Periodic Reporting for period 2 - ETNA4Ryd (Enhancing Tensor Network Approaches for Rydberg Atom Quantum Simulators)
Reporting period: 2025-01-01 to 2025-12-31
Summary of the context and overall objectives of the project
Recent experiments with ultracold atoms, trapped ions, Rydberg atoms, and superconducting circuits succeeded in
realizing quantum many-body states at unprecedented sizes and thus investigating their static and dynamical properties.
These achievements have boosted the search for protocols to observe exotic phases of matter in quantum simulators, as well as deveoloping protocols to solve problems
unaffordable for classical supercomputers. I aim to investigate Rydberg-atom
platforms, improving their efficiency for future quantum simulation and computation tasks, motivated by their versatility
and manipulation capability. Classical numerical simulations are fundamental to developing quantum simulators,
engineering efficient experimental protocols, and benchmarking the results. Still, an exact representation for large
quantum many-body states is highly inefficient and impossible to achieve for the sizes available in current experiments.
I will exploit advanced numerical tensor network methods to simulate the out-of-equilibrium properties of highly
constrained quantum phases, recently realized on Rydberg-atom highdimensional
lattices. Indeed, tensor networks are a balanced approximation between accuracy and computational
resources and are the ideal set of tools to investigate constrained regimes in quantum many-body systems. Realizing
this project first at the Lukin Quantum Optics Group at Harvard University and then at theQuantum Theory Group at
Padova University, I will access world-leading experimental and numerical expertise to perform cutting-edge analytical,
numerical, and experimental investigations on Rydberg atom platforms. In addition, I will acquire experiment modeling
expertise and apply them at the near-future quantum computation laboratory at Padova University. This project is aligned
with the Quantum Technologies Flagship, making me valuable for future innovative research on competitive quantum
technologies applications.
realizing quantum many-body states at unprecedented sizes and thus investigating their static and dynamical properties.
These achievements have boosted the search for protocols to observe exotic phases of matter in quantum simulators, as well as deveoloping protocols to solve problems
unaffordable for classical supercomputers. I aim to investigate Rydberg-atom
platforms, improving their efficiency for future quantum simulation and computation tasks, motivated by their versatility
and manipulation capability. Classical numerical simulations are fundamental to developing quantum simulators,
engineering efficient experimental protocols, and benchmarking the results. Still, an exact representation for large
quantum many-body states is highly inefficient and impossible to achieve for the sizes available in current experiments.
I will exploit advanced numerical tensor network methods to simulate the out-of-equilibrium properties of highly
constrained quantum phases, recently realized on Rydberg-atom highdimensional
lattices. Indeed, tensor networks are a balanced approximation between accuracy and computational
resources and are the ideal set of tools to investigate constrained regimes in quantum many-body systems. Realizing
this project first at the Lukin Quantum Optics Group at Harvard University and then at theQuantum Theory Group at
Padova University, I will access world-leading experimental and numerical expertise to perform cutting-edge analytical,
numerical, and experimental investigations on Rydberg atom platforms. In addition, I will acquire experiment modeling
expertise and apply them at the near-future quantum computation laboratory at Padova University. This project is aligned
with the Quantum Technologies Flagship, making me valuable for future innovative research on competitive quantum
technologies applications.
Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far
The scope of the project is to exploit advanced numerical tensor network methods to simulate the out-of-equilibrium properties of highly constrained phases of Rydberg atom quantum simulators.
Rydberg atom quantum simulators represent one of the most promising platforms for exploring complex quantum many-body physics and developing quantum information processing protocols. These systems utilize neutral atoms excited to high-lying electronic states—known as Rydberg states—which interact with long-range, strong, tunable van der Waals interactions. The key feature playing a predominant role in this project is the so-called Rydberg blockade. Due to their strong interactions, two atoms cannot be simultaneously excited to the Rydberg state if their distance is below a certain radius, called blockade radius. This feature allows the implementation of highly complex and nontrivial quantum phases. These phases have interest per se, as well relevant implications in the exploitation of Rydberg atom quantum simulators to solve real-life problems. Both these aspects have been considered in the outgoing period of my action.
During the outgoing phase, I have been carrying out two projects. The first was about strategies to solve the Maximum Independent Set (MIS) problem with Rydberg atom quantum simulators. The second is about dynamics across quantum phase transitions in a square lattice of Rydberg atoms.
