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1. Spacetime-Efficient Low-Depth Quantum State Preparation with Applications

   https://doi.org/10.22331/q-2024-02-15-1257  

   Gui, Kaiwen; Dalzell, Alexander M.; Achille, Alessandro; Suchara, Martin; Chong, Frederic T. (February 2024, Quantum)

   We propose a novel deterministic method for preparing arbitrary quantum states. When our protocol is compiled into CNOT and arbitrary single-qubit gates, it prepares an $N$-dimensional state in depth $O(\log (N\left)\right)$and $\text{spacetime allocation}$(a metric that accounts for the fact that oftentimes some ancilla qubits need not be active for the entire circuit) $O\left(N\right)$, which are both optimal. When compiled into the $\left\{\mathrm{H},\mathrm{S},\mathrm{T},\mathrm{C}\mathrm{N}\mathrm{O}\mathrm{T}\right\}$gate set, we show that it requires asymptotically fewer quantum resources than previous methods. Specifically, it prepares an arbitrary state up to error $ϵ$with optimal depth of $O(\log (N)+\log (1/ϵ\left)\right)$and spacetime allocation $O(N\log (\log \left(N\right)/ϵ\left)\right)$, improving over $O(\log (N)\log (\log \left(N\right)/ϵ\left)\right)$and $O(N\log (N/ϵ\left)\right)$, respectively. We illustrate how the reduced spacetime allocation of our protocol enables rapid preparation of many disjoint states with only constant-factor ancilla overhead – $O\left(N\right)$ancilla qubits are reused efficiently to prepare a product state of $w$ $N$-dimensional states in depth $O(w+\log (N\left)\right)$rather than $O(w\log (N\left)\right)$, achieving effectively constant depth per state. We highlight several applications where this ability would be useful, including quantum machine learning, Hamiltonian simulation, and solving linear systems of equations. We provide quantum circuit descriptions of our protocol, detailed pseudocode, and gate-level implementation examples using Braket. 

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2. Spectroscopic study of the 4 f ⁷ 6 s ^{2 8} S 7/2∘−4 f ⁷ ( ⁸ S ^∘ )6 s 6 p ( ¹ P ^∘ ) ⁸ P _9/2 transition in neutral europium-151 and europium-153: absolute frequency and hyperfine structure

   https://doi.org/10.1364/JOSAB.467968  

   Herd, M_T; Maruko, C.; Herzog, M_M; Brand, A.; Cannon, G.; Duah, B.; Hollin, N.; Karani, T.; Wallace, A.; Whitmore, M.; et al (September 2022, Journal of the Optical Society of America B)

   We report on spectroscopic measurements on the $4f^{7}6s^2{}^8{S}_{7/2}^{\circ <\#comment/>}\to <\#comment/>4f^7{(}^8{S}^{\circ <\#comment/>}\left)6s6p{(}^1{P}^{\circ <\#comment/>}\right){}^8{P}_{9/2}$transition in neutral europium-151 and europium-153 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 $6s6p{(}^1{P}^{\circ <\#comment/>}){}^8{P}_{9/2}$state were found to be $\mathrm{A}\left(151\right)=-<\#comment/>228.84\left(2\right)\mathrm{M}\mathrm{H}\mathrm{z}$, $\mathrm{B}\left(151\right)=226.9\left(5\right)\mathrm{M}\mathrm{H}\mathrm{z}$and $\mathrm{A}\left(153\right)=-<\#comment/>101.87\left(6\right)\mathrm{M}\mathrm{H}\mathrm{z}$, $\mathrm{B}\left(153\right)=575.4\left(1.5\right)\mathrm{M}\mathrm{H}\mathrm{z}$, 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. 

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3. Bell state analyzer for spectrally distinct photons

   https://doi.org/10.1364/OPTICA.443302  

   Lingaraju, Navin_B; Lu, Hsuan-Hao; Leaird, Daniel_E; Estrella, Steven; Lukens, Joseph_M; Weiner, Andrew_M (March 2022, Optica)

   We demonstrate a Bell state analyzer that operates directly on frequency mismatch. Based on electro-optic modulators and Fourier-transform pulse shapers, our quantum frequency processor design implements interleaved Hadamard gates in discrete frequency modes. Experimental tests on entangled-photon inputs reveal fidelities of $\sim <\#comment/>98\mathrm{\%<\#comment/>}$for discriminating between the $|{\mathrm{\Psi <\#comment/>}}^{+}⟩<\#comment/>$and $|{\mathrm{\Psi <\#comment/>}}^{-<\#comment/>}⟩<\#comment/>$frequency-bin Bell states. Our approach resolves the tension between wavelength-multiplexed state transport and high-fidelity Bell state measurements, which typically require spectral indistinguishability. 

