Search arXiv⌕ Search

arXiv subjects

Steven Rayan

Publications and source records attributed to Steven Rayan.

At least 19 recordsLinked to original sources
arXiv2026

Rethinking Quantum Circuits

These notes develop four interconnected ways of reading a quantum circuit. A circuit for us begins as an operational composition of gates; then, it becomes a diagram whose local equalities may be used as calculations; next, it becomes a protected process once errors, syndromes, and logical degrees of freedom are separated; and finally, it becomes geometric when its connectivity, topology, and boundary data are treated as physical design parameters. The development begins at the level of bits and qubits before appealing to Deutsch's and Grover's algorithms as basic examples of quantum circuits. With the basics in hand, we interpret quantum circuits diagramatically, leading us to compact closed string diagrams and the ZX-calculus. After that, we consider how to correct quantum circuits by introducing the Knill--Laflamme condition, homological surface codes, and related concepts with a view towards thinking of these as operations on diagrams. The lectures eventually arrive at the properties of hyperbolic quantum codes and the prospect of physical superconducting circuits emulating the negatively-curved lattices needed to support those codes. These mathematical ideas and physical experiments, taken together, represent one way to impart a geometric layer onto quantum circuits. By the very end, we bring the ideas nearly full circle by assessing the extent to which these device physics experiments operationalize the basic ZX diagrams encountered much earlier in the story. While the later material reports on original research, and while the discussion becomes increasingly mathematical as the sections progress, no prior knowledge of quantum information, quantum computing, or quantum error correction is actually assumed.

quant-ph↗
arXiv2026

Beyond $K$-Theory: Geometry and Holomorphy in Hyperbolic Band Theory

Topological $K$-theory supplies the decisive stable classification principle for gapped free-fermion phases in Euclidean crystals once symmetry and stabilization are fixed. In hyperbolic band theory, by contrast, it is less decisive for the full band problem: generalized momentum sectors vary through moduli spaces, but passing to ordinary $K$-classes collapses their geometric and holomorphic variation. We distinguish kinematical holomorphy, in which complex geometry organizes the sector spaces, from dynamical holomorphy, in which that geometry informs a Hamiltonian or projector, and formulate the resulting loss as an observable-factorization problem. For a compact hyperbolic surface $X$, we prove nonfactorization in three settings: fibrewise $K^0(X)$, occupied-state $K^0(B)$, and operator-algebraic $K$-theory. The affected quantities include spectra and Higgs spectral curves, the Berry holonomy and the quantum metric, the partially filled Hall response, and the Fermi surface and nodal geometries. We also identify the quantized pairings and local charges retained by topology. Up to an explicit area factor, the Kotani--Sunada bottom-band Hessian is the Hodge inner product on $H^1(X;\mathbb R)$. Together with the integral intersection form, it recovers the homology-marked principally polarized Jacobian and hence, by Torelli and uniformization, the underlying complex and hyperbolic surface, though not a full Teichm\"uller marking. We also separate finite-rank sectors from the thermodynamic bulk. Locally faithful covers reproduce polynomial traces exactly and control continuous spectral observables. In arithmetic congruence towers, analytic observables converge as $O(|G_n|^{-\eta})$ for some $\eta>0$, and $C^s$ observables as $O((\log |G_n|)^{-s})$, while established coherent large-rank limits recover bulk density-of-states moments.

math-ph↗
arXiv2026

Diffeological non-Abelian Hodge theory: relative harmonic metrics and deformation theory

