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arXiv:2505.13937

On Quantum Context-Free Grammars

Merina Aruja, Lisa Mathew, Jayakrishna Vijayakumar

Published May 20, 2025• 0 citations
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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

DOI
10.48550/arxiv.2505.13937

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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
To study quantum computation, Moore and Crutchfield (Theoret. Comput. Sci. vol. 237, pp. 275-306, 2000) proposed quantum versions of finite-state and push-down automata, as well as regular and context-free grammars. In this note, we show that the languages generated by regular quantum grammars are equivalent to the languages generated by regular grammars. Therefore, from the perspective of generating languages, regular quantum grammars are not more powerful than classical regular grammars. This implies that it may be necessary to look for more well-formed definitions of quantum grammars and maybe those of quantum automata.
In this thesis we will study the decidability of the Intersection problem for the measure-once quantum finite automata introduced by Moore and Crutchfield in the 2000s. Specifically, we ask for which families of grammars it is possible to find conditions ensuring that, given a language recognized by such model of computation and a language generated by such families of grammars, it is decidable whether or not they have a nonempty intersection. Throughout the thesis, we will investigate the correlation between this problem and some decidability problems for matrices subsets on the Zariski topology, so as to apply the corresponding results to show the decidability of the problem for languages generated by monoidal context-free grammars and restricted matrix context-free grammars of finite index.
Here we consider some well-known facts in syntax from a physics perspective, allowing us to establish equivalences between both fields with many consequences. Mainly, we observe that the operation MERGE, put forward by N. Chomsky in 1995, can be interpreted as a physical information coarse-graining. Thus, MERGE in linguistics entails information renormalization in physics, according to different time scales. We make this point mathematically formal in terms of language models. In this setting, MERGE amounts to a probability tensor implementing a coarse-graining, akin to a probabilistic context-free grammar. The probability vectors of meaningful sentences are given by stochastic tensor networks (TN) built from diagonal tensors and which are mostly loop-free, such as Tree Tensor Networks and Matrix Product States, thus being computationally very efficient to manipulate. We show that this implies the polynomially-decaying (long-range) correlations experimentally observed in language, and also provides arguments in favour of certain types of neural networks for language processing. Moreover, we show how to obtain such language models from quantum states that can be efficiently prepared on a quantum computer, and use this to find bounds on the perplexity of the probability distribution of words in a sentence. Implications of our results are discussed across several ambits.
The quantum Turing machine (QTM) has been introduced by Deutsch as an abstract model of quantum computation. The transition function of a QTM is linear, and has to be unitary to be a well-formed QTM. This well-formedness condition ensures that the evolution of the machine does not violate the postulates of quantum mechanics. However, we claim in this paper that the well-formedness condition is too strong and we introduce a weaker condition, leading to a larger class of Turing machines called Observable Quantum Turing Machines (OQTMs). We prove that the evolution of such OQTM does not violate the postulates of quantum mechanics while offering a more general abstract model for quantum computing. This novel abstract model unifies classical and quantum computations, since every well-formed QTM and every deterministic TM are OQTMs, whereas a deterministic TM has to be reversible to be a well-formed QTM. In this paper we introduce the fundamentals of OQTM like a well-observed lemma and a completion lemma. The introduction of such an abstract machine allowing classical and quantum computations is motivated by the emergence of models of quantum computation like the one-way model. More generally, the OQTM aims to be an abstract framework for the pragmatic paradigm of quantum computing: quantum data, classical control'. Furthermore, this model allows a formal and rigorous treatment of problems requiring classical interactions, like the halting of QTM. Finally, it opens new perspectives for the construction of a universal QTM.