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Applying the Poincaré theorems and Euler integral in the qubit-based signal and sensing framework of the quantum computing

Chur Chin

Published Apr 18, 2025• 0 citations
Publisher

Abstract

Applying the Poincaré theorems and Euler integral in the qubit based signal and sensing framework can be as Qubit State Space. Defined by rational amplitudes or unit-fraction-based norms. Signal as Path: A time-evolving qubit state (signal) as a path through this discrete manifold.Use Euler integration or Poincaré recurrence to describe or detect repeating patterns, signal stability and minimal sensing requirements.Signal values across this grid can be analyzed via Euler calculus. Signal coverage, gaps, and redundancy are reflected in topological counts and we can apply Euler integrals to compress, summarize, or filter the signal.

DOI
10.31219/osf.io/b8dk7_v1

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Applying the Poincaré theorems and Euler integral in the qubit based signal and sensing framework can be as Qubit State Space. Defined by rational amplitudes or unit-fraction-based norms. Signal as Path: A time-evolving qubit state (signal) as a path through this discrete manifold.Use Euler integration or Poincaré recurrence to describe or detect repeating patterns, signal stability and minimal sensing requirements.Signal values across this grid can be analyzed via Euler calculus. Signal coverage, gaps, and redundancy are reflected in topological counts and we can apply Euler integrals to compress, summarize, or filter the signal.
In this section we introduce the framework of quantum mechanics as it pertains to the types of systems we will consider for quantum computing. Here we also introduce the notion of a quantum bit or ‘qubit’, which is a fundamental concept for quantum computing. At the beginning of the twentieth century, it was believed by most that the laws of Newton and Maxwell were the correct laws of physics. By the 1930s, however, it had become apparent that these classical theories faced serious problems in trying to account for the observed results of certain experiments. As a result, a new mathematical framework for physics called quantum mechanics was formulated, and new theories of physics called quantum physics were developed in this framework. Quantum physics includes the physical theories of quantum electrodynamics and quantum field theory, but we do not need to know these physical theories in order to learn about quantum information. Quantum information is the result of reformulating information theory in this quantum framework. We saw in Section 1.6 an example of a two-state quantum system: a photon that is constrained to follow one of two distinguishable paths. We identified the two distinguishable paths with the 2-dimensional basis vectors and then noted that a general ‘path state’ of the photon can be described by a complex vector with |α0|2 +|α1|2 = 1. This simple example captures the essence of the first postulate, which tells us how physical states are represented in quantum mechanics. Depending on the degree of freedom (i.e. the type of state) of the system being considered, H may be infinite-dimensional. For example, if the state refers to the position of a particle that is free to occupy any point in some region of space, the associated Hilbert space is usually taken to be a continuous (and thus infinite-dimensional) space. It is worth noting that in practice, with finite resources, we cannot distinguish a continuous state space from one with a discrete state space having a sufficiently small minimum spacing between adjacent locations. For describing realistic models of quantum computation, we will typically only be interested in degrees of freedom for which the state is described by a vector in a finite-dimensional (complex) Hilbert space.
We explore Poincar´e recurrence as a mechanism for generating quantum-likestatistics from recurrent classical dynamics. Modeling the universe as a singlerecurrent, ergodic trajectory on a compact phase space, we show that an embedded observer’s ignorance of their location along the recurrent trajectory definesa natural ensemble. Using the Kolmogorov Extension Theorem, we constructa Kolmogorov-consistent, normalizable measure on the induced path space. Wefurther show that recurrent loop topology fixes the complex phase weighting ofhistories, reconstructing the Feynman path integral. The Born rule emerges fromthe same consistency requirements.
The Poincaré recurrence theorem shows that conservative systems in a bounded region of phase space eventually return arbitrarily close to their initial state after a finite amount of time. An analogous behavior occurs in certain quantum systems where quantum states can recur after sufficiently long unitary evolution, a phenomenon known as quantum recurrence. Periodically driven (i.e. Floquet) quantum systems in particular exhibit complex dynamics even in small dimensions, motivating the study of how interactions and Hamiltonian structure affect recurrence behavior. While most studies treat recurrence in an approximate, distance-based sense, here we address the problem of {exact}, state-independent recurrences in a broad class of finite-dimensional Floquet systems, spanning both integrable and non-integrable models. Leveraging techniques from algebraic field theory, we construct an arithmetic framework that identifies all possible recurrence times by analyzing the cyclotomic structure of the Floquet unitary's spectrum. This computationally tractable approach yields both positive results, enumerating all candidate recurrence times, and definitive negative results, rigorously ruling out candidate recurrence times for a given set of Hamiltonian parameters. We further prove that rational Hamiltonian parameters do not, in general, guarantee exact recurrences, revealing a subtle interplay between system parameters and long-time dynamics. Our findings sharpen the theoretical understanding of quantum recurrences, clarify their relationship to quantum chaos, and highlight parameter regimes of special interest for quantum metrology and control.
The rapid advancement of quantum-inspired classical algorithms, particularly Quantics Tensor Trains (QTT), has demonstrated that exponentially large vectors can be manipulated efficiently via Matrix Product States (MPS). Conventionally, the compression of these tensor networks relies exclusively on Singular Value Decomposition (SVD)-based truncation. While SVD is optimal for minimizing the Frobenius norm error, it remains structurally blind to exact algebraic correlations, such as reversible logic gates or Clifford symmetries, that do not require numerical approximation to factor out. In this work, we propose a hybrid compression protocol that integrates the topological rigor of ZX-Calculus with the numerical power of SVD. We construct an explicit isomorphism between Rank-3 MPS tensors and ZX-diagrams, allowing us to subject QTT representations to formal diagrammatic rewriting rules prior to numerical truncation. We establish that this Topological Preconditioning can algebraically collapse the effective bond dimension (χ) and T-count of the underlying tensor network without information loss. By benchmarking the algorithmic implementation against hardware-accelerated SVD on discretized functions and stabilizer states, we demonstrate that ZX-driven algebraic erasure fundamentally bypasses standard [Formula: see text] bottlenecks, yielding up to a computational speedup. Finally, we formalize this advantage by defining the structural complexity class [Formula: see text], proving that for systems characterized by algebraic symmetry, this work reflects our ongoing effort to deepen understanding of Categorical Quantum Mechanics, aiming to contribute to the optimization of high-dimensional linear algebra.
Quantum-Inspired Geometric Modeling for Signal Processing presents a novel approach to signal processing by leveraging quantum-inspired techniques to construct geometric models of signal propagation. The core claim is that this methodology will enable the detection of subtle changes in signal characteristics that classical methods often miss, offering a significant advancement over traditional approaches. This paper explores the fundamental principles behind this approach, detailing the model construction process and the potential for enhanced signal detection. The research will focus on the emergence of patterns and distortions inherent within the signal, leveraging quantum-inspired concepts to provide a more robust and insightful analysis.