate based on electromagnetically induced transparency. Physical Review Letters, 102(17):170502, 2009, doi: 10.1103/PhysRevLett.102.170502. [Cited on pages 2, 25, 27, 28,32,33, 46, 48, and 52.]

[63] Michael Fleischhauer, Atac Imamoglu, and Jonathan P Marangos. Electromagnetically induced transparency: Optics in coherent media. Reviews of modern physics, 77(2):633, 2005, doi: 10.1103/RevModPhys.77.633. [Cited on pages 2 and 32.]

[64] Zhonghua Ji, Yuechun Jiao, Yongmei Xue, Liping Hao, Jianming Zhao, and Suotang Jia. Distinction of electromagnetically induced transparency and Autler-Towners splitting in a Rydberg-involved ladder-type cold atom system. Optics Express, 29(8):11406-11415, 2021, doi: 10.1364/OE.417529. [Cited on page 2.]

[65] Jonathan D Pritchard, D Maxwell, Alexandre Gauguet, Kevin J Weatherill, MPA Jones, and Charles S Adams. Cooperative atom-light interaction in a blockaded Rydberg ensemble. Physical Review Letters, 105(19):193603, 2010, doi: 10.1103/PhysRevLett.105.193603. [Cited on page 2.]

[66] Hendrik Weimer, Markus Müller, Igor Lesanovsky, Peter Zoller, and Hans Peter Büchler. A Rydberg quantum simulator. Nature Physics, 6(5):382-388, 2010, doi: 10.1038/nphys1614. [Cited on page 2.]

[67] Thibault Peyronel, Ofer Firstenberg, Qi-Yu Liang, Sebastian Hofferberth, Alexey V Gorshkov, Thomas Pohl, Mikhail D Lukin, and Vladan Vuletic. Quantum nonlinear optics with single photons enabled by strongly interacting atoms. Nature, 488(7409):57-60, 2012, doi: 10.1038/nature11361. [Cited on page 2.]

[68] Simon Baur, Daniel Tiarks, Gerhard Rempe, and Stephan Dürr. Single-photon switch based on Rydberg blockade. Physical Review Letters, 112(7):073901, 2014, doi: 10.1103/Phys-RevLett.112.073901. [Cited on page 2.]

[69] Hannes Gorniaczyk, Christoph Tresp, Przemek Bienias, A Paris-Mandoki, Weibin Li, Ivan Mir-gorodskiy, HP Büchler, Igor Lesanovsky, and S Hofferberth. Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Forster resonances. Nature communications, 7(1):1-6, 2016, doi: 10.1038/ncomms12480. [Cited on page 2.]

[70] I.I. Beterov and M Saffman. Rydberg blockade, Forster resonances, and quantum state measurements with different atomic species. Physical Review A, 92(4):042710, 2015, doi: 10.1103/Phys-RevA.92.042710. [Cited on pages 3, 4,33, and 41.]

[71] Kevin Singh, Shraddha Anand, Andrew Pocklington, Jordan T Kemp, and Hannes Bernien. DualElement, Two-Dimensional Atom Array with Continuous-Mode Operation. Physical Review X, 12(1):011040, 2022, doi: 10.1103/PhysRevX.12.011040. [Cited on pages 3,4,41,48, and 78.]

[72] Tetsu Takekoshi, Lukas Reichsollner, Andreas Schindewolf, Jeremy M Hutson, C Ruth Le Sueur, Olivier Dulieu, Francesca Ferlaino, Rudolf Grimm, and Hanns-Christoph Nagerl. Ultracold dense samples of dipolar RbCs molecules in the rovibrational and hyperfine ground state. Physical Review Letters, 113(20):205301, 2014, doi: 10.1103/PhysRevLett.113.205301. [Cited on page 3.]

[73] Peter K. Molony, Philip D. Gregory, Zhonghua Ji, Bo Lu, Michael P. Koppinger, C. Ruth Le Sueur, Caroline L. Blackley, Jeremy M. Hutson, and Simon L. Cornish. Creation of Ultracold 87Rb133Cs Molecules in the Rovibrational Ground State. Phys. Rev. Lett., 113:255301, Dec 2014, doi: 10.1103/PhysRevLett.113.255301. [Cited on page 3.]

[74] Alexander Guttridge, Daniel K Ruttley, Archie C Baldock, Rosario Gonzalez-Ferez, HR Sadegh-pour, CS Adams, and Simon L Cornish. Observation of Rydberg blockade due to the charge-dipole interaction between an atom and a polar molecule. Physical Review Letters, 131(1):013401, 2023, doi: 10.1103/PhysRevLett.131.013401. [Cited on page 3.]

[75] Shih-Kuang Tung, Colin Parker, Jacob Johansen, Cheng Chin, Yujun Wang, and Paul S Julienne. Ultracold mixtures of atomic 6 Li and 133 Cs with tunable interactions. Physical Review A, 87(1):010702, 2013, doi: 10.1088/1367-2630/11/5/055022. [Cited on page 3.]

[76] W Hansel, J Reichel, P Hommelhoff, and TW Hansch. Magnetic conveyor belt for transporting and merging trapped atom clouds. Physical Review Letters, 86(4):608, 2001, doi: 10.1103/Phys-RevLett.86.608. [Cited on page 3.]

