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arXiv:1205.2563

Quantum computation from a de Broglie-Bohm perspective

Philipp Roser

Published May 11, 2012• 1 citation
arXivHTMLPDF

Abstract

We consider an example of a quantum algorithm from the point of view of the de Broglie-Bohm formulation of quantum mechanics. For concreteness we look at two particular implementations: one using spin-1/2 particles as described by a simple model due to Bell and the other as energy states in an infinite-well potential. We extend the analysis to encompass a complete set of quantum gates. We conclude by discussing the relevance of our investigations for the debate about the origin of the quantum-computational speed-up.

DOI
10.48550/arxiv.1205.2563

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We consider an example of a quantum algorithm from the point of view of the de Broglie-Bohm formulation of quantum mechanics. For concreteness we look at two particular implementations: one using spin-1/2 particles as described by a simple model due to Bell and the other as energy states in an infinite-well potential. We extend the analysis to encompass a complete set of quantum gates. We conclude by discussing the relevance of our investigations for the debate about the origin of the quantum-computational speed-up.
This paper systematically reviews the theoretical progression from blackbody radiation to the foundations of quantum computing. Beginning with Planck’s energy quantization and the representation of electromagnetic wave electric field components, the wave-particle duality of photons is analogized to introduce the de Broglie relation between momentum and wave vector for matter waves, leading to the establishment of the matter wave function and the Schrödinger equation as its governing partial differential equation. The consistency between the rigorous solution for a one-dimensional infinitely deep rectangular potential well and the conclusions for a one-dimensional hydrogen atom is demonstrated. Subsequently, within the Dirac notation formalism, the wave function is expressed, and the correspondence between spatial differentiation and momentum is derived, formalizing the momentum operator (often referred to as first quantization). To lay the physical groundwork for manipulating two-qubit states in quantum computing, the potential energy function of an electron’s orbital magnetic moment in an external magnetic field is introduced, starting from the torque and work on a current-carrying rectangular coil. This is extended to the magnetic moment and potential energy of a spinning electron, where the orientation of the magnetic field determines the form of the potential and accordingly selects the three components of the Pauli operators. The properties of quantum gates corresponding to the Pauli operators are then examined for single-qubit states. A two-qubit state, constructed via the coupling of two arbitrary single-qubit states represented in a two-level system or electron spin, is shown to be uniquely expressible as a superposition of four eigen two-qubit states. The paper further investigates joint manipulations of two-qubit states, including evolution under X and Z gates conditioned on high- and low-level states, the representation of control-flow circuits (with the low-level as control), and the formulation of measurement and CNOT gates. Throughout this work, essential mathematical tools—including integration, matrix operations, complex Euler’s formula, and solutions to simple second-order ordinary differential equations—are incorporated to provide a coherent theoretical framework for state representation and gate operations in quantum computing.
Much attention has been drawn to quantum computing and the exponential speed-up in computation the technology would be able to provide. Various claims have been made about what aspect of quantum mechanics causes this speed-up. Formulations of quantum computing have traditionally been made in orthodox (Copenhagen) and sometimes many-worlds quantum mechanics. We will aim to understand quantum computing in terms of de Broglie-Bohm Pilot-Wave Theory by considering different simple systems that may function as a basic quantum computer. We will provide a careful discussion of Pilot-Wave Theory and evaluate criticisms of the theory. We will assess claims regarding what causes the exponential speed-up in the light of our analysis and the fact that Pilot-Wave Theory is perfectly able to account for the phenomena involved in quantum computing.
The de Broglie–Bohm interpretation (Bohmian mechanics, hidden-variables theory, causal interpretation or pilot-wave theory) is one of the interpretations of quantum mechanics which opposes to the claim of standard interpretation on the completeness of description of the state of quantummechanical system, that description in Bohmian mechanics is completed by defining positions of the particles of the system i.e. hidden variables and their dynamics. These facts are the basis of motivation for the introduction of a new theory that can describe dubious axiomatic requirements of standard interpretation. The presented theory is deterministic and explicitly nonlocal. Originators of Bohmian mechanics were L. de Broglie (1927.) and D. Bohm (1952.) who presented mathematically more general and conceptual elaborate formulation. Along with the historical overview, a note is given about the main properties of the theory, such as the issue of nonlocality and hidden variables. The aim of this paper is to present and explain the main ideas and properties of the theory as well as to give a systematic overview of present and historical mathematical construction of the theory and most of all to discuss implications of de Broglie–Bohm theory on the ontology of quantum-mechanical picture, showcasing examples of the equivalency of the description of a free particle as a propagation of the plane wave in space or as the ensemble of the trajectories and of the spreading of the Gaussian wave packet. The main example of the obvious conceptual clashing of the interpretations is presented: the double-slit experiment, with a detailed examination of the Bohmian mechanics’ predictions using presented formalism on the two double-slit experiment. The graphs of trajectories for given examples are obtained.