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1. Another view of sequential sampling in the birth process with immigration

   https://doi.org/10.1007/s00285-023-02041-0  

   da Silva, Poly H. ; Jamshidpey, Arash ; Tavaré, Simon ( February 2024 , Journal of Mathematical Biology)

   We explore properties of the family sizes arising in a linear birth process with immigration (BI). In particular, we study the correlation of the number of families observed during consecutive disjoint intervals of time. LettingS(a, b) be the number of families observed in (a, b), we study the expected sample variance and its asymptotics forpconsecutive sequential samples$$S_p =(S(t_0,t_1),\dots , S(t_{p-1},t_p))$$$S_p=(S(t_0,t_1),\cdots ,S(t_{p-1},t_p))$, for$$0=t_0$0=t_0<t_1<\cdots <t_p$. By conditioning on the sizes of the samples, we provide a connection between$$S_p$$$S_p$andpsequential samples of sizes$$n_1,n_2,\dots ,n_p$$$n_1,n_2,\cdots ,n_p$, drawn from a single run of a Chinese Restaurant Process. Properties of the latter were studied in da Silva et al. (Bernoulli 29:1166–1194, 2023.https://doi.org/10.3150/22-BEJ1494). We show how the continuous-time framework helps to make asymptotic calculations easier than its discrete-time counterpart. As an application, for a specific choice of$$t_1,t_2,\dots , t_p$$$t_1,t_2,\cdots ,t_p$, where the lengths of intervals are logarithmically equal, we revisit Fisher’s 1943 multi-sampling problem and give another explanation of what Fisher’s model could have meant in the world of sequential samples drawn from a BI process.

    

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2. Size dependence- and induced transformations- of fractional quantum Hall effects under tilted magnetic fields

   https://doi.org/10.1038/s41598-022-22812-x  

   Wijewardena, U. Kushan ; Nanayakkara, Tharanga R. ; Kriisa, Annika ; Reichl, Christian ; Wegscheider, Werner ; Mani, Ramesh G. ( November 2022 , Scientific Reports)

   Two-dimensional electron systems subjected to high transverse magnetic fields can exhibit Fractional Quantum Hall Effects (FQHE). In the GaAs/AlGaAs 2D electron system, a double degeneracy of Landau levels due to electron-spin, is removed by a small Zeeman spin splitting,$$g \mu _B B$$$g{\mu }_{B}B$, comparable to the correlation energy. Then, a change of the Zeeman splitting relative to the correlation energy can lead to a re-ordering between spin polarized, partially polarized, and unpolarized many body ground states at a constant filling factor. We show here that tuning the spin energy can produce fractionally quantized Hall effect transitions that include both a change in$$\nu$$$\nu$for the$$R_{xx}$$$R_{\mathrm{xx}}$minimum, e.g., from$$\nu = 11/7$$$\nu =11/7$to$$\nu = 8/5$$$\nu =8/5$, and a corresponding change in the$$R_{xy}$$$R_{\mathrm{xy}}$, e.g., from$$R_{xy}/R_{K} = (11/7)^{-1}$$$R_{\mathrm{xy}}/R_K={(11/7)}^{-1}$to$$R_{xy}/R_{K} = (8/5)^{-1}$$$R_{\mathrm{xy}}/R_K={(8/5)}^{-1}$, with increasing tilt angle. Further, we exhibit a striking size dependence in the tilt angle interval for the vanishing of the$$\nu = 4/3$$$\nu =4/3$and$$\nu = 7/5$$$\nu =7/5$resistance minima, including “avoided crossing” type lineshape characteristics, and observable shifts of$$R_{xy}$$$R_{\mathrm{xy}}$at the$$R_{xx}$$$R_{\mathrm{xx}}$minima- the latter occurring for$$\nu = 4/3, 7/5$$$\nu =4/3,7/5$and the 10/7. The results demonstrate both size dependence and the possibility, not just of competition between different spin polarized states at the same$$\nu$$$\nu$and$$R_{xy}$$$R_{\mathrm{xy}}$, but also the tilt- or Zeeman-energy-dependent- crossover between distinct FQHE associated with different Hall resistances.

    

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3. Random Quantum Circuits Transform Local Noise into Global White Noise

   https://doi.org/10.1007/s00220-024-04958-z  

   Dalzell, Alexander M. ; Hunter-Jones, Nicholas ; Brandão, Fernando G. S. L. ( March 2024 , Communications in Mathematical Physics)

