arXiv subjects

Michael Ragone

Publications and source records attributed to Michael Ragone.

6 recordsLinked to original sources

arXiv2026

Spectral Gap of the Davies Generator for the Mean-Field Heisenberg Model

The mean-field Heisenberg ferromagnet is a quantum spin model on the complete graph with isotropic spin-1/2 interactions. This non-commuting Hamiltonian is permutation and $\mathsf{SU}(2)$ invariant, and its Gibbs states undergo an $\mathsf{SU}(2)$ symmetry breaking phase transition at inverse temperature $\beta=2$. We consider the associated Davies generator, a canonical model of open-system thermalization, and prove tight asymptotic estimates for its spectral gap at all noncritical temperatures. For fixed $\beta<2$, the gap as a function of number of qubits $n$ is $\Theta(1)$, while for fixed $\beta>2$ the gap is $\Theta(n^{-1})$. The matching upper bound of the spectral gap is witnessed by the total magnetization order parameter, suggesting that the low-temperature ($\beta>2$) slowdown is associated with broken continuous symmetry. Two key ingredients in our approach are a comparison argument, which introduces auxiliary generators to bound dissipation on nontrivial representations of the symmetry groups $\mathsf{SU}(2)$ and $\mathsf{S}_n$, and a decomposition of the space of observables into spherical tensor operators to reveal a form of monotonicity.

quant-ph↗

arXiv2025

The Many-Body Ground State Manifold of Flat Band Interacting Hamiltonian for Magic Angle Twisted Bilayer Graphene

At a magic relative twist angle, magic angle twisted bilayer graphene (MATBG) has an octet of flat bands that can host strong correlation physics when partially filled. A key theoretical discovery in MATBG is the existence of ferromagnetic Slater determinants as exact ground states of the corresponding flat band interacting (FBI) Hamiltonian. The FBI Hamiltonian describes the behavior of electrons that interact with each other in a high-dimensional space, and is constructed from the band structure of the non-interacting Bistritzer--MacDonald model at the chiral limit. A key property of the FBI Hamiltonian for MATBG is that it is frustration free and can be written as a sum of non-commuting terms. In this work, we provide a complete characterization of the ground state manifold of the FBI Hamiltonian, proving that it is precisely the linear span of such ferromagnetic Slater determinants.

math-ph↗

arXiv2024

SO(n) AKLT Chains as Symmetry Protected Topological Quantum Ground States

This thesis studies a pair of symmetry protected topological (SPT) phases which arise when considering one-dimensional quantum spin systems possessing a natural orthogonal group symmetry. Particular attention is given to a family of exactly solvable models whose ground states admit a matrix product state description and generalize the AKLT chain. We call these models ``$SO(n)$ AKLT chains'' and the phase they occupy the ``$SO(n)$ Haldane phase''. We present new results describing their ground state structure and, when $n$ is even, their peculiar $O(n)$-to-$SO(n)$ symmetry breaking. We also prove that these states have arbitrarily large correlation and injectivity length by increasing $n$, but all have a 2-local parent Hamiltonian, in contrast to the natural expectation that the interaction range of a parent Hamiltonian should diverge as these quantities diverge. We extend Ogata's definition of an SPT index for a split state for a finite symmetry group $G$ to an SPT index for a compact Lie group $G$. We then compute this index, which takes values in the second Borel group cohomology $H^2(SO(n),U(1))$, at a single point in each of the SPT phases. The two points have different indices, confirming the two SPT phases are indeed distinct. Chapter 1 contains an introduction with a detailed overview of the contents of this thesis, which includes several chapters of background information before presenting new results in Chapter 7 and Chapter 8.

math-ph↗

arXiv2023

A Lie Algebraic Theory of Barren Plateaus for Deep Parameterized Quantum Circuits

Variational quantum computing schemes train a loss function by sending an initial state through a parametrized quantum circuit, and measuring the expectation value of some operator. Despite their promise, the trainability of these algorithms is hindered by barren plateaus (BPs) induced by the expressiveness of the circuit, the entanglement of the input data, the locality of the observable, or the presence of noise. Up to this point, these sources of BPs have been regarded as independent. In this work, we present a general Lie algebraic theory that provides an exact expression for the variance of the loss function of sufficiently deep parametrized quantum circuits, even in the presence of certain noise models. Our results allow us to understand under one framework all aforementioned sources of BPs. This theoretical leap resolves a standing conjecture about a connection between loss concentration and the dimension of the Lie algebra of the circuit's generators.

quant-ph↗

arXiv2023

Representation Theory for Geometric Quantum Machine Learning

Recent advances in classical machine learning have shown that creating models with inductive biases encoding the symmetries of a problem can greatly improve performance. Importation of these ideas, combined with an existing rich body of work at the nexus of quantum theory and symmetry, has given rise to the field of Geometric Quantum Machine Learning (GQML). Following the success of its classical counterpart, it is reasonable to expect that GQML will play a crucial role in developing problem-specific and quantum-aware models capable of achieving a computational advantage. Despite the simplicity of the main idea of GQML -- create architectures respecting the symmetries of the data -- its practical implementation requires a significant amount of knowledge of group representation theory. We present an introduction to representation theory tools from the optics of quantum learning, driven by key examples involving discrete and continuous groups. These examples are sewn together by an exposition outlining the formal capture of GQML symmetries via "label invariance under the action of a group representation", a brief (but rigorous) tour through finite and compact Lie group representation theory, a reexamination of ubiquitous tools like Haar integration and twirling, and an overview of some successful strategies for detecting symmetries.

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arXiv2022

Theory for Equivariant Quantum Neural Networks

Quantum neural network architectures that have little-to-no inductive biases are known to face trainability and generalization issues. Inspired by a similar problem, recent breakthroughs in machine learning address this challenge by creating models encoding the symmetries of the learning task. This is materialized through the usage of equivariant neural networks whose action commutes with that of the symmetry. In this work, we import these ideas to the quantum realm by presenting a comprehensive theoretical framework to design equivariant quantum neural networks (EQNN) for essentially any relevant symmetry group. We develop multiple methods to construct equivariant layers for EQNNs and analyze their advantages and drawbacks. Our methods can find unitary or general equivariant quantum channels efficiently even when the symmetry group is exponentially large or continuous. As a special implementation, we show how standard quantum convolutional neural networks (QCNN) can be generalized to group-equivariant QCNNs where both the convolution and pooling layers are equivariant to the symmetry group. We then numerically demonstrate the effectiveness of a SU(2)-equivariant QCNN over symmetry-agnostic QCNN on a classification task of phases of matter in the bond-alternating Heisenberg model. Our framework can be readily applied to virtually all areas of quantum machine learning. Lastly, we discuss about how symmetry-informed models such as EQNNs provide hopes to alleviate central challenges such as barren plateaus, poor local minima, and sample complexity.

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