Quantum mechanics of lattice gas automata. II. Boundary conditions and other inhomogeneities
David A. Meyer
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24 pages, plain TeX, 9 PostScript figures included with epsf.tex (ignore the under/overfull \vbox error messages), 3 additional large figures available upon request or from http://math.ucsd.edu/~dmeyer/papers/papers.html
We continue our analysis of the physics of quantum lattice gas automata (QLGA). Previous work has been restricted to periodic or infinite lattices; simulation of more realistic physical situations requires finite sizes and non-periodic boundary conditions. Furthermore, envisioning a QLGA as a nanoscale computer architecture motivates consideration of inhomogeneities in the `substrate'; this translates into inhomogeneities in the local evolution rules. Concentrating on the one particle sector of the model, we determine the various boundary conditions and rule inhomogeneities which are consistent with unitary global evolution. We analyze the reflection of plane waves from boundaries, simulate wave packet refraction across inhomogeneities, and conclude by discussing the extension of these results to multiple particles.
Using Albert results we argue that we don't need new physics to understand Gödelization. Albert quantum automaton can "understand" both a formal system and a Gödel proposition which can't be obtained within this system. There are two significant conclusions. The first speaks "against Penrose" whereas the second speaks for him.
We introduce an action principle for a class of integer valued cellular automata and obtain Hamiltonian equations of motion. Employing sampling theory, these discrete deterministic equations are invertibly mapped on continuum equations for a set of bandwidth limited harmonic oscillators, which encode the Schrödinger equation. Thus, the linearity of quantum mechanics is related to the action principle of such cellular automata and its conservation laws to discrete ones.
Classical lattice gas automata effectively simulate physical processes such as diffusion and fluid flow (in certain parameter regimes) despite their simplicity at the microscale. Motivated by current interest in quantum computation we recently defined quantum lattice gas automata; in this paper we initiate a project to analyze which physical processes these models can effectively simulate. Studying the single particle sector of a one dimensional quantum lattice gas we find discrete analogues of plane waves and wave packets, and then investigate their behaviour in the presence of inhomogeneous …
We discuss the action principle and resulting Hamiltonian equations of motion for a class of integer-valued cellular automata introduced recently [1]. Employing sampling theory, these deterministic finite-difference equations are mapped reversibly on continuum equations describing a set of bandwidth limited harmonic oscillators. They represent the Schroedinger equation. However, modifications reflecting the bandwidth limit are incorporated, i.e., the presence of a time (or length) scale. When this discreteness scale is taken to zero, the usual results are obtained. Thus, the linearity of quant…
The idea of information encoding on quantum bearers and its quantum-mechanical processing has revolutionized our world and brought mankind on the verge of enigmatic era of quantum technologies. Inspired by this idea, in present paper we search for advantages of quantum information processing in the field of machine learning. Exploiting only basic properties of the Hilbert space, superposition principle of quantum mechanics and quantum measurements, we construct a quantum analog for Rosenblatt's perceptron, which is the simplest learning machine. We demonstrate that the quantum perceptron super…
We consider discrete and integer-valued cellular automata (CA). A particular class of which comprises "Hamiltonian CA" with equations of motion that bear similarities to Hamilton's equations, while they present discrete updating rules. The dynamics is linear, quite similar to unitary evolution described by the Schroedinger equation. This has been essential in our construction of an invertible map between such CA and continuous quantum mechanical models, which incorporate a fundamental discreteness scale. Based on Shannon's sampling theory, it leads, for example, to a one-to-one relation betwee…
Cellular automata can show well known features of quantum mechanics, such as a linear rule according to which they evolve and which resembles a discretized version of the Schroedinger equation. This includes corresponding conservation laws. The class of "natural" Hamiltonian cellular automata is based exclusively on integer-valued variables and couplings and their dynamics derives from an Action Principle. They can be mapped reversibly to continuum models by applying Sampling Theory. Thus, "deformed" quantum mechanical models with a finite discreteness scale $l$ are obtained, which for $l\righ…
Cellular automata can show well known features of quantum mechanics, such as a linear updating rule that resembles a discretized form of the Schrödinger equation together with its conservation laws. Surprisingly, a whole class of "natural" Hamiltonian cellular automata, which are based entirely on integer-valued variables and couplings and derived from an Action Principle, can be mapped reversibly to continuum models with the help of Sampling Theory. This results in "deformed" quantum mechanical models with a finite discreteness scale $l$, which for $l\rightarrow 0$ reproduce the familiar cont…