   We study the distribution over measurement outcomes of noisy random quantum circuits in the regime of low fidelity, which corresponds to the setting where the computation experiences at least one gate-level error with probability close to one. We model noise by adding a pair of weak, unital, single-qubit noise channels after each two-qubit gate, and we show that for typical random circuit instances, correlations between the noisy output distribution$$p_{\text {noisy}}$$$p_{\text{noisy}}$and the corresponding noiseless output distribution$$p_{\text {ideal}}$$$p_{\text{ideal}}$shrink exponentially with the expected number of gate-level errors. Specifically, the linear cross-entropy benchmarkFthat measures this correlation behaves as$$F=\text {exp}(-2s\epsilon \pm O(s\epsilon ^2))$$$F=\text{exp}(-2sϵ\pm O\left(s{ϵ}^2\right))$, where$$\epsilon $$$ϵ$is the probability of error per circuit location andsis the number of two-qubit gates. Furthermore, if the noise is incoherent—for example, depolarizing or dephasing noise—the total variation distance between the noisy output distribution$$p_{\text {noisy}}$$$p_{\text{noisy}}$and the uniform distribution$$p_{\text {unif}}$$$p_{\text{unif}}$decays at precisely the same rate. Consequently, the noisy output distribution can be approximated as$$p_{\text {noisy}}\approx Fp_{\text {ideal}}+ (1-F)p_{\text {unif}}$$$p_{\text{noisy}}\approx F{p}_{\text{ideal}}+(1-F)p_{\text{unif}}$. In other words, although at least one local error occurs with probability$$1-F$$$1-F$, the errors are scrambled by the random quantum circuit and can be treated as global white noise, contributing completely uniform output. Importantly, we upper bound the average total variation error in this approximation by$$O(F\epsilon \sqrt{s})$$$O\left(Fϵ\sqrt{s}\right)$. Thus, the “white-noise approximation” is meaningful when$$\epsilon \sqrt{s} \ll 1$$$ϵ\sqrt{s}\ll 1$, a quadratically weaker condition than the$$\epsilon s\ll 1$$$ϵs\ll 1$requirement to maintain high fidelity. The bound applies if the circuit size satisfies$$s \ge \Omega (n\log (n))$$$s\ge \Omega (nlog(n\left)\right)$, which corresponds to onlylogarithmic depthcircuits, and if, additionally, the inverse error rate satisfies$$\epsilon ^{-1} \ge {\tilde{\Omega }}(n)$$${ϵ}^{-1}\ge \tilde{\Omega }\left(n\right)$, which is needed to ensure errors are scrambled faster thanFdecays. The white-noise approximation is useful for salvaging the signal from a noisy quantum computation; for example, it was an underlying assumption in complexity-theoretic arguments that noisy random quantum circuits cannot be efficiently sampled classically, even when the fidelity is low. Our method is based on a map from second-moment quantities in random quantum circuits to expectation values of certain stochastic processes for which we compute upper and lower bounds.

    

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4. Melting of charge order in the low-temperature state of an electronic ferroelectric-like system

   https://doi.org/10.1038/s41535-020-0217-5  

   Hassan, Nora M. ; Thirunavukkuarasu, Komalavalli ; Lu, Zhengguang ; Smirnov, Dmitry ; Zhilyaeva, Elena I. ; Torunova, Svetlana ; Lyubovskaya, Rimma N. ; Drichko, Natalia ( March 2020 , npj Quantum Materials)

   Strong electronic interactions can drive a system into a state with a symmetry breaking. Lattice frustration or competing interactions tend to prevent symmetry breaking, leading to quantum disordered phases. In spin systems frustration can produce a spin liquid state. Frustration of a charge degree of freedom also can result in various exotic states, however, experimental data on these effects is scarce. In this work we demonstrate how in a Mott insulator on a weakly anisotropic triangular lattice a charge ordered state melts on cooling down to low temperatures. Raman scattering spectroscopy finds that$$\kappa$$$\kappa$-(BEDT-TTF)$${}_{2}$$${}_2$Hg(SCN)$${}_{2}$$${}_2$Cl enters an insulating “dipole solid” state at$$T=30\,{\mathrm{K}}$$$T=30K$, but below$$T=15\,{\mathrm{K}}$$$T=15K$the order melts, while preserving the insulating energy gap. Based on these observations, we suggest a phase diagram relevant to other quantum paraelectric materials.

    

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5. Optomechanical ring resonator for efficient microwave-optical frequency conversion

   https://doi.org/10.1038/s41467-023-43393-x  

   Chen, I-Tung ; Li, Bingzhao ; Lee, Seokhyeong ; Chakravarthi, Srivatsa ; Fu, Kai-Mei ; Li, Mo ( November 2023 , Nature Communications)

   Phonons traveling in solid-state devices are emerging as a universal excitation for coupling different physical systems. Phonons at microwave frequencies have a similar wavelength to optical photons in solids, enabling optomechanical microwave-optical transduction of classical and quantum signals. It becomes conceivable to build optomechanical integrated circuits (OMIC) that guide both photons and phonons and interconnect photonic and phononic devices. Here, we demonstrate an OMIC including an optomechanical ring resonator (OMR), where  co-resonant infrared photons and GHz phonons induce significantly enhanced interconversion. The platform is hybrid, using wide bandgap semiconductor gallium phosphide (GaP) for waveguiding and piezoelectric zinc oxide (ZnO) for phonon generation. The OMR features photonic and phononic quality factors of >1 × 10⁵and 3.2 × 10³, respectively. The optomechanical interconversion between photonic modes achieved an internal conversion efficiency$${\eta }_{i}=(2.1\pm 0.1)\%$$${\eta }_i=\left(2.1\pm 0.1\right)\%$and a total device efficiency$${\eta }_{{tot}}=0.57{\times 10}^{-6}$$${\eta }_{tot}=0.57{\times 10}^{-6}$at a low acoustic pump power of 1.6 mW. The efficient conversion in OMICs enables microwave-optical transduction for quantum information and microwave photonics applications.

    

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