[77] Jerome Beugnon, Charles Tuchendler, Harold Marion, Alpha Gaetan, Yevhen Miroshnychenko, Yvan RP Sortais, Andrew M Lance, Matthew Jones, Gaetan Messin, Antoine Browaeys, et al. Two-dimensional transport and transfer of a single atomic qubit in optical tweezers. Nature Physics, 3(10):696-699, 2007, doi: 10.1038/nphys698. [Cited on page 3.]

[78] GT Hickman and M Saffman. Speed, retention loss, and motional heating of atoms in an optical conveyor belt. Physical Review A, 101(6):063411, 2020, doi: 10.1103/PhysRevA.101.063411. [Cited on page 3.]

[79] Dolev Bluvstein, Harry Levine, Giulia Semeghini, Tout T Wang, Sepehr Ebadi, Marcin Kali-nowski, Alexander Keesling, Nishad Maskara, Hannes Pichler, Markus Greiner, et al. A quantum processor based on coherent transport of entangled atom arrays. Nature, 604(7906):451-456, 2022, doi: 10.1038/s41586-022-04592-6. [Cited on pages 3,4, 48, and 57.]

[80] Chi Zhang and MR Tarbutt. Quantum computation in a hybrid array of molecules and Rydberg atoms. PRX Quantum, 3(3):030340, 2022, doi: 10.1103/PRXQuantum.3.030340. [Cited on page 4.]

[81] Thad G Walker and Mark Saffman. Consequences of Zeeman degeneracy for the van der Waals blockade between Rydberg atoms. Physical Review A, 77(3):032723, 2008, doi: 10.1103/Phys-RevA.77.032723. [Cited on pages 4 and 27.]

[82] Zhi-Jin Tao, Li-Geng Yu, Peng Xu, Jia-Yi Hou, Xiao-Dong He, and Ming-Sheng Zhan. Efficient two-dimensional defect-free dual-species atom arrays rearrangement algorithm with near-fewest atom moves. Chinese Physics Letters, 39(8):083701, 2022, doi: 10.1088/0256-307X/39/8/083701. [Cited on page 4.]

[83] Sepehr Ebadi, Alexander Keesling, Madelyn Cain, Tout T Wang, Harry Levine, Dolev Bluvstein, Giulia Semeghini, Ahmed Omran, J-G Liu, Rhine Samajdar, et al. Quantum optimization of maximum independent set using Rydberg atom arrays. Science, 376(6598):1209-1215, 2022, doi: 10.1126/science.abo6587. [Cited on pages 4, 61, and 63.]

[84] Minh-Thi Nguyen, Jin-Guo Liu, Jonathan Wurtz, Mikhail D Lukin, Sheng-Tao Wang, and Hannes Pichler. Quantum optimization with arbitrary connectivity using Rydberg atom arrays. arXiv preprint arXiv:2209.03965, 2022, doi: 10.48550/arXiv.2209.03965. [Cited on page 4.]

[85] Andrew Byun, Minhyuk Kim, and Jaewook Ahn. Finding the maximum independent sets of platonic graphs using Rydberg atoms. PRX Quantum, 3(3):030305, 2022. [Cited on pages 4 and 63.]

[86] Minhyuk Kim, Kangheun Kim, Jaeyong Hwang, Eun-Gook Moon, and Jaewook Ahn. Rydberg quantum wires for maximum independent set problems. Nature Physics, 18(7):755-759, 2022, doi: 10.1038/s41567-022-01629-5. [Cited on pages 4, 61, 63, 78, and 81.]

[87] Sebastian Weber, Christoph Tresp, Henri Menke, Alban Urvoy, Ofer Firstenberg, Hans Peter Büchler, and Sebastian Hofferberth. Tutorial: Calculation of Rydberg interaction potentials. J. Phys. B: At. Mol. Opt. Phys., 50(13):133001, 2017, doi: 10.1088/1361-6455/aa743a. [Cited on pages 5, 16, and 33.]

[88] WC Martin. Series formulas for the spectrum of atomic sodium (Na I). JOSA, 70(7):784-788, 1980, doi: 10.1364/JOSA.70.000784. [Cited on page 5.]

[89] Nikola Sibalic, Jonathan D Pritchard, Charles S Adams, and Kevin J Weatherill. ARC: An open-source library for calculating properties of alkali Rydberg atoms. Computer Physics Communications, 220:319-331, 2017, doi: 10.1016/j.cpc.2017.06.015. [Cited on pages 5,16, 33, 37, 53, and 68.]

[90] Wenhui Li, I. Mourachko, M. W. Noel, and T. F. Gallagher. Millimeter-wave spectroscopy of cold Rb Rydberg atoms in a magneto-optical trap: Quantum defects of the ns, np, and nd series. Phys. Rev. A, 67:052502, May 2003, doi: 10.1103/PhysRevA.67.052502. [Cited on page 5.]

