The physical Church-Turing thesis and the principles of quantum theory

Notoriously, quantum computation shatters complexity theory, but is innocuous to computability theory. Yet several works have shown how quantum theory as it stands could breach the physical Church-Turing thesis. We draw a clear line as to when this is the case, in a way that is inspired by Gandy. Gandy formulates postulates about physics, such as homogeneity of space and time, bounded density and velocity of information — and proves that the physical Church-Turing thesis is a consequence of these postulates. We provide a quantum version of the theorem. Thus this approach exhibits a formal non-trivial interplay between theoretical physics symmetries and computability assumptions.

💡 Research Summary

The paper revisits the Physical Church‑Turing Thesis (PCTT)—the claim that any function computable by a physical system can be computed by a Turing machine—and asks whether this claim still holds in the context of quantum theory. The authors build on Gandy’s classic 1980 result, which derived PCTT from four physical postulates for classical physics: (i) homogeneity of space, (ii) homogeneity of time, (iii) bounded density of information, and (iv) bounded speed of information propagation together with a quiescent (inactive) background state. Gandy proved that any system satisfying these constraints evolves in a way that can be simulated by a Turing machine, thus establishing the physical Church‑Turing thesis for classical physics.

The central contribution of the paper is a quantum analogue of Gandy’s theorem. The authors first formalize a quantum cellular automaton model: three‑dimensional Euclidean space is discretized into cubic cells, each cell carrying a finite‑dimensional Hilbert space (e.g., a single qubit). The global state is a tensor product of all cell states, and the evolution proceeds in discrete time steps of fixed duration T. The four Gandy postulates are re‑interpreted for this quantum setting:

1. Homogeneity of space – the same local quantum rule (a unitary gate or completely positive map) is applied to every cell; translations of the lattice commute with the global evolution.
2. Homogeneity of time – the same rule is applied at each time step; the evolution operator does not depend on the absolute time.
3. Bounded density of information – although the underlying Hilbert space is infinite‑dimensional in principle, the physically accessible information from any finite region is finite because each cell’s state space is a finite‑dimensional vector space over a field of computable numbers (e.g., rationals). Consequently, any measurement on a finite region yields only finitely many outcomes.
4. Bounded speed of information propagation (causality) – the state of a cell at time t + T depends only on the states of cells within a fixed finite neighbourhood (radius 1) at time t. This captures the relativistic light‑cone constraint in a quantum‑information‑theoretic form.
5. Quiescence – all but finitely many cells are in a distinguished “blank” state at any time; this ensures that the global configuration can be encoded as an infinite string that is constant almost everywhere, and thus as a single natural number.

Under these assumptions the global evolution operator G can be expressed as a finite local function χ that maps the finite tuple of neighbouring cell states to the next state of a given cell. Because χ is a finite table, it is computable; consequently the iterated map k ↦ G^k(initial state) is a computable function from ℕ to ℕ. This yields the Quantum Gandy Theorem: any quantum physical system obeying the above postulates cannot compute more than a Turing machine.

The paper then demonstrates the necessity of each postulate by constructing counter‑examples that violate PCTT when a single postulate is dropped. For instance, without spatial homogeneity one can embed an undecidable set U into the rule applied to each cell, thereby allowing the system to decide membership in U. Without bounded density, a cell’s state space could be the whole set of natural numbers, enabling a direct implementation of a non‑computable function f_U. Similar constructions are given for the other postulates, mirroring Gandy’s original arguments.

A substantial discussion follows on how these quantum‑adapted postulates reconcile with known features of quantum theory. The authors argue that the infinite‑dimensional Hilbert space of a particle does not contradict bounded density, because only finitely many measurement outcomes are physically accessible. They also exclude non‑computable amplitudes by restricting the field of scalars to a computable extension of the rationals, which aligns with the practical limitation that quantum gates are built from a finite universal set. Entanglement and the tensor‑product structure are handled by working with density matrices and partial traces, ensuring that the locality condition still makes sense in the presence of quantum correlations.

In conclusion, the paper shows that if quantum physics respects the quantum‑Gandy postulates, the Physical Church‑Turing Thesis remains valid even in the quantum realm. Quantum computers may still provide exponential or polynomial speed‑ups for certain problems, but they do not enlarge the class of computable functions beyond the Turing‑computable ones. This result reinforces the view that quantum theory, despite its exotic features, does not overturn the fundamental limits of computability, and it provides a rigorous, physics‑grounded justification for the PCTT in the quantum setting. Future work is suggested on extending the framework to continuous‑time, continuous‑space quantum field theories and on exploring whether weaker or alternative physical postulates might still guarantee the thesis.