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4. On the Size of the Singular Set of Minimizing Harmonic Maps

   https://doi.org/10.1090/memo/1519  

   Mazowiecka, Katarzyna; Miśkiewicz, Michał; Schikorra, Armin (October 2024, Memoirs of the American Mathematical Society)

   We consider minimizing harmonic maps $u$from $\mathrm{\Omega <\#comment/>}\subset <\#comment/>{\mathbb{R}}^n$into a closed Riemannian manifold $\mathcal{N}$and prove: 1. an extension to $n\ge <\#comment/>4$of Almgren and Lieb’s linear law. That is, if the fundamental group of the target manifold $\mathcal{N}$is finite, we have\[ ${\mathcal{H}}^{n-<\#comment/>3}(\mathrm{sing}<\#comment/>u)\le <\#comment/>C{\int <\#comment/>}_{\mathrm{\partial <\#comment/>}\mathrm{\Omega <\#comment/>}}|{\mathrm{\nabla <\#comment/>}}_{T}u{|}^{n-<\#comment/>1}\mathrm{d}{\mathcal{H}}^{n-<\#comment/>1};$\]2. an extension of Hardt and Lin’s stability theorem. Namely, assuming that the target manifold is $\mathcal{N}={\mathbb{S}}^2$we obtain that the singular set of $u$is stable under small $W^{1,n-<\#comment/>1}$-perturbations of the boundary data. In dimension $n=3$both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$with $s\in <\#comment/>(\frac{1}{2},1]$and $p\in <\#comment/>[2,\mathrm{\infty <\#comment/>})$satisfying $sp\ge <\#comment/>2$. We also discuss sharpness. 

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5. Algebraic cobordism and a Conner–Floyd isomorphism for algebraic K-theory

   https://doi.org/10.1090/jams/1045  

   Annala, Toni; Hoyois, Marc; Iwasa, Ryomei (March 2024, Journal of the American Mathematical Society)

   We formulate and prove a Conner–Floyd isomorphism for the algebraic K-theory of arbitrary qcqs derived schemes. To that end, we study a stable $\mathrm{\infty <\#comment/>}$-category of non- ${\mathbb{A}}^1$-invariant motivic spectra, which turns out to be equivalent to the $\mathrm{\infty <\#comment/>}$-category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We prove that this $\mathrm{\infty <\#comment/>}$-category satisfies ${\mathbb{P}}^1$-homotopy invariance and weighted ${\mathbb{A}}^1$-homotopy invariance, which we use in place of ${\mathbb{A}}^1$-homotopy invariance to obtain analogues of several key results from ${\mathbb{A}}^1$-homotopy theory. These allow us in particular to define a universal oriented motivic ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-ring spectrum $\mathrm{M}\mathrm{G}\mathrm{L}$. We then prove that the algebraic K-theory of a qcqs derived scheme $X$can be recovered from its $\mathrm{M}\mathrm{G}\mathrm{L}$-cohomology via a Conner–Floyd isomorphism\[ ${\mathrm{M}\mathrm{G}\mathrm{L}}^{\ast <\#comment/>\ast <\#comment/>}\left(X\right){\otimes <\#comment/>}_{\mathrm{L}}\mathbb{Z}\left[{\mathrm{\beta <\#comment/>}}^{\pm <\#comment/>1}\right]\simeq <\#comment/>\mathrm{K}{}^{\ast <\#comment/>\ast <\#comment/>}\left(X\right),$\]where $\mathrm{L}$is the Lazard ring and $\mathrm{K}{}^{p,q}\left(X\right)=\mathrm{K}{}_{2q-<\#comment/>p}\left(X\right)$. Finally, we prove a Snaith theorem for the periodized version of $\mathrm{M}\mathrm{G}\mathrm{L}$. 

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