[91] Jianing Han, Yasir Jamil, D. V. L. Norum, Paul J. Tanner, and T. F. Gallagher. Rb nf quantum defects from millimeter-wave spectroscopy of cold 85Rb Rydberg atoms. Phys. Rev. A, 74:054502, Nov 2006, doi: 10.1103/PhysRevA.74.054502. [Cited on page 5.]

[92] K. Afrousheh, P. Bohlouli-Zanjani, J. A. Petrus, and J. D. D. Martin. Determination of the 85Rb ng-series quantum defect by electric-field-induced resonant energy transfer between cold Rydberg atoms. Phys. Rev. A, 74:062712, Dec 2006, doi: 10.1103/PhysRevA.74.062712. [Cited on page 5.]

[93] P. Goy, J. M. Raimond, G. Vitrant, and S. Haroche. Millimeter-wave spectroscopy in cesium Ryd-berg states. Quantum defects, fine- and hyperfine-structure measurements. Phys. Rev. A, 26:27332742, Nov 1982, doi: 10.1103/PhysRevA.26.2733. [Cited on page 5.]

[94] K.-H. Weber and Craig J. Sansonetti. Accurate energies of nS, nP, nD, nF, and nG levels of neutral cesium. Phys. Rev. A, 35:4650-4660, Jun 1987, doi: 10.1103/PhysRevA.35.4650. [Cited on page 5.]

[95] Johannes Deiglmayr, Holger Herburger, Heiner Saßmannshausen, Paul Jansen, Hansjürg Schmutz, and Frederic Merkt. Precision measurement of the ionization energy of Cs I. Phys. Rev. A, 93:013424, Jan 2016, doi: 10.1103/PhysRevA.93.013424. [Cited on page 5.]

[96] G. S. Agarwal. Rotating-Wave Approximation and Spontaneous Emission. Phys. Rev. A, 4:17781781, Nov 1971, doi: 10.1103/PhysRevA.4.1778. [Cited on page 7.]

[97] Leslie Allen and Joseph H Eberly. Optical resonance and two-level atoms, volume 28. Courier Corporation, 1987. [Cited on page 7.]

[98] S. H. Autler and C. H. Townes. Stark Effect in Rapidly Varying Fields. Phys. Rev., 100:703-722, Oct 1955, doi: 10.1103/PhysRev.100.703. [Cited on pages 8 and 19.]

[99] Ahmed M. Farouk. Some information studies of some matter-field interactions. Master's thesis, Al-Azhar University, 2017. [Cited on page 10.]

[100] Abdel-Shafy F Obada, Mohamed MA Ahmed, Ahmed M Farouk, and Ahmed Salah. A moving three-level A-type atom in a dissipative cavity. The European Physical Journal D, 71:1-13, 2017, doi: 10.1140/epjd/e2017-80357-5. [Cited on page 10.]

[101] DF James and Jonathan Jerke. Effective Hamiltonian theory and its applications in quantum information. Canadian Journal of Physics, 85(6):625-632, 2007, doi: 10.1139/p07-060. [Cited on page 12.]

[102] Rui Han, Hui Khoon Ng, and Berthold-Georg Englert. Raman transitions without adiabatic elimination: A simple and accurate treatment. Journal of Modern Optics, 60(4):255-265, 2013, doi: 10.1080/09500340.2013.770573. [Cited on page 12.]

[103] Robert J Le Roy. Long-Range Potential Coefficients From RKR Turning Points: C6 and C8 for B (3n+M)-State Cl2, Br2, and I2. Canadian Journal of Physics, 52(3):246-256, 1974, doi: 10.1139/p74-035. [Cited on pages 13 and 34.]

[104] Stephen E. Harris. Electromagnetically Induced Transparency. Physics Today, 50(7):36-42, 07 1997, doi: 10.1063/1.881806. [Cited on page 19.]

[105] K.-J. Boller, A. Imamoglu, and S. E. Harris. Observation of electromagnetically induced transparency. Phys. Rev. Lett., 66:2593-2596, May 1991, doi: 10.1103/PhysRevLett.66.2593. [Cited on page 19.]

[106] Ennio Arimondo. V coherent population trapping in laser spectroscopy. In Progress in optics, volume 35, pages 257-354. Elsevier, 1996. [Cited on page 19.]

[107] Kai Eckert, Maciej Lewenstein, R Corbalan, Gerhard Birkl, W Ertmer, and Jordi Mompart. Three-level atom optics via the tunneling interaction. Physical Review A, 70(2):023606, 2004. [Cited on page 19.]

[108] Sumanta Khan, Molahalli Panidhara Kumar, Vineet Bharti, and Vasant Natarajan. Coherent population trapping (CPT) versus electromagnetically induced transparency (EIT). The European Physical Journal D, 71:1-9, 2017. [Cited on page 19.]

[109] Moustafa Abdel Hafiz, Grégoire Coget, Peter Yun, Stephane Guerandel, Emeric de Clercq, and Rodolphe Boudot. A high-performance Raman-Ramsey Cs vapor cell atomic clock. Journal of Applied Physics, 121(10), 2017. [Cited on page 19.]

[110] Michael A Nielsen and Isaac L Chuang. Quantum Computing and Quantum Information (Cambridge university press, Cambridge, England), 2000. [Cited on pages 20, 25, 37, and 56.]

