Can a problem undecidable with classical resources be decidable with quantum ones? The answer expected is no; as both being Turing theories, they should not solve the Halting problem - a problem unsolvable by any Turing machine. Yet, we provide an affirmative answer to the aforesaid question. We come up with a novel logical structure to formulate infinitely many such problems for any pair of distinct Turing theories, including but not limited to the classical and quantum theories. Importantly, a class of other decision problems, such as the Halting one, remains unsolvable in all those theories. The apparent paradoxical situation gets resolved once it is perceived that the reducibility of Halting problem changes with varying resources available for computations in different theories. In the end, we propose a multi-agent game where winnability of the player having access to only classical resources is undecidable while quantum resources provide a perfect winning strategy.
Can a problem undecidable with classical resources be decidable with quantum ones? The answer expected is no; as both being Turing theories, they should not solve the Halting problem - a problem unsolvable by any Turing machine. Yet, we provide an affirmative answer to the aforesaid question. We come up with a novel logical structure to formulate infinitely many such problems for any pair of distinct Turing theories, including but not limited to the classical and quantum theories. Importantly, a class of other decision problems, such as the Halting one, remains unsolvable in all those theories. The apparent paradoxical situation gets resolved once it is perceived that the reducibility of Halting problem changes with varying resources available for computations in different theories. In the end, we propose a multi-agent game where winnability of the player having access to only classical resources is undecidable while quantum resources provide a perfect winning strategy.
Quantum computing allows us to solve some problems much faster than existing classical algorithms. Yet, the quantum computer has been believed to be no more powerful than the most general computing model—the Turing machine. Undecidable problems, such as the halting problem, and unrecognizable inputs, such as the real numbers, are beyond the theoretical limit of the Turing machine. I suggest a model for a quantum computer, which is less general than the Turing machine, but may solve the halting problem for any task programmable on it. Moreover, inputs unrecognizable by the Turing machine can be recognized by the model, thus breaking the theoretical limit for a computational task. A quantum computer is not just a successful design of the Turing machine as it is widely perceived now, but is a different, less general but more powerful model for computing, the practical realization of which may need different strategies than those in use now.
Incompleteness and undecidability theorems have to be revised in view of quantum information and computation theory. qrt.tex 1 As has already been pointed out in Gödel’s centennial paper on the incompleteness af arithmetic [1], the classical undecidability theorems of formal logic [2] and the theory of computable functions [4, 5] are based on semantical pardoxes such as the liar [6] or Richard’s paradox. The method of diagonalization, which was first applied by Cantor for a proof of the undenumerability of real numbers [7], has been applied by Turing for a proof of the recursive undecidability of the halting problem [8]. The halting problem is the problem of whether or not an arbitrary algorithm terminates or produces a particular output and terminates. Assume that the halting problem is decidable. Turing [8] proved that this assumption yields a contradiction. To construct the contradiction, consider
FINDING: Undecidable problems extend from computation to quantum field theory and many-body physics, revealing fundamental limits on what can be known or computed about physical systems. | MATH: Halting problem: no Turing machine can decide if an arbitrary program halts (Turing 1936). Gödel incompleteness: any consistent formal system containing arithmetic has true but unprovable statements. Reducibility: problem A reduces to B if solving B solves A (A ≤ B). In QFT, spectral gap problem is undecidable (Cubitt, Perez-Garcia, Wolf 2015): no algorithm can determine if a quantum many-body system has a nonzero energy gap. | CONNECTION: Undecidability in physics often arises from encoding Turing machines into Hamiltonian lattice systems — this maps computational logic onto geometric lattice structures (e.g., 2D spin lattices with local interactions). The spectral gap problem's undecidability proof uses aperiodic tilings and translation-invariant Hamiltonians, linking to crystallographic symm Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: The halting problem's undecidability is foundational to computation limits; recent MIP*=RE result links it to quantum entanglement and nonlocal games, while a 2024 paper nuances Turing's original attribution. | MATH: Undecidability proof via diagonalization: assume HALT exists, construct D(x) = loop if HALT(x,x) halts, else halt; contradiction. MIP*=RE: the class of languages decidable by multiprover interactive proofs with quantum entanglement equals recursively enumerable sets — equivalently, the commuting-operator and tensor-product models diverge (Connes embedding conjecture false). No classical constants; structure is ordinal/recursion-theoretic (ω₁^CK, Turing degrees). | CONNECTION: No direct geometric ratio (0.382, 0.618, etc.) appears. However, the MIP*=RE result relies on *lattice-like* constructions of nonlocal games (Tsirelson bounds, operator algebras) and the *root system* of quantum correlations — the set of quantum strategies forms a convex body with extreme poi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
The halting problem is a decision problem first posed and proved by Alan Turing in 1936. With the recent surge of interest in quantum computation, one is led to ask if the problem can also be considered for a quantum computer. It is reported that the halting problem may not be solved consistently in both the Schrodinger and Heisenberg pictures of quantum dynamics. The assumption of the existence of the quantum halting machine leads to a contradiction when a vector representing an observable is the system that is to be unitarily evolved in both pictures.
Abstract In the classical theory of computation, e.g. , in the Turing Machine model, computational processes employ only local space and time resources, and their resource usage can be accurately measured by us as users. General relativity and quantum theory, however, introduce the possibility of computational processes that employ nonlocal spatial or temporal resources, raising the question of how these relate to classical resources. Operational, i.e. , device-independent, protocols to certify the use of entanglement as a resource are well known. We prove, however, that the independence of spatially separated systems cannot be operationally certified. The verifier (C) in a multiple interactive provers with shared entanglement (MIP*) protocol cannot, therefore, operationally demonstrate that the “multiple” provers are independent, i.e. , cannot operationally distinguish a MIP* machine from a monolithic quantum computer. Thus C cannot operationally distinguish a MIP* machine from a quantum TM, and hence cannot operationally demonstrate the solution to arbitrary problems in RE. Any claim that a MIP* machine has solved a TM-undecidable problem, e.g. , that of Ji Z. et al. , Commun. ACM , 64 (2020) 131, is therefore operationally circular, as the problem of deciding whether a physical system is a MIP* machine is itself TM-undecidable. We then prove a similar result showing that whether a system employs a closed time-like curve (CTC) as a resource is operationally undecidable. In such settings, therefore, theoretical analyses of resource usage cease to be reliable indicators of practical computational capability.