arXiv subjects

Rui Soares Barbosa

Publications and source records attributed to Rui Soares Barbosa.

At least 19 recordsLinked to original sources

arXiv2026

Algebraic paradoxes in adaptive quantum computation

Measurement-based quantum computation (MBQC) is a universal model of quantum computation whose full power requires adaptivity. Contextuality is known to power quantum advantage in MBQC, yet it has resisted algebraic analysis in the adaptive setting. We show that if an adaptive $\mathbb{Z}_2$-linear measurement-based quantum computing protocol deterministically computes a non-affine Boolean function, then the underlying quantum resource satisfies an inconsistent set of linear equations. This witnesses an algebraic form of strong contextuality generalising Mermin's All-versus-Nothing arguments. Such algebraic contextuality can be detected cohomologically, resolving an open question posed by Raussendorf, who had established cohomological witnesses of contextuality for non-adaptive protocols, but left the adaptive case open. We prove this result constructively: we model adaptive measurement protocols as ordinary measurements on a larger scenario of tree-like measurements, and explicitly build the inconsistent equations inductively.

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arXiv2026

A refined Frauchiger--Renner paradox based on strong contextuality

The Frauchiger--Renner paradox derives an inconsistency when quantum theory is used to describe the use of itself, by means of a scenario where agents model other agents quantumly and reason about each other's knowledge. We observe that logical contextuality (à la Hardy) is the key ingredient of the FR paradox, and we provide a stronger paradox based on the strongly contextual GHZ--Mermin scenario. In contrast to the FR paradox, this GHZ--FR paradox neither requires post-selection nor any reasoning by observers who are modelled quantumly. If one accepts the universality of quantum theory including superobservers, we propose a natural extension of Peres's dictum to resolve these extended Wigner's friend paradoxes.

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arXiv2025

Contextuality with Pauli observables in cycle scenarios

Contextuality is a fundamental marker of quantum non-classicality, which has emerged as a key resource for quantum computational advantage in multiple settings. Many such results hinge specifically on contextuality witnessed through Pauli measurements. In this work, we initiate a systematic study of (state-dependent) Pauli contextuality, focusing on cycle measurement scenarios, the simplest scenarios capable of exhibiting contextual behaviour. First, we study realizability of cycle scenarios with multi-qubit Pauli observables: we show that the maximum size of a cycle faithfully realizable by $m$-qubit Paulis is upper bounded by $3m$, while we construct explicit realizations of cycles of size $2m-1$ or $2m$, depending on whether $m \not\equiv 1 \pmod{3}$ or $m \equiv 1 \pmod{3}$. Then, we investigate the presence of contextuality: we prove that no $n$-cycle Pauli realization for $n > 4$ can witness contextuality (on any quantum state), whereas for $n = 4$ every Pauli realization exhibits contextuality, attaining the quantum bound for all noncontextuality inequalities on some pure state. Finally, we discuss arbitrary Pauli scenarios in light of Vorob'ev's theorem, and show that, contrary to what our cycle characterization might suggest, the presence of $4$-cycles is not necessary for witnessing contextuality in general Pauli scenarios.

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arXiv2025

Exclusivity principle, Ramsey theory, and $n$-cycle PR boxes

The exclusivity principle (E-principle) states that the sum of probabilities of pairwise exclusive events cannot exceed 1. Unlike other principles proposed to characterize quantum correlations, its intrinsically non-bipartite formulation enables its application in more general contextuality scenarios. Although equivalent to the no-signalling condition for any bipartite Bell scenario, this equivalence breaks down for two independent copies of the same scenario. Such violation of the E-principle due to multiple copies, known as its activation effect, was studied in [Nat Commun 4, 2263 (2013)] for the nonlocal extremal boxes of $(2,m,2)$, $(2,2,d)$, and $(3,2,2)$ Bell scenarios. The authors mapped the problem of exhibiting activation effects to finding certain cliques inside joint exclusivity graphs. In this work, we refine the joint exclusivity structure to be an edge-colored exclusivity multigraph. This allows us to draw a novel connection to Ramsey theory, which guarantees the existence of certain monochromatic subgraphs in sufficiently large edge-colored cliques, providing a powerful tool for ruling out E-principle violations. We then exploit this connection, drawing on Ramsey-theoretic results to study violations of the E-principle by multiple copies of the contextual extremal boxes of $n$-cycle scenarios, called $n$-cycle PR boxes. For the usual ($n=4$) PR box we show that the known E-principle violation of $5/4$ is the maximal achievable using two copies, and that this same upper bound applies to two copies of the KCBS ($n = 5$) PR box. We then prove that $n \geq 6$-cycle PR boxes, unlike the extremal boxes of the aforementioned Bell scenarios, do not exhibit activation effects with two or three copies. Finally, for any number of independent copies $k$, we establish a lower bound on $n$ above which $n$-cycle PR boxes do not exhibit activation effects with $k$ copies.

