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1. Quadratic Chabauty for modular curves: algorithms and examples

   https://doi.org/10.1112/S0010437X23007170  

   Balakrishnan, Jennifer S. ; Dogra, Netan ; Müller, J. Steffen ; Tuitman, Jan ; Vonk, Jan ( June 2023 , Compositio Mathematica)

   We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus$g>1$whose Jacobians have Mordell–Weil rank$g$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients$X_0^+(N)$of prime level$N$, the curve$X_{S_4}(13)$, as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve$X_{\scriptstyle \mathrm { ns}} ^+ (17)$.

    

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2. Global solutions for 1D cubic defocusing dispersive equations: Part I

   https://doi.org/10.1017/fmp.2023.30  

   Ifrim, Mihaela ; Tataru, Daniel ( January 2023 , Forum of Mathematics, Pi)

   This article is devoted to a general class of one-dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions for such problems has attracted a lot of attention in recent years, and many global well-posedness results have been proved for a number of models under the assumption that the initial data are bothsmallandlocalized. However, except for the completely integrable case, no such results have been known for small but not necessarily localized initial data.

   In this article, we introduce a new, nonperturbative method to prove global well-posedness and scattering for$L^2$initial data which aresmallandnonlocalized. Our main structural assumption is that our nonlinearity isdefocusing. However, we do not assume that our problem has any exact conservation laws. Our method is based on a robust reinterpretation of the idea of Interaction Morawetz estimates, developed almost 20 years ago by the I-team.

   In terms of scattering, we prove that our global solutions satisfy both global$L^6$Strichartz estimates and bilinear$L^2$bounds. This is a Galilean invariant result, which is new even for the classical defocusing cubic NLS.¹There, by scaling, our result also admits a large data counterpart.

    

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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. Efficient symmetry-preserving state preparation circuits for the variational quantum eigensolver algorithm

   https://doi.org/10.1038/s41534-019-0240-1  

   Gard, Bryan T. ; Zhu, Linghua ; Barron, George S. ; Mayhall, Nicholas J. ; Economou, Sophia E. ; Barnes, Edwin ( January 2020 , npj Quantum Information)

   The variational quantum eigensolver is one of the most promising approaches for performing chemistry simulations using noisy intermediate-scale quantum (NISQ) processors. The efficiency of this algorithm depends crucially on the ability to prepare multi-qubit trial states on the quantum processor that either include, or at least closely approximate, the actual energy eigenstates of the problem being simulated while avoiding states that have little overlap with them. Symmetries play a central role in determining the best trial states. Here, we present efficient state preparation circuits that respect particle number, total spin, spin projection, and time-reversal symmetries. These circuits contain the minimal number of variational parameters needed to fully span the appropriate symmetry subspace dictated by the chemistry problem while avoiding all irrelevant sectors of Hilbert space. We show how to construct these circuits for arbitrary numbers of orbitals, electrons, and spin quantum numbers, and we provide explicit decompositions and gate counts in terms of standard gate sets in each case. We test our circuits in quantum simulations of the$${H}_{2}$$$H_2$and$$LiH$$$LiH$molecules and find that they outperform standard state preparation methods in terms of both accuracy and circuit depth.

    

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5. Move schedules: fast persistence computations in coarse dynamic settings

   https://doi.org/10.1007/s41468-023-00156-3  

   Piekenbrock, Matthew ; Perea, Jose A. ( January 2024 , Journal of Applied and Computational Topology)

   Matrix reduction is the standard procedure for computing the persistent homology of a filtered simplicial complex withmsimplices. Its output is a particular decomposition of the total boundary matrix, from which the persistence diagrams and generating cycles are derived. Persistence diagrams are known to vary continuously with respect to their input, motivating the study of their computation for time-varying filtered complexes. Computing persistence dynamically can be reduced to maintaining a valid decomposition under adjacent transpositions in the filtration order. Since there are$$O(m^2)$$$O\left(m^2\right)$such transpositions, this maintenance procedure exhibits limited scalability and is often too fine for many applications. We propose a coarser strategy for maintaining the decomposition over a 1-parameter family of filtrations. By reduction to a particular longest common subsequence problem, we show that the minimal number of decomposition updatesdcan be found in$$O(m \log \log m)$$$O(mloglogm)$time andO(m) space, and that the corresponding sequence of permutations—which we call aschedule—can be constructed in$$O(d m \log m)$$$O(dmlogm)$time. We also show that, in expectation, the storage needed to employ this strategy is actually sublinear inm. Exploiting this connection, we show experimentally that the decrease in operations to compute diagrams across a family of filtrations is proportional to the difference between the expected quadratic number of states and the proposed sublinear coarsening. Applications to video data, dynamic metric space data, and multiparameter persistence are also presented.

    

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