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Quantum phase estimation requires simulating the evolution of the Hamiltonian, for which product formulas are attractive due to their smaller qubit cost and ease of implementation. However, the estimation of the error incurred by product formulas is usually pessimistic and task-agnostic, which poses problems for assessing their performance in practice for problems of interest. In this work, we study the error of product formulas for the specific task of quantum energy estimation. To this end, we employ the theory of Trotter error with a Magnus-based expansion of the effectively simulated Hamiltonian. The result is a generalization of previous energy estimation error analysis of gapped eigenstates to arbitrary order product formulas. As an application, we discover a 9-term second-order product formula with an energy estimation error that is quadratically better than Trotter-Suzuki. Furthermore, by leveraging recent work on low-energy dynamics of product formulas, we provide tighter bounds for energy estimation error in the low-energy subspace. We show that for Hamiltonians with some locality and positivity properties, the cost can achieve up to a quadratic asymptotic speedup in terms of the target error.

This paper presents a review of the NTIRE 2024 challenge on night photography rendering. The goal of the challenge was to find solutions that process raw camera images taken in nighttime conditions, and thereby produce a photo-quality output images in the standard RGB (sRGB) space. Unlike the previous year's competition, the challenge images were collected with a mobile phone and the speed of algorithms was also measured alongside the quality of their output. To evaluate the results, a sufficient number of viewers were asked to assess the visual quality of the proposed solutions, considering the subjective nature of the task. There were 2 nominations: quality and efficiency. Top 5 solutions in terms of output quality were sorted by evaluation time (see Fig. 1). The top ranking participants' solutions effectively represent the state-of-the-art in nighttime photography rendering. More results can be found at https://nightimaging.org., Comment: 10 pages, 10 figures

Response theory has a successful history of connecting experimental observations with theoretical predictions. Of particular interest is the optical response of matter, from which spectroscopy experiments can be modelled. However, the calculation of response properties for quantum systems is often prohibitively expensive, especially for nonlinear spectroscopy, as it requires access to either the time evolution of the system or to excited states. In this work, we introduce a generalized quantum phase estimation framework designed for multi-variate phase estimation. This allows the treatment of general correlation functions enabling the recovery of response properties of arbitrary orders. The generalized quantum phase estimation circuit has an intuitive construction that is linked with a physical process of interest, and can directly sample frequencies from the distribution that would be obtained experimentally. In addition, we provide a single-ancilla modification of the new framework for early fault-tolerant quantum computers. Overall, our framework enables the efficient simulation of spectroscopy experiments beyond the linear regime, such as Raman spectroscopy. This opens up an exciting new field of applications for quantum computers with potential technological impact., Comment: 16 pages, 3 figures, 1 table

There is a pressing need for more accurate computational simulations of the opto-electronic properties of defects in materials to aid in the development of quantum sensing platforms. In this work, we explore how quantum computers could be effectively utilized for this purpose. Specifically, we develop fault-tolerant quantum algorithms to simulate optically active defect states and their radiative emission rates. We employ quantum defect embedding theory to translate the Hamiltonian of a defect-containing supercell into a smaller, effective Hamiltonian that accounts for dielectric screening effects. Our approach integrates block-encoding of the dipole operator with quantum phase estimation to selectively sample the optically active excited states that exhibit the largest dipole transition amplitudes. We also provide estimates of the quantum resources required to simulate a negatively-charged boron vacancy in a hexagonal boron nitride cluster. We conclude by offering a forward-looking perspective on the potential of quantum computers to enhance quantum sensor capabilities and identify specific scenarios where quantum computing can resolve problems traditionally challenging for classical computers., Comment: 18 pages, 4 figures, 2 tables

The on-chip measurement of polarization states plays an increasingly crucial role in modern sensing and imaging applications. While high-performance monolithic linearly polarized photodetectors have been extensively studied, integrated circularly polarized light (CPL) photodetectors are still hindered by inadequate discrimination capability. In this study, we employ achiral all-dielectric nanostructures to develop a broadband CPL photodetector with an impressive discrimination ratio of ~107 at the wavelength of 405 nm, significantly surpassing its counterparts by two orders of magnitude. Our device shows outstanding CPL discrimination capability across the visible band without requiring intensity calibration. Its function mechanism is based on the CPL-dependent near-field modes within achiral structures: under left or right CPL illumination, distinct near-field modes are excited, resulting in asymmetric irradiation of the two electrodes and generating a photovoltage with directions determined by the chirality of the incident light field. The proposed design strategy facilitates the realization of ultra-compact CPL detection across diverse materials, structures, and spectral ranges, presenting a novel avenue for achieving high-performance monolithic CPL detection., Comment: 20 pages, 4 figures

