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arXiv:2510.04598

Novel frame changes for quantum physics

Pierre-Louis Giscard, Omid Faizy Namarvar, Christian Bonhomme

Published Oct 6, 2025• 0 citations
arXivHTMLPDF

Abstract

We present novel, exotic types of frame changes for the calculation of quantum evolution operators. We detail in particular the biframe, in which a physical system's evolution is seen in an equal mixture of two different standard frames at once. We prove that, in the biframe, convergence of all series expansions of the solution is quadratically faster than in `conventional' frames. That is, if in laboratory frame or after a standard frame change the error at order $n$ of some perturbative series expansion of the evolution operator is on the order of $ε^n$, $0<ε<1$, for a computational cost $C(n)$ then it is on the order of $ε^{2n+1}$ in the biframe for the same computational cost. We demonstrate that biframe is one of an infinite family of novel frames, some of which lead to higher accelerations but require more computations to set up initially, leading to a trade-off between acceleration and computational burden.

DOI
10.48550/arxiv.2510.04598

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This paper presents a novel approach to quantum computing leveraging multi-scale modeling and simulation. The core claim is to develop a method capable of simultaneously simulating quantum systems at both microscopic and macroscopic levels, thereby enhancing the precision and efficiency of quantum computations. This is achieved through the construction of a multi-scale model integrating quantum mechanics and classical mechanics, coupled with advanced numerical simulation techniques. The proposed framework addresses the limitations of current quantum simulation methods by offering a more holistic representation of quantum systems. This allows for a deeper understanding of quantum phenomena and ultimately contributes to the advancement of quantum computing technologies. The key mathematical components utilized within this framework include the Schrödinger equation, perturbation theory, and various numerical integration schemes. Specifically, the time evolution of a quantum state is governed by the equation: (iħ ∂/∂t) |ψ(t)> = H |ψ(t)> where |ψ(t)> is the quantum state at time *t*, *ħ* is the reduced Planck constant, and *H* is the Hamiltonian operator. Perturbation theory, used to approximate the solution when the Hamiltonian is not solvable exactly, relies on the expansion: |ψperturbed>(t) = |ψunperturbed>(t) + ∑n=1∞ (-1)n+1 *An* |ψn>(t) where *An* are the perturbation coefficients. Numerical integration techniques, such as the Runge-Kutta method, are employed to approximate the time evolution operator. The accuracy of the simulation is critically dependent on the chosen time step, Δt, and the order of the integration method. A smaller Δt generally leads to higher accuracy but requires more computational resources. The overall approach seeks to bridge the gap between theoretical quantum mechanics and practical quantum computation, offering a viable path towards building more sophisticated and reliable quantum computers.
We study the quantum Wasserstein distances of De Palma and Trevisan associated with quadratic costs generated by self-adjoint observables in arbitrary finite dimension. We first show that any positive semidefinite cost operator is determined by the corresponding distance on pairs of pure states. For nonzero observable-generated quadratic costs, we then prove that the full isometry monoid consists exactly of unitary and antiunitary conjugations if and only if the distance is invariant under unitary conjugations on pure states. These conditions are equivalent to isotropy of the cost on the traceless subspace. They are also equivalent to a geometric characterization: two states attain the Wasserstein diameter exactly when their supports are orthogonal. It already suffices to impose this condition on pairs of pure states. We construct explicit mutually inverse maps between observable-generated quadratic costs and Hilbert--Schmidt frame-type operators formed from the traceless parts of their generators. This correspondence also expresses the transport variational problem directly in terms of the frame operator. For every such cost CCC on a ddd-dimensional Hilbert space, the minimal number of self-adjoint generators equals the rank of its recovered frame operator Θ(C)Θ(C)Θ(C), and is therefore at most d2−1d^2-1d2−1. Nonzero isotropic costs require exactly d2−1d^2-1d2−1 generators, attaining the largest minimal generator number that any observable-generated quadratic cost can require. Their arbitrary generating families give tight frames, while their minimal realizations give scaled Hilbert--Schmidt orthonormal bases of the traceless subspace. These symmetry, frame, and diametral characterizations determine the same one-parameter family of quantum Wasserstein distances.
The real time evolution of quantum field theory models can be calculated order by order in perturbation theory. For $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ models, the perturbative series have a zero radius of convergence which in part motivated the design of digitized versions suitable for quantum computing. In agreement with general arguments suggesting that a large field cutoff modifies Dyson's reasoning and improves convergence properties, we show that the harmonic digitizations of $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ lattice field theories lead to weak coupling expansions with a finite radius of convergence. Similar convergence properties are found for strong coupling expansions. We compare the resources needed to calculate the real-time evolution of the digitized models with perturbative expansions to those needed to do so with universal quantum computers. Unless new approximate methods can be designed to calculate long perturbative series for large systems efficiently, it appears that the use of universal quantum computers with digitizations involving a few qubits per site has the potential for more efficient calculations of the real-time evolution for large systems at intermediate coupling.