Let $X$ be a compact K\"ahler manifold. In prior work, we constructed diffeological moduli stacks of Higgs and flat bundles on $X$, related by extension completion of smooth harmonic families. Here, we develop the relative analytic theory. On Sobolev completions over arbitrary plots, we prove that every smooth stable Higgs family satisfying the numerical conditions admits a global smooth harmonic metric. Fixing a Hermitian--Einstein determinant metric removes scalar freedom, and then elliptic regularity and normalized gluing yield plotwise smoothness. The theorem holds at every finite parameter regularity $C^d$ and on reduced singular parameter spaces with ambient extensions. For a Higgs deformation $\eta$, the normalized metric variation satisfies $L_hs=-\mathcal S_h(\eta)$ and $s=-G_h\mathcal S_h(\eta)$ up to an independent rank-one determinant term for $\mathrm{GL}_r$. This computes the plotwise differential and recovers the classical comparison. Locally split, constant-type polystable families admit smooth harmonic metrics. Real-analytic examples show general polystable families may have neither continuous harmonic metrics nor relative harmonic filtrations and may lie outside every $C^d$ extension-generated locus. In one example a singular harmonic reduction produces a continuous adjoint Higgs field and a flat family with semisimple slices. This defines a weak $C^0$ operator-level harmonic mediator, strictly larger than the metric-regular one, whose endpoint images after finite extension completion and stackification satisfy $\mathscr M_{\mathrm{Dol},0}^{\mathrm{wk}\mathcal H}(X)\simeq\mathscr M_{\mathrm{dR},0}^{\mathrm{wk}\mathcal H}(X)$. We characterize the extension-generated stack by relative harmonic filtrations, develop their obstruction theory, analyze the loss of extension data under heat flow, and construct the smooth Hodge $\lambda$-family on the stable locus.

math.DG↗
arXiv2026

A diagrammatic field theory of quantum error correction

We develop a field-theoretic framework for quantum error correction centred on fusion-space codes in unitary fusion categories. Admissible clusters determine total-charge sectors and orthogonal footprint projectors recording locally visible data left by error histories. The central distinction is between diagnostic footprint algebras and syndrome-admissible commuting algebras: the latter can be measured without revealing logical information and resolve chosen error representatives into measured sectors. For such algebras, exact correctability is equivalent to fibrewise Knill--Laflamme conditions, yielding a measure-then-recover factorization. Under a contractible-vacuum locality hypothesis, closed neutral composites give a categorical sufficient criterion for scalar action on the code. In the Ising theory, four $\sigma$ punctures show that pair-charge footprints can be complementary logical diagnostics and realize an exact one-qubit Clifford shadow. A proper six-$\sigma$ code instead admits a syndrome-admissible pair-charge measurement and exact recovery from an explicit Majorana bilinear error. A second bilinear has the same measured footprint but differs by a logical bit flip, producing a concrete nontrivial footprint fibre and genuine decoding ambiguity. We also formulate conformal-block likelihood data and compute geometry-dependent Ising four-point weights. For growing code families, we prove a conditional Peierls-type threshold theorem: bounded connected-region growth, local stochastic noise, local neutralizability of small residual components, and componentwise decoder balance imply $\Pr_L(\mathrm{fail})\le C|\Omega_L|e^{-cL}$ below a nonzero constant error rate. We conclude with representation-theoretic and algebro-geometric directions involving tube and Hopf algebras, Yangian-type structures, Higgs bundles, spectral curves, Jacobians, and abelian varieties.

quant-ph↗
arXiv2026

Exploring the Effects of Entanglement on Quantum Machine Learning of Pathogen Epitope-Receptor Binding

Parameterized quantum circuits (PQCs) provide a flexible substrate for hybrid quantum machine learning (QML), but their practical value on Noisy Intermediate-Scale Quantum (NISQ) devices remains an empirical question, especially because training depth and scale can introduce optimization challenges such as barren plateaus. Here we study how the number and topology of two-qubit entangling gates in the feature-map stage influence a fixed hybrid QNN workflow for classifying strong versus weak epitope-receptor binding in Porcine Reproductive and Respiratory Syndrome (PRRS) vaccine design. The dataset consists of docking-derived binding affinities for N=80 9-mer epitopes, labeled as Strong or Weak binding, and partitioned into training, validation, and test subsets using a 40:30:30 split. We compare a classical CNN benchmark with a hybrid Embedding-QNN architecture under four feature-map configurations: a non-entangling Z feature map, an all-to-all high-entanglement ZZ feature map, and two interleaved nearest-neighbour entanglement patterns of low and high depth. Among the configurations tested, the high-entanglement ZZ feature map is seen to provide the strongest evidence of reduced training-set overfit, with a lower training area under the accuracy curve (AUAC) and the highest test/training AUAC ratio, while preserving competitive test-set accuracy. These results do not establish a general QML advantage, but they suggest that feature-map entanglement topology is a meaningful design variable for sparse biological screening tasks and warrants further evaluation with additional metrics, larger datasets, and noise-aware or hardware-based experiments.