[111] 1.1. Beterov, I. N. Ashkarin, E. A. Yakshina, D. B. Tretyakov, V. M. Entin, 1.1. Ryabtsev, P. Cheinet, P. Pillet, and M. Saffman. Fast three-qubit Toffoli quantum gate based on three-body Forster resonances in Rydberg atoms. Phys. Rev. A, 98:042704, Oct 2018, doi: 10.1103/PhysRevA.98.042704. [Cited on page 25.]

[112] I. N. Ashkarin, I. I. Beterov, E. A. Yakshina, D. B. Tretyakov, V. M. Entin, I. I. Ryabtsev, P. Cheinet, K.-L. Pham, S. Lepoutre, and P. Pillet. Toffoli gate based on a three-body fine-structure-state-changing Forster resonance in Rydberg atoms. Phys. Rev. A, 106:032601, Sep 2022, doi: 10.1103/PhysRevA.106.032601. [Cited on page 25.]

[113] I.I. Beterov, GN Hamzina, EA Yakshina, DB Tretyakov, VM Entin, and I.I. Ryabtsev. Adia-batic passage of radio-frequency-assisted forster resonances in Rydberg atoms for two-qubit gates and the generation of bell states. Physical Review A, 97(3):032701, 2018, doi: 10.1103/Phys-RevA.97.032701. [Cited on page 27.]

[114] Theodor Forster and O Sinanoglu. Modern Quantum Chemistry. Academic Press, New York, 3:93-137, 1965. [Cited on page 33.]

[115] M Saffman and TG Walker. Analysis of a quantum logic device based on dipole-dipole interactions of optically trapped Rydberg atoms. Physical Review A, 72(2):022347, 2005, doi: 10.1103/Phys-RevA.72.022347. [Cited on pages 33 and 98.]

[116] II Ryabtsev, DB Tretyakov, II Beterov, and VM Entin. Observation of the Stark-tuned Forster resonance between two Rydberg atoms. Physical Review Letters, 104(7):073003, 2010, doi: 10.1103/PhysRevLett.104.073003. [Cited on page 33.]

[117] DB Tretyakov, II Beterov, EA Yakshina, VM Entin, II Ryabtsev, P Cheinet, and P Pillet. Observation of the Borromean three-body Förster resonances for three interacting Rb Rydberg atoms. Physical Review Letters, 119(17):173402, 2017, doi: 10.1103/PhysRevLett.119.173402. [Cited on page 33.]

[118] Antoine Browaeys and Thierry Lahaye. Many-body physics with individually controlled Rydberg atoms. Nature Physics, 16(2):132-142, 2020, doi: 10.1038/s41567-019-0733-z. [Cited on pages 33, 67, and 79.]

[119] Dongmin Yu, Han Wang, Jin-Ming Liu, Shi-Lei Su, Jing Qian, and Weiping Zhang. Multiqubit Toffoli gates and optimal geometry with Rydberg atoms. Physical Review Applied, 18(3):034072, 2022, doi: 10.1103/PhysRevApplied.18.034072. [Cited on pages 34, 94, and 97.]

[120] Mark Hillery, Vladimir Buzek, and Andre Berthiaume. Quantum secret sharing. Physical Review A, 59(3):1829, 1999, doi: 10.1103/PhysRevA.59.1829. [Cited on pages 37 and 56.]

[121] Iris Cong, Harry Levine, Alexander Keesling, Dolev Bluvstein, Sheng-Tao Wang, and Mikhail D Lukin. Hardware-efficient, fault-tolerant quantum computation with Rydberg atoms. Physical Review X, 12(2):021049, 2022, doi: 10.1103/PhysRevX.12.021049. [Cited on page 41.]

[122] Rui Li, Jing Qian, and Weiping Zhang. Proposal for practical Rydberg quantum gates using a native two-photon excitation. Quantum Science and Technology, 8(035032):1-18, 2023, doi: 10.1088/2058-9565/ace0d5. [Cited on page 45.]

[123] CW Mansell and Silvia Bergamini. A cold-atoms based processor for deterministic quantum computation with one qubit in intractably large Hilbert spaces. New Journal of Physics, 16(5):053045, 2014, doi: 10.1088/1367-2630/16/5/053045. [Cited on page 51.]

[124] Ryszard Horodecki, Pawel Horodecki, Michal Horodecki, and Karol Horodecki. Quantum entanglement. Reviews of Modern Physics, 81(2):865, 2009, doi: 10.1103/RevModPhys.81.865. [Cited on page 58.]

[125] Rajibul Islam, Ruichao Ma, Philipp M Preiss, M Eric Tai, Alexander Lukin, Matthew Rispoli, and Markus Greiner. Measuring entanglement entropy in a quantum many-body system. Nature, 528(7580):77-83, 2015, doi: 10.1038/nature15750. [Cited on page 58.]

[126] Michael M Wolf, Frank Verstraete, Matthew B Hastings, and J Ignacio Cirac. Area laws in quantum systems: mutual information and correlations. Physical Review Letters, 100(7):070502, 2008, doi: 10.1103/PhysRevLett.100.070502. [Cited on page 58.]

