aqcmp Annals of Quantum and Condensed Matter Physics Ann. Quantum Condens. Matter Phys. 2511-9877 2511-9958 Meridian Academic Press 10.58303/4r5cee25 Ultrafast control of superconducting qubits observed by time-resolved spectroscopy Takahashi Sakura Writing – review & editing Project administration Software Validation School of Environmental Sciences, Anatolian Technical University, Turkey

One author is a consultant for a company that manufactures related equipment. The company had no role in the design of the study or the interpretation of the results.

 11 02 2024 4 1 e5639 04 08 2023 10 12 2023 10 01 2024 © 2024 Meridian Academic Press 2024 Meridian Academic Press https://creativecommons.org/licenses/by/4.0/

Despite considerable progress in recent years, several fundamental questions about superconducting qubits remain open. We observe good agreement between exact diagonalization and the measured spectra, and we show that the result is stable under a range of perturbations of the assumptions. In contrast to earlier studies, our approach accounts for heterogeneity and avoids the usual simplifying assumptions. We observe a crossover between two distinct regimes controlled by the coupling strength. Together, these results contribute to a more complete picture and open the way to further applications.

 entanglement condensed matter quantum information quantum optics This work was supported by the Japan Society for the Promotion of Science (KAKENHI JP22H01234). Introduction

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with Lipschitz boundary and let $p \in (1,\infty)$. We consider the boundary value problem1

\begin{aligned} -\operatorname{div}\left(|\nabla u|^{p-2}\nabla u\right) &= f \quad \text{in } \Omega,\\ u &= 0 \quad \text{on } \partial\Omega. \end{aligned}

Existence of weak solutions in $W_0^{1,p}(\Omega)$ follows from the direct method of the calculus of variations. Our interest here is in sharp quantitative bounds that do not depend on the dimension.

 Main result

Theorem 1. Suppose $f \in L^{q}(\Omega)$ with $q > d/p$. Then there is a constant $C = C(d,p,q)$ such that

\|u\|_{L^\infty(\Omega)} \le C\,|\Omega|^{\frac{1}{d}-\frac{1}{p}+\frac{1}{q}}\,\|f\|_{L^q(\Omega)}^{\frac{1}{p-1}}.

The proof combines a truncation argument with the Sobolev inequality $\|v\|_{p^*} \le S\|\nabla v\|_p$. A second estimate, for the gradient, is given by

\int_\Omega |\nabla u|^p\,dx ;\le; \sum_{k=1}^{\infty} 2^{-k}\,\lambda_k^{-1}\,\|f\|_{q}^{p'}, \qquad p' = \tfrac{p}{p-1}. Numerical illustration

Table 1 reports computed values of the constant in (2) for several choices of the parameters, and Fig. 1 shows the decay of the error under mesh refinement.

Computed values of the constant $C(d,p,q)$

dpqC (computed)C (bound)
2220.0810.125
2340.1470.210
3230.0520.083
3450.2090.341

Error versus mesh size on a logarithmic scale; the dashed line has slope 2.

 Discussion

The bound (2) is sharp up to the value of the constant, as the radial example $u(x) = (1-|x|^{p/(p-1)})_+$ shows. Extensions to systems and to variable exponents will be treated elsewhere.

Here the divergence and gradient are taken in the sense of distributions.

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All data supporting the findings are available in the supplementary material and in a public repository.