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arXiv2024

Matchgate hierarchy: A Clifford-like hierarchy for deterministic gate teleportation in matchgate circuits

The Clifford hierarchy, introduced by Gottesman and Chuang in 1999, is an increasing sequence of sets of quantum gates crucial to the gate teleportation model for fault-tolerant quantum computation. Gates in the hierarchy can be deterministically implemented, with increasing complexity, via gate teleportation using (adaptive) Clifford circuits with access to magic states. We propose an analogous gate teleportation protocol and a related hierarchy in the context of matchgate circuits, another restricted class of quantum circuits that can be efficiently classically simulated but are promoted to quantum universality via access to `matchgate-magic' states. The protocol deterministically implements any $n$-qubit gate in the hierarchy using adaptive matchgate circuits with magic states, with the level in the hierarchy indicating the required depth of adaptivity and thus number of magic states consumed. It also provides a whole family of novel deterministic matchgate-magic states. We completely characterise the gates in the matchgate hierarchy for two qubits, with the consequence that, in this case, the required number of resource states grows linearly with the target gate's level in the hierarchy. For an arbitrary number of qubits, we propose a characterisation of the matchgate hierarchy by leveraging the fermionic Stone$\unicode{x2013}$von Neumann theorem. It places a polynomial upper bound on the space requirements for representing gates at each level.

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arXiv2024

Lifting noncontextuality inequalities

Kochen-Specker contextuality is a fundamental feature of quantum mechanics and a crucial resource for quantum computational advantage and reduction of communication complexity. Its presence is witnessed in empirical data by the violation of noncontextuality inequalities. However, all known noncontextuality inequalities corresponding to facets of noncontextual polytopes are either Bell inequalities or refer to cyclic or state-independent contextuality scenarios. We introduce a general method for lifting noncontextuality inequalities, deriving facets of noncontextual polytopes for more complex scenarios from known facets of simpler subscenarios. Concretely, starting from an arbitrary scenario, the addition of a new measurement or a new outcome preserves the facet-defining nature of any noncontextuality inequality. This extends the results of Pironio [J. Math. Phys. 46, 062112 (2005)] from Bell nonlocality scenarios to contextuality scenarios, unifying liftings of Bell and noncontextuality inequalities. Our method produces facet-defining noncontextuality inequalities in all scenarios with contextual correlations, and we present examples of facet-defining noncontextuality inequalities for scenarios where no examples were known. Our results shed light on the structure of noncontextuality polytopes and the relationship between such polytopes across different scenarios.

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arXiv2024

Combining contextuality and causality: a game semantics approach

We develop an approach to combining contextuality with causality, which is general enough to cover causal background structure, adaptive measurement-based quantum computation, and causal networks. The key idea is to view contextuality as arising from a game played between Experimenter and Nature, allowing for causal dependencies in the actions of both the Experimenter (choice of measurements) and Nature (choice of outcomes).

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arXiv2024

Closing Bell: Boxing black box simulations in the resource theory of contextuality

This chapter contains an exposition of the sheaf-theoretic framework for contextuality emphasising resource-theoretic aspects, as well as some original results on this topic. In particular, we consider functions that transform empirical models on a scenario S to empirical models on another scenario T, and characterise those that are induced by classical procedures between S and T corresponding to 'free' operations in the (non-adaptive) resource theory of contextuality. We construct a new 'hom' scenario built from S and T, whose empirical models induce such functions. Our characterisation then boils down to being induced by a non-contextual model. We also show that this construction on scenarios provides a closed structure on the category of measurement scenarios.