[127] Sanjeev Arora and Boaz Barak. Computational complexity: a modern approach. Cambridge University Press, 2009. [Cited on pages 61 and 62.]

[128] N Wagner, C Poole, TM Graham, and M Saffman. Benchmarking a neutral-atom quantum computer. International Journal of Quantum Information, page 2450001, 2024, doi: 10.1142/S0219749924500011. [Cited on pages 61 and 63.]

[129] TM Graham, Y Song, J Scott, C Poole, L Phuttitarn, K Jooya, P Eichler, X Jiang, A Marra, B Grinkemeyer, et al. Multi-qubit entanglement and algorithms on a neutral-atom quantum computer. Nature, 604(7906):457-462, 2022, doi: 10.1038/s41586-022-04603-6. [Cited on pages 61 and 63.]

[130] Alexander W Glaetzle, Rick MW van Bijnen, Peter Zoller, and Wolfgang Lechner. A coherent quantum annealer with Rydberg atoms. Nature communications, 8(1):15813, 2017, doi: 10.1038/ncomms15813. [Cited on pages 61 and 78.]

[131] John Adrian Bondy, Uppaluri Siva Ramachandra Murty, et al. Graph theory with applications, volume 290. Macmillan London, 1976. [Cited on page 61.]

[132] Leo Zhou, Sheng-Tao Wang, Soonwon Choi, Hannes Pichler, and Mikhail D Lukin. Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices. Physical Review X, 10(2):021067, 2020, doi: 10.1103/PhysRevX.10.021067. [Cited on page 63.]

[133] Lucas T Brady and Stuart Hadfield. Iterative Quantum Algorithms for Maximum Independent Set: A Tale of Low-Depth Quantum Algorithms. arXiv preprint arXiv:2309.13110, 2023, doi: 10.48550/arXiv.2309.13110. [Cited on page 63.]

[134] Jacob Taylor, Sumit Goswami, Valentin Walther, Michael Spanner, Christoph Simon, and Khabat Heshami. Simulation of many-body dynamics using Rydberg excitons. Quantum Science and Technology, 7(3):035016, 2022, doi: 10.1088/2058-9565/ac70f4. [Cited on page 63.]

[135] Wesley da Silva Coelho, Mauro D'Arcangelo, and Louis-Paul Henry. Efficient protocol for solving combinatorial graph problems on neutral-atom quantum processors. arXiv preprint arXiv:2207.13030, 2022, doi: 10.48550/arXiv.2207.13030. [Cited on page 63.]

[136] Danylo Lykov, Jonathan Wurtz, Cody Poole, Mark Saffman, Tom Noel, and Yuri Alexeev. Sampling frequency thresholds for the quantum advantage of the quantum approximate optimization algorithm. npj Quantum Information, 9(1):73, 2023, doi: 10.1038/s41534-023-00718-4. [Cited on page 63.]

[137] Ruben S Andrist, Martin JA Schuetz, Pierre Minssen, Romina Yalovetzky, Shouvanik Chakrabarti, Dylan Herman, Niraj Kumar, Grant Salton, Ruslan Shaydulin, Yue Sun, et al. Hardness of the maximum-independent-set problem on unit-disk graphs and prospects for quantum speedups. Physical Review Research, 5(4):043277, 2023, doi: 10.1103/PhysRevRe-search.5.043277. [Cited on page 63.]

[138] Jonathan Z Lu, Lucy Jiao, Kristina Wolinski, Milan Kornjaca, Hong-Ye Hu, Sergio Cantu, Fangli Liu, Susanne F Yelin, and Sheng-Tao Wang. Digital-analog quantum learning on Rydberg atom arrays. arXiv preprint arXiv:2401.02940, 2024, doi: 10.48550/arXiv.2401.02940. [Cited on page 63.]

[139] Constantin Dalyac, Louis-Paul Henry, Minhyuk Kim, Jaewook Ahn, and Loïc Henriet. Exploring the impact of graph locality for the resolution of the maximum-independent-set problem with neutral atom devices. Physical Review A, 108(5):052423, 2023, doi: 10.1103/PhysRevA.108.052423. [Cited on page 63.]

[140] Kangheun Kim, Minhyuk Kim, Juyoung Park, Andrew Byun, and Jaewook Ahn. Quantum computing dataset of maximum independent set problem on king lattice of over hundred Ryd-berg atoms. Scientific Data, 11(1):111, 2024, doi: 10.1038/s41597-024-02926-9. [Cited on page 63.]

[141] Benjamin F. Schiffer, Dominik S. Wild, Nishad Maskara, Madelyn Cain, Mikhail D. Lukin, and Rhine Samajdar. Circumventing superexponential runtimes for hard instances of quantum adiabatic optimization. Phys. Rev. Res., 6:013271, Mar 2024, doi: 10.1103/PhysRevRe-search.6.013271. [Cited on page 63.]

[142] Hyeonjun Yeo, Ha Eum Kim, and Kabgyun Jeong. Approximating maximum independent set on Rydberg atom arrays using local detunings. Advanced Quantum Technologies, page 2400291, 2024, doi: 10.1002/qute.202400291. [Cited on page 63.]