quant-ph↗
arXiv2026

Transformer-Based Active Learning for Data-Efficient Vaccine Epitope Selection in PRRS

High-fidelity molecular docking simulations can produce biologically relevant estimates of epitope-receptor binding affinity but are computationally expensive and therefore limit the number of candidates that can be screened for vaccine design. In this work, we evaluate machine learning (ML) approaches where variants of active learning are used to classify instances of high binding affinity between 9-mer epitopes and a well-conserved swine leukocyte antigen (SLA) receptor in the context of Porcine Reproductive and Respiratory Syndrome (PRRS). We use an internally generated dataset of 80 epitope-SLA docking affinities, each requiring more than 48 hours of high-performance computing (HPC). Multiple model families (linear, MLP, CNN, and a small transformer) are trained under strict low-data conditions within a pool-based active learning loop. In each case, optimal model configurations are identified by conducting large-scale hyperparameter optimization over the combined space of model architecture, training configuration, acquisition policy, and ensemble decision rules. To mitigate the effects of data subsample selection, each candidate configuration is evaluated by averaging performance over many randomized and balanced training and validation data subsets. Across experiments, transformer-based sequence models consistently emerged as the best-performing architecture, with active incremental learning yielding significant improvement over a baseline random sample acquisition strategy. Under moderate training data availability (N=30), the optimized ML-model configuration outperforms a standard baseline trained on twice the amount of data. Under higher training data availability (N=60), the same configuration achieves a peak accuracy of 86.8%, consistent with an upper bound of 85% classification accuracy based on two independent estimates of conformational noise.

q-bio.BM↗
arXiv2026

A QUBO Formulation for Nowhere-Zero $k$-Flows

We consider the encoding of graph problems as Quadratic Unconstrained Binary Optimization (QUBO) problems, which are solvable by either quantum or classical annealers. Yet, the class of problems encoded as QUBO problems has not previously included nowhere-zero flows. Nowhere-zero flows are related to Tutte's $5$-flow conjecture and appear in many contexts in graph theory. We provide an encoding of nowhere-zero flows as a QUBO Hamiltonian and prove the correctness of the construction. Our construction yields a Hamiltonian $H_{\mathrm{mod},k}$ whose ground state has zero energy if and only if the graph $G$ has a nowhere-zero $\mathbb Z_k$-flow. By Tutte's equivalence theorem, zero ground energy is equivalent to $\varphi(G)\le k$, and the zero-energy degeneracy is given by the flow polynomial $F(G;k)$. In particular, when the ground-state energy is zero, this is also the ground-state degeneracy. The construction uses one-hot variables to represent the edge flow residues modulo $k$ and auxiliary variables to represent the per-vertex modular quotient. We prove that the correctness of the construction is independent of the choice of orientation, root vertex, and positive penalty weights. We verify the construction on $59$ examples of graphs and values of $k$ that include both yes-instances and no-instances. We exhaustively sweep orientations and root choices on selected robustness instances and test a finite suite of positive penalty weights. The resulting Hamiltonian is implemented using the dimod.BinaryQuadraticModel class, which is compatible with the D-Wave Ocean SDK. Quantum-hardware runs and claims about potential speedup using these devices are left to follow-up work.

