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arXiv:2005.03401 · Frontiers in Physics

Discrete-Event Simulation of Quantum Walks

Madita Willsch, Dennis Willsch, Kristel F. L. Michielsen, Hans A. De Raedt

Published May 7, 2020• 4 citations
arXivHTMLPDF

Abstract

We use discrete-event simulation on a digital computer to study two different models of experimentally realizable quantum walks. The simulation models comply with Einstein locality, are as ``realistic'' as the one of the simple random walk in that the particles follow well-defined trajectories, are void of concepts such as particle-wave duality and wave-function collapse, and reproduce the quantum-theoretical results by means of a cause-and-effect, event-by-event process.

DOI
10.3389/fphy.2020.00145

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We use discrete-event simulation on a digital computer to study two different models of experimentally realizable quantum walks. The simulation models comply with Einstein locality, are as ``realistic'' as the one of the simple random walk in that the particles follow well-defined trajectories, are void of concepts such as particle-wave duality and wave-function collapse, and reproduce the quantum-theoretical results by means of a cause-and-effect, event-by-event process.
We review the data gathering and analysis procedure used in real Einstein-Podolsky-Rosen-Bohm experiments with photons and we illustrate the procedure by analyzing experimental data. Based on this analysis, we construct event-based computer simulation models in which every essential element in the experiment has a counterpart. The data is analyzed by counting single-particle events and two-particle coincidences, using the same procedure as in experiments. The simulation models strictly satisfy Einstein's criteria of local causality, do not rely on any concept of quantum theory or probability theory, and reproduce all results of quantum theory for a quantum system of two S=1/2S=1/2S=1/2 particles. We present a rigorous analytical treatment of these models and show that they may yield results that are in exact agreement with quantum theory. The apparent conflict with the folklore on Bell's theorem, stating that such models are not supposed to exist, is resolved. Finally, starting from the principles of probable inference, we derive the probability distributions of quantum theory of the Einstein-Podolsky-Rosen-Bohm experiment without invoking concepts of quantum theory.
We present a computer simulation model that is a one-to-one copy of a quantum eraser experiment with photons (P. D. D. Schwindt et al., Phys. Rev. A 60, 4285 (1999)). The model is solely based on experimental facts, satisfies Einstein's criterion of local causality and does not require knowledge of the solution of a wave equation. Nevertheless, the simulation model reproduces the averages as obtained from the wave mechanical description of the quantum eraser experiment, proving that it is possible to give a particle-only description of quantum eraser experiments with photons. We demonstrate that although the visibility can be used as a measure for the interference, it cannot be used to quantify the wave character of a photon. The classical particle-like simulation model renders the concept of wave-particle duality, used to explain the outcome of the quantum eraser experiment with photons, superfluous.
The Dirac equation can be modelled as a quantum walk (QW), whose main features are being: discrete in time and space (i.e. a unitary evolution of the wave-function of a particle on a lattice); homogeneous (i.e. translation-invariant and time-independent) and causal (i.e. information propagates at a bounded speed, in a strict sense). This link, which was proposed already by Succi and Benzi, Bialynicki-Birula and Meyer, is shown to hold for Bargmann–Wigner equations and symmetric hyperbolic systems in general. We then analytically prove the convergence of the solution of the QW to the solution of the Cauchy problem for the Dirac equation. We do so by adapting a powerful method from standard numerical analysis, which is of general interest to the field of quantum simulation. At the practical level, this result provides precise error bounds and convergence rates, thereby validating the QW as a quantum simulation scheme. At the theoretical level, it reinforces the status of this QW as a simple, discrete toy model of relativistic particles.
Description “Discrete detection does not imply discrete existence.” For over a century, the photoelectric effect has been cited as the decisive experimental result that forced physics to abandon wave-only models of light and accept wave–particle duality. Einstein’s 1905 interpretation argued that because electrons are ejected instantaneously and in discrete events, light itself must consist of discrete particles. This paper shows that this conclusion is not logically required. We present a reinterpretation of the photoelectric effect based on the Total Wave Modified Schrödinger Equation (TWMSE), a deterministic wave-interference framework with threshold-based collapse. Within this model, the observed discreteness arises from the detection process, not from the ontology of light itself. Key Findings The “Photon” Is an Event, Not an ObjectThe photon is shown to be a retrospective label applied to a discrete wave-collapse event. The event is discrete; the source field (light) remains continuous and wave-like. Threshold-Based Collapse Explains EmissionElectron emission occurs when the curvature density of the interfering electromagnetic wave field exceeds a material-specific stability threshold. Below this threshold, no emission occurs regardless of intensity. Resolution of Classical ParadoxesThe model reproduces all defining features of the photoelectric effect—instantaneous emission, threshold frequency, and the linear dependence of electron kinetic energy on frequency—without invoking particle ontology. Mathematical Equivalence Without Particle AssumptionsA direct mapping is provided showing that the TWMSE collapse formulation reproduces Einstein’s photoelectric equation purely through wave curvature dynamics. Why This Matters By resolving the photoelectric effect entirely within a deterministic wave framework, this work removes the primary historical justification for wave–particle duality. It clarifies that discrete experimental outcomes do not require discrete underlying entities and supports a unified field interpretation of quantum measurement consistent with prior TWMSE results in electromagnetism, gravity, and macroscopic systems. Included in This Upload Full theoretical and mathematical derivation Conceptual comparison: Einstein’s particle interpretation vs. TWMSE wave-collapse interpretation Mapping table between classical photoelectric variables and TWMSE collapse parameters