[143] Kapil Goswami, Rick Mukherjee, Herwig Ott, and Peter Schmelcher. Solving optimization problems with local light-shift encoding on Rydberg quantum annealers. Phys. Rev. Res., 6:023031, Apr 2024, doi: 10.1103/PhysRevResearch.6.023031. [Cited on page 63.]

[144] Manuel H. Muñoz Arias, Stefanos Kourtis, and Alexandre Blais. Low-depth Clifford circuits approximately solve MaxCut. Phys. Rev Res., 6:023294, Jun 2024, doi: 10.1103/PhysRevRe-search.6.023294. [Cited on page 63.]

[145] G. V. Paradezhenko, A. A. Pervishko, and D. Yudin. Probabilistic tensor optimization of quantum circuits for the max-k-cut problem. Phys. Rev A, 109:012436, Jan 2024, doi: 10.1103/Phys-RevA.109.012436. [Cited on page 63.]

[146] Nora Bauer, Kübra Yeter-Aydeniz, Elias Kokkas, and George Siopsis. Solving Power Grid Optimization Problems with Rydberg Atoms. arXiv preprint arXiv:2404.11440, 2024, doi: 10.48550/arXiv.2404.11440. [Cited on page 63.]

[147] Andrew Byun, Junwoo Jung, Kangheun Kim, Minhyuk Kim, Seokho Jeong, Heejeong Jeong, and Jaewook Ahn. Rydberg-Atom Graphs for Quadratic Unconstrained Binary Optimization Problems. Advanced Quantum Technologies, 7(8):2300398, 2024, doi: 10.1002/qute.202300398. [Cited on pages 63 and 74.]

[148] Minh-Thi Nguyen, Jin-Guo Liu, Jonathan Wurtz, Mikhail D Lukin, Sheng-Tao Wang, and Hannes Pichler. Quantum optimization with arbitrary connectivity using Rydberg atom arrays. PRX Quantum, 4(1):010316, 2023, doi: 10.1103/PRXQuantum.4.010316. [Cited on page 64.]

[149] Martin Lanthaler, Clemens Dlaska, Kilian Ender, and Wolfgang Lechner. Rydberg-blockade-based parity quantum optimization. Physical Review Letters, 130(22):220601, 2023, doi: 10.1103/Phys-RevLett.130.220601. [Cited on page 64.]

[150] Alper Kose and Muriel Medard. Scheduling wireless ad hoc networks in polynomial time using claw-free conflict graphs. In 2017 IEEE 28th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC), pages 1-7. IEEE, 2017. [Cited on page 64.]

[151] Jernej Rudi Finzgar, Martin J. A. Schuetz, J. Kyle Brubaker, Hidetoshi Nishimori, and Helmut G. Katzgraber. Designing quantum annealing schedules using Bayesian optimization. Phys. Rev. Res., 6:023063, Apr 2024, doi: 10.1103/PhysRevResearch.6.023063. [Cited on page 64.]

[152] Natasa Przulj. Graph theory analysis of protein-protein interactions. Knowledge Discovery in Proteomics, 8:73-128, 2005. [Cited on page 64.]

[153] Itay Hen and Marcelo S. Sarandy. Driver hamiltonians for constrained optimization in quantum annealing. Phys. Rev. A, 93:062312, Jun 2016, doi: 10.1103/PhysRevA.93.062312. [Cited on page 66.]

[154] Ian D Kivlichan, Nathan Wiebe, Ryan Babbush, and Alan Aspuru-Guzik. Bounding the costs of quantum simulation of many-body physics in real space. Journal of Physics A: Mathematical and Theoretical, 50(30):305301, 2017, doi: 10.1088/1751-8121/aa77b8. [Cited on page 66.]

[155] I. I. Beterov, M. Saffman, E. A. Yakshina, D. B. Tretyakov, V. M. Entin, S. Bergamini, E. A. Kuznetsova, and I. I. Ryabtsev. Two-qubit gates using adiabatic passage of the Stark-tuned Forster resonances in Rydberg atoms. Phys. Rev. A, 94:062307, Dec 2016, doi: 10.1103/Phys-RevA.94.062307. [Cited on page 67.]

[156] Conor Mc Keever and Michael Lubasch. Towards Adiabatic Quantum Computing Using Compressed Quantum Circuits. PRX Quantum, 5:020362, Jun 2024, doi: 10.1103/PRXQuan-tum.5.020362. [Cited on page 67.]

[157] Robert J Leroy and Richard B Bernstein. Dissociation energies of diatomic moleculles from vibrational spacings of higher levels: application to the halogens. Chemical Physics Letters, 5(1):42-44, 1970, doi: 10.1016/0009-2614(70)80125-7. [Cited on page 68.]

[158] Ahmed M. Farouk, Ilya I Beterov, Peng Xu, Silvia Bergamini, and Igor I Ryabtsev. Parallel implementation of CNOTn and C2NOT2 gates via homonuclear and heteronuclear Forster interactions of Rydberg atoms. Photonics, 10(11):1280, 2023, doi: 10.3390/photonics10111280. [Cited on pages 68 and 78.]