A classical local cellular automaton can describe an interacting quantum field theory for fermions. We construct a simple classical automaton for a particular version of the Thirring model with imaginary coupling. This interacting fermionic quantum field theory obeys a unitary time evolution and shows all properties of quantum mechanics. Classical cellular automata with probabilistic initial conditions admit a description in the formalism of quantum mechanics. Our model exhibits interesting features as spontaneous symmetry breaking or solitons. The same model can be formulated as a generalized…
From celestial mechanics to quantum theory of atoms and molecules, perturbation theory has played a central role in natural sciences. Particularly in quantum mechanics, the amount of information needed for specifying the state of a many-body system commonly scales exponentially as the system size. This poses a fundamental difficulty in using perturbation theory at arbitrary order. As one computes the terms in the perturbation series at increasingly higher orders, it is often important to determine whether the series converges and if so, what is an accurate estimation of the total error that co…
After leading to a new axiomatic derivation of quantum theory, the new informational paradigm is entering the domain of quantum field theory, suggesting a quantum automata framework that can be regarded as an extension of quantum field theory to including an hypothetical Planck scale, and with the usual quantum field theory recovered in the relativistic limit of small wave-vectors. Being derived from simple principles (linearity, unitarity, locality, homogeneity, isotropy, and minimality of dimension), the automata theory is quantum ab-initio, and does not assume Lorentz covariance and mechani…
Recent advances on quantum foundations achieved the derivation of free quantum field theory from general principles, without referring to mechanical notions and relativistic invariance. From the aforementioned principles a quantum cellular automata (QCA) theory follows, whose relativistic limit of small wave-vector provides the free dynamics of quantum field theory. The QCA theory can be regarded as an extended quantum field theory that describes in a unified way all scales ranging from an hypothetical discrete Planck scale up to the usual Fermi scale. The present paper reviews the elementary …
We examine the impact of the intrinsic molecular reorganization energy on switching in two-state quantum-dot cellular automata (QCA) cells. Switching a bit involves an electron transferring between charge centers within the molecule. This in turn causes the other atoms in the molecule to rearrange their positions in response. We capture this in a model that treats the electron motion quantum-mechanically, but the motion of nuclei semiclassically. This results in a non-linear Hamiltonian for the electron system. Interaction with a thermal environment is included by solving the Lindblad equation…
A synopsis is offered of the properties of discrete and integer-valued, hence "natural", cellular automata (CA). A particular class comprises the "Hamiltonian CA" with discrete updating rules that resemble Hamilton's equations. The resulting dynamics is linear like the unitary evolution described by the Schrödinger equation. Employing Shannon's Sampling Theorem, we construct an invertible map between such CA and continuous quantum mechanical models which incorporate a fundamental discreteness scale $l$. Consequently, there is a one-to-one correspondence of quantum mechanical and CA conservatio…
A class of fermionic quantum field theories with interactions is shown to be equivalent to probabilistic cellular automata, namely cellular automata with a probability distribution for the initial states. Probabilistic cellular automata on a one-dimensional lattice are equivalent to two - dimensional quantum field theories for fermions. They can be viewed as generalized Ising models on a square lattice and therefore as classical statistical systems. As quantum field theories they are quantum systems. Thus quantum mechanics emerges from classical statistics. As an explicit example for an intera…
Based on the concept of ontological states and their dynamical evolution by permutations, as assumed in the Cellular Automaton Interpretation (CAI) of quantum mechanics, we address the issue whether quantum-classical hybrids can be described consistently in this framework. We consider chains of 'classical' two-state Ising spins and their discrete deterministic dynamics as an ontological model with an unitary evolution operator generated by pair exchange interactions. A simple error mechanism is identified, which turns them into quantum mechanical objects, chains of qubits. Consequently, an int…
In a series of recent papers it has been shown how free quantum field theory can be derived without using mechanical primitives (including space-time, special relativity, quantization rules, etc.), but only considering the easiest quantum algorithm encompassing a countable set of quantum systems whose network of interactions satisfies the simple principles of unitarity, homogeneity, locality, and isotropy. This has opened the route to extending the axiomatic information-theoretic derivation of the quantum theory of abstract systems to include quantum field theory. The inherent discrete nature …
Linear quantum cellular automata were introduced recently as one of the models of quantum computing. A basic postulate of quantum mechanics imposes a strong constraint on any quantum machine: it has to be unitary, that is its time evolution operator has to be a unitary transformation. In this paper we give an efficient algorithm to decide if a linear quantum cellular automaton is unitary. The complexity of the algorithm is O(n^((3r-1)/(r+1))) = O(n^3) in the algebraic computational model if the automaton has a continuous neighborhood of size r, where $n$ is the size of the input.
