Quantum AI: Mathematising Language
I want to start with a few premises of quantum computing that may help us think differently about next-generation AI.
Quantum computers do not succeed because each computational step is faster. Where they offer an advantage, it is because certain problems can be reformulated so that they require a fundamentally different sequence of operations. In classical systems, solving a difficult problem can mean exploring a very large number of possibilities, with increasing complexity producing more steps and therefore more time. Quantum systems work differently. Through superposition, entanglement and interference, quantum algorithms can represent relations among many possible states simultaneously and manipulate their probability amplitudes so that some possible solutions are amplified while others are suppressed.
It is sometimes said that a quantum computer simply ‘tries every answer at once’. This is misleading. Measurement does not allow us to read out all the possibilities contained within a superposition. The computational achievement lies instead in designing an algorithm in which interference changes the distribution of possible outcomes so that useful answers become more likely. Quantum advantage, where it exists, therefore concerns the structure of a computational problem rather than simply an increase in processing speed.
This leads to a second distinction. We need to think about quantum computing and Big Data differently: volume is not the same problem as state complexity. A common misconception about quantum computing is that it is designed simply to process massive datasets faster than Big Data platforms. It is not.
The two technologies address fundamentally different challenges. Big Data systems are built to collect, store, process and analyse enormous volumes of structured and unstructured information. Distributed architectures, parallel processing and scalable storage make them the backbone of modern analytics, AI pipelines, IoT platforms, fraud detection and enterprise reporting.
Quantum computing addresses a different problem. Its potential lies particularly in computational problems where the number of possible configurations, states or relationships grows combinatorially or exponentially. Instead of merely managing more data, quantum algorithms may offer ways of navigating or manipulating complex state spaces differently. Optimization, molecular simulation and certain forms of linear algebra are familiar examples.
The distinction is important. Big Data scales by adding storage, processors and infrastructure. Quantum computing potentially targets forms of complexity that remain difficult even when enormous amounts of classical computational infrastructure are available. This is why the future is unlikely to be a choice between Big Data and quantum computing. It is much more likely to involve hybrid architectures: classical systems handling storage, retrieval, conventional computation and large-scale datasets, while quantum processors are used selectively for computationally difficult sub-problems.
This is already an important qualification. There is no established quantum advantage for general AI, let alone for large language models. Loading large quantities of classical data into quantum states itself presents a potentially serious computational bottleneck, and contemporary Quantum Machine Learning remains highly experimental. Current research repeatedly stresses that any claimed speed-up has to include the cost of state preparation rather than treating access to quantum-encoded data as free.
Hence, the distinction between data volume and state complexity suggests a different question about AI. Perhaps we have been looking in the wrong place when asking what quantum computing might contribute.
Recursive self-improvement
Let us combine these observations with one of the key current debates in AI: recursive self-improvement, and the striking difference between domains such as mathematics and code, on the one hand, and ordinary language on the other.
The following argument is prompted by Alberto Romero’s excellent article, “The Actual Reason Why Google ‘Fell Out’ of the AI Race Changes Everything”. Romero frames his discussion primarily around the corporate strategies of Google, OpenAI and Anthropic, which is interesting in itself. But within that discussion he draws out a much broader conceptual problem concerning recursive self-improvement.
In section II of his article, “The Mystery of Recursive Self-Improvement,” Romero writes:
Between 2016 and 2019, AlphaGo and AlphaZero showed that reinforcement learning through self-play could produce superhuman performance. Given a clear reward signal — win or lose — repeated trial mattered more than human heuristics. The hope was that the same principle could extend beyond games, into language and science.
So far, we have seen something resembling this happening most convincingly with coding and mathematics. Romero continues:
If you can guide the model through a learning maze with obscene amounts of high-quality reasoning data, they learn. Winning a chess match or making code that compiles or solving an Erdős conjecture is suddenly possible. […] The next step to achieve RSI is to take this set of ideas and apply them to the AI systems themselves, which then could make the next version better. And so on until AGI.
