arXiv subjects

Timothy Stroschein

Publications and source records attributed to Timothy Stroschein.

4 recordsLinked to original sources

arXiv2026

After 100 Years of Quantum Mechanics: Toward a Constructive Observation-Centered Perspective

Quantum mechanics owes much of its extraordinary success to a Hilbertian program of mathematical formalization. Yet, the formalism remains poorly aligned with the practical limitations of computations in finite dimensions and under finite accuracy. In this perspective, we argue that this mismatch points to the need for a new mathematical program: a rigorous constructive theory for effective descriptions to identify essential degrees of freedom. We propose an observation-centered point of view in which signals are treated as the primary objects of analysis, while wave functions and Hamiltonians are reconstructed as auxiliary structures to rationalize the observed data. Our starting point is a signal-based spectral equation that reformulates frequency analysis as an operator problem. We connect this point of view to results on prolate Fourier theory, spectral analysis with finite observation time, and short-time quantum simulation. We highlight a sharp accuracy transition relating necessary observation time to the effective spectral density of a signal for achieving accurate resolution. The resulting framework integrates approximation as a fundamental necessity more directly into the foundations of quantum mechanics and points toward a broader program for the effective description of complex quantum systems, such as those found in the molecular sciences.

quant-ph↗

arXiv2025

Ground and excited-state energies with analytic errors and short time evolution on a quantum computer

Accurately solving the Schr\"odinger equation remains a central challenge in computational physics, chemistry, and materials science. Here, we propose an alternative eigenvalue problem based on a system's autocorrelation function, avoiding direct reference to a wave function. In particular, we develop a rigorous approximation framework that enables precise frequency estimation from a finite number of signal samples. Our analysis builds on new results involving prolate spheroidal wave functions and yields error bounds that reveal a sharp accuracy transition governed by the observation time and spectral density of the signal. These results are very general and thus carry far. As one important example application we consider the quantum computation for molecular systems. By combining our spectral method with a quantum subroutine for signal generation, we define quantum prolate diagonalization (QPD) - a hybrid classical-quantum algorithm. QPD simultaneously estimates ground and excited state energies within chemical accuracy at the Heisenberg limit. An analysis of different input states demonstrates the robustness of the method, showing that high precision can be retained even under imperfect state preparation.

quant-ph↗

arXiv2025

Local spectral approximation of unbounded operators: non-asymptotic and unified error quantification for subspace methods

We introduce a framework for subspace methods which approximate the spectra of self-adjoint, unbounded operators in a local region. Using the projection-valued measure, we derive integrated spectral inequalities that also apply to unbounded operators. Our framework is non-asymptotic, gap-independent, and enables a unified error quantification of numerical routines subject to multiple error sources. Furthermore, we formalize the class of methods applicable to our framework, and establish a rigorous foundation for dimension detection in the presence of noise as solution to frequent numerical artifacts such as spectral pollution. The practical relevance of this non-asymptotic analysis is substantiated by its recent application to sampled prolate filter diagonalization, where it successfully predicted a sharp accuracy transition linking spectral density to the minimal observation time required to decompose a signal.

math.NA↗

arXiv2024

Prolate Spheroidal Wave Functions and the Accuracy and Dimensionality of Spectral Analysis

The main result of this thesis is an efficient protocol to determine the frequencies of a signal $C(t)= \sum_k |a_k|^2 e^{i \omega_k t}$, which is given for a finite time, to a high degree of precision. Specifically, we develop a theorem that provides a fundamental precision guarantee. Additionally, we establish an approximation theory for spectral analysis through low-dimensional subspaces that can be applied to a wide range of problems. The signal processing routine relies on a symmetry between harmonic analysis and quantum mechanics. In this context, prolate spheroidal wave functions (PSWF) are identified as the optimal information processing basis. To establish rigorous precision guarantees, we extend the concentration properties of PSWFs to a supremum bound and an $\ell_2$ bound on their derivatives. The new bounds allow us to refine the truncation estimates for the prolate sampling formula. We also provide a new geometrical insight into the commutation relation between an integral operator and a differential operator, both of which have PSWFs as eigenfunctions.

math-ph↗