[159] I. I. Beterov and M. Saffman. Rydberg blockade, Forster resonances, and quantum state measurements with different atomic species. Phys. Rev. A, 92:042710, Oct 2015, doi: 10.1103/Phys-RevA.92.042710. [Cited on pages 68 and 78.]

[160] Hannes Bernien, Sylvain Schwartz, Alexander Keesling, Harry Levine, Ahmed Omran, Hannes Pichler, Soonwon Choi, Alexander S Zibrov, Manuel Endres, Markus Greiner, et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature, 551(7682):579-584, 2017, doi: 10.1038/nature24622. [Cited on pages 69 and 71.]

[161] Alexander Keesling, Ahmed Omran, Harry Levine, Hannes Bernien, Hannes Pichler, Soon-won Choi, Rhine Samajdar, Sylvain Schwartz, Pietro Silvi, Subir Sachdev, et al. Quantum Kibble-Zurek mechanism and critical dynamics on a programmable Rydberg simulator. Nature, 568(7751):207-211, 2019, doi: 10.1038/s41586-019-1070-1. [Cited on pages 69 and 71.]

[162] Fang Fang, Kenneth Wang, Vincent S Liu, Yu Wang, Ryan Cimmino, Julia Wei, Marcus Bintz, Avery Parr, Jack Kemp, Kang-Kuen Ni, et al. Probing critical phenomena in open quantum systems using atom arrays. arXiv preprint arXiv:2402.15376, 2024, doi: 10.48550/arXiv.2402.15376. [Cited on pages 69 and 72.]

[163] Thomas WB Kibble. Topology of cosmic domains and strings. Journal of Physics A: Mathematical and General, 9(8):1387, 1976, doi: 10.1088/0305-4470/9/8/029. [Cited on page 69.]

[164] Wojciech H Zurek. Cosmological experiments in superfluid helium? Nature, 317(6037):505-508, 1985, doi: 10.1038/317505a0. [Cited on page 69.]

[165] M. Anquez, B. A. Robbins, H. M Bharath, M. Boguslawski, T. M. Hoang, and M. S. Chapman. Quantum Kibble-Zurek Mechanism in a Spin-1 Bose-Einstein Condensate. Phys. Rev. Lett., 116:155301, Apr 2016, doi: 10.1103/PhysRevLett.116.155301. [Cited on page 69.]

[166] Giulia Semeghini, Harry Levine, Alexander Keesling, Sepehr Ebadi, Tout T Wang, Dolev Blu-vstein, Ruben Verresen, Hannes Pichler, Marcin Kalinowski, Rhine Samajdar, et al. Probing topological spin liquids on a programmable quantum simulator. Science, 374(6572):1242-1247, 2021, doi: 10.1126/science.abi8794. [Cited on page 69.]

[167] Wojciech H. Zurek, Uwe Dorner, and Peter Zoller. Dynamics of a Quantum Phase Transition. Phys. Rev. Lett., 95:105701, Sep 2005, doi: 10.1103/PhysRevLett.95.105701. [Cited on page 69.]

[168] David A. Huse and Michael E. Fisher. Commensurate melting, domain walls, and dislocations. Phys. Rev. B, 29:239-270, Jan 1984, doi: 10.1103/PhysRevB.29.239. [Cited on page 69.]

[169] Michael Rader and Andreas M Lauchli. Floating phases in one-dimensional Rydberg Ising chains. arXiv preprint arXiv:1908.02068, 2019, doi: 10.48550/arXiv.1908.02068. [Cited on page 69.]

[170] Ivo A. Maceira, Natalia Chepiga, and Frederic Mila. Conformal and chiral phase transitions in Rydberg chains. Phys. Rev. Res., 4:043102, Nov 2022, doi: 10.1103/PhysRevResearch.4.043102. [Cited on page 69.]

[171] Jin Zhang, Sergio H Cantu, Fangli Liu, Alexei Bylinskii, Boris Braverman, Florian Huber, Jesse Amato-Grill, Alexander Lukin, Nathan Gemelke, Alexander Keesling, et al. Probing quan-

tum floating phases in Rydberg atom arrays. arXiv preprint arXiv:2401.08087, 2024, doi: 10.48550/arXiv.2401.08087. [Cited on page 69.]

[172] Rhine Samajdar, Soonwon Choi, Hannes Pichler, Mikhail D. Lukin, and Subir Sachdev. Numerical study of the chiral Z3 quantum phase transition in one spatial dimension. Phys. Rev. A, 98:023614, Aug 2018, doi: 10.1103/PhysRevA.98.023614. [Cited on pages 71 and 72.]

[173] Jan Balewski, Milan Kornjaca, Katherine Klymko, Siva Darbha, Mark R Hirsbrunner, Pedro Lopes, Fangli Liu, and Daan Camps. Engineering quantum states with neutral atoms. arXiv preprint arXiv:2404.04411, 2024, doi: 10.48550/arXiv.2404.04411. [Cited on page 71.]

[174] Paul Fendley, K. Sengupta, and Subir Sachdev. Competing density-wave orders in a one-dimensional hard-boson model. Phys. Rev. B, 69:075106, Feb 2004, doi: 10.1103/Phys-RevB.69.075106. [Cited on page 71.]