The attraction is clear. Recursive self-improvement requires some way for an AI system to identify whether its modification of another AI system (or of itself) is actually an improvement. Games, code and significant parts of mathematics possess clear verification mechanisms. A chess game terminates in a result. A program can be compiled and tested. A mathematical proof may be checked against formal constraints. The system can generate an answer, receive feedback and use that feedback as a training signal. But, unsurprisingly, this turns out not to generalise straightforwardly. As Romero puts it, “not all fields are equally tractable with this approach: the intelligence profile of the best AI models remains surprisingly jagged.”
Despite their commitments to scaling and reinforcement learning, Anthropic and OpenAI, “remain convinced it will work, but,” Romero argues, “the evidence is thinner than they’d like.” DeepSeek provides a useful case in point:
The Chinese AI startup DeepSeek tried to apply the ‘no human supervision’ AlphaZero approach to language with DeepSeek-R1 Zero, but that didn’t quite work either. Language doesn’t provide a clear reward signal for what ‘a good sentence’ is, and so the models simply mimic fan-fiction style with all the unnecessary em dashes and the weirdly ungrounded metaphors. In chess, you either win or lose. Your code either compiles or doesn’t. Your math solution is either correct or incorrect. But language — and like language, much else — just doesn’t work that way. […] The big question is, thus: is the ability to do successful AI research more like language or more like games, code, and math?
The underlying technical problem is real. DeepSeek-R1-Zero demonstrated that reinforcement learning based substantially upon rule-based rewards could induce striking reasoning behaviour, particularly in domains such as mathematics, coding and logical reasoning. But DeepSeek’s subsequent work also required additional mechanisms concerned with linguistic consistency, readability and alignment. The distinction reflects precisely the asymmetry Romero identifies: correctness is considerably easier to formalise in some domains than quality or meaning is in natural language.
The more recent successes of Anthropic, OpenAI and other frontier-model companies (and the enormous valuations that have followed) arguably owe a great deal to the growing ability of models to code, use tools, perform mathematical reasoning and increasingly participate in the production of their own technical infrastructure. We hear rather less about an equivalent breakthrough in language understood in the ordinary sense, i.e. simply grammatical fluency, but judgement, ambiguity, metaphor, implication, rhetoric, cultural difference and meaning.
But perhaps Romero’s final question can be turned around. What if the important question is not whether AI research is more like language or more like games, code and mathematics, but whether we should return to a much older question: To what extent is language itself mathematisable?
Mathematising language
This question takes us directly back to the intellectual conditions from which computing and artificial intelligence originally emerged.
The attempt to give mathematical or formal descriptions of thought predates computers by centuries. Leibniz imagined a characteristica universalis, a symbolic language through which disputes might eventually be settled by calculation. George Boole’s nineteenth-century Laws of Thought transformed logical operations into an algebra. Frege developed a formal system for logic. Russell and Whitehead attempted to ground mathematics formally in Principia Mathematica. Gödel then demonstrated fundamental limits to formal systems, while Turing transformed questions about formal procedure into the concept of a universal computing machine. Tom Griffiths’ recent book The Laws of Thought: The Quest for a Mathematical Theory of the Mind is particularly insightful on this longer history. Griffiths organises the modern attempt to formalise cognition around three major traditions — rules and symbols, neural networks, and probability and statistics — showing how contemporary AI inherits all three rather than emerging from machine learning alone.
Language occupies a fascinating position within this history because it repeatedly appears to invite formalisation while simultaneously exceeding it. Claude Shannon’s information theory provided a mathematical account of communication based upon probabilities and uncertainty. His famous experiments with approximations to English showed that statistical regularities could produce increasingly language-like sequences without requiring a model of meaning. In retrospect, this is an extraordinary precursor to the statistical conception of language underlying contemporary language modelling.
Another tradition approached the problem through structure. Noam Chomsky’s early work is especially important because it developed alongside the formative years of artificial intelligence and theoretical computer science. His 1956 paper “Three Models for the Description of Language” explicitly considered whether linguistic structure could be generated by different classes of formal systems. He argued that simple finite-state Markov processes were inadequate as grammars for natural language and developed progressively richer models of linguistic structure. The point was not that language could not be formalised. Quite the opposite. Chomsky’s intervention was that the formal machinery had to be adequate to the complexity of the object.
