Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2404.07016

Variational Quantum Crank-Nicolson and Method of Lines for the Solution of Initial Value Problems

Francisco Guzman-Cajica·Francisco S. Guzman
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we use a Variational Quantum Algorithm to solve Initial Value Problems with the Implicit Crank-Nicolson and the Method of Lines (MoL) evolution schemes. The unknown functions use a spectral decomposition with the Fourier basis. The examples developed to illustrate the implementation are the Advection equation, the wave equation written as a system of first order coupled equations and the viscous Burgers equation as a non-linear case. The problems are solved using: i) standard Finite Differences as the solution to compare with, ii) the State Vector Formalism (SVF), and iii) the Sampling Error Formalism (SEF). The contributions of this paper include: 1) cost functions for generic first order in time PDEs using the implicit Crank-Nicholson and the MoL, 2) detailed convergence or self-convergence tests are presented for all the equations solved, 3) a system of three coupled PDEs is solved, 4) solutions using sampling error are presented, which highlights the importance of simulating the sampling process and 5) a fast version of the SVF and SEF was developed which can be used to test different optimizers faster.

Explore related subjects

quant-phphysics.comp-ph

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Francisco Guzman-Cajica, Francisco S. Guzman. 2024-10-03. Variational Quantum Crank-Nicolson and Method of Lines for the Solution of Initial Value Problems. https://doi.org/10.1103/physreva.110.042415

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Enhanced measurements on quantum computers via the simultaneous probing of non-commuting Pauli operators

Measuring the state of quantum computers is a highly non-trivial task, with implications for virtually all quantum algorithms. A promising avenue is multi-copy schemes, where identical copies of a quantum state are measured jointly so that all Pauli operators within the considered observable can be simultaneously assessed. Here, we present a first implementation of such a two-copy scheme in a measurement protocol. Based on Bayesian statistics, it accurately estimates not only the average of the desired observable but also the error en route. This enables an adaptive shot-allocation algorithm that preferentially samples the most uncertain Pauli terms. In regimes with many non-commuting Pauli operators, this ``double'' scheme can outperform the state-of-the-art measurement protocol in minimizing total shots for a given precision. We also numerically confirm the finding in previous theoretical works that the two-copy scheme incurs an overhead due to the square-root relationship between the variance of measured quantities and the number of measurement shots.

quant-ph↗
arXiv2026

Thermodynamics of a phaseonium-driven optomechanical Otto engine

We study an optomechanical Otto engine whose working medium is a single-mode cavity driven by beams of coherently prepared three-level phaseonium atoms. The atoms are not thermal reservoirs in the Gibbs sense; rather, their populations and ground-state coherence set the detailed-balance ratio of the cavity collision map, so that the field relaxes to a Gibbs state at an operational apparent temperature. We combine the finite-time collision-model dynamics with radiation-pressure work extraction and compare three reservoir preparations: a thermal reference at the same apparent temperatures, an incoherent atomic beam with the same populations, and the coherent phaseonium beam. We show that the phaseonium isochore charges the cavity passively: the cavity ergotropy and energy-basis coherence remain zero up to numerical precision, while the state converges to the Gibbs fixed point selected by the apparent detailed balance. We further estimate lower bounds on the cost of preparing the atomic populations and coherence, showing that the relevant advantage of phaseonium is a resource-preparation tradeoff rather than a cost-free enhancement over a thermal bath at the same temperature. Finally, we assess the finite-time performance of a two-cavity cascade with additive mechanical work accounting. Over the investigated coherence-phase range, the cascade produces approximately $47\%$--$52\%$ more power than the single-cavity engine while requiring only $65\%$--$68\%$ of the hot and cold phaseonium atoms needed by two independent engines, resulting in a $9\%$--$15\%$ enhancement of power per injected atom over a complete cycle.

quant-ph↗
arXiv2026

Dynamical Correlation of the Post-quench Non-thermal Equilibrium State

After a quantum quench, the integrable system is expected to relax to a non-thermal equilibrium state (NTES) whose local properties are believed to be governed by a generalized Gibbs ensemble (GGE). Combining quench action and the form factor approach, we compute the field-field correlation in the NTES produced by an interaction quench of the Lieb-Liniger model. The spectral distribution is shown to be qualitatively different from that of a thermal equilibrium state (TES): a new dispersion branch appears whose microscopic mechanism can be traced to the algebraic decaying tail for the root density distribution function, and indicates the existence of a broader family of NTES featuring similar spectral property.

quant-ph↗