Gurevich's Quizani Dialogs as an Example of Explainable Mathematics, and How This Is Related to Quantum Space-Time Ideas that Can Speed Up Computations
Everyone talks about the need for Explainable AI -- when, to supplement a long difficult-to-understand sequence of computational steps leading to AI's decision, we are looking for a shorter and understandable more-informal explanation for this decision. In this paper, we argue that this need is a particular case of what we call Explainable Mathematics -- when we want to supplement a long sequence of arguments and/or computations with a shorter and understandable more-informal explanation. Important instances of Explainable Mathematics are Yuri Gurevich's Quizani dialogs that help explain complex results from theoretical computer science and physicists' more-informal explanations of complex physical phenomena. We explain that in the physics' case, since -- according to most physicists -- all physical theories are approximate, the use of approximate more-informal methods often makes more sense that the use of rigorous methods that implicitly assume that the current theories are absolute correct. We then apply this argument to one of the common uses of physics in theory of computation -- that limitation by the speed of light limits the computation speed. Specifically, we show that quantum space-time ideas potentially allow computations at the micro-level speed of light which can be higher than its usual macro-level value. This potential increase in possible communication speed can speed up computations.
Everyone talks about the need for Explainable AI -- when, to supplement a long difficult-to-understand sequence of computational steps leading to AI's decision, we are looking for a shorter and understandable more-informal explanation for this decision. In this paper, we argue that this need is a particular case of what we call Explainable Mathematics -- when we want to supplement a long sequence of arguments and/or computations with a shorter and understandable more-informal explanation. Important instances of Explainable Mathematics are Yuri Gurevich's Quizani dialogs that help explain complex results from theoretical computer science and physicists' more-informal explanations of complex physical phenomena. We explain that in the physics' case, since -- according to most physicists -- all physical theories are approximate, the use of approximate more-informal methods often makes more sense that the use of rigorous methods that implicitly assume that the current theories are absolute correct. We then apply this argument to one of the common uses of physics in theory of computation -- that limitation by the speed of light limits the computation speed. Specifically, we show that quantum space-time ideas potentially allow computations at the micro-level speed of light which can be higher than its usual macro-level value. This potential increase in possible communication speed can speed up computations.
The increasing deployment of Artificial Intelligence (AI) systems across various domains necessitates a deeper understanding of their decision-making processes. Current explainable AI (XAI) techniques often struggle to provide truly transparent and reliable explanations, particularly for complex models like deep neural networks. This paper proposes a novel approach to algorithmic transparency and XAI leveraging concepts from quantum information theory. Specifically, we explore the potential of quantum entanglement and mutual information to represent and analyze AI model decisions in a fundamentally different way. Our framework represents model outputs as quantum states and investigates the correlations between these states and the input data. This analysis aims to identify the key factors driving the model's decision and provide a more intuitive and understandable explanation. We demonstrate that utilizing quantum information theory can offer a richer and more robust approach to XAI than traditional methods, potentially addressing limitations related to interpretability and trust in AI systems. This work lays the groundwork for a new paradigm in AI transparency, utilizing the unique properties of quantum mechanics to enhance the explainability of complex algorithms.
In the last two decades, the scientific community has witnessed a surge in activity, interesting results, and notable progress in our conceptual understanding of computing and information based on the laws of quantum theory. One of the significant aspects of these developments has been an integration of several fields of inquiry that not long ago appeared to be evolving, more or less, along narrow disciplinary paths without any major overlap with each other. In the resulting body of work, investigators have revealed a deeper connection among the ideas and techniques of (apparently) disparate fields. As is evident from the title of this volume, logic, mathematics, physics, computer science and information theory are intricately involved in this fascinating story. The inquisitive reader might focus, perhaps, on the marriage of the most unlikely and intriguing fields of quantum theory and logic and ask: Why quantum logic? By many, “logic” is deemed to be panacea for faulty intuition. It is often associated with the rules of correct thinking and decision-making, but not necessarily in its most sublime role as a deep intellectual subject underlying the validity of mathematical structures and worthy of investigation and discovery in its own right. Indeed, within the realm of the classical theories of nature, one may encounter situations that defy comprehension, should one hold to the intuition developed through experiencing familiar macroscopic scenarios in our routine impressions of natural phenomena. One such example is a statement within the special theory of relativity that the speed of light is the same in all inertial frames. It certainly defies the common intuition regarding the observation of velocities of familiar objects in relative motion. One might be tempted to dismiss it as contrary to observation. However, while analyzing natural phenomena for objects moving close to the speed of light and, therefore, unfamiliar in the range of velocities we are normally accustomed to, logical deductions based on the postulates of the special relativity theory lead to the correct predictions of experimental observations. There exists an undeniable interconnection between the deepest theories of nature and mathematical reasoning, famously stated by Eugene Wigner as the unreasonable efficacy of mathematics in physical theories.