This distinction matters enormously today. A language contains a finite vocabulary at any given moment, yet its combinatorial possibilities appear effectively unbounded. Chomsky famously characterised language in terms of the capacity to generate an indefinite number of sentences from finite means. The problem of language therefore already involves a tension between the finite and the apparently infinite. Computational linguistics grew partly from attempts to understand that tension.
Early AI inherited symbolic models in which intelligence could be represented through rules, symbols and transformations. Connectionism offered a competing account in terms of distributed activation and learned relations. Statistical approaches subsequently shifted emphasis toward probability. Contemporary Large Language Models (LLMs) combine several aspects of this history: their architecture is neural, their training is statistical, while their outputs frequently exhibit capacities previously associated with symbolic manipulation.
The historical debate was therefore never simply between “language” and “mathematics.” It concerned which mathematics was adequate to language. That question deserves reopening.
From sequence to relational state
Contemporary language models predominantly treat language computationally as sequences of tokens. Transformers dramatically improved the representation of relationships across those sequences through attention. A word need no longer be processed merely in relation to its immediate predecessor; attention mechanisms allow a token to be evaluated in relation to many other tokens within a context.
This matters because meaning is relational.
The insight is much older than AI. Saussure’s structural linguistics insisted that linguistic units acquire value through differences within a system rather than through simple correspondence between words and things. Structuralism subsequently extended this principle across anthropology, psychoanalysis, literary theory and semiotics. Language is therefore not merely a very long sequence of symbols. It is a structure of relations between possible relations. This is where the question of quantum computation becomes interesting again.
If quantum advantage is potentially concerned not principally with volume but with state complexity, then language presents an intriguing candidate problem. Natural language constantly maintains multiple potential interpretations before context reduces them. Words are polysemous. Sentences depend upon relationships between distant elements. Metaphor transfers structures between semantic domains. Pragmatic meaning depends upon speaker, audience and situation. Cultural meanings depend upon systems of difference that are not contained within individual words.
None of this means that language is literally quantum mechanical. There is no reason to suppose that linguistic ambiguity depends upon physical quantum superposition in the brain, and invoking quantum mechanics merely as a metaphor for ambiguity would add little. The more interesting possibility is computational: Could quantum representations provide useful mathematical machinery for representing and transforming some of these highly relational semantic state spaces?
There is already a field concerned with precisely this possibility: Quantum Natural Language Processing (QNLP). One important tradition develops from categorical models of compositional meaning, particularly the DisCoCat framework associated with Bob Coecke, Mehrnoosh Sadrzadeh and Stephen Clark. Here, grammatical composition and distributional representations of word meaning can be described within a shared mathematical framework that happens also to map naturally onto quantum formalism. Contemporary QNLP research has begun implementing versions of these ideas using variational quantum circuits and hybrid quantum-classical systems, although current experiments remain small and there is as yet no demonstrated general quantum advantage for natural-language processing.
That last qualification is important. The claim we might make is not that quantum computers will imminently run LLMs faster. Rather, it is more conceptual. Argubly, the relationship between Quantum AI and language requires us to identifywhat kind of computational complexity meaning actually presents.
Meaning & second-order signification
Here semiotics becomes significant. One of the limitations of treating language as something that simply requires a correct answer is that much linguistic and cultural meaning operates at what semioticians describe as a second order of signification.
Roland Barthes’ account of myth remains the familiar example. At a first level, a sign consists of a relation between signifier and signified. But that completed sign can itself become the signifier within another signifying system. A photograph can denote a person performing an action while simultaneously connoting nationality, heroism, modernity, class, gender or political ideology. Meaning does not stop when a referent has been identified. Barthes drew this conception from Louis Hjelmslev, whose distinction between denotation and connotation allowed semiotics to describe signifying systems built upon other signifying systems.
This poses a considerably more difficult computational problem.
