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1. Nonlinear and non-CP gates for Bloch vector amplification

   https://doi.org/10.1088/1572-9494/acf304  

   Geller, Michael R (September 2023, Communications in Theoretical Physics)

   Abstract Any stater= (x,y,z) of a qubit, written in the Pauli basis and initialized in the pure stater= (0, 0, 1), can be prepared by composing three quantum operations: two unitary rotation gates to reach a pure state $\mathit{r}={\left(x^2+y^2+z^2\right)}^{-\frac{1}{2}}\times (x,y,z)$on the Bloch sphere, followed by a depolarization gate to decrease ∣r∣. Here we discuss the complementary state-preparation protocol for qubits initialized at the center of the Bloch ball,r=0, based on increasing or amplifying ∣r∣ to its desired value, then rotating. Bloch vector amplification increases purity and decreases entropy. Amplification can be achieved with a linear Markovian completely positive trace-preserving (CPTP) channel by placing the channel’s fixed point away fromr=0, making it nonunital, but the resulting gate suffers from a critical slowing down as that fixed point is approached. Here we consider alternative designs based on linear and nonlinear Markovian PTP channels, which offer benefits relative to linear CPTP channels, namely fast Bloch vector amplification without deceleration. These gates simulate a reversal of the thermodynamic arrow of time for the qubit and would provide striking experimental demonstrations of non-CP dynamics. 

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2. Proposal for a Lorenz qubit

   https://doi.org/10.1038/s41598-023-40893-0  

   Geller, Michael R. (August 2023, Scientific Reports)

   Abstract Nonlinear qubit master equations have recently been shown to exhibit rich dynamical phenomena such as period doubling, Hopf bifurcation, and strange attractors usually associated with classical nonlinear systems. Here we investigate nonlinear qubit models that support tunable Lorenz attractors. A Lorenz qubit could be realized experimentally by combining qubit torsion, generated by real or simulated mean field dynamics, with linear amplification and dissipation. This would extend engineered Lorenz systems to the quantum regime, allowing for their direct experimental study and possible application to quantum information processing. 

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3. InGaP χ(2) integrated photonics platform for broadband, ultra-efficient nonlinear conversion and entangled photon generation

   https://doi.org/10.1038/s41377-024-01653-5  

   Akin, Joshua; Zhao, Yunlei; Misra, Yuvraj; Haque, A_K_M Naziul; Fang, Kejie (December 2024, Light: Science & Applications)

   Abstract Nonlinear optics plays an important role in many areas of science and technology. The advance of nonlinear optics is empowered by the discovery and utilization of materials with growing optical nonlinearity. Here we demonstrate an indium gallium phosphide (InGaP) integrated photonics platform for broadband, ultra-efficient second-order nonlinear optics. The InGaP nanophotonic waveguide enables second-harmonic generation with a normalized efficiency of 128, 000%/W/cm²at 1.55μm pump wavelength, nearly two orders of magnitude higher than the state of the art in the telecommunication C band. Further, we realize an ultra-bright, broadband time-energy entangled photon source with a pair generation rate of 97 GHz/mW and a bandwidth of 115 nm centered at the telecommunication C band. The InGaP entangled photon source shows high coincidence-to-accidental counts ratio CAR > 10⁴and two-photon interference visibility > 98%. The InGaP second-order nonlinear photonics platform will have wide-ranging implications for non-classical light generation, optical signal processing, and quantum networking. 

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4. Efficient quantum algorithm for dissipative nonlinear differential equations

   https://doi.org/10.1073/pnas.2026805118  

   Liu, Jin-Peng; Kolden, Herman Øie; Krovi, Hari K.; Loureiro, Nuno F.; Trivisa, Konstantina; Childs, Andrew M. (August 2021, Proceedings of the National Academy of Sciences)

   null (Ed.)

   Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming R < 1 , where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity T 2 q   poly ( log ⁡ T , log ⁡ n , log ⁡ 1 / ϵ ) / ϵ , where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for R ≥ 2 . Finally, we discuss potential applications, showing that the R < 1 condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R. 

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5. Nonlinear functional network connectivity in resting functional magnetic resonance imaging data

   https://doi.org/10.1002/hbm.25972  

   Motlaghian, Sara M; Belger, Aysenil; Bustillo, Juan R; Ford, Judith M; Iraji, Armin; Lim, Kelvin; Mathalon, Daniel H; Mueller, Bryon A; O'Leary, Daniel; Pearlson, Godfrey; et al (October 2022, Human Brain Mapping)

   Abstract In this work, we focus on explicitly nonlinear relationships in functional networks. We introduce a technique using normalized mutual information (NMI) that calculates the nonlinear relationship between different brain regions. We demonstrate our proposed approach using simulated data and then apply it to a dataset previously studied by Damaraju et al. This resting‐state fMRI data included 151 schizophrenia patients and 163 age‐ and gender‐matched healthy controls. We first decomposed these data using group independent component analysis (ICA) and yielded 47 functionally relevant intrinsic connectivity networks. Our analysis showed a modularized nonlinear relationship among brain functional networks that was particularly noticeable in the sensory and visual cortex. Interestingly, the modularity appears both meaningful and distinct from that revealed by the linear approach. Group analysis identified significant differences in explicitly nonlinear functional network connectivity (FNC) between schizophrenia patients and healthy controls, particularly in the visual cortex, with controls showing more nonlinearity (i.e., higher normalized mutual information between time courses with linear relationships removed) in most cases. Certain domains, including subcortical and auditory, showed relatively less nonlinear FNC (i.e., lower normalized mutual information), whereas links between the visual and other domains showed evidence of substantial nonlinear and modular properties. Overall, these results suggest that quantifying nonlinear dependencies of functional connectivity may provide a complementary and potentially important tool for studying brain function by exposing relevant variation that is typically ignored. Beyond this, we propose a method that captures both linear and nonlinear effects in a “boosted” approach. This method increases the sensitivity to group differences compared to the standard linear approach, at the cost of being unable to separate linear and nonlinear effects. 

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