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In this work, we explore a deep learning based automated visual inspection and verification algorithm, based on the Siamese Neural Network architecture. Consideration is also given to how the input pairs of images can affect the performance of the Siamese Neural Network. The Siamese Neural Network was explored alongside Convolutional Neural Networks. In addition to investigating these model architectures, additional methods are explored including transfer learning and ensemble methods, with the aim of improving model performance. We develop a novel voting scheme specific to the Siamese Neural Network which sees a single model vote on multiple reference images. This differs from the typical ensemble approach of multiple models voting on the same data sample. The results obtained show great potential for the use of the Siamese Neural Network for automated visual inspection and verification tasks when there is a scarcity of training data available. The additional methods applied, including the novel similarity voting, are also seen to significantly improve the performance of the model. We apply the publicly available omniglot dataset to validate our approach. According to our knowledge, this is the first time a detailed study of this sort has been carried out in the automatic verification of installed brackets in the aerospace sector via Deep Neural Networks.

Floquet quantum circuits are able to realise a wide range of non-equilibrium quantum states, exhibiting quantum chaos, topological order and localisation. In this work, we investigate the stability of operator localisation and emergence of chaos in random Floquet-Clifford circuits subjected to unitary perturbations which drive them away from the Clifford limit. We construct a nearest-neighbour Clifford circuit with a brickwork pattern and study the effect of including disordered non-Clifford gates. The perturbations are uniformly sampled from single-qubit unitaries with probability $p$ on each qubit. We show that the interacting model exhibits strong localisation of operators for $0 \le p < 1$ that is characterised by the fragmentation of operator space into disjoint sectors due to the appearance of wall configurations. Such walls give rise to emergent local integrals of motion for the circuit that we construct exactly. We analytically establish the stability of localisation against generic perturbations and calculate the average length of operator spreading tunable by $p$. Although our circuit is not separable across any bi-partition, we further show that the operator localisation leads to an entanglement bottleneck, where initially unentangled states remain weakly entangled across typical fragment boundaries. Finally, we study the spectral form factor (SFF) to characterise the chaotic properties of the operator fragments and spectral fluctuations as a probe of non-ergodicity. In the $p = 1$ model, the emergence of a fragmentation time scale is found before random matrix theory sets in after which the SFF can be approximated by that of the circular unitary ensemble. Our work provides an explicit description of quantum phases in operator dynamics and circuit ergodicity which can be realised on current NISQ devices., Comment: 21 pages, 14 figures, 2 appendices

Let $R$ be the class of regular cardinals which are not hyperinaccessible. We show that $L[R]$, and similar inner models in the $\alpha$-inaccessible hierarchy, can be generated by iterating a small "machete" mouse up through all the ordinals, and then taking a generic extension by a hyperclass Magidor iteration of Prikry forcings. We then show that such simple mice are themselves elements of $L[\mathsf{Reg}]$., Comment: 21 pages

Nonintegrable many-body quantum systems typically thermalize at long times through the mechanism of quantum chaos. However, some exceptional systems, such as those harboring quantum scars, break thermalization, serving as testbeds for foundational problems of quantum statistical physics. Here, we investigate a class of nonintegrable bond-staggered models that is endowed with a large number of zero-energy eigenstates and possesses a non-Abelian internal symmetry. We use character theory to give a lower bound on the zero-energy degeneracy, which matches exact diagonalization results, and is found to grow exponentially with the system size. We also show that few-magnon zero-energy states have an exact analytical description, allowing us to build a basis of low-entangled fixed-separation states, which is stable to most perturbations found in experiments. This remarkable dynamical stability of special states elucidates our understanding of nonequilibrium processes in non-Abelian chaotic quantum models., Comment: 10 pages, 7 figures

Topology and many-body localization (MBL) have opened new avenues for preserving quantum information at finite energy density. Resonant delocalization plays a crucial role in destabilizing these phenomena. In this work, we study the statistical properties of many-body resonances in a disordered interacting Ising model - which can host symmetry protected topological order - using a Clifford circuit encoding of the real space renormalization group which allows the resonant properties of the wave functions to be efficiently characterized. Our findings show that both the trivial and topologically ordered MBL phases remain stable to the resonances, but in the vicinity of the transition between them localization is destabilized by resonance proliferation. Diverging susceptibility towards the development of an avalanche instability suggests an intervening ergodic phase. We are also able to access the local integrals of motion in the MBL phases and identify the topological edge-mode operators in the ordered phase. Our results have important implications for the stability of MBL and phase transitions between distinct MBL phases with and without symmetries., Comment: 13 pages, 11 figures

We introduce two schemes of quantifiers analogous to $I$ and $Q^\text{e.c.}$, which tell us about regular cardinals of small Cantor-Bendixson rank. We examine how the L\"owenheim-Skolem-Tarski numbers of these quantifiers interact with one another, and with those of $I$ and $Q^{\text{e.c.}}$. We then find the exact lower bound for each of the LST numbers, assuming the consistency of supercompacts.

