arXiv subjects

Cihan Okay

Publications and source records attributed to Cihan Okay.

At least 19 recordsLinked to original sources

arXiv2026

Possibilistic collapse and extremality of simplicial distributions

Consistent families of locally defined probability distributions that do not admit a joint global distribution are known as contextual, with primary examples arising in quantum theory. In this paper, we study such families of distributions using the theory of simplicial distributions, and further develop the theory for possibilistic distributions defined over the Boolean semiring. We characterize possibilistic collapses of simplicial distributions geometrically using bundle scenarios. Using this characterization together with a new connectivity condition on the total space of a bundle scenario, we provide a criterion for detecting extremal simplicial distributions. In parallel, we develop an analogous theory for presheaves on simplicial complexes, describe possibilistic collapses of empirical models on them using event scenarios together with a categorical extremality condition, and relate the two frameworks via a comparison result. We provide examples of contextual simplicial distributions that arise from our criteria on scenarios of interest in quantum foundations, such as Bell scenarios and boundaries of standard simplices, the latter connecting to Vorob'ev's classical theorem on acyclic complexes.

math.CT↗

arXiv2026

Vertex structure of fiber products of probability polytopes

We develop tools for characterizing vertices of fiber products of polytopes and apply them to simplicial distribution polytopes, a class of probability polytopes arising in quantum foundations and quantum information. In the theory of simplicial distributions, a pair of simplicial sets encoding measurement and outcome spaces determines a convex polytope of compatible probability assignments. Our first results give geometric criteria for detecting vertices of fiber products in terms of support data. These results are obtained in the more general framework of inverse limits of diagrams of polytopes in standard form, and they translate to corresponding criteria for simplicial distributions on arbitrary colimits of measurement spaces. We then focus on one-dimensional measurement spaces, where simplicial distributions recover and generalize local marginal polytopes in graphical models. In this setting, our sharpest results concern dipole graphs, for which we obtain a complete characterization of vertices and refine it to a graph-theoretic criterion. These characterizations are reminiscent of the classical support-graph criteria for transportation polytopes, but they arise in a richer class of polytopes in which vertex structure depends not only on support acyclicity but also on additional geometric compatibility data. Using the collapsing method from simplicial topology, we transfer the dipole characterization to rose graphs and obtain analogous results there. Finally, we apply collapsing to complete bipartite graphs, which encode physically relevant bipartite Bell scenarios, and more generally to arbitrary connected graphs, yielding lower bounds on the number of vertices.

math.CO↗

arXiv2026

No quantum solutions to linear constraint systems from monomial measurement-based quantum computation in odd prime dimension

We combine the study of resources in measurement-based quantum computation (MBQC) with that of quantum solutions to linear constraint systems (LCS). Contextuality of the input state in MBQC has been identified as a key resource for quantum advantage, and in a stronger form, underlies algebraic relations between (measurement) operators which obey classically unsatisfiable (linear) constraints. Here, we compare these two perspectives on contextuality, and study to what extent they are related. More precisely, we associate a LCS to certain MBQC which exhibit strong forms of state-dependent contextuality, and ask if the measurement operators in such MBQC give rise to state-independent contextuality in the form of quantum solutions of its associated LCS. Our main result rules out such quantum solutions for a large class of MBQC. This both sharpens the distinction between state-dependent and state-independent forms of contextuality, and further generalises results on the non-existence of quantum solutions to LCS in finite odd (prime) dimension.

quant-ph↗

arXiv2025

Double categories for adaptive quantum computation

Quantum computation admits several models that emphasize different computational primitives and forms of classical control. We develop a unified double categorical framework for describing these models and the conversions between them. The syntax is provided by double port graphs, whose horizontal wires carry quantum information and whose vertical wires carry classical information and control. For each set of port labels, these graphs form a double category, and this construction is functorial in the label set. The semantics is given by the one-object double category of adaptive instruments. Its associated horizontal and vertical monoidal categories recover, respectively, quantum channels and stochastic maps. An assignment of an adaptive instrument to each primitive label therefore extends canonically to a double functor on labeled double port graphs, providing their computational semantics. We apply this framework to prominent models of quantum computation, including the circuit model, measurement-based quantum computation, quantum computation with magic states, and measurement-based Pauli computation. Gadget constructions from quantum computing that implement conversions between these models become double functors. Finally, we show that the interaction between quantum operations and affine classical control in measurement-based Pauli computation realizes every Boolean function in the vertical direction, thereby providing the non-affine classical operations required for its simulation of the circuit model.