A sentence may be grammatically correct and factually accurate yet sarcastic. An image may accurately depict its subject while simultaneously functioning as propaganda. A phrase can acquire an entirely different significance when quoted. A metaphor is meaningful precisely because literal substitution is insufficient. Cultural signs acquire additional meanings through histories of previous use. There is therefore no single equivalent of “the code compiled.” Even the idea of a “good sentence” is insufficient. What matters is what the sentence does within a field of relations.
This may explain part of the jaggedness Romero identifies. Current reinforcement-learning systems perform particularly well when the problem admits a relatively inexpensive verifier. Language often does not. Evaluating language may require reconstructing multiple levels of context, intention, genre, historical association, cultural convention and interpretation. The apparent fuzziness of language may therefore conceal an extraordinarily difficult state-space problem.
Finite, immense relational spaces
This brings us back to the relationship between the finite and infinite.
Language appears infinite to human beings because it is generative. We routinely encounter and produce sentences that have never previously existed. Yet this apparent infinity emerges from finite alphabets, finite vocabularies, finite grammatical resources and finite cognitive systems. Perhaps what matters computationally is therefore neither infinity nor sheer quantity. It is combinatorial relational complexity.
Even a moderately sized collection of signs can participate in an enormous number of possible relationships. Add syntax, semantic association, discourse context, speaker intention, cultural convention and second-order signification, and the relevant state space becomes vast. Classical AI deals with this through large vector spaces, statistical approximation, attention mechanisms and enormous quantities of training data. Scaling has proved remarkably effective. But scaling may not be the only possible response (or at least we begin to hit up against significant issues with the energy required to make calculations).
What if the capacity of quantum computation to represent and manipulate complex state spaces becomes relevant here? Not because a quantum computer would magically “understand” language. Nor because language can simply be converted into qubits. And not because superposition means that every interpretation is computed simultaneously and then the right one appears.
Rather, the research question would be whether particular aspects of linguistic and semiotic complexity can be reformulated as problems for which quantum representations or quantum algorithms provide computationally useful structure. The distinction is subtle but crucial. We might be led to ask:
Are there structures of linguistic meaning whose computational difficulty lies in the relationships among possible states rather than in the amount of linguistic data itself?
From verification to transformation
Hence, Quantum AI begins to look different and potentially suggests another way of thinking about recursive self-improvement. The current approach is governed by verification. An AI produces something; an external mechanism determines whether it is better; the model learns from the result; this process is repeated. As already noted, for chess, code and formal mathematics, this works because powerful objective functions can be constructed. Language may require something different.
Recursive improvement in linguistic intelligence cannot proceed by simply discovering a universal scalar reward for meaning. It requires increasingly sophisticated models capable of representing transformations between levels and systems of meaning. This is where the notion of second-order signification becomes more than a theoretical embellishment. It points toward a specific computational problem.
Can an AI represent not only that x signifies y, but that the relation x → y subsequently functions as an element within another signifying relation? And can it recognise transformations across such levels?
signifier → signified becomes [signifier → signified] → second-order signified
Phrased in this way, semiotics begins to look surprisingly computational. What has traditionally been regarded as interpretative complexity might also be understood as the iterative construction of relational state spaces.
There are obvious issues at stake. Formalisation can strip cultural meaning of history, embodiment and power. Structuralism itself was criticised precisely for producing systems that appeared too closed, synchronic or autonomous. Post-structuralism subsequently emphasised instability, difference, iterability and the impossibility of finally closing the signifying process. But those criticisms need not terminate the computational project. They may instead specify its difficulty (see Manghani, ‘Preparatory Space: Roland Barthes and Large Language Models‘, Barthes Studies, Vol. 10, 2024).
The lesson from the history of linguistics is not that language escaped mathematics.It is that every proposed mathematics disclosed another layer of the problem.
Research programme for Quantum AI
A rather different Quantum AI research agenda emerges from the familiar promise that quantum computers will simply make machine learning faster.
A opening question would be:
Which forms of intelligence are difficult because they involve large quantities of data, and which are difficult because they involve extraordinarily complex spaces of possible relations?
For the first class of problem, classical computation may remain overwhelmingly preferable. For the second, quantum computation may eventually become interesting.