We investigate a variety of cut and choose games, their relationship with (generic) large cardinals, and show that they can be used to characterize a number of properties of ideals and of partial orders: certain notions of distributivity, strategic closure, and precipitousness., Comment: 31 pages

We study principles of the form: if a name $\sigma$ is forced to have a certain property $\varphi$, then there is a ground model filter $g$ such that $\sigma^g$ satisfies $\varphi$. We prove a general correspondence connecting these name principles to forcing axioms. Special cases of the main theorem are: Any forcing axiom can be expressed as a name principle. For instance, $\mathsf{PFA}$ is equivalent to a principle for rank $1$ names (equivalently, nice names) for subsets of $\omega_1$, and a principle for rank $2$ names for sets of reals. Moreover, $\lambda$-bounded forcing axioms are equivalent to name principles. Bagaria's characterisation of $\mathsf{BFA}$ via generic absoluteness is a corollary. We further systematically study name principles where $\varphi$ is a notion of largeness for subsets of $\omega_1$ (such as being unbounded, stationary or in the club filter) and corresponding forcing axioms.

Motivated by recent observations of ergodicity breaking due to Hilbert space fragmentation in 1D Fermi-Hubbard chains with a tilted potential [Scherg et al., arXiv:2010.12965], we show that the same system also hosts quantum many-body scars in a regime $U\approx \Delta \gg J$ at electronic filling factor $\nu=1$. We numerically demonstrate that the scarring phenomenology in this model is similar to other known realisations such as Rydberg atom chains, including persistent dynamical revivals and ergodicity-breaking many-body eigenstates. At the same time, we show that the mechanism of scarring in the Fermi-Hubbard model is different from other examples in the literature: the scars originate from a subgraph, representing a free spin-1 paramagnet, which is weakly connected to the rest of the Hamiltonian's adjacency graph. Our work demonstrates that correlated fermions in tilted optical lattices provide a platform for understanding the interplay of many-body scarring and other forms of ergodicity breaking, such as localisation and Hilbert space fragmentation., Comment: 13 pages, 15 figures

The theory of quantum scarring -- a remarkable violation of quantum unique ergodicity -- rests on two complementary pillars: the existence of unstable classical periodic orbits and the so-called quasimodes, i.e., the non-ergodic states that strongly overlap with a small number of the system's eigenstates. Recently, interest in quantum scars has been revived in a many-body setting of Rydberg atom chains. While previous theoretical works have identified periodic orbits for such systems using time-dependent variational principle (TDVP), the link between periodic orbits and quasimodes has been missing. Here we provide a conceptually simple analytic construction of quasimodes for the non-integrable Rydberg atom model, and prove that they arise from a "requantisation" of previously established periodic orbits when quantum fluctuations are restored to all orders. Our results shed light on the TDVP classical system simultaneously playing the role of both the mean-field approximation and the system's classical limit, thus allowing us to firm up the analogy between the eigenstate scarring in the Rydberg atom chains and the single-particle quantum systems., Comment: 22 pages, 13 figures, accepted for publication

Motivated by recent experimental observations of coherent many-body revivals in a constrained Rydberg atom chain, we construct a weak quasi-local deformation of the Rydberg blockade Hamiltonian, which makes the revivals virtually perfect. Our analysis suggests the existence of an underlying non-integrable Hamiltonian which supports an emergent SU(2)-spin dynamics within a small subspace of the many-body Hilbert space. We show that such perfect dynamics necessitates the existence of atypical, nonergodic energy eigenstates - quantum many-body scars. Furthermore, using these insights, we construct a toy model that hosts exact quantum many-body scars, providing an intuitive explanation of their origin. Our results offer specific routes to enhancing coherent many-body revivals, and provide a step towards establishing the stability of quantum many-body scars in the thermodynamic limit., Comment: 6+11 pages, 3+6 figures

Quantum many-body systems realise many different phases of matter characterised by their exotic emergent phenomena. While some simple versions of these properties can occur in systems of free fermions, their occurrence generally implies that the physics is dictated by an interacting Hamiltonian. The interaction distance has been successfully used to quantify the effect of interactions in a variety of states of matter via the entanglement spectrum [Nat. Commun. 8, 14926 (2017), arXiv:1705.09983]. The computation of the interaction distance reduces to a global optimisation problem whose goal is to search for the free-fermion entanglement spectrum closest to the given entanglement spectrum. In this work, we employ techniques from machine learning in order to perform this same task. In a supervised learning setting, we use labelled data obtained by computing the interaction distance and predict its value via linear regression. Moving to a semi-supervised setting, we train an auto-encoder to estimate an alternative measure to the interaction distance, and we show that it behaves in a similar manner., Comment: 9 pages, 6 figures, conference "Quantum Techniques in Machine Learning"

Certain wave functions of non-interacting quantum chaotic systems can exhibit "scars" in the fabric of their real-space density profile. Quantum scarred wave functions concentrate in the vicinity of unstable periodic classical trajectories. We introduce the notion of many-body quantum scars which reflect the existence of a subset of special many-body eigenstates concentrated in certain parts of the Hilbert space. We demonstrate the existence of scars in the Fibonacci chain -- the one- dimensional model with a constrained local Hilbert space realized in the 51 Rydberg atom quantum simulator [H. Bernien et al., arXiv:1707.04344]. The quantum scarred eigenstates are embedded throughout the thermalizing many-body spectrum, but surprisingly lead to direct experimental signatures such as robust oscillations following a quench from a charge-density wave state found in experiment. We develop a model based on a single particle hopping on the Hilbert space graph, which quantitatively captures the scarred wave functions up to large systems of L = 32 atoms. Our results suggest that scarred many-body bands give rise to a new universality class of quantum dynamics, which opens up opportunities for creating and manipulating novel states with long-lived coherence in systems that are now amenable to experimental study., Comment: 8 pages, 4 figures; updated references