X-ray absorption spectroscopy is a crucial experimental technique for elucidating the mechanisms of structural degradation in battery materials. However, extracting information from the measured spectrum is challenging without high-quality simulations. In this work, we propose simulating near-edge X-ray absorption spectra as a promising application for quantum computing. It is attractive due to the ultralocal nature of X-ray absorption that significantly reduces the sizes of problems to be simulated, and because of the classical hardness of simulating spectra. We describe three quantum algorithms to compute the X-ray absorption spectrum and provide their asymptotic cost. One of these is a Monte-Carlo based time-domain algorithm, which is cost-friendly to early fault-tolerant quantum computers. We then apply the framework to an industrially relevant example, a CAS(22e,18o) active space for an O-Mn cluster in a Li-excess battery cathode, showing that practically useful simulations could be obtained with much fewer qubits and gates than ground-state energy estimation of the same material.

Quantum algorithms for probing ground-state properties of quantum systems require good initial states. Projection-based methods such as eigenvalue filtering rely on inputs that have a significant overlap with the low-energy subspace, which can be challenging for large, strongly-correlated systems. This issue has motivated the study of physically-inspired dynamical approaches such as thermodynamic cooling. In this work, we introduce a ground-state preparation algorithm based on the simulation of quantum dynamics. Our main insight is to transform the Hamiltonian by a shifted sign function via quantum signal processing, effectively mapping eigenvalues into positive and negative subspaces separated by a large gap. This automatically ensures that all states within each subspace conserve energy with respect to the transformed Hamiltonian. Subsequent time-evolution with a perturbed Hamiltonian induces transitions to lower-energy states while preventing unwanted jumps to higher energy states. The approach does not rely on a priori knowledge of energy gaps and requires no additional qubits to model a bath. Furthermore, it makes $\tilde{\mathcal{O}}(d^{\,3/2}/\epsilon)$ queries to the time-evolution operator of the system and $\tilde{\mathcal{O}}(d^{\,3/2})$ queries to a block-encoding of the perturbation, for $d$ cooling steps and an $\epsilon$-accurate energy resolution. Our results provide a framework for combining quantum signal processing and Hamiltonian simulation to design heuristic quantum algorithms for ground-state preparation.

Hamiltonian simulation is arguably the most fundamental application of quantum computers. The Magnus operator is a popular method for time-dependent Hamiltonian simulation in computational mathematics, yet its usage requires the implementation of exponentials of commutators, which has previously made it unappealing for quantum computing. The development of commutator-free quasi-Magnus operators (CFQMs) circumvents this obstacle, at the expense of a lack of provable global numeric error bounds. In this work, we establish one such error bound for CFQM-based time-dependent quantum Hamiltonian simulation by carefully estimating the error of each step involved in their definition. This allows us to compare its cost with the alternatives, and show that CFQMs are often the most efficient product-formula technique available by more than an order of magnitude. As a result, we find that CFQMs may be particularly useful to simulate time-dependent Hamiltonians on early fault-tolerant quantum computers., Comment: 32 pages, 7 figures

Product formulas are one of the main approaches for quantum simulation of the Hamiltonian dynamics of a quantum system. Their implementation cost is computed based on error bounds which are often pessimistic, resulting in overestimating the total runtime. In this work, we rigorously consider the error induced by product formulas when the state undergoing time evolution lies in the low-energy sector with respect to the Hamiltonian of the system. We show that in such a setting, the usual error bounds based on the operator norm of nested commutators can be replaced by those restricted to suitably chosen low-energy subspaces, yielding tighter error bounds. Furthermore, under some locality and positivity assumptions, we show that the simulation of generic product formulas acting on low-energy states can be done asymptotically more efficiently when compared with previous results.

