On Quantum Context-Free Grammars
Merina Aruja, Lisa Mathew, Jayakrishna Vijayakumar
TL;DR
This paper establishes necessary and sufficient unitary conditions for quantum context-free grammars (QCFGs) to be well-formed, addressing the issue that prior quantum grammars did not guarantee probability normalization. It introduces evolution matrices $U_a$ acting on the sentential-form Hilbert space and derives three well-formedness criteria—row-norm, separability, and length-orthogonality—to ensure each $U_a$ is unitary. The authors prove that a QCFG is unitary iff these criteria hold and illustrate the framework with a concrete 4-state example, showing probability bounds $f(w)<1$ for multi-step derivations. The results connect to the broader quantum automata literature (Golovkins) and offer a rigorous foundation for unitary quantum grammars in language generation. This advances the theoretical understanding of quantum language processing and provides a pathway for constructing well-formed QCFGs in practice.
Abstract
Quantum computing is a relatively new field of computing, which utilises the fundamental concepts of quantum mechanics to process data. The seminal paper of Moore et al. [2000] introduced quantum grammars wherein a set of amplitudes was attached to each production. However they did not study the final probability of the derived word. Aruja et al. [2025] considered conditions for the well-formedness of quantum context-free grammars (QCFGs), in order to ensure that the probabilty of the derived word does not exceed one. In this paper we propose certain necessary and sufficient conditions (also known as unitary conditions) for well-formedness of QCFGs