quant-ph↗
arXiv2026

A diffeological perspective on non-Abelian Hodge theory

We construct diffeological moduli stacks $\mathscr{M}_{Dol}(X)$ and $\mathscr{M}_{dR}(X)$ parametrizing smooth families of Higgs bundles and those of flat bundles, respectively, on a compact K\"ahler manifold $X$. We then establish an equivalence of stacks between diffeological substacks $\mathscr{M}^\mathscr{H}_{Dol} \subset \mathscr{M}_{Dol}(X)$ and $\mathscr{M}^\mathscr{H}_{dR}(X) \subset \mathscr{M}_{dR}(X)$, whose fibres over the point are the categories of semistable Higgs bundles, with the usual condition on Chern classes, and of flat bundles on $X$, respectively. $\mathscr{M}^\mathscr{H}_{Dol}(X)$ contains families of semistable Higgs bundles to points of which the classical correspondence of coarse moduli spaces does not extend continuously. This shows that the equivalence we provide is, in a sense, a common extension, in the context of diffeological moduli stacks, of both the homeomorphism of coarse moduli spaces, and of the equivalence of categories between semistable Higgs bundles and arbitrary flat bundles.

math.DG↗
arXiv2026

Hyperbolic Cluster States for Fault-Tolerant Measurement-Based Quantum Computing

Fault-tolerant measurement-based quantum computing (MBQC) provides a compelling framework for fault-tolerant quantum computation, in which quantum information is processed through single-qubit measurements on a three-dimensional entangled resource known as cluster state. To date, this resource has been predominantly studied on Euclidean lattices, most notably in the Raussendorf-Harrington-Goyal (RHG) construction, which underlies topological fault tolerance in MBQC. In this work, we introduce the hyperbolic cluster state, a generalization of the three-dimensional cluster state to negatively curved geometries, obtained via the foliation of periodic hyperbolic lattices. We present an explicit construction of hyperbolic cluster states and investigate their fault-tolerant properties under a realistic circuit-level depolarizing noise model. Using large-scale numerical simulations, we perform memory experiments to characterize their logical error rates and decoding performance. Our results demonstrate that hyperbolic cluster states exhibit a fault-tolerance threshold comparable to that of the Euclidean RHG cluster state, while simultaneously supporting a constant encoding rate in the thermodynamic limit. This represents a substantial improvement in qubit overhead relative to conventional cluster-state constructions. These findings establish hyperbolic geometry as a powerful and experimentally relevant resource for scalable, fault-tolerant MBQC and open new avenues for leveraging negative curvature in quantum information processing.

quant-ph↗
arXiv2026

Quantum Entanglement, Stratified Spaces, and Topological Matter: Towards Entanglement-Sensitive Langlands Data

Using the spinless Haldane model, we study the witness-filtered Berry curvature, quantum geometric tensor, and quantum Fisher information on the gapped strata of the parameter space and evaluate them through the Fukui-Hatsugai-Suzuki discretization. The filtered quantities isolate the part of the geometric response carried by sublattice coherence: they suppress contributions from regions where the occupied Bloch state is locally A/B-separable and emphasize regions where curvature and coherence coexist. We derive exact lattice identities, reconstruction formulas for the curvature-weighted coherence, and bounds relating the filtered quantum geometric tensor and quantum Fisher information to single-particle mode entanglement. Across the gap-closing stratum, the quantized response changes admit a natural description in terms of Hecke modifications. We elicit a corresponding Langlands viewpoint -- not as a full correspondence, but as an organizational principle and as the mathematical shadow of these physical geometric constructions.

quant-ph↗
arXiv2025

Moduli stacks of quiver connections and non-Abelian Hodge theory

In arXiv:2407.11958, a moduli stack parametrizing $I$--indexed diagrams of Higgs bundles over a base stack $X$ was constructed for any finite simplicial set $I$, inspiring speculations about extending the non-Abelian Hodge correspondence to these moduli stacks. In the present work, we formalize the de Rham side of this conjectural extension. We construct moduli stacks parametrizing diagrams of bundles with $\lambda$--connections over a base prestack $X$, where $\lambda$ can be a fixed number or a parameter. Taking $\lambda$ to be $1$ gives a moduli stack parametrizing diagrams of bundles with connection, while taking it to be a parameter gives a version of Simpson's non-Abelian Hodge filtration for digrams of bundles with connection. We show that when $X$ is a smooth and projective scheme over an algebraically closed field $k$ of characteristic $0$, these moduli stacks are algebraic and locally of finite presentation, and have affine diagonal.