The fractal dimension provides a statistical index of object complexity by studying how the pattern changes with the measuring scale. Although useful in several classification tasks, the fractal dimension is under-explored in deep learning applications. In this work, we investigate the features that are learned by deep models and we study whether these deep networks are able to encode features as complex and high-level as the fractal dimensions. Specifically, we conduct a correlation analysis experiment to show that deep networks are not able to extract such a feature in none of their layers. We combine our analytical study with a human evaluation to investigate the differences between deep learning networks and models that operate on the fractal feature solely. Moreover, we show the effectiveness of fractal features in applications where the object structure is crucial for the classification task. We empirically show that training a shallow network on fractal features achieves performance comparable, even superior in specific cases, to that of deep networks trained on raw data while requiring less computational resources. Fractals improved the accuracy of the classification by 30% on average while requiring up to 84% less time to train. We couple our empirical study with a complexity analysis of the computational cost of extracting the proposed fractal features, and we study its limitation.

In this paper, we prove the first Bayesian regret bounds for Thompson Sampling in reinforcement learning in a multitude of settings. We simplify the learning problem using a discrete set of surrogate environments, and present a refined analysis of the information ratio using posterior consistency. This leads to an upper bound of order $\widetilde{O}(H\sqrt{d_{l_1}T})$ in the time inhomogeneous reinforcement learning problem where $H$ is the episode length and $d_{l_1}$ is the Kolmogorov $l_1-$dimension of the space of environments. We then find concrete bounds of $d_{l_1}$ in a variety of settings, such as tabular, linear and finite mixtures, and discuss how how our results are either the first of their kind or improve the state-of-the-art., Comment: 37th Conference on Neural Information Processing Systems (NeurIPS 2023)

Quantum algorithms for ground-state energy estimation of chemical systems require a high-quality initial state. However, initial state preparation is commonly either neglected entirely, or assumed to be solved by a simple product state like Hartree-Fock. Even if a nontrivial state is prepared, strong correlations render ground state overlap inadequate for quality assessment. In this work, we address the initial state preparation problem with an end-to-end algorithm that prepares and quantifies the quality of initial states, accomplishing the latter with a new metric -- the energy distribution. To be able to prepare more complicated initial states, we introduce an implementation technique for states in the form of a sum of Slater determinants that exhibits significantly better scaling than all prior approaches. We also propose low-precision quantum phase estimation (QPE) for further state quality refinement. The complete algorithm is capable of generating high-quality states for energy estimation, and is shown in select cases to lower the overall estimation cost by several orders of magnitude when compared with the best single product state ansatz. More broadly, the energy distribution picture suggests that the goal of QPE should be reinterpreted as generating improvements compared to the energy of the initial state and other classical estimates, which can still be achieved even if QPE does not project directly onto the ground state. Finally, we show how the energy distribution can help in identifying potential quantum advantage.

In this work, we assess several deep learning strategies for hyperspectral pansharpening. First, we present a new dataset with a greater extent than any other in the state of the art. This dataset, collected using the ASI PRISMA satellite, covers about 262200 km2, and its heterogeneity is granted by randomly sampling the Earth's soil. Second, we adapted several state of the art approaches based on deep learning to fit PRISMA hyperspectral data and then assessed, quantitatively and qualitatively, the performance in this new scenario. The investigation has included two settings: Reduced Resolution (RR) to evaluate the techniques in a supervised environment and Full Resolution (FR) for a real-world evaluation. The main purpose is the evaluation of the reconstruction fidelity of the considered methods. In both scenarios, for the sake of completeness, we also included machine-learning-free approaches. From this extensive analysis has emerged that data-driven neural network methods outperform machine-learning-free approaches and adapt better to the task of hyperspectral pansharpening, both in RR and FR protocols., Comment: Project page: https://thezino.github.io/HSbenchmarkPRISMA/

Ionic pseudopotentials are widely used in classical simulations of materials to model the effective potential due to the nucleus and the core electrons. Modeling fewer electrons explicitly results in a reduction in the number of plane waves needed to accurately represent the states of a system. In this work, we introduce a quantum algorithm that uses pseudopotentials to reduce the cost of simulating periodic materials on a quantum computer. We use a qubitization-based quantum phase estimation algorithm that employs a first-quantization representation of the Hamiltonian in a plane-wave basis. We address the challenge of incorporating the complexity of pseudopotentials into quantum simulations by developing highly-optimized compilation strategies for the qubitization of the Hamiltonian. This includes a linear combination of unitaries decomposition that leverages the form of separable pseudopotentials. Our strategies make use of quantum read-only memory subroutines as a more efficient alternative to quantum arithmetic. We estimate the computational cost of applying our algorithm to simulating lithium-excess cathode materials for batteries, where more accurate simulations are needed to inform strategies for gaining reversible access to the excess capacity they offer. We estimate the number of qubits and Toffoli gates required to perform sufficiently accurate simulations with our algorithm for three materials: lithium manganese oxide, lithium nickel-manganese oxide, and lithium manganese oxyfluoride. Our optimized compilation strategies result in a pseudopotential-based quantum algorithm with a total Toffoli cost four orders of magnitude lower than the previous state of the art for a fixed target accuracy.

