Quantum mechanics of lattice gas automata. I. One particle plane waves and potentials
David A. Meyer
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19 pages, plain TeX, 14 PostScript figures included with epsf.tex (ignore the under/overfull \vbox error messages), two additional large figures available upon request
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 potentials.
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.
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…
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…
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 consisten…
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…
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…
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…
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…
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…
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…
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.
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…
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 …
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…
Quantum-dot cellular automata (QCA) is a low-power, non-von-Neumann, general-purpose paradigm for classical computing using transistor-free logic. An elementary QCA device called a "cell" is made from a system of coupled quantum dots with a few mobile charges. The cell's charge configuration encodes a bit, and quantum charge tunneling within a cell enables device switching. Arrays of cells networked locally via the electrostatic field form QCA circuits, which mix logic, memory and interconnect. A molecular QCA implementation promises ultra-high device densities, high switching speeds, and room…
On the base of symplectic quantum tomogram we define a probability distribution on the plane. The dual map transfers all observables which are polynomials of the position and momentum operators to the set of polynomials of two variables. In this representation the average values of observables can be calculated by means of integration over all the plane.
The detailed study of a quantum free particle on a pointed plane is performed. It is shown that there is no problem with a mysterious ``quantum anticentrifugal force" acting on a free particle on a plane discussed in a very recent paper: M. A. Cirone et al, Phys. Rev. A 65, 022101 (2002), but we deal with a purely topological efect related to distinguishing a point on a plane. The new results are introduced concerning self-adjoint extensions of operators describing the free particle on a pointed plane as well as the role played by discrete symmetries in the analysis of such extensions.
It is shown fields that cannot be represented over one complex plane can be further decomposed for representation over multiple complex planes. This finding is demonstrated here by solving of the Schrödinger equation for the hydrogen atom in a complex space containing three complex planes. The complex coordinate system is generated from real coordinates using an isometric transformation. One plane is applied to mix energy and time; the other two planes are used to represent the z-component of angular momentum of the electron. The eigensolutions of the Schrödinger equation are shown to be holom…
Quantum mechanics is often developed in the position representation, but this is not necessary, and one can perform calculations in a representation-independent fashion, even for wavefunctions. In this work, we illustrate how one can determine wavefunctions, aside from normalization, using only operators and how those operators act on state vectors. To do this in plane polar and spherical coordinates requires one to convert the translation operator into those coordinates. As examples of this approach, we illustrate the solution of the Coulomb problem in two and three dimensions without needing…
The rapid development of high-Q macroscopic mechanical resonators has enabled great advances in optomechanics. Further improvements could allow for quantum-limited or quantum-enhanced applications at ambient temperature. Some of the remaining challenges include the integration of high-Q structures on a chip, while simultaneously achieving large coupling strengths through an optical read-out. Here, we present a versatile fabrication method, which allows us to build fully integrated optomechanical structures. We place a photonic crystal cavity directly above a mechanical resonator with high-Q fu…
The standard momentum operator $-i \nabla$ can not be accepted as observable in relativistic quantum mechanics of the Majorana particle. Instead, one can use axial momentum operator recently proposed in Phys. Lett. A {\bf383}, 1242 (2019). In the present paper we report several new results related to the axial momentum which elucidate its usability. First, a new motivation for the axial momentum is given, and the Heisenberg uncertainty relation checked. Next, we show that the general solution of time evolution equation in the axial momentum basis has a connection with quaternions. Single trave…
The correspondence principle asserts that quantum mechanics resembles classical mechanics in the high-quantum-number limit. In the past few years many papers have been published on the extension of both quantum mechanics and classical mechanics into the complex domain. However, the question of whether complex quantum mechanics resembles complex classical mechanics at high energy has not yet been studied. This paper introduces the concept of a local quantum probability density $ρ(z)$ in the complex plane. It is shown that there exist infinitely many complex contours $C$ of infinite length on wh…
Borel summable semiclassical expansions in 1D quantum mechanics are considered. These are the Borel summable expansions of fundamental solutions and of quantities constructed with their help. An expansion, called topological,is constructed for the corresponding Borel functions. Its main property is to order the singularity structure of the Borel plane in a hierarchical way by an increasing complexity of this structure starting from the analytic one. This allows us to study the Borel plane singularity structure in a systematic way. Examples of such structures are considered for linear, harmonic…
If the systems of quantum theory are thought of as elementary information carriers in the first place, rather than elementary constituents of matter, and their connections are logical connections within a given algorithm, rather than space-time relations, then we need to find the origin of mechanical concepts---that characterise quantum mechanics as a theory of physical systems. To this end, we will illustrate how physical laws can be viewed as algorithms for the update of memory registers that make a physical system. Imposing the characteristic properties of physical laws to such an algorithm…
Probability is an important question in the ontological interpretation of quantum mechanics. It has been discussed in some trajectory interpretations such as Bohmian mechanics and stochastic mechanics. New questions arise when the probability domain extends to the complex space, including the generation of complex trajectory, the definition of the complex probability, and the relation of the complex probability to the quantum probability. The complex treatment proposed in this article applies the optimal quantum guidance law to derive the stochastic differential equation governing a particle r…