math.AG↗
arXiv2025

A Scalable Superconducting Circuit Framework for Emulating Physics in Hyperbolic Space

Theoretical studies and experiments in the last six years have revealed the potential for novel behaviours and functionalities in device physics through the synthetic engineering of negatively-curved spaces. For instance, recent developments in hyperbolic band theory have unveiled the emergence of higher-dimensional eigenstates -- features fundamentally absent in conventional Euclidean systems. At the same time, superconducting quantum circuits have emerged as a leading platform for quantum analogue emulations and digital simulations in scalable architectures. Here, we introduce a scalable superconducting circuit framework for the analogue quantum emulation of tight-binding models on hyperbolic and kagome-like lattices. Using this approach, we experimentally realize three distinct lattices, including, for the first time to our knowledge, a hyperbolic lattice whose unit cell resides on a genus-3 Riemann surface. Our method encodes the hyperbolic metric directly into capacitive couplings between high-quality superconducting resonators, enabling tenable reproduction of spectral and localization properties while overcoming major scalability and spectral resolution limitations of previous designs. These results set the stage for large-scale experimental studies of hyperbolic materials in condensed matter physics and lay the groundwork for realizing hyperbolic quantum processors, with potential implications for both fundamental physics and quantum computing

quant-ph↗
arXiv2025

Reconstructing Biological Pathways by Applying Selective Incremental Learning to (Very) Small Language Models

The use of generative artificial intelligence (AI) models is becoming ubiquitous in many fields. Though progress continues to be made, general purpose large language AI models (LLM) show a tendency to deliver creative answers, often called "hallucinations", which have slowed their application in the medical and biomedical fields where accuracy is paramount. We propose that the design and use of much smaller, domain and even task-specific LM may be a more rational and appropriate use of this technology in biomedical research. In this work we apply a very small LM by today's standards to the specialized task of predicting regulatory interactions between molecular components to fill gaps in our current understanding of intracellular pathways. Toward this we attempt to correctly posit known pathway-informed interactions recovered from manually curated pathway databases by selecting and using only the most informative examples as part of an active learning scheme. With this example we show that a small (~110 million parameters) LM based on a Bidirectional Encoder Representations from Transformers (BERT) architecture can propose molecular interactions relevant to tuberculosis persistence and transmission with over 80% accuracy using less than 25% of the ~520 regulatory relationships in question. Using information entropy as a metric for the iterative selection of new tuning examples, we also find that increased accuracy is driven by favoring the use of the incorrectly assigned statements with the highest certainty (lowest entropy). In contrast, the concurrent use of correct but least certain examples contributed little and may have even been detrimental to the learning rate.

q-bio.MN↗
arXiv2025

Spectral coverings without embeddings

In this article, we investigate a weakened version of the spectral correspondence for twisted Higgs bundles. Namely, we construct twisted Higgs bundles from a finite covering map and a vector bundle on that covering but without requiring that they match the eigen-data for some fixed twisted Higgs bundle. We investigate stability for twisted Higgs bundles constructed in this way, and compare our covering data to that of the traditional spectral cover.

math.AG↗
arXiv2025

GPU-accelerated Modeling of Biological Regulatory Networks

The complex regulatory dynamics of a biological network can be succinctly captured using discrete logic models. Given even sparse time-course data from the system of interest, previous work has shown that global optimization schemes are suitable for proposing logic models that explain the data and make predictions about how the system will behave under varying conditions. Considering the large scale of the parameter search spaces associated with these regulatory systems, performance optimizations on the level of both hardware and software are necessary for making this a practical tool for in silico pharmaceutical research. We show here how the implementation of these global optimization algorithms in a GPU-computing environment can accelerate the solution of these parameter search problems considerably. We carry out parameter searches on two model biological regulatory systems that represent almost an order of magnitude scale-up in complexity, and we find the gains in efficiency from GPU to be a 33%-43% improvement compared to multi-thread CPU implementations and a 33%-1866% increase compared to CPU in serial. These improvements make global optimization of logic model identification a far more attractive and feasible method for in silico hypothesis generation and design of experiments.