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arXiv2024

Inequalities witnessing coherence, nonlocality, and contextuality

Quantum coherence, nonlocality, and contextuality are key resources for quantum advantage in metrology, communication, and computation. We introduce a graph-based approach to derive classicality inequalities that bound local, non-contextual, and coherence-free models, offering a unified description of these seemingly disparate quantum resources. Our approach generalizes recently proposed basis-independent coherence witnesses, and recovers all non-contextuality inequalities of the exclusivity graph approach. Moreover, violations of certain classicality inequalities witness preparation contextuality. We describe an algorithm to find all such classicality inequalities, and use it to analyze some of the simplest scenarios.

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arXiv2023

Quantum circuits for measuring weak values, Kirkwood--Dirac quasiprobability distributions, and state spectra

Weak values and Kirkwood--Dirac (KD) quasiprobability distributions have been independently associated with both foundational issues in quantum theory and advantages in quantum metrology. We propose simple quantum circuits to measure weak values, KD distributions, and spectra of density matrices without the need for post-selection. This is achieved by measuring unitary-invariant, relational properties of quantum states, which are functions of Bargmann invariants, the concept that underpins our unified perspective. Our circuits also enable experimental implementation of various functions of KD distributions, such as out-of-time-ordered correlators (OTOCs) and the quantum Fisher information in post-selected parameter estimation, among others. An upshot is a unified view of nonclassicality in all those tasks. In particular, we discuss how negativity and imaginarity of Bargmann invariants relate to set coherence.

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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↗

arXiv2022

Continuous-variable nonlocality and contextuality

Contextuality is a non-classical behaviour that can be exhibited by quantum systems. It is increasingly studied for its relationship to quantum-over-classical advantages in informatic tasks. To date, it has largely been studied in discrete-variable scenarios, where observables take values in discrete and usually finite sets. Practically, on the other hand, continuous-variable scenarios offer some of the most promising candidates for implementing quantum computations and informatic protocols. Here we set out a framework for treating contextuality in continuous-variable scenarios. It is shown that the Fine--Abramsky--Brandenburger theorem extends to this setting, an important consequence of which is that Bell nonlocality can be viewed as a special case of contextuality, as in the discrete case. The contextual fraction, a quantifiable measure of contextuality that bears a precise relationship to Bell inequality violations and quantum advantages, is also defined in this setting. It is shown to be a non-increasing monotone with respect to classical operations that include binning to discretise data. Finally, we consider how the contextual fraction can be formulated as an infinite linear program. Through Lasserre relaxations, we are able to express this infinite linear program as a hierarchy of semi-definite programs that allow to calculate the contextual fraction with increasing accuracy.

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arXiv2022

Sheaf representation of monoidal categories

Every small monoidal category with universal finite joins of central idempotents is monoidally equivalent to the category of global sections of a sheaf of local monoidal categories on a topological space. Every small stiff monoidal category monoidally embeds into such a category of global sections. An infinitary version of these theorems also holds in the spatial case. These representation results are functorial and subsume the Lambek-Moerdijk-Awodey sheaf representation for toposes, the Stone representation of Boolean algebras, and the Takahashi representation of Hilbert modules as continuous fields of Hilbert spaces. Many properties of a monoidal category carry over to the stalks of its sheaf, including having a trace, having exponential objects, having dual objects, having limits of some shape, and the central idempotents forming a Boolean algebra.

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arXiv2020

The logic of contextuality

Contextuality is a key signature of quantum non-classicality, which has been shown to play a central role in enabling quantum advantage for a wide range of information-processing and computational tasks. We study the logic of contextuality from a structural point of view, in the setting of partial Boolean algebras introduced by Kochen and Specker in their seminal work. These contrast with traditional quantum logic à la Birkhoff and von Neumann in that operations such as conjunction and disjunction are partial, only being defined in the domain where they are physically meaningful. We study how this setting relates to current work on contextuality such as the sheaf-theoretic and graph-theoretic approaches. We introduce a general free construction extending the commeasurability relation on a partial Boolean algebra, i.e. the domain of definition of the binary logical operations. This construction has a surprisingly broad range of uses. We apply it in the study of a number of issues, including: - establishing the connection between the abstract measurement scenarios studied in the contextuality literature and the setting of partial Boolean algebras; - formulating various contextuality properties in this setting, including probabilistic contextuality as well as the strong, state-independent notion of contextuality given by Kochen-Specker paradoxes, which are logically contradictory statements validated by partial Boolean algebras, specifically those arising from quantum mechanics; - investigating a Logical Exclusivity Principle, and its relation to the Probabilistic Exclusivity Principle widely studied in recent work on contextuality as a step towards closing in on the set of quantum-realisable correlations; - developing some work towards a logical presentation of the Hilbert space tensor product, using logical exclusivity to capture some of its salient quantum features.