Current interpretability methods focus on explaining a particular model's decision through present input features. Such methods do not inform the user of the sufficient conditions that alter these decisions when they are not desirable. Contrastive explanations circumvent this problem by providing explanations of the form "If the feature $X>x$, the output $Y$ would be different''. While different approaches are developed to find contrasts; these methods do not all deal with mutability and attainability constraints. In this work, we present a novel approach to locally contrast the prediction of any classifier. Our Contrastive Entropy-based explanation method, CEnt, approximates a model locally by a decision tree to compute entropy information of different feature splits. A graph, G, is then built where contrast nodes are found through a one-to-many shortest path search. Contrastive examples are generated from the shortest path to reflect feature splits that alter model decisions while maintaining lower entropy. We perform local sampling on manifold-like distances computed by variational auto-encoders to reflect data density. CEnt is the first non-gradient-based contrastive method generating diverse counterfactuals that do not necessarily exist in the training data while satisfying immutability (ex. race) and semi-immutability (ex. age can only change in an increasing direction). Empirical evaluation on four real-world numerical datasets demonstrates the ability of CEnt in generating counterfactuals that achieve better proximity rates than existing methods without compromising latency, feasibility, and attainability. We further extend CEnt to imagery data to derive visually appealing and useful contrasts between class labels on MNIST and Fashion MNIST datasets. Finally, we show how CEnt can serve as a tool to detect vulnerabilities of textual classifiers.

In this paper we compute the automorphism group of the curves $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$ introduced in Tafazolian et al. in 2016 as new examples of maximal curves which cannot be covered by the Hermitian curve. They arise as subcovers of the first generalized GK curve (GGS curve). As a result, a new characterization of the GK curve, as a member of this family, is obtained.

Contrastive explanation methods go beyond transparency and address the contrastive aspect of explanations. Such explanations are emerging as an attractive option to provide actionable change to scenarios adversely impacted by classifiers' decisions. However, their extension to textual data is under-explored and there is little investigation on their vulnerabilities and limitations. This work motivates textual counterfactuals by laying the ground for a novel evaluation scheme inspired by the faithfulness of explanations. Accordingly, we extend the computation of three metrics, proximity,connectedness and stability, to textual data and we benchmark two successful contrastive methods, POLYJUICE and MiCE, on our suggested metrics. Experiments on sentiment analysis data show that the connectedness of counterfactuals to their original counterparts is not obvious in both models. More interestingly, the generated contrastive texts are more attainable with POLYJUICE which highlights the significance of latent representations in counterfactual search. Finally, we perform the first semantic adversarial attack on textual recourse methods. The results demonstrate the robustness of POLYJUICE and the role that latent input representations play in robustness and reliability.

While there has been a recent explosion of work on ExplainableAI ExAI on deep models that operate on imagery and tabular data, textual datasets present new challenges to the ExAI community. Such challenges can be attributed to the lack of input structure in textual data, the use of word embeddings that add to the opacity of the models and the difficulty of the visualization of the inner workings of deep models when they are trained on textual data. Lately, methods have been developed to address the aforementioned challenges and present satisfactory explanations on Natural Language Processing (NLP) models. However, such methods are yet to be studied in a comprehensive framework where common challenges are properly stated and rigorous evaluation practices and metrics are proposed. Motivated to democratize ExAI methods in the NLP field, we present in this work a survey that studies model-agnostic as well as model-specific explainability methods on NLP models. Such methods can either develop inherently interpretable NLP models or operate on pre-trained models in a post-hoc manner. We make this distinction and we further decompose the methods into three categories according to what they explain: (1) word embeddings (input-level), (2) inner workings of NLP models (processing-level) and (3) models' decisions (output-level). We also detail the different evaluation approaches interpretability methods in the NLP field. Finally, we present a case-study on the well-known neural machine translation in an appendix and we propose promising future research directions for ExAI in the NLP field.

