We explore in the framework of Quantum Computation the notion of Computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert’s tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm could be physically implemented, as much as it is valid in principle—that is, if certain hamiltonian and its ground state can be physically constructed according to the proposal—quantum computability would surpass classical computability as delimited by the Church-Turing thesis. It is thus argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles. 1

We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic which is equivalent to the Turing halting problem and known to be mathematically noncomputable. Generalised quantum algorithms are also considered for some other mathematical noncomputables in the same and of different noncomputability classes. The key element of all these algorithms is the measurability of both the values of physical observables and of the quantum-mechanical probability distributions for these values. It is argued that computability, and thus the limits of Mathematics, ought to be determined not

Abstract. We give an overview of a quantum adiabatic algorithm for Hilbert’s tenth problem, including some discussions on its fundamental aspects and the emphasis on the probabilistic correctness of its findings. For the purpose of illustration, the numerical simulation results of some simple Diophantine equations are presented. We also discuss some prejudicial misunderstandings as well as some plausible difficulties faced by the algorithm in its physical implementations. “To believe otherwise is merely to cling to a prejudice which only gives rise to further prejudices... ” 1

Abstract. We study a set of truncated matrices, given by Smith [8], in connection to an identification criterion for the ground state in our proposed quantum adiabatic algorithm for Hilbert’s tenth problem. We identify the origin of the trouble for this truncated example and show that for a suitable choice of some parameter it can always be removed. We also argue that it is only an artefact of the truncation of the underlying Hilbert spaces, through showing its sensitivity to different boundary conditions available for such a truncation. It is maintained that the criterion, in general, should be applicable provided certain conditions are satisfied. We also point out that, apart from this one, other criteria serving the same identification purpose may also be available. In a proposal of a quantum adiabatic algorithm for Hilbert’s tenth problem [5], we employ an adiabatic process with a time-dependent Hamiltonian (1) H(t) = (1 − t/T)HI + (t/T)HP. Here t is time and this Hamiltonian metamorphoses from HI when t = 0 to HP when t = T. The final Hamiltonian HP encodes the Diophantine equation in consideration, while the initial HI is universal and independent of the Diophantine equation, except only on its number of variables K. The process is captured by the Schrödinger equation (2) ∂t|ψ(t)〉

A theoretical model of truly autonomic computing systems (ACS), with infinitely many constraints, is proposed. An argument similar to Turing’s for the unsolvability of the halting problem, which is permitted in classical logic, shows that such systems cannot exist. Turing’s argument fails in the recently proposed non-Aristotelian finitary logic (NAFL), which permits the existence of ACS. NAFL also justifies quantum superposition and entanglement, which are essential ingredients of quantum algorithms, and resolves the Einstein-Podolsky-Rosen (EPR) paradox in favour of quantum mechanics and non-locality. NAFL requires that the autonomic manager (AM) must be conceptually and architecturally distinct from the managed element, in order for the ACS to exist as a nonself-referential entity. Such a scenario is possible if the AM uses quantum algorithms and is protected from all problems by (unbreakable) quantum encryption, while the managed element remains classical. NAFL supports such a link between autonomic and quantum computing, with the AM existing as a metamathematical entity. NAFL also allows quantum algorithms to access truly random elements and thereby supports non-standard models of quantum (hyper-) computation that permit infinite parallelism. 1.

The term ‘hypermachine ’ denotes any data processing device (theoretical or that can be implemented) capable of carrying out tasks that cannot be performed by a Turing machine. We present a possible quantum algorithm for a classically non-computable decision problem, Hilbert’s tenth problem; more specifically, we present a possible hypercomputation model based on quantum computation. Our algorithm is inspired by the one proposed by Tien D. Kieu, but we have selected the infinite square well instead of the (one-dimensional) simple harmonic oscillator as the underlying physical system. Our model exploits the quantum adiabatic process and the characteristics of the representation of the dynamical Lie algebra su(1, 1) associated to the infinite square well.

To explore the limitation of a class of quantum algorithms originally proposed for the Hilbert’s tenth problem, we consider two further classes of mathematically nondecidable problems, those of a modified version of the Hilbert’s tenth problem and of the computation of the Chaitin’s Ω number, which is a representation of the Gödel’s Incompletness theorem. Some interesting connection to Quantum Field Theory is pointed out, but a direct generalisation of the quantum algorithms cannot satisfy, among others, the requirement of finite energy.

Abstract. Inspired by Quantum Mechanics, we reformulate Hilbert’s tenth problem in the domain of integer arithmetics into problems involving either a set of infinitely-coupled non-linear differential equations or a class of linear Schrödinger equations with some appropriate timedependent Hamiltonians. We then raise the questions whether these two classes of differential equations are computable or not in some computation models of computable analysis. These are non-trivial and important questions given that: (i) not all computation models of computable analysis are equivalent, unlike the case with classical recursion theory; (ii) and not all models necessarily and inevitably reduce computability of real functions to discrete computations on Turing machines. However unlikely the positive answers to our computability questions, their existence should deserve special attention and be satisfactorily settled since such positive answers may also have interesting logical consequence back in the classical recursion theory for the Church-Turing thesis.

Abstract. For a given k-dimensional subspace V0 in a Hilbert space H and a unitary transformation g0: V0 − → V0, we find a path in the Grassmann manifold the monodromy of which coincides with g0. Let H be a finite-dimensional Hilbert space; U(H) be the Lie group of unitary transformations of H and u(H) be the corresponding Lie algebra. For any positive integer k, the Grassmann manifold Grk(H) is defined as the set of all k-dimensional subspaces of H. This manifold can also be dscribed as the set of corresponding orthogonal projectors Grk(H) = P: H − → H | P is linear, P †} = P, tr(P) = k. As it is well-known for any given Hamiltonian H ∈ u(H) the corresponding Schrödinger equation is defined as the equation of the form (1) ˙ ψ(t) = H(ψ(t)), ψ(t) ∈ H, t ∈ R, ψ(0) = ψ0, and ΦH = {exp(tH) | t ∈ R} is the corresponding one-parameter family of unitary transformations of H Obviously, the equation (1) defines a dynamical system on the Grassmann manifold Grk(H): (2) ˙ P(t) = [H,P(t)], t ∈ R and the corresponding one-parameter group of diffeomorphisms of Grk(H) is defined by the action of the group ΦH on Grk(H). The action of the group ΦH for the projector representation of Grk(H), is P ↦ → exp(tH)P exp(−tH). For a given k-dimensional subspace V0 ∈ H, we are interested in Hamiltonians H ∈ u(H) such that, after the time period t = 1, the one-parameter group ΦH brings V0 to itself. In other words, for a given point P0 ∈ Grk(H) we are looking for Hamiltonians H ∈ u(H) such that the trajectory of the equation (2) through the point P0 is closed: exp(H)P0 exp(−H) = P0.

Abstract. The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations are investigated. These methods are shown to be very useful for the problem of constructing an universal set of gates for quantum computations: the well-known result that the set of all one-bit gates together with almost any one two-bit gate is universal is considered from the control theory viewpoint. 1.