q-bio.MN↗
arXiv2025

Identifying Protein Co-regulatory Network Logic by Solving B-SAT Problems through Gate-based Quantum Computing

There is growing awareness that the success of pharmacologic interventions on living organisms is significantly impacted by context and timing of exposure. In turn, this complexity has led to an increased focus on regulatory network dynamics in biology and our ability to represent them in a high-fidelity way, in silico. Logic network models show great promise here and their parameter estimation can be formulated as a constraint satisfaction problem (CSP) that is well-suited to the often sparse, incomplete data in biology. Unfortunately, even in the case of Boolean logic, the combinatorial complexity of these problems grows rapidly, challenging the creation of models at physiologically-relevant scales. That said, quantum computing, while still nascent, facilitates novel information-processing paradigms with the potential for transformative impact in problems such as this one. In this work, we take a first step at actualizing this potential by identifying the structure and Boolean decisional logic of a well-studied network linking 5 proteins involved in the neural development of the mammalian cortical area of the brain. We identify the protein-protein connectivity and binary decisional logic governing this network by formulating it as a Boolean Satisfiability (B-SAT) problem. We employ Grover's algorithm to solve the NP-hard problem faster than the exponential time complexity required by deterministic classical algorithms. Using approaches deployed on both quantum simulators and actual noisy intermediate scale quantum (NISQ) hardware, we accurately recover several high-likelihood models from very sparse protein expression data. The results highlight the differential roles of data types in supporting accurate models; the impact of quantum algorithm design as it pertains to the mutability of quantum hardware; and the opportunities for accelerated discovery enabled by this approach.

quant-ph↗
arXiv2025

Systematic Approach to Hyperbolic Quantum Error Correction Codes

Quantum error correction codes defined on hyperbolic lattices leverage the unique geometric properties of the hyperbolic space to enhance the performance of quantum error correction. By embedding qubits in hyperbolic lattices, these codes achieve higher encoding rates and lower qubit overhead compared to those defined on conventional Euclidean lattices. Building on recent advances in hyperbolic crystallography, we introduce a unified framework for the systematic construction and scalable benchmarking of CSS quantum error correction codes on hyperbolic lattices. A central component of this framework is the Hyperbolic Cycle Basis algorithm, which employs graph-theoretic methods to efficiently identify all plaquette cycles (parity-check supports) and nontrivial cycles (logical operators). This enables scalable and automated benchmarking of a broad class of CSS codes defined on hyperbolic geometries. We apply this framework to construct and simulate two representative hyperbolic quantum error correction codes (HQECCs), evaluating key performance metrics such as encoding rate, error threshold, and code distance for different sublattices. While HQECCs serve as concrete examples, the framework can be adapted to a wide range of CSS codes, including those with more intricate stabilizer structures such as Floquet codes. This work establishes a foundation for systematic exploration and benchmarking of CSS codes on hyperbolic lattices, paving the way toward practical, high-performance quantum error correction.

quant-ph↗
arXiv2025

Direct entanglement ansatz learning (DEAL) with ZNE on error-prone superconducting qubits

Quantum combinatorial optimization algorithms typically face challenges due to complex optimization landscapes featuring numerous local minima, exponentially scaling latent spaces, and susceptibility to quantum hardware noise. In this study, we introduce Direct Entanglement Ansatz Learning (DEAL), wherein we employ a direct mapping from quadratic unconstrained binary problem parameters to quantum ansatz angles for cost and mixer hamiltonians, which improves the convergence rate towards the optimal solution. Our approach exploits a quantum entanglement-based ansatz to effectively explore intricate latent spaces and zero noise extrapolation (ZNE) to greatly mitigate the randomness caused by crosstalk and coherence errors. Our experimental evaluation demonstrates that DEAL increases the success rate by up to 14% compared to the classic quantum approximation optimization algorithm while also controlling the error variance. In addition, we demonstrate the capability of DEAL to provide near optimum ground energy solutions for travelling salesman, knapsack, and maxcut problems, which facilitates novel paradigms for solving relevant NP-hard problems and extends the practical applicability of quantum optimization using noisy quantum hardware.

quant-ph↗