Parameterized quantum circuits are central to near-term quantum algorithms, and serve as the foundation for many quantum machine learning frameworks, which should be robust to noise, maintain trainability, and exhibit sufficient expressibility. Here we introduce 2MC-OBPPP, a polynomial-time classical estimator that quantifies the three aforementioned diagnostics for a given parameterized quantum circuit. As a demonstration of the power of our approach, we show that moderate amplitude damping noise can reduce the severity of vanishing gradients, but at the cost of lowered expressibility. In particular, our approach can yield a spatiotemporal "noise-hotspot" map that pinpoints the most noise-sensitive qubits/gates in parameterized quantum circuits. Applying this map to a specific circuit, we show that implementing interventions on fewer than 2% of the qubits is sufficient to mitigate up to 90% of the errors. Therefore, 2MC-OBPPP is not only an efficient, hardware-independent tool for pre-execution circuit evaluation but also enables targeted strategies that significantly reduce the cost of noise suppression.

Two classically equivalent expressions of mutual information of probability distributions (classical bipartite states) diverge when extended to quantum systems, and this difference has been employed to define quantum discord, a quantifier of quantum correlations beyond entanglement. Similarly, equivalent expressions of classical Fisher information of parameterized probability distributions diverge when extended to quantum states, and this difference may be exploited to characterize the complex nature of quantum states. By complexity of quantum states, we mean some hybrid nature which intermingles the classical and quantum features. It is desirable to quantify complexity of quantum states from various perspectives. In this work, we pursue the idea of discord and introduce an information-theoretic quantifier of complexity for quantum states (relative to the Hamiltonian that drives the evolution of quantum systems) via the notion of Fisher discord, which is defined by the difference between two important versions of quantum Fisher information: the quantum Fisher information defined via the symmetric logarithmic derivatives and the Wigner-Yanase skew information defined via the square roots of quantum states. We reveal basic properties of the quantifier of complexity, and compare it with some other quantifiers of complexity. In particular, we show that equilibrium states (or stable states, which commute with the Hamiltonian of the quantum system) and all pure states exhibit zero complexity in this setting. As illustrations, we evaluate the complexity for various prototypical states in both discrete and continuous-variable quantum systems.

This work analyzes the inverse optimal transport (IOT) problem under Bregman regularization. We establish well-posedness results, including existence, uniqueness (up to equivalence classes of solutions), and stability, under several structural assumptions on the cost matrix. On the computational side, we investigate the existence of solutions to the optimization problem with general constraints on the cost matrix and provide a sufficient condition guaranteeing existence. In addition, we propose an inexact block coordinate descent (BCD) method for the problem with a strongly convex penalty term. In particular, when the penalty is quadratic, the subproblems admit a diagonal Hessian structure, which enables highly efficient element-wise Newton updates. We establish a linear convergence rate for the algorithm and demonstrate its practical performance through numerical experiments, including the validation of stability bounds, the investigation of regularization effects, and the application to a marriage matching dataset.

In this paper, we introduce and study two cyclotomic level maps defined respectively on the set of nilpotent orbits $\underline{\mathcal{N}}$ in a complex semi-simple Lie algebra $\mathfrak{g}$ and the set of conjugacy classes $\underline{W}$ in its Weyl group, with values in positive integers. We show that these maps are compatible under Lusztig's map $\underline{W} \to \underline{\mathcal{N}}$, which is also the minimal reduction type map as shown by Yun. We also discuss their relationship with two-sided cells in affine Weyl groups. We use these maps to formulate a conjecture on the associated varieties of simple affine vertex algebras attached to $\mathfrak{g}$ at non-admissible integer levels, and provide some evidence for this conjecture.

