Visual Diagrams Unlock New Links Between Quantum Physics and Computing

Muhammad Hamza Waseem, affiliated with the University of Oxford, and colleagues present a new application of string diagrams to fields including quantum foundations, computing, and natural language processing. The team formalises constructor theory as a process theory and explores the implications of rejecting locality within quantum physics, proposing a connection between categorical quantum mechanics and constructor theory. They also introduce a formalism for wave-based logic circuits, such as spin-wave circuits, and demonstrate its effectiveness in Boolean logic circuit design and optimisation. This extends to natural language processing, where the team develops a hybrid grammar for Urdu and demonstrates a surprising equivalence between Urdu and English text circuits within their DisCoCirc framework, supporting a process-relational perspective across scientific disciplines.

Urdu and English sentence meaning now mapped to equivalent gate-level electrical circuits

Up to gate-level translation, Urdu and English text circuits are now identical, a result previously unattainable due to limitations in representing complex linguistic structures. Detailed in arXiv:2605.11417v1, this equivalence surpasses prior methods of language translation by utilising a new hybrid grammar for Urdu and a framework called DisCoCirc, which represents sentence meaning as electrical circuits. The DisCoCirc framework, built upon the foundations of applied category theory, allows for the encoding of semantic relationships as signal flow within these circuits. This is achieved by representing linguistic constituents, words, phrases, and clauses, as specific circuit components, with the connections between these components reflecting the grammatical dependencies and semantic roles. The team’s approach moves beyond traditional symbolic representations of language, offering a potentially more robust and computationally tractable model. Prior attempts at computational linguistics often struggle with ambiguity and context-sensitivity; the DisCoCirc approach aims to address these challenges by grounding meaning in the physical properties of the circuit.

The work formalises constructor theory as a process theory, resolving conflicts between its principles of locality and composition, and consequently allowing for a deeper understanding of quantum interactions. Constructor theory, initially proposed by David Deutsch and Chiara Marletto, posits that what is physically possible is determined not by the laws of motion, but by the laws of what transformations are possible. Formalising this within a process theory framework, utilising string diagrams, allows for a rigorous mathematical treatment of these transformations and their composition. The inherent tension between locality, the idea that an object is only directly influenced by its immediate surroundings, and composition, the ability to combine simpler processes to create more complex ones, is brought into sharp relief. This conflict is not necessarily a contradiction, but a fundamental aspect of the theory that requires careful consideration.

A formalism for wave-based logic circuits, exemplified by spin-wave circuits, has been developed, demonstrating its utility in Boolean logic circuit design and optimisation. Spin-wave circuits offer potential advantages over conventional CMOS circuits in terms of energy efficiency and speed, as information is encoded and processed using spin waves rather than electron flow. The developed formalism allows for the systematic design and analysis of these circuits, enabling the creation of more efficient and compact logic gates. This is particularly relevant in the context of emerging nano-electronic devices where minimising power consumption is crucial. The team’s approach leverages the mathematical structure of symmetric monoidal categories to represent the composition of these wave-based components, facilitating optimisation and verification. Modelling linguistic structures with electrical circuits is now possible, opening avenues for interdisciplinary research, although currently limited to a restricted fragment of language. This achievement is underpinned by applied category theory and string diagrams, offering a new approach to language representation. The team’s new hybrid grammar for Urdu allows its representation within the DisCoCirc framework, a system modelling sentence meaning as circuits. Scaling to full, complex language, however, remains a key hurdle to practical application, requiring further development of both the hybrid grammar and the DisCoCirc framework to accommodate more nuanced linguistic structures. Specifically, handling phenomena such as metaphor, irony, and presupposition will require significant advancements in the underlying formalisms.

Locality and composition clash within formalised constructor theory

Applied category theory offers a powerful, visual language for modelling complex systems, but fully embracing its potential requires confronting fundamental tensions. The work formalises constructor theory, a framework defining what is physically possible, as a process theory using string diagrams, revealing an inherent conflict between its principles of locality and composition. Acknowledging this conflict between established principles is itself a valuable outcome. String diagrams provide a graphical notation for representing categorical constructions, allowing for a more intuitive understanding of complex mathematical relationships. They are particularly well-suited for representing processes and their composition, as the diagrams directly reflect the flow of information or energy between different components.

Mathematical tools for modelling processes and their composition are supplied by applied category theory. Symmetric monoidal categories, involving series and parallel composition, are particularly suited for describing the composition of processes in space and time. These categories provide a formal framework for reasoning about the structure of complex systems, allowing for the identification of common patterns and the development of general principles. Series composition corresponds to the sequential execution of processes, while parallel composition represents the simultaneous execution of independent processes. These, also termed process theories, utilise string diagrams, a visually intuitive, mathematically strong, expressive and flexible syntax applicable to diverse scientific domains. The expressive power of string diagrams stems from their ability to capture both the local interactions between components and the global structure of the system. This thesis employs string diagrams to investigate topics in quantum foundations, computing, and natural language processing, with constructor theory formalised as a process theory.

Within quantum physics, a conflict between constructor-theoretic principles of locality and composition is demonstrated, and if locality is rejected, categorical quantum mechanics can be considered a constructor theory of quantum physics. The rejection of locality, while seemingly radical, is consistent with many interpretations of quantum mechanics, such as the many-worlds interpretation. Categorical quantum mechanics provides a mathematical framework for describing quantum systems and processes using the language of category theory. By framing quantum mechanics within constructor theory, the team suggests a deeper connection between the fundamental laws of physics and the principles of information processing. A formalism for wave-based logic circuits with phase encoding is developed, motivated by spin-wave circuits and demonstrated in the design, analysis and optimisation of Boolean logic circuits. Investigation into the elimination of inter-language grammatical bureaucracy within the DisCoCirc framework is also presented, including a hybrid grammar for Urdu and a demonstration of the equivalence of Urdu and English text circuits up to gate-level translation. This work supports a process-relational outlook in science, well-supported by applied category-theoretic tools, particularly string diagrams. This process-relational perspective emphasizes the importance of relationships and interactions between entities, rather than focusing solely on the entities themselves. It offers a potentially unifying framework for understanding diverse scientific phenomena, from quantum physics to linguistics.

The research successfully formalised constructor theory as a process theory using string diagrams, a visually intuitive mathematical tool. This work demonstrates a conflict between constructor-theoretic principles of locality and composition within quantum physics, suggesting categorical quantum mechanics may offer a resolution if locality is not upheld. Furthermore, a formalism for wave-based logic circuits and a method for simplifying grammatical structures in natural language processing, exemplified by Urdu, were developed and tested. The study supports a process-relational perspective, highlighting the importance of interactions between entities across diverse scientific fields.