Unitarity is a pillar of quantum theory. Nevertheless, it is also a source of several of its conceptual problems. We note that in a world where measurements are relational, as is the case in gravitation, quantum mechanics exhibits a fundamental level of loss of coherence. This can be the key to solving, among others, the puzzles posed by the black hole information paradox, the formation of inhomogeneities in cosmology and the measurement problem in quantum mechanics.
According to the inflationary scenario for the very early Universe, all inhomogeneities in the Universe are of genuine quantum origin. On the other hand, looking at these inhomogeneities and measuring them, clearly no specific quantum mechanical properties are observed. We show how the transition from their inherent quantum gravitational nature to classical behaviour comes about -- a transition whereby none of the successful quantitative predictions of the inflationary scenario for the present-day universe is changed. This is made possible by two properties. First, the quantum state for the sp…
We construct zero-error quantum finite automata (QFAs) for promise problems which cannot be solved by bounded-error probabilistic finite automata (PFAs). Here is a summary of our results: - There is a promise problem solvable by an exact two-way QFA in exponential expected time, but not by any bounded-error sublogarithmic space probabilistic Turing machine (PTM). - There is a promise problem solvable by an exact two-way QFA in quadratic expected time, but not by any bounded-error $ o(\log \log n) $-space PTMs in polynomial expected time. The same problem can be solvable by a one-way Las Vegas …
Quantum-dot Cellular Automata (QCA) may offer a viable alternative of traditional transistor-based technology at the nanoscale. When modeling a QCA circuit, the number of degrees of freedom necessary to describe the quantum mechanical state increases exponentially making modeling even modest size cell arrays difficult. The intercellular Hartree approximation largely reduces the number of state variables and still gives good results especially when the system remains near ground state. This suggests that large part of the correlation degrees of freedom are not essential from the point of view o…
Detailed measurements of the cosmic microwave background indicate the large-scale homogeneity of the universe. On very small scales, we observe however inhomogeneities such as galaxies, stars, planets and ourselves. In the context of hot Big-Bang cosmology, these inhomogeneities are often explained as the remains of quantum fluctuations at very early times, enlarged to observable scales through the process of inflation. In this dissertation, I examine two important questions surrounding this scenario: a) How do inherently quantal fluctuations transition to the observed inhomogeneities, which b…
In this thesis we will work under the premises of the Cellular Automata Interpretation of QM, by Gerard 't Hooft, according to whom particles evolve following the rules of Cellular Automata (CA), a mathematical model consisting of discrete units that evolve following deterministic laws in discrete space and time. The states of a Cellular Automaton are, by definition, classical and thus deterministic and do not form superpositions. Since it is not possible to know how to demonstrate the underlying classical deterministic structure and dynamics at the smallest microscopic scales at present, what…
A quantum-mechanical analysis of hyper-fast (faster than ballistic) diffusion of a quantum wave packet in random optical lattices is presented. The main motivation of the presented analysis is experimental demonstrations of hyper-diffusive spreading of a wave packet in random photonic lattices [L. Levi \textit{et al.}, Nature Phys. \textbf{8}, 912 (2012)]. A rigorous quantum-mechanical calculation of the mean probability amplitude is suggested, and it is shown that the power law spreading of the mean squared displacement (MSD) is $< x^2(t)>\sim t^α$, where $2<α\leq 3$. The values of the transp…