A connection between the completion of quantum mechanics featuring events and the theory of emergent spacetime in quantum gravity where von Neumann algebra plays a vital role is established. In thermal equilibrium, we show that the Principle of Diminishing Potentialities (PDP) holds for the large $N$ algebra of $\mathcal{N}=4$ Super Yang-Mills (SYM) theory with gauge group $SU(N)$ when the temperature is higher than Hawking-Page temperature. Below Hawking-Page transition and for the case of zero temperature, PDP does not hold. Since the centralizer of thermofield double state on the large $N$ algebra of $\mathcal{N}=4$ SYM theory coincides with the center of the large $N$ algebra which is trivial, we extend the large $N$ algebra by performing crossed product by the maximal abelian subgroup $H$ of the compact symmetry group $G$ of the two-sided eternal black hole. In this case, the centralizer of an extension of thermofield double state is non-trivial and it is given by the action of the maximal abelian subgroup $H$ on the Hilbert space $\mathcal{H}_{TFD} \otimes L^{2} (H)$. This centralizer is by itself commutative and it coincides with its own center. This implies that the first actual event that initiate the ``Events-Trees-Histories'' dynamical evolution in this framework is given by the spectral projectors associated to the action of the Cartan subalgebra $\mathfrak{h}$ of the Lie algebra $\mathfrak{g}$ associated to the group $G$ on the Hilbert space $\mathcal{H}_{TFD} \otimes L^{2} (H)$.

By using the Wigner transform, it is shown that the nonlinear Schr$\ddot{\textmd{o}}$dinger equation can be described, in phase space, by a kinetic theory similar to the Vlasov equation which is used for describing a classical collisionless plasma. In this paper we mainly show Landau damping in the quantum sense, namely,quantum Landau damping exists for the Wigner-Poisson system. At the same time, we also prove the existence and the stability of the nonlinear Schr$\ddot{\textmd{o}}$dinger equation under the quantum stability assumption.

We study thermodynamic aspects of a tractable toy model of holography for extremal Kerr black holes proposed in [arXiv:1806.10127]. On the gravity side, the theory can be described by the worldsheet action of string theory on a warped AdS$_3$ background supported by NS-NS flux. Once we turn on temperature, the deformed background is described by a black string solution of type IIB supergravity that features a locally warped AdS$_3$ factor. The dual field theory is conjectured to be a single-trace version of a $J\bar{T}$-deformed CFT at finite temperature. As evidence for the correspondence we show that the spectrum of strings winding on the deformed background agrees with the spectrum of $J\bar{T}$-deformed CFTs. Furthermore, we show that the gravitational charges of the black string match the averaged charges of a thermal ensemble in the dual field theory. Finally, we reproduce the Bekenstein-Hawking entropy of the black string from the microscopic density of states of $J\bar{T}$-deformed CFTs.

Parameter shift rules (PSRs) are useful methods for computing arbitrary-order derivatives of the cost function in parameterized quantum circuits. The basic idea of PSRs is to evaluate the cost function at different parameter shifts, then use specific coefficients to combine them linearly to obtain the exact derivatives. In this work, we propose an extended parameter shift rule (EPSR) which generalizes a broad range of existing PSRs and has the following two advantages. First, EPSR offers an infinite number of possible parameter shifts, allowing the selection of the optimal parameter shifts to minimize the final derivative variance and thereby obtaining the more accurate derivative estimates with limited quantum resources. Second, EPSR extends the scope of the PSRs in the sense that EPSR can handle arbitrary Hermitian operator $H$ in gate $U(x) = \exp (iHx)$ in the parameterized quantum circuits, while existing PSRs are valid only for simple Hermitian generators $H$ such as simple Pauli words. Additionally, we show that the widely used ``general PSR'', introduced by Wierichs et al. (2022), is a special case of our EPSR, and we prove that it yields globally optimal shifts for minimizing the derivative variance under the weighted-shot scheme. Finally, through numerical simulations, we demonstrate the effectiveness of EPSR and show that the usage of the optimal parameter shifts indeed leads to more accurate derivative estimates.

In IIB string theory on AdS$_3$ background with NS-NS fluxes, we show that Brown-Henneaux asymptotic Killing vectors can be derived by requiring both the worldsheet equations of motion and Virasoro constraints are preserved near the asymptotic boundary of the target spacetime. The charges on the worldsheet that generate the corresponding transformations can be written down in both the Lagrangian formalism and Hamiltonian formalism. This provides a method of studying asymptotic symmetry of the target spacetime directly from worldsheet string theories, without using results from the supergravity limit. As an example, we apply this method to flat spacetime in three dimensions and obtain the BMS$_3$ generators on the worldsheet theory.