The first project explored how Rydberg atom arrays can be used to tackle complex combinatorial optimization problems, with a focus on the Maximum Independent Set (MIS) problem. The standard approach that uses quantum simulators to solve this problems is to use adiabatic quantum protocols, which rely on slowly evolving a quantum system to reach the desired solution. However, these methods face serious limitations due to the long timescales required for large systems, making them impractical for many real-world applications. To address these challenges, this research investigated alternative strategies that are inspired by recent theoretical developments but have not yet been experimentally implemented. One line of investigation examined how to extend the class of problem instances (i.e. graphs) that can be addressed using Rydberg atom arrays. Starting from known theoretical mappings, the practical feasibility of these mappings was assessed. Several limitations were identified, particularly regarding their efficiency when applied to realistic experimental setups. This investigation led to a collaborative exchange with a research group at the Institute for Quantum Optics and Quantum Information (IQOQI) in Innsbruck. While that group focused on the foundational elements of the mapping approach, leading to a scientific publication, this project contributed critical insights into the feasibility and limitations of the broader framework. A second area of research focused on improving the performance of adiabatic protocols by testing faster, approximate techniques based on the so-called Laplacian operator. These methods aim to spread the quantum state across many possibilities more effectively, helping the system find the optimal solution more quickly. Although the results are currently limited to a specific class of graphs, they represent promising directions and will be further developed in future research. Finally, the project examined the use of "counterdiabatic" driving techniques, which offer a theoretical advantage by bypassing the need to follow slow adiabatic evolution. These methods are not constrained by the minimum energy gap of the system, potentially allowing much faster computations. However, the practical implementation of counterdiabatic potentials was found to be highly demanding, requiring a large number of complex interactions that scale exponentially with the system size. While this result limits the immediate applicability of the method, it provides a rigorous theoretical benchmark that fills an important gap in the current understanding of quantum optimization strategies. Overall, the project produced meaningful insights into the capabilities and limitations of Rydberg atom arrays for quantum optimization, identifying both promising paths forward and key challenges that must be addressed for future experimental applications.
The second project focused on understanding a quantum phase observed in the two-dimensional Rydberg atom square lattice, the striated phase. The striated phase represents a form of spatial ordering in quantum systems, and studying its formation and behavior can provide valuable insights into the nature of quantum phase transitions. In this work, both numerical simulations and experimental approaches were used to investigate how this phase emerges under different lattice sizes and boundary conditions.
The first part of the research explored the equilibrium phase diagram of the system using advanced computational techniques based on tensor networks. By studying lattices of varying linear dimensions (even and odd sizes) with open boundaries, important differences in the system’s behavior were identified. Specifically, the results suggest that for odd-size lattices, the transition from a disordered state to a striated phase is compatible with a second-order phase transition, characterized by a smooth and continuous change. In contrast, for even-size lattices, the transition appears to be of first order, involving a more abrupt and discontinuous change in the system’s state.
In the second phase of the project, the dynamics of the system were simulated using state-of-the-art tensor network methods. Simulations were performed on two-dimensional lattices of up to 16 x 16 atoms, using adiabatic protocols designed to prepare the striated phase. These simulations revealed that the system's behavior during this preparation process is strongly influenced by the size of the lattice, with distinct and unexpected differences depending on whether the lattice dimensions are even or odd.
Building on these numerical results, the same out-of-equilibrium protocol was tested in an experimental setup. The experimental observations showed notable agreement with the numerical simulations, validating key aspects of the theoretical predictions. Interestingly, the experiment also revealed the presence of a quantum state that could not be fully captured by the simulations due to the complexity and richness of quantum correlations in the system. This highlights the limitations of current numerical tools and underscores the importance of experimental validation.
In support of the experimental work, a dedicated software library was developed to analyze the results. This tool was designed to process experimental data while accounting for realistic imperfections, such as system defects and measurement errors, providing a more accurate interpretation of the experimental outcomes.
During the return phase, I carried out two projects, both of which are currently going on.
In the first, I am studying the transpilation problem in quantum circuits. Transpilation is the fundamental process of transforming an abstract unitary transformation into a set of gates that can be efficiently implemented on a real quantum computer. Transpilation must take into account realistic sources of error that can occur during a quantum computing protocol. For example, minimizing the total computation time reduces the decoherence of the system. On the other, gates applied in parallel are subject to spurious correlations. We examined the interplay between these two main sources of error with an approach at the boundary with statistical mechanics, observing unexpected phase transitions.
In the second, I am working with members of the return phase host institution to develop novel protocols to compute dynamics on quantum many-body systems, leveraging on the efficiency of tensor network ansatze.
Overall, the project contributed new insights into the interplay between system size, boundary conditions, and quantum phase transitions in Rydberg atom arrays. It demonstrated the complementary power of numerical and experimental techniques in exploring complex quantum systems and set the foundation for future investigations into nonequilibrium quantum dynamics.
