We report on spectroscopic measurements on the
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$4{f}^{7}6{s}^{2}{\phantom{\rule{thickmathspace}{0ex}}}^{8}{\phantom{\rule{negativethinmathspace}{0ex}}S}_{7/2}^{\circ <\#comment/>}\to <\#comment/>4{f}^{7}{(}^{8}\phantom{\rule{negativethinmathspace}{0ex}}{S}^{\circ <\#comment/>})6s6p{(}^{1}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{P}^{\circ <\#comment/>}){\phantom{\rule{thinmathspace}{0ex}}}^{8}\phantom{\rule{negativethinmathspace}{0ex}}{P}_{9/2}$ transition in neutral europium151 and europium153 at 459.4 nm. The center of gravity frequencies for the 151 and 153 isotopes, reported for the first time in this paper, to our knowledge, were found to be 652,389,757.16(34) MHz and 652,386,593.2(5) MHz, respectively. The hyperfine coefficients for the$6s6p{(}^{1}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{P}^{\circ <\#comment/>}){\phantom{\rule{thinmathspace}{0ex}}}^{8}\phantom{\rule{negativethinmathspace}{0ex}}{P}_{9/2}$ state were found to be$\mathrm{A}(151)=<\#comment/>228.84(2)\phantom{\rule{thickmathspace}{0ex}}\mathrm{M}\mathrm{H}\mathrm{z}$ ,$\mathrm{B}(151)=226.9(5)\phantom{\rule{thickmathspace}{0ex}}\mathrm{M}\mathrm{H}\mathrm{z}$ and$\mathrm{A}(153)=<\#comment/>101.87(6)\phantom{\rule{thickmathspace}{0ex}}\mathrm{M}\mathrm{H}\mathrm{z}$ ,$\mathrm{B}(153)=575.4(1.5)\phantom{\rule{thickmathspace}{0ex}}\mathrm{M}\mathrm{H}\mathrm{z}$ , which all agree with previously published results except for A(153), which shows a small discrepancy. The isotope shift is found to be 3163.8(6) MHz, which also has a discrepancy with previously published results.Free, publiclyaccessible full text available September 9, 2023 
We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speedups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{10} + sqrt{n} epsilon^{12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{1}), with B an upper bound on the tracenorm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r. We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(logmore »