Quality-sensitive applications of machine learning (ML) require quality assurance (QA) by humans before the predictions of an ML model can be deployed. QA for ML (QA4ML) interfaces require users to view a large amount of data and perform many interactions to correct errors made by the ML model. An optimized user interface (UI) can significantly reduce interaction costs. While UI optimization can be informed by user studies evaluating design options, this approach is not scalable, because there are typically numerous small variations that can affect the efficiency of a QA4ML interface. Hence, we propose using simulation to evaluate and aid the optimization of QA4ML interfaces. In particular, we focus on simulating the combined effects of human intelligence in initiating appropriate interaction commands and machine intelligence in providing algorithmic assistance for accelerating QA4ML processes. As QA4ML is usually labor-intensive, we use the simulated task completion time as the metric for UI optimization under different interface and algorithm setups. We demonstrate the usage of this UI design method in several QA4ML applications.

The ZX-calculus is a universal graphical language for qubit quantum computation, meaning that every linear map between qubits can be expressed in the ZX-calculus. Furthermore, it is a complete graphical rewrite system: any equation involving linear maps that is derivable in the Hilbert space formalism for quantum theory can also be derived in the calculus by rewriting. It has widespread usage within quantum industry and academia for a variety of tasks such as quantum circuit optimisation, error-correction, and education.The ZW-calculus is an alternative universal graphical language that is also complete for qubit quantum computing. In fact, its completeness was used to prove that the ZX-calculus is universally complete. This calculus has advanced how quantum circuits are compiled into photonic hardware architectures in the industry.Recently, by combining these two calculi, a new calculus has emerged for qubit quantum computation, the ZXW-calculus. Using this calculus, graphical-differentiation, -integration, and -exponentiation were made possible, thus enabling the development of novel techniques in the domains of quantum machine learning and quantum chemistry.Here, we generalise the ZXW-calculus to arbitrary finite dimensions, that is, to qudits. Moreover, we prove that this graphical rewrite system is complete for any finite dimension. This is the first completeness result for any universal graphical language beyond qubits.

Quantum Natural Language Processing (QNLP) deals with the design and implementation of NLP models intended to be run on quantum hardware. In this paper, we present results on the first NLP experiments conducted on Noisy Intermediate-Scale Quantum (NISQ) computers for datasets of size greater than 100 sentences. Exploiting the formal similarity of the compositional model of meaning by Coecke, Sadrzadeh, and Clark (2010) with quantum theory, we create representations for sentences that have a natural mapping to quantum circuits. We use these representations to implement and successfully train NLP models that solve simple sentence classification tasks on quantum hardware. We conduct quantum simulations that compare the syntax-sensitive model of Coecke et al. with two baselines that use less or no syntax; specifically, we implement the quantum analogues of a “bag-of-words” model, where syntax is not taken into account at all, and of a word-sequence model, where only word order is respected. We demonstrate that all models converge smoothly both in simulations and when run on quantum hardware, and that the results are the expected ones based on the nature of the tasks and the datasets used. Another important goal of this paper is to describe in a way accessible to AI and NLP researchers the main principles, process and challenges of experiments on quantum hardware. Our aim in doing this is to take the first small steps in this unexplored research territory and pave the way for practical Quantum Natural Language Processing.

In many applications, while machine learning (ML) can be used to derive algorithmic models to aid decision processes, it is often difficult to learn a precise model when the number of similar data points is limited. One example of such applications is data reconstruction from historical visualizations, many of which encode precious data, but their numerical records are lost. On the one hand, there is not enough similar data for training an ML model. On the other hand, manual reconstruction of the data is both tedious and arduous. Hence, a desirable approach is to train an ML model dynamically using interactive classification, and hopefully, after some training, the model can complete the data reconstruction tasks with less human interference. For this approach to be effective, the number of annotated data objects used for training the ML model should be as small as possible, while the number of data objects to be reconstructed automatically should be as large as possible. In this article, we present a novel technique for the machine to initiate intelligent interactions to reduce the user’s interaction cost in interactive classification tasks. The technique of machine-initiated intelligent interaction (MI3) builds on a generic framework featuring active sampling and default labeling. To demonstrate the MI3 approach, we use the well-known cholera map visualization by John Snow as an example, as it features three instances of MI3 pipelines. The experiment has confirmed the merits of the MI3 approach.

We introduce a general categorical framework to reason about quantum theory and other process theories living in spacetimes where chronology violation can occur-e.g. in the form of Closed Timelike Curves (CTCs)-allowing resources to travel back in time and provide computational speedups. Our framework is based on a weakening of the definition of traced symmetric monoidal categories, obtained by dropping the yanking axiom and the requirement that the trace be defined on all morphisms. We show that the two leading models for quantum theory with closed timelike curves-namely the P-CTC model of Lloyd et al. and the D-CTC model of Deutsch-are captured by our framework, and in doing so we provide the first compositional description of the D-CTC model. Our description of the D-CTC model results in a process theory which respects the constraints of relativistic causality: this is in direct contrast to the P-CTC model, where CTCs are implemented by a trace and allow post-selection to be performed deterministically.

Within the categorical compositional distributional model of meaning, we provide semantic interpretations for the subject and object roles of the possessive relative pronoun ‘whose’. This is done in terms of Frobenius algebras over compact closed categories. These algebras and their diagrammatic language expose how meanings of words in relative clauses interact with each other. We show how our interpretation is related to Montague-style semantics and provide a truth-theoretic interpretation. We also show how vector spaces provide a concrete interpretation and provide preliminary corpus-based experimental evidence. In a prequel to this article, we used similar methods and dealt with the case of subject and object relative pronouns.