Rydberg atom quantum simulators represent one of the most promising platforms for exploring complex quantum many-body physics and developing quantum information processing protocols. These systems utilize neutral atoms excited to high-lying electronic states—known as Rydberg states—which interact with long-range, strong, tunable van der Waals interactions. The key feature playing a predominant role in this project is the so-called Rydberg blockade. Due to their strong interactions, two atoms cannot be simultaneously excited to the Rydberg state if their distance is below a certain radius, called blockade radius. This feature allows the implementation of highly complex and nontrivial quantum phases. These phases have interest per se, as well relevant implications in the exploitation of Rydberg atom quantum simulators to solve real-life problems. Both these aspects have been considered in the outgoing period of my action.
During the outgoing phase, I have been carrying out two projects. The first was about strategies to solve the Maximum Independent Set (MIS) problem with Rydberg atom quantum simulators. The second is about dynamics across quantum phase transitions in a square lattice of Rydberg atoms.
The first project explored how Rydberg atom arrays can be used to tackle complex combinatorial optimization problems, with a focus on the Maximum Independent Set (MIS) problem. The standard approach that uses quantum simulators to solve this problems is to use adiabatic quantum protocols, which rely on slowly evolving a quantum system to reach the desired solution. However, these methods face serious limitations due to the long timescales required for large systems, making them impractical for many real-world applications. To address these challenges, this research investigated alternative strategies that are inspired by recent theoretical developments but have not yet been experimentally implemented. One line of investigation examined how to extend the class of problem instances (i.e. graphs) that can be addressed using Rydberg atom arrays. Starting from known theoretical mappings, the practical feasibility of these mappings was assessed. Several limitations were identified, particularly regarding their efficiency when applied to realistic experimental setups. This investigation led to a collaborative exchange with a research group at the Institute for Quantum Optics and Quantum Information (IQOQI) in Innsbruck. While that group focused on the foundational elements of the mapping approach, leading to a scientific publication, this project contributed critical insights into the feasibility and limitations of the broader framework. A second area of research focused on improving the performance of adiabatic protocols by testing faster, approximate techniques based on the so-called Laplacian operator. These methods aim to spread the quantum state across many possibilities more effectively, helping the system find the optimal solution more quickly. Although the results are currently limited to a specific class of graphs, they represent promising directions and will be further developed in future research. Finally, the project examined the use of "counterdiabatic" driving techniques, which offer a theoretical advantage by bypassing the need to follow slow adiabatic evolution. These methods are not constrained by the minimum energy gap of the system, potentially allowing much faster computations. However, the practical implementation of counterdiabatic potentials was found to be highly demanding, requiring a large number of complex interactions that scale exponentially with the system size. While this result limits the immediate applicability of the method, it provides a rigorous theoretical benchmark that fills an important gap in the current understanding of quantum optimization strategies. Overall, the project produced meaningful insights into the capabilities and limitations of Rydberg atom arrays for quantum optimization, identifying both promising paths forward and key challenges that must be addressed for future experimental applications.
The second project focused on understanding a quantum phase observed in the two-dimensional Rydberg atom square lattice, the striated phase. The striated phase represents a form of spatial ordering in quantum systems, and studying its formation and behavior can provide valuable insights into the nature of quantum phase transitions. In this work, both numerical simulations and experimental approaches were used to investigate how this phase emerges under different lattice sizes and boundary conditions.
The first part of the research explored the equilibrium phase diagram of the system using advanced computational techniques based on tensor networks. By studying lattices of varying linear dimensions (even and odd sizes) with open boundaries, important differences in the system’s behavior were identified. Specifically, the results suggest that for odd-size lattices, the transition from a disordered state to a striated phase is compatible with a second-order phase transition, characterized by a smooth and continuous change. In contrast, for even-size lattices, the transition appears to be of first order, involving a more abrupt and discontinuous change in the system’s state.
In the second phase of the project, the dynamics of the system were simulated using state-of-the-art tensor network methods. Simulations were performed on two-dimensional lattices of up to 16 x 16 atoms, using adiabatic protocols designed to prepare the striated phase. These simulations revealed that the system's behavior during this preparation process is strongly influenced by the size of the lattice, with distinct and unexpected differences depending on whether the lattice dimensions are even or odd.
Building on these numerical results, the same out-of-equilibrium protocol was tested in an experimental setup. The experimental observations showed notable agreement with the numerical simulations, validating key aspects of the theoretical predictions. Interestingly, the experiment also revealed the presence of a quantum state that could not be fully captured by the simulations due to the complexity and richness of quantum correlations in the system. This highlights the limitations of current numerical tools and underscores the importance of experimental validation.
In support of the experimental work, a dedicated software library was developed to analyze the results. This tool was designed to process experimental data while accounting for realistic imperfections, such as system defects and measurement errors, providing a more accurate interpretation of the experimental outcomes.
During the return phase, I carried out two projects, both of which are currently going on.