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arXiv2020

Compiling quantamorphisms for the IBM Q Experience

Based on the connection between the categorical derivation of classical programs from specifications and the category-theoretic approach to quantum physics, this paper contributes to extending the laws of classical program algebra to quantum programming. This aims at building correct-by-construction quantum circuits to be deployed on quantum devices such as those available at the IBM Q Experience. Quantum circuit reversibility is ensured by minimal complements, extended recursively. Measurements are postponed to the end of such recursive computations, termed "quantamorphisms", thus maximising the quantum effect. Quantamorphisms are classical catamorphisms which, extended to ensure quantum reversibility, implement quantum cycles (vulg. for-loops) and quantum folds on lists. By Kleisli correspondence, quantamorphisms can be written as monadic functional programs with quantum parameters. This enables the use of Haskell, a monadic functional programming language, to perform the experimental work. Such calculated quantum programs prepared in Haskell are pushed through Quipper to the Qiskit interface to IBM Q quantum devices. The generated quantum circuits - often quite large - exhibit the predicted behaviour. However, running them on real quantum devices incurs into a significant amount of errors. As quantum devices are constantly evolving, an increase in reliability is likely in the near future, allowing for our programs to run more accurately.

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arXiv2019

A comonadic view of simulation and quantum resources

We study simulation and quantum resources in the setting of the sheaf-theoretic approach to contextuality and non-locality. Resources are viewed behaviourally, as empirical models. In earlier work, a notion of morphism for these empirical models was proposed and studied. We generalize and simplify the earlier approach, by starting with a very simple notion of morphism, and then extending it to a more useful one by passing to a co-Kleisli category with respect to a comonad of measurement protocols. We show that these morphisms capture notions of simulation between empirical models obtained via `free' operations in a resource theory of contextuality, including the type of classical control used in measurement-based quantum computation schemes.

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arXiv2018

Contextuality and noncommutative geometry in quantum mechanics

Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcomes; dually, states can be modelled as functions from the algebra of observables to outcomes. The probabilistic predictions of quantum physics are contextual in that they preclude this classical assumption of reality: noncommuting observables, which are not assumed to be comeasurable, cannot be consistently ascribed deterministic values even if one enriches the description of a quantum state. Here, we consider the geometrically dual objects of noncommutative algebras of observables as being generalisations of classical state spaces to the quantum setting and argue that these generalised geometric spaces represent the objects of study of noncommutative operator geometry. By adapting the spectral presheaf of Hamilton-Isham-Butterfield, a formulation of quantum state space that collates contextual data, we reconstruct tools of noncommutative geometry in an explicitly geometric fashion. In this way, we bridge the foundations of quantum mechanics with the foundations of noncommutative geometry à la Connes et al. To each unital $C^*$-algebra we associate a geometric object acting as a generalised Gel'fand spectrum. We show how any functor $F$ from compact Hausdorff spaces to a suitable target category can be applied directly to these geometric objects to automatically yield an extension $\tilde{F}$ acting on all unital $C^*$-algebras. This procedure is used to give a novel formulation of the operator $K_0$-functor in terms of the topological $K$-functor. We then delineate a $C^*$-algebraic conjecture that the extension of the functor that assigns to a topological space its lattice of open sets assigns to a unital $C^*$-algebra its lattice of closed, two-sided ideals. We prove the von Neumann algebraic analogue of this conjecture.

math.OA↗

arXiv2017

Minimum quantum resources for strong non-locality

We analyse the minimum quantum resources needed to realise strong non-locality, as exemplified e.g. by the classical GHZ construction. It was already known that no two-qubit system, with any finite number of local measurements, can realise strong non-locality. For three-qubit systems, we show that strong non-locality can only be realised in the GHZ SLOCC class, and with equatorial measurements. However, we show that in this class there is an infinite family of states which are pairwise non-LU-equivalent that realise strong non-locality with finitely many measurements. These states have decreasing entanglement between one qubit and the other two, necessitating an increasing number of local measurements on the latter.

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