Current Explainable AI (ExAI) methods, especially in the NLP field, are conducted on various datasets by employing different metrics to evaluate several aspects. The lack of a common evaluation framework is hindering the progress tracking of such methods and their wider adoption. In this work, inspired by offline information retrieval, we propose different metrics and techniques to evaluate the explainability of SA models from two angles. First, we evaluate the strength of the extracted "rationales" in faithfully explaining the predicted outcome. Second, we measure the agreement between ExAI methods and human judgment on a homegrown dataset1 to reflect on the rationales plausibility. Our conducted experiments comprise four dimensions: (1) the underlying architectures of SA models, (2) the approach followed by the ExAI method, (3) the reasoning difficulty, and (4) the homogeneity of the ground-truth rationales. We empirically demonstrate that anchors explanations are more aligned with the human judgment and can be more confident in extracting supporting rationales. As can be foreseen, the reasoning complexity of sentiment is shown to thwart ExAI methods from extracting supporting evidence. Moreover, a remarkable discrepancy is discerned between the results of different explainability methods on the various architectures suggesting the need for consolidation to observe enhanced performance. Predominantly, transformers are shown to exhibit better explainability than convolutional and recurrent architectures. Our work paves the way towards designing more interpretable NLP models and enabling a common evaluation ground for their relative strengths and robustness., Comment: 13 pages, 3 figures, conference (AIAI - springer)

The co-emergence of locality between the Hamiltonian and initial state of the universe is studied in a simple toy model. We hypothesize a fundamental loss functional for the combined Hamiltonian and quantum state and minimize it by gradient descent. This minimization yields a tensor product structure simultaneously respected by both the Hamiltonian and the state, suggesting that locality can emerge by a process analogous to spontaneous symmetry breaking. We discuss the relevance of this program to the arrow of time problem. In our toy model, we interpret the emergence of a tensor factorization as the appearance of individual degrees of freedom within a previously undifferentiated (raw) Hilbert space. Earlier work [5, 6] looked at the emergence of locality in Hamiltonians only, and found strong numerical confirmation of that raw Hilbert spaces of $\dim = n$ are unstable and prefer to settle on tensor factorization when $n=pq$ is not prime, and in [6] even primes were seen to "factor" after first shedding a small summand, e.g. $7=1+2\cdot 3$. This was found in the context of a rather general potential functional $F$ on the space of metrics $\{g_{ij}\}$ on $\mathfrak{su}(n)$, the Lie algebra of symmetries. This emergence of qunits through operator-level spontaneous symmetry breaking (SSB) may help us understand why the world seems to consist of myriad interacting degrees of freedom. But understanding why the universe has an initial Hamiltonian $H_0$ with a many-body structure is of limited conceptual value unless the initial state, $|\psi_0\rangle$, is also structured by this tensor decomposition. Here we adapt $F$ to become a functional on $\{g,|\psi_0\rangle\}=(\text{metrics})\times (\text{initial states})$, and find SSB now produces a conspiracy between $g$ and $|\psi_0\rangle$, where they simultaneously attain low entropy by settling on the same qubit decomposition.

Scattered polynomials over finite fields attracted an increasing attention in the last years. One of the reasons is their deep connection with Maximum Rank Distance (MRD) codes. Known classification results for exceptional scattered polynomials, i.e. polynomials which are scattered over infinite field extensions, are limited to the cases where their index $\ell$ is small, or a prime number larger than the $q$-degree $k$ of the polynomial, or an integer smaller than the $k$ in the case where $k$ is a prime. In this paper we completely classify exceptional scattered polynomials when the maximum between $\ell$ and $k$ is odd, and give partial results when it is even, extending a result of Ferraguti and Micheli in 2021.

