BO
B.C. Oudejans
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In this thesis, we derive a lower bound on a quantity appearing in a Fourier multiplier inequality on solvable Lie groups.
In Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi (2022), a classical result by de Leeuw about the restriction of Fourier multipliers on $\mathbb{R}^n$ to a discrete subgroup is extended to a noncommutative setting. It is shown that a compactly supported $p$-multiplier $m$ on a locally compact group $G$ has the following relation to its restriction to a discrete subgroup $\Gamma$:
$$c(\operatorname{supp}(m|_{\Gamma})) \norm{T_{m|_{\Gamma}}}_p \leq \norm{T_m}_p.$$
Here $c(U) = \inf\left\{ \sqrt{\delta_F} \mid F\subseteq U \mathrm{ finite} \right\}$, where $\delta_F$ is a quantity that determines to what extent small neighborhoods of the identity in $G$ are left invariant by conjugation by elements of $F$. In this thesis, we estimate $\delta_F$ for connected solvable Lie groups.\\
Our main result is theorem 9, which states that for a connected solvable Lie group $G$ with Lie algebra $\g$, if $\lambda_1,\ldots,\lambda_n\colon \g_{\mathbb{C}}\to\mathbb{C}$ are the generalized weights of the complexification $\g_{\mathbb{C}}$, there exist unique homomorphisms $\chi_1,\ldots,\chi_n\colon G\to\mathbb{R}_{>0}$ such that $\chi_i = \mathrm{d}\lambda_i$, and $$\delta_F \geq \prod\limits_{i=1}^n \inf\limits_{g\in F}\chi_i(g).$$ ...
In Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi (2022), a classical result by de Leeuw about the restriction of Fourier multipliers on $\mathbb{R}^n$ to a discrete subgroup is extended to a noncommutative setting. It is shown that a compactly supported $p$-multiplier $m$ on a locally compact group $G$ has the following relation to its restriction to a discrete subgroup $\Gamma$:
$$c(\operatorname{supp}(m|_{\Gamma})) \norm{T_{m|_{\Gamma}}}_p \leq \norm{T_m}_p.$$
Here $c(U) = \inf\left\{ \sqrt{\delta_F} \mid F\subseteq U \mathrm{ finite} \right\}$, where $\delta_F$ is a quantity that determines to what extent small neighborhoods of the identity in $G$ are left invariant by conjugation by elements of $F$. In this thesis, we estimate $\delta_F$ for connected solvable Lie groups.\\
Our main result is theorem 9, which states that for a connected solvable Lie group $G$ with Lie algebra $\g$, if $\lambda_1,\ldots,\lambda_n\colon \g_{\mathbb{C}}\to\mathbb{C}$ are the generalized weights of the complexification $\g_{\mathbb{C}}$, there exist unique homomorphisms $\chi_1,\ldots,\chi_n\colon G\to\mathbb{R}_{>0}$ such that $\chi_i = \mathrm{d}\lambda_i$, and $$\delta_F \geq \prod\limits_{i=1}^n \inf\limits_{g\in F}\chi_i(g).$$ ...
In this thesis, we derive a lower bound on a quantity appearing in a Fourier multiplier inequality on solvable Lie groups.
In Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi (2022), a classical result by de Leeuw about the restriction of Fourier multipliers on $\mathbb{R}^n$ to a discrete subgroup is extended to a noncommutative setting. It is shown that a compactly supported $p$-multiplier $m$ on a locally compact group $G$ has the following relation to its restriction to a discrete subgroup $\Gamma$:
$$c(\operatorname{supp}(m|_{\Gamma})) \norm{T_{m|_{\Gamma}}}_p \leq \norm{T_m}_p.$$
Here $c(U) = \inf\left\{ \sqrt{\delta_F} \mid F\subseteq U \mathrm{ finite} \right\}$, where $\delta_F$ is a quantity that determines to what extent small neighborhoods of the identity in $G$ are left invariant by conjugation by elements of $F$. In this thesis, we estimate $\delta_F$ for connected solvable Lie groups.\\
Our main result is theorem 9, which states that for a connected solvable Lie group $G$ with Lie algebra $\g$, if $\lambda_1,\ldots,\lambda_n\colon \g_{\mathbb{C}}\to\mathbb{C}$ are the generalized weights of the complexification $\g_{\mathbb{C}}$, there exist unique homomorphisms $\chi_1,\ldots,\chi_n\colon G\to\mathbb{R}_{>0}$ such that $\chi_i = \mathrm{d}\lambda_i$, and $$\delta_F \geq \prod\limits_{i=1}^n \inf\limits_{g\in F}\chi_i(g).$$
In Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi (2022), a classical result by de Leeuw about the restriction of Fourier multipliers on $\mathbb{R}^n$ to a discrete subgroup is extended to a noncommutative setting. It is shown that a compactly supported $p$-multiplier $m$ on a locally compact group $G$ has the following relation to its restriction to a discrete subgroup $\Gamma$:
$$c(\operatorname{supp}(m|_{\Gamma})) \norm{T_{m|_{\Gamma}}}_p \leq \norm{T_m}_p.$$
Here $c(U) = \inf\left\{ \sqrt{\delta_F} \mid F\subseteq U \mathrm{ finite} \right\}$, where $\delta_F$ is a quantity that determines to what extent small neighborhoods of the identity in $G$ are left invariant by conjugation by elements of $F$. In this thesis, we estimate $\delta_F$ for connected solvable Lie groups.\\
Our main result is theorem 9, which states that for a connected solvable Lie group $G$ with Lie algebra $\g$, if $\lambda_1,\ldots,\lambda_n\colon \g_{\mathbb{C}}\to\mathbb{C}$ are the generalized weights of the complexification $\g_{\mathbb{C}}$, there exist unique homomorphisms $\chi_1,\ldots,\chi_n\colon G\to\mathbb{R}_{>0}$ such that $\chi_i = \mathrm{d}\lambda_i$, and $$\delta_F \geq \prod\limits_{i=1}^n \inf\limits_{g\in F}\chi_i(g).$$
Variational Quantum Algorithms
For Optimizing Probe States
As quantum computers are developing, they are beginning to become useful for practical applications, for example in the field of quantum metrology. In this work, a variational quantum algorithm is used to find an optimal probe state for measuring parameters in a noisy environment. This is achieved by optimizing a cost on a quantum computer, based on the Fisher information of the parameters to be estimated. These parameters are then estimated using maximum likelihood estimators. In a simulation, a probe state was found that performed better than the best possible state for noiseless measurements, although this could not be reproduced on an actual quantum computer.
...
As quantum computers are developing, they are beginning to become useful for practical applications, for example in the field of quantum metrology. In this work, a variational quantum algorithm is used to find an optimal probe state for measuring parameters in a noisy environment. This is achieved by optimizing a cost on a quantum computer, based on the Fisher information of the parameters to be estimated. These parameters are then estimated using maximum likelihood estimators. In a simulation, a probe state was found that performed better than the best possible state for noiseless measurements, although this could not be reproduced on an actual quantum computer.