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Combinatorial Analysis of the Exponential Map for Time-Evolution Operator

Ali Mert CEYLAN, Vecdi Aytaç

Published Sep 18, 2023• 0 citations
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Abstract

Abstract In this study, we present a novel formalization for exponential approx-imation. This formalization is for discrete analysis of continuous time-dependent perturbation operator of quantum mechanics, com-pared to multiplying exponentials. This leads us to Galton arrays for better representation and algebraic operations over exponential mul-tiplications. Rotations on Galton arrays allow us to perform various algebraic operations and intuitively explain commutative and non-commutative operators equality in exponential approximations. Pro-posed algebraic operations over Galton arrays will provide an easier understanding and a novel way to formalize and visualize multi-plying exponentials. But most importantly, it focused on the time-evolution operator to be written in the proposed formalization of time-ordered multiplication of exponentials with noncommuting operators.

DOI
10.21203/rs.3.rs-2271337/v3

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Abstract In this study, we present a novel formalization for exponential approx-imation. This formalization is for discrete analysis of continuous time-dependent perturbation operator of quantum mechanics, com-pared to multiplying exponentials. This leads us to Galton arrays for better representation and algebraic operations over exponential mul-tiplications. Rotations on Galton arrays allow us to perform various algebraic operations and intuitively explain commutative and non-commutative operators equality in exponential approximations. Pro-posed algebraic operations over Galton arrays will provide an easier understanding and a novel way to formalize and visualize multi-plying exponentials. But most importantly, it focused on the time-evolution operator to be written in the proposed formalization of time-ordered multiplication of exponentials with noncommuting operators.
Abstract In this study, we present a novel formalization for exponential approximation. This formalization is for discrete analysis of continuous time-dependent perturbation operator of quantum mechanics, compared to multiplying exponentials. This leads us to Galton arrays for better representation and algebraic operations over exponential multiplications. Rotations on Galton arrays allow us to perform various algebraic operations and intuitively explain commutative and noncommutative operators equality in exponential approximations. Proposed algebraic operations over Galton arrays will provide an easier understanding and a novel way to formalize and visualize multiplying exponentials. But most importantly, it focused on the timeevolution operator to be written in the proposed formalization of timeordered multiplication of exponentials with noncommuting operators.
Abstract In this study, we present a novel formalization for exponential approximation. This formalization is for discrete analysis of continuous time-dependent perturbation operator of quantum mechanics, compared to multiplying exponentials. This leads us to Galton arrays for better representation and algebraic operations over exponential multiplications. Rotations on Galton arrays allow us to perform various algebraic operations and intuitively explain commutative and noncommutative operators equality in exponential approximations. Proposedalgebraic operations over Galton arrays will provide an easier understanding and a novel way to formalize and visualize multiplying exponentials. But most importantly, it focused on the time evolution operator to be written in the proposed formalization of time ordered multiplication of exponentials with noncommuting operators. MSC Classification(2020): 81Q05 , 81Q15 , 81R60 , 81S05 , 81T75 , 82C99 , 52B70
We present a novel combinatorial proof of the \textbf{Time-Ordered Exponential} formula for the \textbf{Time-Evolution Operator} based on the \textbf{Heaviside Formulation Of The Time-Ordering Operator}. Unlike conventional derivations that rely on induction our approach decomposes the \textbf{Symmetric Group} into permutations that produce decreasing time sequences and those that do not using a simple discriminant function. This decomposition together with an \textbf{Interchange Rule } between time-ordering and multiple integrals yields the \textbf{Dyson–Chen Identity} directly. The proof is self-contained and avoids recursion entirely. It appears in the literature as a problem in Advanced Quantum Mechanics (Springer, 2012, Problem 13.1).
This paper introduces a novel temporal logic formalism, termed Temporal Quantum Logic (TQL), specifically tailored for representing and reasoning about the continuous evolution of quantum states. The core challenge in applying temporal logic to quantum mechanics lies in effectively capturing the inherent complexities of superposition, entanglement, and decoherence, particularly the non-commutativity of quantum operations. Existing temporal logics lack the necessary tools to accurately model these phenomena. TQL addresses this limitation by employing a multi-valued logic built upon tensor networks and process algebra concepts. This approach enables a precise and explicit representation of quantum operations, measurement outcomes, and their temporal dependencies, incorporating probabilistic semantics directly into the logic. The formalization utilizes a state representation based on tensor network contractions, allowing for the dynamic tracking of quantum state evolution over time. The logic incorporates operators that explicitly handle non-commutativity and decoherence, offering a framework for formally verifying the correctness of quantum algorithms and protocols. The key innovation is the combination of tensor network formalism with temporal logic, providing a rigorous method for describing the evolution of quantum systems and enabling a deeper understanding of their behavior. This work lays the foundation for future research in quantum verification, quantum algorithm design, and the development of quantum software.
The approximation of a short-time evolution operator for a molecular system with the Hamiltonian is constructed in terms of the evolution operator of an unperturbed system, a perturbative potential V and commutators of and V . A description of a powerful method for such a construction based on the Lie-algebra technique is given. It is demonstrated that efficient representation for the evolution operator can be obtained using approximations like where . This relation is accurate up to the third order in time. The simplest approximations of this type are listed.
An alternative to the Magnus expansion in time-dependent quantum mechanical perturbation theory is presented. An exponential form of the time-development operator, given as the exponential of a sum of products of integrals well known in time-dependent quantum mechanics, is derived from the standard perturbation expansion. The derivation is simple and the form of the terms in the exponential is so straightforward that the Nth order term can be written by inspection. The standard Magnus terms can be obtained by rearrangement of the terms of the new expansion probably easier than they can be obtained by iteration of the original equations. A proof by induction is presented to establish the form of the Nth term and it is shown how the new expansion terms relate to the Magnus terms. Some apparent differences between Magnus terms in the literature are resolved.