There is a pressing need to develop new rechargeable battery technologies that can offer higher energy storage, faster charging, and lower costs. Despite the success of existing methods for the simulation of battery materials, they can sometimes fall short of delivering accurate and reliable results. Quantum computing has been discussed as an avenue to overcome these issues, but only limited work has been done to outline how they may impact battery simulations. In this work, we provide a detailed answer to the following question: how can a quantum computer be used to simulate key properties of a lithium-ion battery? Based on recently-introduced first-quantization techniques, we lay out an end-to-end quantum algorithm for calculating equilibrium cell voltages, ionic mobility, and thermal stability. These can be obtained from ground-state energies of materials, which is the core calculation executed by the quantum computer using qubitization-based quantum phase estimation. The algorithm includes explicit methods for preparing approximate ground states of periodic materials in first quantization. We bring these insights together to perform the first estimation of the resources required to implement a quantum algorithm for simulating a realistic cathode material, dilithium iron silicate., Comment: 31 pages, 14 figures

We introduce a camera pipeline for rendering visually pleasing photographs in low light conditions, as part of the NTIRE2022 Night Photography Rendering challenge. Given the nature of the task, where the objective is verbally defined by an expert photographer instead of relying on explicit ground truth images, we design an handcrafted solution, characterized by a shallow structure and by a low parameter count. Our pipeline exploits a local light enhancer as a form of high dynamic range correction, followed by a global adjustment of the image histogram to prevent washed-out results. We proportionally apply image denoising to darker regions, where it is more easily perceived, without losing details on brighter regions. The solution reached the fifth place in the competition, with a preference vote count comparable to those of other entries, based on deep convolutional neural networks. Code is available at www.github.com/AvailableAfterAcceptance.

In this paper we investigate some properties of ideals in group algebras of finite groups over fields. First, we highlight an important link between their dimension, their minimal Hamming distance and the group order. This is a generalized version of an uncertainty principle shown in 1992 by Meshulam. Secondly, we introduce the notion of the Schur product of ideals in group algebras and investigate the module structure and the dimension of the Schur square. We give a structural result on ideals that coincide with their Schur square, and we provide conditions for an ideal to be such that its Schur square has the projective cover of the trivial module as a direct summand. This has particularly interesting consequences for group algebras of p-groups over fields of characteristic p., Comment: 12 pages

Several recent works on self-supervised learning are trained by mapping different augmentations of the same image to the same feature representation. The data augmentations used are of crucial importance to the quality of learned feature representations. In this paper, we analyze how the color jitter traditionally used in data augmentation negatively impacts the quality of the color features in learned feature representations. To address this problem, we propose a more realistic, physics-based color data augmentation - which we call Planckian Jitter - that creates realistic variations in chromaticity and produces a model robust to illumination changes that can be commonly observed in real life, while maintaining the ability to discriminate image content based on color information. Experiments confirm that such a representation is complementary to the representations learned with the currently-used color jitter augmentation and that a simple concatenation leads to significant performance gains on a wide range of downstream datasets. In addition, we present a color sensitivity analysis that documents the impact of different training methods on model neurons and shows that the performance of the learned features is robust with respect to illuminant variations., Comment: Accepted at Eleventh International Conference on Learning Representations (ICLR 2023)

The tensor rank of some Gabidulin codes of small dimension is investigated. In particular, we determine the tensor rank of any rank metric code equivalent to an $8$-dimensional $\mathbb{F}_q$-linear generalized Gabidulin code in $\mathbb{F}_{q}^{4\times4}$. This shows that such a code is never minimum tensor rank. In this way, we detect the first infinite family of Gabidulin codes which are not minimum tensor rank.

In parts I [6] and II [7] we studied how metrics $g_{ij}$ on $\mathfrak{su}(n)$ may spontaneously break symmetry and $\textit{crystallize}$ into a form which is kaq, $\textit{knows about qubits}$. We did this for $n=2^N$ and then away from powers of 2. Here we address the Fermionic version and find kam metrics, these $\textit{know about Majoranas}$. That is, there is a basis of principal axes $\{H_k\}$ of which is of homogeneous Majorana degree. In part I, we searched unsuccessfully for functional minima representing crystallized metrics exhibiting the Brown-Susskind penalty schedule, motivated by their study of black hole scrambling time. Here, by segueing to the Fermionic setting we find, to good approximation, kam metrics adhering to this schedule on both $\mathfrak{su}(4)$ and $\mathfrak{su}(8)$. Thus, with this preliminary finding, our toy model exhibits two of the three features required for the spontaneous emergence of spatial structure: 1. localized degrees of freedom and, 2. a preference for low body-number (or low Majorana number) interactions. The final feature, 3. constraints on who may interact with whom, i.e. a neighborhood structure, must await an effective analytic technique, being entirely beyond what we can approach with classical numerics.