In this paper, we study moduli spaces of sextic curves with simple singularities. Through period maps of K3 surfaces with ADE singularities, we prove such moduli spaces admit algebraic open embeddings into arithmetic quotients of type IV domains. For all cases, we prove the identifications of GIT compactifications and Looijenga compactifications. We also describe Picard lattices in an explicit way for many cases and apply this to study the relation of orbifold structures on two sides of the period map.

$T\bar T$ deformed CFTs with positive deformation parameter have been proposed to be holographically dual to Einstein gravity in a glue-on $\mathrm{AdS}_3$ spacetime. The latter is constructed from AdS$_3$ by gluing a patch of an auxiliary AdS$_3^*$ spacetime to its asymptotic boundary. In this work, we propose a glue-on version of the Ryu-Takayanagi formula, which is given by the signed area of an extremal surface. The extremal surface is anchored at the endpoints of an interval on a cutoff surface in the glue-on geometry. It consists of an RT surface lying in the AdS$_3$ part of the spacetime and its extension to the AdS$_3^*$ region. The signed area is the length of the RT surface minus the length of the segments in AdS$_3^*$. We find that the Ryu-Takayanagi formula with the signed area reproduces the entanglement entropy of a half interval for $T\bar T$-deformed CFTs on the sphere. We then study the properties of extremal surfaces on various glue-on geometries, including Poincaré $\mathrm{AdS}_3$, global $\mathrm{AdS}_3$, and the BTZ black hole. When anchored on multiple intervals at the boundary, the signed area of the minimal surfaces undergoes phase transitions with novel properties. In all of these examples, we find that the glue-on extremal surfaces exhibit a minimum length related to the deformation parameter of $T\bar T$-deformed CFTs.

6d superconformal field theories (SCFTs) are the SCFTs in the highest possible dimension. They can be geometrically engineered in F-theory by compactifying on non-compact elliptic Calabi-Yau manifolds. In this paper we focus on the class of SCFTs whose base geometry is determined by $-2$ curves intersecting according to ADE Dynkin diagrams and derive the corresponding mirror Calabi-Yau manifold. The mirror geometry is uniquely determined in terms of the mirror curve which has also an interpretation in terms of the Seiberg-Witten curve of the four-dimensional theory arising from torus compactification. Adding the affine node of the ADE quiver to the base geometry, we connect to recent results on SYZ mirror symmetry for the $A$ case and provide a physical interpretation in terms of little string theory. Our results, however, go beyond this case as our construction naturally covers the $D$ and $E$ cases as well.

We study 2d and 3d gravity theories on spacetimes with causal (timelike or null) codimension one boundaries while allowing for variations in the position of the boundary. We construct the corresponding solution phase space and specify boundary degrees freedom by analysing boundary (surface) charges labelling them. We discuss Y and W freedoms and change of slicing in the solution space. For D dimensional case we find D+1 surface charges, which are generic functions over the causal boundary. We show that there exist solution space slicings in which the charges are integrable. For the 3d case there exists an integrable slicing where charge algebra takes the form of Heisenberg \oplus\ {\cal A}_3 where {\cal A}_3 is two copies of Virasoro at Brown-Henneaux central charge for AdS_3 gravity and BMS_3 for the 3d flat space gravity.

We extend the kernel-differentiation method (or likelihood-ratio method) for the linear response of random dynamical systems, after revisit its derivation in a microscopic view via transfer operators. First, for the linear response of physical measures, we extend the method to an ergodic version, which is sampled by one infinitely-long sample path; moreover, the magnitude of the integrand is bounded by a finite decorrelation step number. Second, when the noise and perturbation are along a given foliation, we show that the method is still valid for both finite and infinite time. We numerically demonstrate the ergodic version on a tent map, whose linear response of the physical measure does not exist; but we can add some noise and compute an approximate linear response. We also demonstrate on a chaotic neural network with 51 layers $\times$ 9 neurons, whose perturbation is along a given foliation; we show that adding foliated noise incurs a smaller error than noise in all directions. We give a rough error analysis of the method and show that it can be expensive for small-noise. We propose a potential future program unifying the three popular linear response methods.

