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Symmetric Folding Principle for Quantum Propagator Synthesis

Vishesh Kaushik

Published Jun 1, 2026• 0 citations
Publisher

Abstract

Author-prepared preprint, Version 1. This manuscript presents a symmetry-based framework for quantum propagator synthesis, called the Symmetric Folding Principle. The framework uses folding symmetries to reformulate certain unitary propagator-synthesis problems as structured state/fiber steering problems, followed by a prescribed folded continuation of the control sequence. This version may differ from later submitted, accepted, or published versions of the manuscript. Copyright © 2026 Vishesh Kaushik. The author retains copyright in this preprint. Rights in any future peer-reviewed published version will be determined by the corresponding publishing agreement with the relevant journal or publisher.

DOI
10.5281/zenodo.20494054

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Author-prepared preprint, Version 1. This manuscript presents a symmetry-based framework for quantum propagator synthesis, called the Symmetric Folding Principle. The framework uses folding symmetries to reformulate certain unitary propagator-synthesis problems as structured state/fiber steering problems, followed by a prescribed folded continuation of the control sequence. This version may differ from later submitted, accepted, or published versions of the manuscript. Copyright © 2026 Vishesh Kaushik. The author retains copyright in this preprint. Rights in any future peer-reviewed published version will be determined by the corresponding publishing agreement with the relevant journal or publisher.
We introduce the symmetric-folding principle, a geometric quantum-control framework that lifts quantum state steering to exact propagator synthesis. The required structure is a folding symmetry: a unitary transformation that reverses the drift Hamiltonian while preserving the admissible control space. The method recasts propagator synthesis on the unitary group as steering of a chosen quantum state variable between fibers of an equivariant quotient map, followed by a prescribed folded continuation of the controls. We prove that control inputs satisfying the resulting fiber-transfer condition, when extended by the prescribed folded continuation, generate the desired unitary propagator in the associated symmetry-selected target family. This gives two key structural advantages. The fiber-transfer condition constrains the midpoint propagator only to a coset of the folding symmetry's centralizer. The prescribed fold cancels this residual freedom exactly. Thus a full group-level endpoint constraint is replaced by a lower-codimension fiber/coset condition, giving additional freedom in control design. For quantum ensembles, different members may reach different midpoint propagators within the same coset, while the folded sequence still yields a common final propagator for all ensemble members. This parameter-dependent freedom can be especially significant for robust ensemble-propagator design. We demonstrate the construction on an SO(3) Bloch ensemble and an SU (4) scalar-coupled two-spin system.
We develop a constructive control framework for distance-and configuration-selective operations in neutral-atom arrays. The starting point is the symmetric folding principle, which reformulates propagator synthesis as a structured state-or fiber-steering problem followed by a symmetry-prescribed folded continuation of the control sequence. We specialize this principle to spectrally programmed (SU(2)) spin propagators whose action is prescribed as a function of the spectral detuning. For a driven target atom coupled to an undriven controller, the Rydberg interaction decomposes the two-atom Hamiltonian into invariant target subspaces with effective detunings (∆) and (∆ -V (ρ)). Interatomic distance is thereby converted into a spectral control parameter. Distance-selective gates can consequently be formulated as passband/stopband designs in which a chosen target rotation is produced over a prescribed distance range and the identity is produced over a separated stopband, with a finite transition region. The framework extends to one driven target and multiple conserved controller configurations, provided that the corresponding effective spectral detunings remain distinguishable.
Elucidating quantum systems theory in terms of symmetry principles has triggered us in a number of recent advances: (1) it leads to a new handy controllability criterion, (2) it guides the design of quantum hardware, (3) it governs which quantum system can simulate another one given, and (4) it specifies the limit between time-optimal control and relaxation-optimised control of open systems. How principles turn into practice has been illustrated in the talk by a plethora of examples showing practical applications in solid-state devices, circuit-QED, and in spin systems. The algorithmic tools have been presented in a unified programming framework.
This work presents an axiomatic and structural reconstruction of core elements of quantum mechanics. Starting from minimal principles — symmetry under vanishing preference, normalization, consistency of probability assignments, and coherence-preserving dynamics — we derive several central features of the quantum formalism rather than postulating them. Key results include: A geometric derivation of the constants 1/21/\sqrt{2}1/2 and 1/21/21/2 from symmetry and normalization. A theorem-level derivation of the Born rule (including the qubit case) from noncontextuality and additivity. A reconstruction of Schrödinger dynamics from reversible norm-preserving structure. A derivation of the tensor-product structure for composite systems from composition axioms and local tomography, with entanglement emerging as a necessary consequence. Formal treatment of Bell–CHSH nonlocality, Tsirelson’s bound, and no-signalling. A conditional reconstruction of complex Hilbert space structure from intrinsic phase dynamics and compositional consistency. The manuscript is self-contained and framed in standard mathematical-physics language. It does not propose modifications to quantum mechanics, but instead clarifies why many of its defining structures arise inevitably from deeper structural constraints. Prior computational and model-based work by the authors (including diagnostic toy models exploring phase locking, coherence stabilization, and recursive-time dynamics) is acknowledged but intentionally not reproduced here, in order to keep the present work focused on axiomatic structure rather than model-specific architecture. This upload is intended as a foundations and reconstruction contribution, suitable for readers interested in the mathematical and conceptual structure of quantum theory.