Originally inspired by categorical quantum mechanics (Abramsky and Coecke, LiCS'04), the categorical compositional distributional model of natural language meaning of Coecke, Sadrzadeh and Clark provides a conceptually motivated procedure to compute the meaning of a sentence, given its grammatical structure within a Lambek pregroup and a vectorial representation of the meaning of its parts. Moreover, just like CQM allows for varying the model in which we interpret quantum axioms, one can also vary the model in which we interpret word meaning. In this paper we show that further developments in categorical quantum mechanics are relevant to natural language processing too. Firstly, Selinger's CPM-construction allows for explicitly taking into account lexical ambiguity and distinguishing between the two inherently different notions of homonymy and polysemy. In terms of the model in which we interpret word meaning, this means a passage from the vector space model to density matrices. Despite this change of model, standard empirical methods for comparing meanings can be easily adopted, which we demonstrate by a small-scale experiment on real-world data. Secondly, commutative classical structures as well as their non-commutative counterparts that arise in the image of the CPM-construction allow for encoding relative pronouns, verbs and adjectives, and finally, iteration of the CPM-construction, something that has no counterpart in the quantum realm, enables one to accommodate both entailment and ambiguity.

This paper develops a compositional vector-based semantics of subject and object relative pronouns within a categorical framework. Frobenius algebras are used to formalise the operations required to model the semantics of relative pronouns, including passing information between the relative clause and the modified noun phrase, as well as copying, combining, and discarding parts of the relative clause. We develop two instantiations of the abstract semantics, one based on a truth-theoretic approach and one based on corpus statistics.

We show that an orthogonal basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative †-Frobenius monoid in the category FdHilb, which has finite-dimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal structure. The basis is normalised exactly when the corresponding commutative †-Frobenius monoid is special. Hence, both orthogonal and orthonormal bases are characterised without mentioning vectors, but just in terms of the categorical structure: composition of operations, tensor product and the †-functor. Moreover, this characterisation can be interpreted operationally, since the †-Frobenius structure allows the cloning and deletion of basis vectors. That is, we capture the basis vectors by relying on their ability to be cloned and deleted. Since this ability distinguishes classical data from quantum data, our result has important implications for categorical quantum mechanics.

Categorical quantum mechanics studies quantum theory in the framework of dagger-compact closed categories. Using this framework, we establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, we establish a direct connection between Mermin-type non-locality scenarios, which we generalise to an arbitrary number of parties, using systems of arbitrary dimension, and performing arbitrary measurements, and a new stronger notion of complementarity which we introduce here. Our derivation of the fact that strong complementarity is a necessary condition for a Mermin scenario provides a crisp operational interpretation for strong complementarity. We also provide a complete classification of strongly complementary observables for quantum theory, something which has not yet been achieved for ordinary complementarity. Since our main results are expressed in the (diagrammatic) language of dagger-compact categories, they can be applied outside of quantum theory, in any setting which supports the purely algebraic notion of strongly complementary observables. We have therefore introduced a method for discussing non-locality in a wide variety of models in addition to quantum theory. The diagrammatic calculus substantially simplifies (and sometimes even trivialises) many of the derivations, and provides new insights. In particular, the diagrammatic computation of correlations clearly shows how local measurements interact to yield a global overall effect. In other words, we depict non-locality.

We provide an algebraic semantics together with a sound and complete sequent calculus for information update due to epistemic actions. This semantics is flexible enough to accommodate incomplete as well as wrong information e.g. due to secrecy and deceit, as well as nested knowledge. We give a purely algebraic treatment of the muddy children puzzle, which moreover extends to situations where the children are allowed to lie and cheat. Epistemic actions, that is, information exchanges between agents A,B,…∈A⁠, are modeled as elements of a quantale. The quantale (Q,⋁,•) acts on an underlying Q-right module(M,⋁) of epistemic propositions and facts. The epistemic content is encoded by appearance maps, one pair fMA:M→M and fQA:Q→Q of (lax) morphisms for each agent A∈A⁠, which preserve the module and quantale structure respectively. By adjunction, they give rise to epistemic modalities, capturing the agents' knowledge on propositions and actions. The module action is epistemic update and gives rise to dynamic modalities—cf. weakest precondition. This model subsumes the crucial fragment of Baltag, Moss and Solecki's dynamic epistemic logic, abstracting it in a constructive fashion while introducing resource-sensitive structure on the epistemic actions.

Particular focus in this paper is on quantum information protocols, which exploit quantum-mechanical effects in an essential way. The particular examples we shall use to illustrate our approach will be teleportation (Benett et al., 1993), logic-gate teleportation (Gottesman and Chuang,1999), and entanglement swapping (Zukowski et al., 1993). The ideas illustrated in these protocols form the basis for novel and potentially very important applications to secure and fault-tolerant communication and computation (2001,1999,2000).

We establish the following equation: Quantitative Probability = Logic + Partiality of Knowledge + Entropy I.e.: 1. A finitary probability space Δn (= all probability measures on {1,…, n}) can be fully and faithfully represented by the pair consisting of the abstraction Dn (= the object up to isomorphism) of the partially ordered set (Δn, ⊑) introduced in [3], and, Shannon entropy; 2. Dn itself can be obtained via a systematic purely order-theoretic procedure (which embodies introduction of partiality of knowledge) on an (algebraic) logic. This procedure applies to any poset A; DA ≅ (Δn ,⊑) when A is the n-element powerset and DA ≅ (Ωn, ⊑), the domain of mixed quantum states also introduced in [3], when A is the lattice of subspaces of a Hilbert space.