Yuexin Liao
California Institute of Technology
Have you ever asked why AI systems that appear intelligent can nonetheless make absurd decisions and confidently produce nonsense? Why do they hallucinate? The root cause is that they do not possess intrinsic understanding; they merely model statistical correlations in language.
You may also wonder: in the post-GPT era, what is the next inflection point for AI? In this essay I offer a perspective: bring quantum computing to bear for fine-grained control and breakthroughs in combinatorial optimization, thereby reshaping AI's rational structure. The path from mimetic pattern-matcher to genuinely intelligent reasoner is the inevitable trajectory of AI's evolution. The future belongs to verifiable, interpretable, and trustworthy intelligence.
Pure large language models (LLMs) are, in the long run, on a path to obsolescence. Contemporary deep learning—especially LLMs—is, at its core, a probability-maximizing system built on massive data statistics. It excels at predicting the next token from context, but that skill marks it as an empiricist imitator (a miner of similarities), not a rationalist reasoner. The correlation-driven mechanism that powered LLMs' early success also imposes conspicuous limits. If those limits remain unaddressed, LLMs will stall at clever mimicry—confined to a simulacrum of understanding—and will struggle to support high-reliability, high-explainability tasks that demand deep reasoning. This essay analyzes the structural bottlenecks of the deep-learning paradigm and argues that only by importing formal axioms and high-precision control structures can AI cross the gap from statistical imitation to robust logical inference. I then explain how quantum computing can serve as the technical fulcrum for that transition.
First, LLMs lack real constraints. They optimize co-occurrence probabilities of words while lacking grounding in facts and physical law, which is why they sometimes emit content that defies common sense or even the laws of physics. They do not "understand" the world; they fit patterns present in the training corpus. Complex-systems theory speaks of causal emergence: macro-level causal structure can be stronger than micro-level correlational structure. As theoretical neuroscientist Erik Hoel has argued, coarse-graining can reveal macro-scale variables with greater causal efficacy than their micro-scale constituents. Applied to LLMs: without macro-level axiomatic constraints, a system assembled from micro-level correlations will struggle to generate inferences that are genuinely causal and withstand scrutiny.
Second, LLM outputs are not reliably predictable. In domains requiring absolute correctness (e.g., financial risk control), a probabilistically "often correct" answer is categorically insufficient. For such tasks, we must ensure that the model's trajectory is dynamically confined to a submanifold defined by pre-specified axioms. Each output step must be equivalent to a verifiable derivation that preserves required invariants. A system of this kind behaves not chaotically or merely statistically, but logically, with predictability, verifiability, and auditability.
Third, large predictive models are prone to shortcut learning: they seize on spurious correlations rather than grasping causal structure. A model can score highly by exploiting superficial regularities that do not generalize. By analogy with statistical physics, standard deep-learning training resembles a zero-temperature quench, whose dynamics make it easy to become trapped in the many local minima of an energy landscape—minima that encode spurious associations.
A more robust paradigm, conceptually akin to simulated annealing, requires a controlled source of stochasticity (an effective "temperature") to explore globally early on, escape local traps, and then settle into global optima that reflect genuine regularities as temperature is lowered. Modern techniques—SGD noise, dropout, learning-rate schedules—can be viewed heuristically as injecting such an "effective temperature," helping an optimizer hop between minima. Yet they remain fundamentally limited, for two reasons:
1. They do not alter the energy landscape itself. The local minima representing spurious associations still exist and remain energetically admissible. These methods modify the dynamics of optimization, not the declarative definition of the problem. They are procedural tweaks, not axiomatic constraints.
2. Their perturbations are (largely) isotropic. They heat the system uniformly, allowing random motion in any direction, without embedding prior knowledge of which directions are correct. A genuine global constraint—i.e., an axiom—acts not as undirected heat but as a strongly directed potential. For example, an energy-conservation axiom erects an infinite barrier that excludes all nonconserving states from the reachable set. It provides sharp directional guidance: "these regions are forbidden." Heuristic noise lacks such intelligence and directionality.
