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Russian Microelectronics

Amplitude quanta in multiparticle-system simulation

Yuri Ozhigov

Published Jan 1, 2006• 12 citations
Publisher

Abstract

An algorithmic approach to theoretical quantum physics is described postulating that any multiparticle quantum evolution can be simulated by classical algorithms of polynomial complexity. Its implementation is presented that is based on the concept of an amplitude quantum, i.e., the smallest carrier of amplitude. As a consequence, the Born rule for quantum probability is derived within the classical urn scheme. Decoherence is treated as stemming from the inherently limited amount of computational resources available for multiparticle simulation. The algorithmic approach denies the possibility of creating a scalable quantum computer. It is therefore refutable by construction of such a computer.

DOI
10.1134/s1063739706010070

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An algorithmic approach to theoretical quantum physics is described postulating that any multiparticle quantum evolution can be simulated by classical algorithms of polynomial complexity. Its implementation is presented that is based on the concept of an amplitude quantum, i.e., the smallest carrier of amplitude. As a consequence, the Born rule for quantum probability is derived within the classical urn scheme. Decoherence is treated as stemming from the inherently limited amount of computational resources available for multiparticle simulation. The algorithmic approach denies the possibility of creating a scalable quantum computer. It is therefore refutable by construction of such a computer.
Algorithmic approach is based on the assumption that any quantum evolution of many particle system can be simulated on a classical computer with the polynomial time and memory cost. Algorithms play the central role here but not the analysis, and a simulation gives a "film" which visualizes many particle quantum dynamics and is demonstrated to a user of the model. Restrictions following from the algorithm theory are considered on a level of fundamental physical laws. Born rule for the calculation of quantum probability as well as the decoherence is derived from the existence of a nonzero minimal value of amplitude module - a grain of amplitude. The limitation on the classical computational resources gives the unified description of quantum dynamics that is not divided to the unitary dynamics and measurements and does not depend on the existence of observer. It is proposed the description of states based on the nesting of particles in each other that permits to account the effects of all levels in the same model. Algorithmic approach admits the possibility of refutation, because it forbids the creation of a scalable quantum computer that is allowed in the conventional quantum formalism.
A straightforward algorithm is proposed for simulating and visualizing nonrelativistic quantum dynamics in terms of the collective behavior of classical particles. In this approach, a quantum particle is treated as a swarm (collection) of its classical samples interacting with one another by simple rules involving the emission and absorption of associated photons. The quantum dynamics is regarded as resulting from the collective behavior of the swarm, and the eigenstates are treated as equilibrium states of the process of photon emission and absorption. Entanglement is seen as classical correlation between samples from distinct swarms; the information on this correlation is stored in a spatiotemporal section of the model inaccessible to the user. Amplitudes are invariably treated as consisting of amplitude quanta. A Coulomb field experienced by quantum particles is represented in a similar manner to the free spreading of a wave packet, by point interaction between the particle samples and the samples of scalar photons propagating by diffusion. This provides a square-root gam in computational speed against the direct method involving item-by-item examination. The approach presented covers decoherence and allows natural generalization to multiparticle quantum electrodynamics, the amount of memory required growing linearly.
Algorithmic approach to quantum theory is considered. It is based on the supposition that every evolution of many particle system can be simulated by classical algorithms of polynomial complexity. This hypothesis agrees with all known experiments but it presumes the principle cut-off of quantum formalism because it excludes a scalable quantum computer. Algorithmic approach describes quantum evolution uniformly, without separation of measurements from the unitary dynamics; it is shown how Bohrn rule for quantum probability follows from the basic principles of this approach. The radical difference of algorithmic approach from the standard and its perspectives are discussed.
This paper presents a novel model for assessing algorithm security based on the principles of quantum decoherence. Current approaches to algorithm security often rely on classical computational assumptions, which may be vulnerable to future advancements in quantum computing. This model leverages the inherent vulnerability of quantum systems – decoherence – to provide a more robust and realistic security assessment. We construct a simulation environment that mimics a quantum computing scenario, allowing us to evaluate the resilience of algorithms against decoherence-induced errors. The model incorporates key concepts such as quantum gates, qubit states, and decoherence rates, represented mathematically as: * State of a qubit: |ψ⟩ = ∑i ci |i⟩ * Quantum gate operation: U = ∑i,j ci cj† |i⟩⟨j| * Decoherence rate: Γ (where Γ represents the probability of a qubit losing its quantum state per unit time) * Probability of error: Pe = exp(-Γt) , where t is the computation time. The core of the model is a simulation that propagates the state of a quantum system through a series of quantum gates, incorporating decoherence effects at each step. The model's output provides a quantifiable measure of an algorithm's security against decoherence, offering a valuable tool for designing and evaluating algorithms in the face of real-world quantum computing limitations. This approach moves beyond purely theoretical security assumptions and offers a more practically relevant framework for algorithm design and security analysis. ---
The mathematical and physical foundations of the Born rule remain open questions, largely because the wave function lacks a mathematical identity and a physical origin. Here we uncover a mathematical structure of the wave function by treating the phase as a circular variable. This structure reveals that the wave function is the pullback of the phase probability-density amplitude to position space via the phase field. Because the phase probability distribution forms an equal-area contour in the complex plane, its probability-area element yields the modulus squared as the position probability density, revealing the Born rule as a geometric relation rooted in the circular topology of the phase variable. The same structure governs the phase-energy distribution of electromagnetic waves. By quantizing phase energy into energy quanta, in the single-quantum limit the energy-quantum density reduces to the probability density, and its amplitude becomes the probability-density amplitude. This therefore identifies the energy-quantum density amplitude in the single-quantum limit as the physical origin of the wave function. To verify this, we derive the time-evolution equation for the energy-quantum density amplitude from five physical principles. This yields the Gross-Pitaevskii equation for many-body systems and, in the single-quantum limit, the Schrödinger equation. The framework also recovers Maxwell's equations in the classical limit. Overall, this study introduces the energy-quantum density amplitude from the classical phase-energy distribution, and derives the three core postulates of quantum mechanics: the wave function, the Born rule, and the Schrödinger equation, thereby identifying the position probability density as the energy-quantum density in the single-quantum limit.