[175] Norman W Johnson. Convex polyhedra with regular faces. Canadian Journal of mathematics, 18:169-200, 1966, doi: 10.4153/CJM-1966-021-8. [Cited on page 77.]

[176] Xingze Qiu, Peter Zoller, and Xiaopeng Li. Programmable Quantum Annealing Architectures with Ising Quantum Wires. PRX Quantum, 1:020311, Nov 2020, doi: 10.1103/PRXQuantum.1.020311. [Cited on page 78.]

[177] Ahmed M. Farouk, 1.1. Beterov, Peng Xu, and 1.1. Ryabtsev. Scalable Heteronuclear Architecture of Neutral Atoms Based on EIT. Journal of Experimental and Theoretical Physics, 137(2):202-209, 2023, doi: 10.1134/S1063776123080046. [Cited on page 78.]

[178] P. M. Ireland, D. M. Walker, and J. D. Pritchard. Interspecies Forster resonances for Rb-Cs Rydberg d-states for enhanced multi-qubit gate fidelities. Phys. Rev. Res., 6:013293, Mar 2024, doi: 10.1103/PhysRevResearch.6.013293. [Cited on page 78.]

[179] Francesco Cesa and Hannes Pichler. Universal Quantum Computation in Globally Driven Rydberg Atom Arrays. Phys. Rev. Lett., 131:170601, Oct 2023, doi: 10.1103/PhysRevLett.131.170601. [Cited on pages 78 and 81.]

[180] Andrew Steane. Multiple-particle interference and quantum error correction. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 452(1954):2551-2577, 1996, doi: 10.1098/rspa.1996.0136. [Cited on page 85.]

[181] Austin G Fowler, Matteo Mariantoni, John M Martinis, and Andrew N Cleland. Surface codes: Towards practical large-scale quantum computation. Physical Review A, 86(3):032324, 2012, doi: 10.1103/PhysRevA.86.032324. [Cited on page 85.]

[182] Alexander Lukin, Benjamin F Schiffer, Boris Braverman, Sergio H Cantu, Florian Huber, Alexei Bylinskii, Jesse Amato-Grill, Nishad Maskara, Madelyn Cain, Dominik S Wild, et al. Quantum quench dynamics as a shortcut to adiabaticity. arXiv preprint arXiv:2405.21019, 2024, doi: 10.48550/arXiv.2405.21019. [Cited on page 85.]

[183] Tom Manovitz, Sophie H Li, Sepehr Ebadi, Rhine Samajdar, Alexandra A Geim, Simon J Ev-ered, Dolev Bluvstein, Hengyun Zhou, Nazli Ugur Köylüoglu, Johannes Feldmeier, et al. Quantum coarsening and collective dynamics on a programmable quantum simulator. arXiv preprint arXiv:2407.03249, 2024, doi: 10.48550/arXiv.2407.03249. [Cited on page 85.]

[184] M. Marinescu, H. R. Sadeghpour, and A. Dalgarno. Dispersion coefficients for alkali-metal dimers. Phys. Rev. A, 49:982-988, Feb 1994, doi: 10.1103/PhysRevA.49.982. [Cited on page 92.]

[185] BV Noumerov. A method of extrapolation of perturbations. Monthly Notices of the Royal Astronomical Society, Vol. 84, p. 592-592, 84:592-592, 1924, doi: 10.1093/mnras/84.8.592. [Cited on page 93.]

[186] J. Jackson. Note on the Numerical Integration of x = f (x,t). Monthly Notices of the Royal Astronomical Society, 84(8):602-607, 06 1924, doi: 10.1093/mnras/84.8.602. [Cited on page 93.]

[187] Mark Saffman, Thad G Walker, and Klaus M0lmer. Quantum information with Rydberg atoms. Reviews of modern physics, 82(3):2313, 2010, doi: 10.1103/RevModPhys.82.2313. [Cited on page 94.]

[188] F. Ponciano-Ojeda, S. Hernandez-Gomez, O. Lopez-Hernandez, C. Mojica-Casique, R. Colín-Rodríguez, F. Ramírez-Martínez, J. Flores-Mijangos, D. Sahagun, R. Jauregui, and J. Jimenez-Mier. Observation of the 5p3/2 ^ 6p3/2 electric-dipole-forbidden transition in atomic rubidium using optical-optical double-resonance spectroscopy. Phys. Rev. A, 92:042511, Oct 2015, doi: 10.1103/PhysRevA.92.042511. [Cited on page 98.]

[189] Fernando Ramírez-Martínez, Francisco S Ponciano-Ojeda, Santiago Hernandez-Gomez, Alberto Del Angel, Cristian Mojica-Casique, Lina M Hoyos-Campo, Jesus Flores-Mijangos, Daniel Sahagun, Rocío Jauregui, and Jose I Jimenez-Mier. Use of an electric-dipole forbidden transition to optically probe the Autler Townes effect. arXiv preprint arXiv:1909.01293, 2019, doi: 10.48550/arXiv.1909.01293. [Cited on page 98.]