quant-ph↗

arXiv2025

Polyhedral Classical Simulators for Quantum Computation

Quantum advantage in computation refers to the existence of computational tasks that can be performed efficiently on a quantum computer but cannot be efficiently simulated on any classical computer. Identifying the precise boundary of efficient classical simulability is a central challenge and motivates the development of new simulation paradigms. In this paper, we introduce polyhedral classical simulators, a framework for classical simulation grounded in polyhedral geometry. This framework encompasses well-known methods such as the Gottesman-Knill algorithm, while also extending naturally to more recent models of quantum computation, including those based on magic states and measurement-based quantum computation. We show how this framework unifies and extends existing simulation methods while at the same time providing a geometric roadmap for pushing the boundary of efficient classical simulation further.

quant-ph↗

arXiv2025

Phase space tableau simulation for quantum computation

We introduce a novel tableau-based classical simulation method for quantum computation, formulated within the phase space framework of the extended stabilizer theory of closed non-contextual operators. This method enables the efficient classical simulation of a broader class of quantum circuits beyond the stabilizer formalism. We implement the simulator and benchmark its performance on basic quantum algorithms, including the hidden shift and Deutsch--Jozsa algorithms.

quant-ph↗

arXiv2025

Simplicial methods in the resource theory of contextuality

We develop a resource theory of contextuality within the framework of symmetric monoidal categories, extending recent simplicial approaches to quantum contextuality. Building on the theory of simplicial distributions, which integrates homotopy-theoretic structures with probability, we introduce event scenarios as a functorial generalization of presheaf-theoretic measurement scenarios and prove their equivalence to bundle scenarios via the Grothendieck construction. We define symmetric monoidal structures on these categories and extend the distribution functor to a stochastic setting, yielding a resource theory that generalizes the presheaf-theoretic notion of simulations. Our main result characterizes convex maps between simplicial distributions in terms of non-contextual distributions on a corresponding mapping scenario, enhancing and extending prior results in categorical quantum foundations.

quant-ph↗

arXiv2025

Simplicial effects and weakly associative partial groups

In this paper, we introduce a new category of simplicial effects that extends the categories of effect algebras and their multi-object counterpart, effect algebroids. Our approach is based on relaxing the associativity condition satisfied by effect algebras and, more generally, partial monoids. Within this framework, simplicial effects and weakly associative partial groups arise as two extreme cases in the category of weak partial monoids. Our motivation is to capture simplicial structures from the theory of simplicial distributions and measurements that behave like effects.

math.CT↗

arXiv2024

Classical simulation of universal measurement-based quantum computation using multipartite Bell scenarios

We introduce a new classical simulation algorithm based on non-signaling polytopes of multipartite Bell scenarios, capable of simulating universal measurement-based quantum computation with single-qubit Pauli measurements. In our model, the simultaneous presence of non-stabilizerness and entanglement is necessary for quantum speedup. The region of quantum states that can be efficiently simulated includes the Bell polytope and extends beyond what is currently achievable by sampling algorithms based on phase space methods.

quant-ph↗

arXiv2024

Extremal simplicial distributions on cycle scenarios with arbitrary outcomes

Cycle scenarios are a significant class of contextuality scenarios, with the Clauser-Horne-Shimony-Holt (CHSH) scenario being a notable example. While binary outcome measurements in these scenarios are well understood, the generalization to arbitrary outcomes remains less explored, except in specific cases. In this work, we employ homotopical methods in the framework of simplicial distributions to characterize all contextual vertices of the non-signaling polytope corresponding to cycle scenarios with arbitrary outcomes. Additionally, our techniques utilize the bundle perspective on contextuality and the decomposition of measurement spaces. This enables us to extend beyond scenarios formed by gluing cycle scenarios and describe contextual extremal simplicial distributions in these generalized contexts.

quant-ph↗

arXiv2024

The Bloch--Kato conjecture, decomposing fields, and generating cohomology in degree one

The famous Bloch--Kato conjecture implies that for a field $F$ containing a primitive $p$th root of unity, the cohomology ring of the absolute Galois group $G_F$ of $F$ with $\mathbb{F}_p$ coefficients is generated by degree one elements. We investigate other groups with this property and characterize all such groups that are finite. Restricting to the case of $p$-groups, our work answers a question of Quadrelli, Snopce and Vanacci posed in 2022. As a further step in this program, we study implications of the Bloch--Kato conjecture to cohomological invariants of finite field extensions. Conversely, these cohomological invariants have implications for refining the Bloch--Kato conjecture. In service of such a refinement, we define the notion of a decomposing field for a cohomology class of a finite field extension and study minimal decomposing fields of degree two cohomology classes arising from degree $p$ extensions. We illustrate this refinement by explicitly computing the cohomology rings of superpythagorean fields and $p$-rigid fields. Finally, we construct nontrivial examples of cohomology classes and their decomposing fields, which rely on computations by David Benson in the appendix.