We revisit the spectrum of linear axisymmetric gravitational perturbations of the (near-)extreme Kerr black hole. Our aim is to characterise those perturbations that are responsible for the deviations away from extremality, and to contrast them with the linearized perturbations treated in the Newman-Penrose formalism. For the near horizon region of the (near-)extreme Kerr solution, i.e. the (near-)NHEK background, we provide a complete characterisation of axisymmetric modes. This involves an infinite tower of propagating modes together with the much subtler low-lying mode sectors that contain the deformations driving the black hole away from extremality. Our analysis includes their effects on the line element, their contributions to Iyer-Wald charges around the NHEK geometry, and how to reconstitute them as gravitational perturbations on Kerr. We present in detail how regularity conditions along the angular variables modify the dynamical properties of the low-lying sector, and in particular their role in the new developments of nearly-AdS$_2$ holography.

In this paper, we present a nonlocal model for Poisson equation and corresponding eigenproblem with Dirichlet boundary condition. In the direct derivation of the nonlocal model, normal derivative is required which is not known for Dirichlet boundary. To overcome this difficulty, we treat the normal derivative as an auxiliary variable and derive corresponding nonlocal approximation of the boundary condition. For this specifically designed nonlocal mode, we can prove its well-posedness and convergence to the counterpart continuous model. The nonlocal model is carefully designed such that coercivity and symmetry are preserved. Based on these good properties, we can prove the nonlocal model converges with first order rate in $H^1$ norm. Our model can be naturally extended to Poisson problems with Robin boundary and corresponding eigenvalue problem.

We investigate the existence of $\frac18$-BPS black hole microstates in the $\mathfrak{su}(1,1|2)$ sector of Type IIB string theory on $\mathrm{AdS}_5 \times \mathrm{S}^5$. As will be explained, these states are in one-to-one correspondence with the Schur operators comprising the chiral algebra of $\mathcal{N}=4$ super-Yang-Mills, and a conjecture of Beem et al. implies that the Schur sector only contains graviton operators and hence $\frac18$-BPS black holes do not exist. We scrutinize this conjecture from multiple angles. Concerning the macroscopic counting, we rigorously prove that the flavored Schur index cannot exhibit black hole entropy growth, and provide numerical evidence that the flavored MacDonald index also does not exhibit such growth. Next, we go beyond counting to examine the algebraic structure, beginning by presenting evidence for the well-definedness of the super-$\mathcal{W}$ algebra of Beem et al., then using modular differential equations to argue for an upper bound on the lightest non-graviton operator if existent, and finally performing a systematic construction of cohomologies to recover only gravitons. Along the way, we clarify key aspects of the 4d/2d correspondence using the formalism of the holomorphic topological twist.

Previous work has given proof and evidence that BPS states in local Calabi-Yau 3-folds can be described and counted by exponential networks on the punctured plane, with the help of a suitable non-abelianization map to the mirror curve. This provides an appealing elementary depiction of moduli of special Lagrangian submanifolds, but so far only a handful of examples have been successfully worked out in detail. In this note, we exhibit an explicit correspondence between torus fixed points of the Hilbert scheme of points on $\mathbb C^2\subset\mathbb C^3$ and anomaly free exponential networks attached to the quadratically framed pair of pants. This description realizes an interesting, and seemingly novel, "age decomposition" of linear partitions. We also provide further details about the networks' perspective on the full D-brane moduli space.

Inspired by the relation between deep neural network (DNN) and partial differential equations (PDEs), we study the general form of the PDE models of deep neural networks. To achieve this goal, we formulate DNN as an evolution operator from a simple base model. Based on several reasonable assumptions, we prove that the evolution operator is actually determined by convection-diffusion equation. This convection-diffusion equation model gives mathematical explanation for several effective networks. Moreover, we show that the convection-diffusion model improves the robustness and reduces the Rademacher complexity. Based on the convection-diffusion equation, we design a new training method for ResNets. Experiments validate the performance of the proposed method.