In brief, heuristics aim to make us better guessers, whereas many real-world tasks require us to be reasoners constrained by task-intrinsic axioms. Consequently, axiomatically constraining AI outputs emerges as a front-line direction for addressing the limitations above. Here, "axioms" encompass any formalizable domain laws we desire the AI to obey. We analyze their properties and explain why AI stands to benefit profoundly from them.
Axioms arise at multiple levels:
(1) Physical axioms: conservation of energy and momentum, Maxwell's equations, etc. In AI-for-Science applications—say, molecular structure prediction—we should require outputs that do not violate physics or chemistry, just as nature never violates conservation laws. Imposing physical axioms prevents physically nonsensical proposals.
(2) Mathematical/Logical axioms: Euclid's postulates, algebraic rules, formal logic. For theorem proving, program synthesis, or complex reasoning, each inferential step must be valid. The reasoning chain should advance with the rigor of a formal system, ensuring conclusions follow from premises via sanctioned rules. Research lines have explored embedding logical constraints into neural architectures—for instance, representing rules as tensors integrated with networks—so that learning and reasoning co-habit.
(3) Economic/Social axioms: rationality assumptions, market-clearing, game-theoretic constraints, fairness principles. In economic decision-making or social simulation, we can require decisions to align with specified theoretical frameworks. In automated market analysis, a no-arbitrage axiom can rule out predictions that violate basic financial logic. In multi-agent systems and game-playing, strategy-compatibility and fairness constraints can deter short-sighted, non-robust policies.
(4) Ethical/Value axioms: non-maleficence, justice, etc.—central in AI safety and alignment. Anthropic's "Constitutional AI" is illustrative: a curated set of constitutional-style principles (e.g., avoiding harmful or discriminatory outputs) guides a model's behavior via explicit rules rather than solely via human-feedback gradients.
In all these ways, axioms seek to upgrade AI from an empiricist imitator to a rationalist reasoner. It is not enough that outputs match the data distribution; they must satisfy the axioms, yielding guaranteed-correct, rule-conforming answers. Conceptually, we require the AI to justify the correctness or legality of its answers as it generates them.
Recent work shows how data and theory can be combined. Cristina Cornelio et al., Nature Communications (2023), propose AIDescartes: Combining data and theory for derivable scientific discovery. Their method first uses symbolic regression to generate candidate formulas from limited experimental data, then applies automated theorem proving or formal-logic inference to filter those candidates against background axioms, retaining only those derivable from the axioms. They recover, among others, Kepler's third law, Einstein's time-dilation formula, and the Langmuir adsorption isotherm, and provide formal derivations showing these follow from the stipulated axioms. The lesson is clear: an AI that leverages axioms to guide and prune hypotheses is more efficient at discovering genuinely meaningful regularities than one that fits data alone.
Of course, making AI outputs strictly axiom-compliant entails formidable technical challenges. All such constraints must be mathematically formalized to be embedded in learning. On one hand, rules must be rendered machine-interpretable (logical formulas, hard constraints, or penalty terms in the loss). On the other hand, this often transmutes prediction into constrained combinatorial optimization: the model must fit data in a continuous parameter space and search a discrete rule space for solutions satisfying all axioms. This is like solving a constrained optimization problem: maximize likelihood (or minimize loss) subject to multiple constraints. Such problems are typically NP-hard; a classical computer must explore a combinatorially exploding state space to find global optima. The situation resembles confining a dynamical system to a thin manifold carved out by multiple conserved quantities: any initial state must "slide" along narrow, branching channels to reach the global attractor. Compared with unconstrained descent along a single energy gradient, searching for the lowest-energy state on a constrained manifold is both narrower and more tortuous, riddled with bifurcations and barriers—demanding more careful trajectory design and finer perturbation control.
Here, the importance of quantum computing as an instrument of fine-grained control comes into sharp relief. If classical computation (semiconductors) embodies precision over bits, quantum computation embodies precision over quantum states, reaching down to the elementary fabric of physical reality. In that sense, quantum computing is the apex form of fine control—both physically and algorithmically.
