The time evolution operator for any quantum-mechanical computer is diagonalizable, but to obtain the diagonal decomposition of a program state of the computer is as hard as actually performing the computation corresponding to the program. In particular, if a quantum-mechanical system is capable of universal computation, then the diagonal decomposition of program states is uncomputable. As a result, in a universe in which local variables support universal computation, a quantum-mechanical theory for that universe that supplies its spectrum cannot supply the spectral decomposition of the computational variables. A ``theory of everything'' can be simultaneously correct and fundamentally incomplete.
Necessary and sufficient conditions are given for a quantummechanical system to possess a coordinate system with respect to which its behaviour at discrete times is that of a universal digital computer. The form of the diagonal representation for the unitary time evolution operator for quantum universal computers is derived; aspects of the transformation between the diagonal representation and the computational representation are shown to be uncomputable. A quantum-mechanical treatment of macroscopic, dissipative computers is given.
We present a structural formulation of physical propagation in which dynamics are described by a unitary operator
W^=f(H^), associated with a self-adjoint structural operator
H^, and by its bilocal representation
W(x,y). Time evolution is formulated as the iterative application of this operator, without assuming differential equations as a fundamental starting point. It is shown that the spectral decomposition of
H^ induces a complete organization of the evolution into independent eigenmodes, each governed by a spectral factor
f(λn). From this structure, it is demonstrated that the bilocal propagation kernel is entirely reconstructed from the spectrum and the eigenvector basis of the structural operator. In the regime of translational homogeneity, it is proven that the propagation operator is diagonalizable in a basis compatible with translations, resulting in the absence of mixing between spectral modes. Outside this regime, it is shown that the support of the kernel
W(x,y) completely determines the connectivity of propagation, while its off-diagonal structure encodes relations between distinct nodal states. Additionally, it is established that quantities defined as functions of the structural operator can be evaluated directly in the spectral basis, providing an intrinsic characterization of the dynamics in terms of the spectrum of
H^. The results obtained do not constitute a complete physical theory nor do they establish a direct identification between the operator's spectrum and specific empirical observables. The objective of this work is to make explicit the structural level at which propagation is organized by unitary operators and by their spectral decomposition, providing a coherent basis for further developments.
We present a structural formulation of physical dynamics based on discrete unitary propagation, in which the time evolution is directly described by a unitary operator
W^=f(H^) associated with a self-adjoint structural operator
H^. Starting from the relationship between action and phase,
ϕ=S/ℏ, and the variational construction of structural operators, we show that the propagation can be written exactly as the action of a bilocal kernel
W(x,y) on the state, in continuity with recent developments of the Action-Phase program. We show that the spectral decomposition of
H^ induces a diagonalization of the unitary propagation, in which the state evolution becomes multiplicative in the eigenvector basis of the structural operator. Consequently, the spectral structure organizes the propagation modes of the system, providing a description in which the dynamics is expressed in a decoupled manner between modes. It is evidenced that the distinction between nodal representation and spectral representation emerges naturally from the operator structure, corresponding to different decompositions of the same state in distinct bases of the same Hilbert space. In the case of translational invariance, the decomposition into Fourier modes is recovered as a particular situation. The results presented do not constitute a complete reconstruction of quantum mechanics nor a general gravitational theory. In particular, aspects of measurement, multiparticle structures, or complete covariant formulations are not treated. The objective of this work is more restricted and precise: to establish a formulation in which the dynamics is determined by the propagation operator and organized by its spectral decomposition, providing a structural layer from which more complete physical descriptions can be developed.
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.
We apply our new approach of quantum Separation of Variables (SoV) to the complete characterization of the transfer matrix spectrum of quantum integrable lattice models associated to \bm{gl_n} 𝐠 𝐥 𝐧 -invariant \bm{R} 𝐑 -matrices in the fundamental representations. We consider lattices with \bm{N} 𝐍 -sites and general quasi-periodic boundary conditions associated to an arbitrary twist matrix \bm{K} 𝐊 having simple spectrum (but not necessarily diagonalizable). In our approach the SoV basis is constructed in an universal manner starting from the direct use of the conserved charges of the models, e.g. from the commuting family of transfer matrices. Using the integrable structure of the models, incarnated in the hierarchy of transfer matrices fusion relations, we prove that our SoV basis indeed separates the spectrum of the corresponding transfer matrices. Moreover, the combined use of the fusion rules, of the known analytic properties of the transfer matrices and of the SoV basis allows us to obtain the complete characterization of the transfer matrix spectrum and to prove its simplicity. Any transfer matrix eigenvalue is completely characterized as a solution of a so-called quantum spectral curve equation that we obtain as a difference functional equation of order \bm{n} 𝐧 . Namely, any eigenvalue satisfies this equation and any solution of this equation having prescribed properties that we give leads to an eigenvalue. We construct the associated eigenvector, unique up to normalization, of the transfer matrices by computing its decomposition on the SoV basis that is of a factorized form written in terms of the powers of the corresponding eigenvalues. Finally, if the twist matrix \bm{K} 𝐊 is diagonalizable with simple spectrum we prove that the transfer matrix is also diagonalizable with simple spectrum. In that case, we give a construction of the Baxter \bm{Q} 𝐐 -operator and show that it satisfies a \bm{T} 𝐓 - \bm{Q} 𝐐 equation of order \bm{n} 𝐧 , the quantum spectral curve equation, involving the hierarchy of the fused transfer matrices.