In the first, I am studying the transpilation problem in quantum circuits. Transpilation is the fundamental process of transforming an abstract unitary transformation into a set of gates that can be efficiently implemented on a real quantum computer. Transpilation must take into account realistic sources of error that can occur during a quantum computing protocol. For example, minimizing the total computation time reduces the decoherence of the system. On the other, gates applied in parallel are subject to spurious correlations. We examined the interplay between these two main sources of error with an approach at the boundary with statistical mechanics, observing unexpected phase transitions.
In the second, I am working with members of the return phase host institution to develop novel protocols to compute dynamics on quantum many-body systems, leveraging on the efficiency of tensor network ansatze.
Overall, the project contributed new insights into the interplay between system size, boundary conditions, and quantum phase transitions in Rydberg atom arrays. It demonstrated the complementary power of numerical and experimental techniques in exploring complex quantum systems and set the foundation for future investigations into nonequilibrium quantum dynamics.
Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)
The research carried out during this project led to relevant advancements beyond the current state of the art both in quantum optimization and in the study of quantum phase transitions using Rydberg atom quantum simulators.
In the context of quantum optimization, the project aimed to go beyond standard adiabatic protocols by exploring faster, approximate techniques based on the Laplacian operator. These methods, while applied to a specific class of graphs, demonstrated a superlinear speedup in distributing the quantum state across potential solutions—marking a departure from traditional slow-evolution strategies. Furthermore, the theoretical investigation of counterdiabatic driving revealed new benchmarks for quantum optimization, showing that although the method is currently impractical for large systems due to exponential scaling, it offers a rigorous framework to evaluate the ultimate limits of quantum speedups.
The work also extended the experimental reach of quantum simulation. The project contributed original insights into the feasibility of implementing broader classes of combinatorial problems, enriching theoretical mappings with concrete evaluations under realistic constraints. This led to international collaboration and a peer-reviewed publication, strengthening the practical relevance of theoretical proposals.
On the front of quantum phase transitions, simulations on large two-dimensional lattices (up to 16×16 atoms) using state-of-the-art tensor network methods yielded unprecedented insights into how system size and parity affect phase behavior—distinguishing between first- and second-order transitions. Crucially, the project's numerical predictions were directly validated through experiments, which not only confirmed the main features but also uncovered complex quantum states beyond current simulation capabilities. This dual approach bridged theoretical and experimental efforts and pushed forward the understanding of nonequilibrium quantum dynamics in constrained systems.
The study on transpilation of quantum circuits is leading to a publication under realization and to possible collaborations with experimental groups that work with superconducting qubits, as the approach we develop can be relevant to characterize the perfomances of their machines and improve the efficiency of quantum computation protocols.
The investigation of improved protocols to compute the dynamics of quantum many body systems with tensor networks has led to a fruitful collaboration with other groups interested in pursuing the same goal. When concluded, this new protocol will allow more robust numerical benchmarking of quantum simulation experimental protocols.
In the context of quantum optimization, the project aimed to go beyond standard adiabatic protocols by exploring faster, approximate techniques based on the Laplacian operator. These methods, while applied to a specific class of graphs, demonstrated a superlinear speedup in distributing the quantum state across potential solutions—marking a departure from traditional slow-evolution strategies. Furthermore, the theoretical investigation of counterdiabatic driving revealed new benchmarks for quantum optimization, showing that although the method is currently impractical for large systems due to exponential scaling, it offers a rigorous framework to evaluate the ultimate limits of quantum speedups.
The work also extended the experimental reach of quantum simulation. The project contributed original insights into the feasibility of implementing broader classes of combinatorial problems, enriching theoretical mappings with concrete evaluations under realistic constraints. This led to international collaboration and a peer-reviewed publication, strengthening the practical relevance of theoretical proposals.
On the front of quantum phase transitions, simulations on large two-dimensional lattices (up to 16×16 atoms) using state-of-the-art tensor network methods yielded unprecedented insights into how system size and parity affect phase behavior—distinguishing between first- and second-order transitions. Crucially, the project's numerical predictions were directly validated through experiments, which not only confirmed the main features but also uncovered complex quantum states beyond current simulation capabilities. This dual approach bridged theoretical and experimental efforts and pushed forward the understanding of nonequilibrium quantum dynamics in constrained systems.
The study on transpilation of quantum circuits is leading to a publication under realization and to possible collaborations with experimental groups that work with superconducting qubits, as the approach we develop can be relevant to characterize the perfomances of their machines and improve the efficiency of quantum computation protocols.
The investigation of improved protocols to compute the dynamics of quantum many body systems with tensor networks has led to a fruitful collaboration with other groups interested in pursuing the same goal. When concluded, this new protocol will allow more robust numerical benchmarking of quantum simulation experimental protocols.