On the physical side, quantum computation directly manipulates the quantum states of particles (e.g., spin/energy levels of single photons or ions), at unprecedented spatiotemporal resolution. Tiny perturbations—environmental noise, phase drift—can collapse superpositions and spoil computation. Reliable quantum processing thus requires near-theoretical-limit control over system evolution. In effect, a quantum computer is one of humanity's most precise control systems: we demand atomic-scale logical operations, surpassing the precision with which we steer charge through silicon transistors. If making AI obey axioms is a demand for control over outputs, quantum computing exemplifies control at the physical substrate.
On the computational side, quantum computing introduces new algorithmic paradigms that tackle complexity in ways unavailable classically. Classical algorithms treat uncertainty as noise to suppress; quantum algorithms treat probability amplitudes and coherence as resources. With superposition, an n-qubit device can implicitly explore 2n configurations in a single unitary evolution. With interference, we can design dynamics so that paths corresponding to constraint-violating solutions cancel, while paths corresponding to axiom-satisfying solutions reinforce—amplifying the probability amplitude of desired outcomes. Grover's algorithm exemplifies this idea, reducing unstructured search from O(N) to O(√N) by coherently "interfering away" wrong answers. Likewise, the Quantum Approximate Optimization Algorithm (QAOA) leverages parameterized quantum evolutions for hard combinatorial optimization; it naturally accommodates multiple constraints, aligning quantum dynamics with axiomatic feasibility regions.
I therefore contend that quantum + LLM will become a new focal point in AI. As models scale to meet real-world complexity, parameter and data sizes verge on classical hardware limits; simultaneously, embedding rigorous rules further increases computational burden. Quantum computing can relieve both pressures: new compute resources and more efficient algorithms open room to explore larger models and deeper rule integration. Early industrial experiments are suggestive. For example, IonQ has reported quantum-enhanced fine-tuning of LLMs by inserting trainable quantum circuit layers into classical Transformer stacks, with certain configurations surpassing purely classical baselines in tasks like sentiment analysis—evidence that quantum modules can capture subtle correlations elusive to classical methods. Other proposals envisage quantum realizations of attention, potentially reducing the quadratic cost of self-attention and enabling longer contexts with fewer resources. These explorations hint that replacing select modules with quantum circuits, or redesigning architectures around quantum subroutines, could help leap beyond current bottlenecks in both quality and efficiency.
More importantly, quantum computing and axiom-constrained AI are naturally aligned. Both emphasize meticulous control and judicious exploitation of the solution space: quantum mechanics guarantees reliable and efficient computation through physical law, while axioms guarantee correctness and consistency of inference. When we require LLMs to obey axioms, we must sift a vast hypothesis space for outputs that satisfy both data fit and rule compliance—precisely the kind of coherent selection that quantum interference implements over exponentially large state spaces.
One can envisage hybrid architectures: a classical neural network learns patterns from data and proposes candidate solutions; a quantum module then precision-filters those candidates, rapidly evaluating—within a superposed state space—which solutions satisfy all constraints, canceling noncompliant options via interference. Such quantum–classical hybrid AI marries intuitive pattern recognition with rigorous axiomatic verification, achieving complementary strengths.
In sum, the logic chain is clear: to break current limits, AI must progress from empirical statistics to axiomatic constraint; to remain precise and efficient amid the combinatorial explosion of rule spaces, quantum computing offers a uniquely powerful technological lever and paradigm. I therefore anticipate quantum computing will become a pivotal hotspot for the future of AI—especially for the next generation of LLMs. Every cutting-edge step we take today—whether embedding conservation laws and first principles into AI, or using quantum algorithms to enhance model capability—advances a common agenda: to build intelligent systems that are both capable and trustworthy, fusing the rule-governed structure of human knowledge with the unbounded potential of computation, and thereby making AI rational and reliable.