Language provides an especially important test case because it sits exactly at the intersection of formal structure and interpretative openness. It is rule-governed without being reducible to rules; probabilistic without being reducible to probability; combinatorial without being merely combinatorics; symbolic while also depending upon context, embodiment, history and culture.
The immediate research programme would therefore not be “put an LLM on a quantum computer.” Instead, we might isolate specific components of linguistic intelligence: compositional semantics; ambiguity resolution; contextual state representation; analogy; metaphor; second-order signification; transformations between semantic frames; and perhaps eventually the modelling of recursive relations among these processes.
Some of these may prove perfectly manageable classically. Some may receive useful quantum-inspired formulations without requiring quantum hardware. Some may eventually map onto quantum algorithms capable of manipulating structured spaces more efficiently than classical alternatives. Many will probably yield no quantum advantage at all. But, that is precisely why the research needs to be done.
The possibility becomes particularly interesting when returning to recursive self-improvement. If AI development increasingly depends upon systems evaluating and improving other AI systems, the decisive question may not simply be whether an AI can write better code for itself. It may be whether it can construct increasingly adequate representations of what better means when no simple verifier exists.
Language is the paradigmatic case. And, here, a historical reading here matters. The developments in logic, linguistics, psychology, information theory, cybernetics and artificial intelligence were once considerably more entangled than our contemporary disciplinary divisions suggest. Chomsky’s formal linguistics, Shannon’s information theory, Turing’s computation, early neural networks and structuralist accounts of signification all emerged within overlapping attempts to ask how apparently complex human activities might possess an underlying formal organisation. Large language models have returned us to that question from an unexpected direction. They demonstrate that astonishing linguistic capacities can emerge from mathematical operations over immense statistical structures. Yet their limitations also show that statistical fluency and meaning cannot simply be equated.
Quantum computing might now allow us to ask the old question again under different mathematical conditions. We can move beyond asking how machines can process more language (as per the scaling debates), and instead ask: What kind of mathematical object is meaning?
If language is generated from finite means yet unfolds across vast spaces of relational possibility; if meaning operates recursively, with signs becoming components of further signs; if intelligence depends not only upon storing information but navigating such structures; then the future significance of Quantum AI may lie less in computing more data than in finding new ways to compute relations. The wager is not that quantum computing will make language behave like code. By changing what kinds of complex state spaces can be represented and transformed computationally, quantum computing might force us to reconsider why we assumed that language and code were computationally different in the first place.
The route to next-generation AI can look beyond larger models, larger datasets or faster chips; to return to one of the oldest ambitions in the history of computation: the mathematisation of thought itself.
Selected references
Barthes, Roland (1957/1972), Mythologies, trans. Annette Lavers. New York: Hill and Wang.
Chomsky, Noam (1956), “Three Models for the Description of Language,” IRE Transactions on Information Theory, 2(3), 113–124.
Chomsky, Noam (1957), Syntactic Structures. The Hague: Mouton.
Coecke, Bob, Mehrnoosh Sadrzadeh and Stephen Clark (2010), “Mathematical Foundations for a Compositional Distributional Model of Meaning,” Linguistic Analysis, 36, 345–384.
Griffiths, Tom (2026), The Laws of Thought: The Quest for a Mathematical Theory of the Mind. London: William Collins / New York: Henry Holt.
Hjelmslev, Louis (1943/1961), Prolegomena to a Theory of Language, trans. Francis J. Whitfield. Madison: University of Wisconsin Press.
Romero, Alberto (2026), “The Actual Reason Why Google ‘Fell Out’ of the AI Race Changes Everything,” The Algorithmic Bridge, 28 July 2026. (thealgorithmicbridge.com)
Saussure, Ferdinand de (1916), Course in General Linguistics.
Shannon, Claude E. (1948), “A Mathematical Theory of Communication,” Bell System Technical Journal, 27, 379–423, 623–656.
Turing, Alan M. (1936), “On Computable Numbers, with an Application to the Entscheidungsproblem,” Proceedings of the London Mathematical Society, 42, 230–265.