When we want to predict the future, we compute it from what we know about the present. Specifically, we take a mathematical representation of observed reality, plug it into some dynamical equations, and then map the time-evolved result back to real-world predictions. But while this computational process can tell us what we want to know, we have taken this procedure too literally, implicitly assuming that the universe must compute itself in the same manner. Physical theories that do not follow this computational framework are deemed illogical, right from the start. But this anthropocentric assumption has steered our physical models into an impossible corner, primarily because of quantum phenomena. Meanwhile, we have not been exploring other models in which the universe is not so limited. In fact, some of these alternate models already have a well-established importance, but are thought to be mathematical tricks without physical significance. This essay argues that only by dropping our assumption that the universe is a computer can we fully develop such models, explain quantum phenomena, and understand the workings of our universe. (This essay was awarded third prize in the 2012 FQXi essay contest; a new afterword compares and contrasts this essay with Robert Spekkens' first prize entry.)
The rapid advancement of Artificial Intelligence (AI) has heightened concerns about the opacity of decision-making processes in complex models, particularly deep neural networks. This research explores a novel paradigm that integrates quantum computing principles into Explainable AI (XAI) to enhance interpretability without compromising predictive accuracy. The study introduces a Quantum-inspired Interpretable Model (QIIM) framework that leverages Hilbert space embeddings, unitary transformations, and operator-based learning to represent knowledge in a mathematically transparent form. The proposed model computes interpretability through the expectation values of observable operators, enabling the decomposition of decisions into quantifiable, human-understandable components. To capture hierarchical relationships, a Tensor Network Interpretable Model (TNIM) is further developed, offering scalable insights into complex dependencies among features. Experimental evaluations—performed on benchmark datasets for decision-making tasks—demonstrate that the quantum-inspired models achieve competitive accuracy while significantly improving local and global interpretability metrics compared to classical XAI techniques. The findings underscore the potential of quantum formalism as a new foundation for transparent AI systems, bridging the gap between computational efficiency and explainability. This study contributes to both theoretical understanding and practical advancement in interpretable machine learning, paving the way for ethically aligned, transparent, and human-trustworthy AI-driven decision support systems.
. Quantum computers take advantage of interfering quantum alternatives in order to handle problems that might be too time consuming with algorithms based on classical logic. Developing quantum computers requires new ways of thinking beyond those in the familiar classical world. To help in this thinking, we give a description of the foundational ideas that hold in all of our successful physical models, including quantum theory. Our emphasis will be on the proper interpretation of our theories, and not just their statements. Our tact will be to build on the concept of information, which lies central to the operation of not just computers, but the Universe. For application to quantum computing, the essence of quantum theory is given, together with special precautions and limitations. Introduction. Having a grasp on the ideas behind a theory helps to apply it correctly, to understand its limitations, and to generate new ideas. Getting a firm hold on quantum theory is not an easy task, because our experiences and even our genetic predispositions have been developed in a world in which quantum effects are largely washed out. Remarkably, our predilection for finding logic behind the behavior of what we observe, including that of electrons and atoms, has led us to quantum theory, a description of nature that is hard for us to conceptualize, but is logical, accurate, and explains a wide variety of phenomena with only a few statements and input. As background to quantum theory and quantum computing, an attempt is made here to give the primitive notions and essential observations that underlie current physical theories, so that foundational ideas are explicit, and a common language is established. In our description, information storage and transfer is made central. A short description of quantum theory follows, and then applied to quantum computing, focusing on what the theory says, and particularly does not say, in areas where conceptual difficulties have arisen. Physical theory and reality. A physical theory is a logical model capable of making predictions of what we observe. It is judged by its accuracy in matching measurements, and by its economy, i.e. whether the proposed theory has only a few relationships and input data needed for its ability to explain observations over a wide realm.
Quantum physicist, New York Times bestselling author, and BBC host Jim Al-Khalili offers a fascinating and illuminating look at what physics reveals about the world Shining a light on the most profound insights revealed by modern physics, Jim Al-Khalili invites us all to understand what this crucially important science tells us about the universe and the nature of reality itself. Al-Khalili begins by introducing the fundamental concepts of space, time, energy, and matter, and then describes the three pillars of modern physics—quantum theory, relativity, and thermodynamics—showing how all three must come together if we are ever to have a full understanding of reality. Using wonderful examples and thought-provoking analogies, Al-Khalili illuminates the physics of the extreme cosmic and quantum scales, the speculative frontiers of the field, and the physics that underpins our everyday experiences and technologies, bringing the reader up to speed with the biggest ideas in physics in just a few sittings. Physics is revealed as an intrepid human quest for ever more foundational principles that accurately explain the natural world we see around us, an undertaking guided by core values such as honesty and doubt. The knowledge discovered by physics both empowers and humbles us, and still, physics continues to delve valiantly into the unknown. Making even the most enigmatic scientific ideas accessible and captivating, this deeply insightful book illuminates why physics matters to everyone and calls one and all to share in the profound adventure of seeking truth in the world around us.