math.NT↗

arXiv2024

Twisted simplicial distributions

We introduce a theory of twisted simplicial distributions on simplicial principal bundles, which allow us to capture Bell's non-locality, and the more general notion of quantum contextuality. We leverage the classical theory of simplicial principal bundles, as well as structures on categories of such bundles, to provide powerful computational tools for analyzing twisted distributions in terms of both direct constructions in simplicial sets and techniques from homological algebra. We use these techniques to analyze our key examples: quantum distributions and operator-theoretic polytopes used in the classical simulation of quantum computation.

quant-ph↗

arXiv2024

The operadic theory of convexity

In this article, we characterize convexity in terms of algebras over a PROP, and establish a tensor-product-like symmetric monoidal structure on the category of convex sets. Using these two structures, and the theory of $\scr{O}$-monoidal categories, we state and prove a Grothendieck construction for lax $\scr{O}$-monoidal functors into convex sets. We apply this construction to the categorical characterization of entropy of Baez, Fritz, and Leinster, and to the study of quantum contextuality in the framework of simplicial distributions.

math.CT↗

arXiv2023

On the rank of two-dimensional simplicial distributions

Simplicial distributions provide a framework for studying quantum contextuality, a generalization of Bell's non-locality. Understanding extremal simplicial distributions is of fundamental importance with applications to quantum computing. We introduce a rank formula for twisted simplicial distributions defined for $2$-dimensional measurement spaces and provide a systematic approach for describing extremal distributions.

quant-ph↗

arXiv2023

The degenerate vertices of the $2$-qubit $\Lambda$-polytope and their update rules

Recently, a class of objects, known as $\Lambda$-polytopes, were introduced for classically simulating universal quantum computation with magic states. In $\Lambda$-simulation, the probabilistic update of $\Lambda$ vertices under Pauli measurement yields dynamics consistent with quantum mechanics. Thus, an important open problem in the study of $\Lambda$-polytopes is characterizing its vertices and determining their update rules. In this paper, we obtain and describe the update of all degenerate vertices of $\Lambda_{2}$, the $2$-qubit $\Lambda$ polytope. Our approach exploits the fact that $\Lambda_{2}$ projects to a well-understood polytope $\text{MP}$ consisting of distributions on the Mermin square scenario. More precisely, we study the ``classical" polytope $\overline{\text{MP}}$, which is $\text{MP}$ intersected by the polytope defined by a set of Clauser-Horne-Shimony-Holt (CHSH) inequalities. Owing to a duality between CHSH inequalities and vertices of $\text{MP}$ we utilize a streamlined version of the double-description method for vertex enumeration to obtain certain vertices of $\overline{\text{MP}}$.

quant-ph↗

arXiv2023

Homotopical characterization of strongly contextual simplicial distributions on cone spaces

This paper offers a novel homotopical characterization of strongly contextual simplicial distributions with binary outcomes, specifically those defined on the cone of a 1-dimensional space. In the sheaf-theoretic framework, such distributions correspond to non-signaling distributions on measurement scenarios where each context contains 2 measurements with binary outcomes. To establish our results, we employ a homotopical approach that includes collapsing measurement spaces and introduce categories associated with simplicial distributions that can detect strong contextuality.

math.AT↗

arXiv2023

Equivariant simplicial distributions and quantum contextuality

We introduce an equivariant version of contextuality with respect to a symmetry group, which comes with natural applications to quantum theory. In the equivariant setting, we construct cohomology classes that can detect contextuality. This framework is motivated by the earlier topological approach to contextuality producing cohomology classes that serve as computational primitives in measurement-based quantum computing.

quant-ph↗

arXiv2023

A bundle perspective on contextuality: Empirical models and simplicial distributions on bundle scenarios

This paper provides a bundle perspective to contextuality by introducing new categories of contextuality scenarios based on bundles of simplicial complexes and simplicial sets. The former approach generalizes earlier work on the sheaf-theoretic perspective on contextuality, and the latter extends simplicial distributions, a more recent approach to contextuality formulated in the language of simplicial sets. After constructing our bundle categories, we also construct functors that relate them and natural isomorphisms that allow us to compare the notions of contextuality formulated in two languages. We are motivated by applications to the resource theory of contextuality, captured by the morphisms in these categories. In this paper, we develop the main formalism and leave applications to future work.

math.CT↗