Papers

Barcode scans, clear phone calls, reliable data storage, satellite communication, and large-scale quantum computation are all made possible by error correction. We present a handbook version of The Error Correction Zoo, a curated reference of methods for protecting classical or quantum information from errors during storage and transmission. The handbook includes descriptions of these error-correcting codes and a classification according to the symbols they use. It also catalogues relations among codes and related objects such as sphere packings, lattices, designs, groups, and classical and quantum phases of matter. The collection is intended both as a rigorous reference and as a practical aid for tracing the web of code relationships and uncovering new connections.

Efficiently decomposing finite element stiffness matrices into the Pauli basis is challenging due to the exponential growth of Pauli strings with problem size. A naive Pauli expansion requires $\Theta(8^{\lceil \log_2 N \rceil})$ operations, where $N$ denotes the number of degrees of freedom, rendering direct decomposition infeasible for large systems. Existing approaches exploit algebraic sparsity or operator structure but do not incorporate the geometric organization intrinsic to finite element discretizations, and consequently exhibit poor scaling for stiffness matrices. To address this problem, we introduce PHASE, a hierarchical, geometry-aware Pauli decomposition algorithm that leverages recursive mesh partitioning to organize element contributions across multiple spatial scales. PHASE employs a hybrid strategy that combines full- and reduced-space Tensorized Pauli Decomposition with Fast Walsh-Hadamard Transform-based aggregation to assemble global Pauli coefficients efficiently. We show that this approach yields a dimension-dependent reduction in the exponential scaling exponent of Pauli assembly asymptotic complexity relative to existing methods, reducing the cost from $2^{2{\lceil \log_2 N \rceil}}$ to $2^{\gamma_d{\lceil \log_2 N \rceil}}$ with $\gamma_d < 2$ under standard mesh regularity and balanced partition assumptions. These results substantially improve the feasibility of quantum-compatible operator synthesis for large-scale finite element models.

The Linear Combination of Hamiltonian Simulation (LCHS) framework simulates dissipative linear dynamics by representing time evolution as an integral over unitary operators, which is discretized by quadrature and implemented via Hamiltonian simulation. While existing analyses achieve near-optimal scaling in time and precision using norm-based quantities of the dissipative generator, we show that implementing the Hamiltonian simulation steps with Multi-Product Formulas (MPFs) yields commutator-sensitive error and complexity bounds. We demonstrate that the quadrature rule affects not only discretization error but also commutator structure and query complexity. This dependence is quantified through post-quadrature analysis for abstract MPF error profiles and for general time-independent and local Hamiltonians using known commutator-sensitive MPF error estimates. We compare uniform trapezoidal and free-scale sinh–sinh quadrature, showing improved quadrature-cardinality scaling for the latter, and illustrate the framework with applications to fractional diffusion, advection–diffusion, and open quantum systems.

We study the task of density estimation, where we hope to accurately estimate a probability density from $n$ samples. A textbook method for density estimation in total variation distance is the minimum-distance estimator approach, where we conclude both the algorithm and the analysis merely from bounding the VC dimension of a particular concept class (the so-called Yatracos class). While this technique has originally yielded sharp guarantees primarily for total variation distance, in this work we extend the minimum-distance estimator approach for learning within Hellinger distance. Our main observation is that we may produce an analogous recipe for Hellinger (where we only require bounding the VC dimension of a related concept class) by drawing connections to recent results yielding reverse data processing inequalities. This recipe is flexible enough to accommodate fast algorithms originally designed for total variation distance; by modifying the approach of Acharya et al. (2017) we conclude the first near-linear time algorithm for learning classes including univariate mixtures of log-concave densities and mixtures of Gaussians (with arbitrary variances), with near-optimal sample complexity.

In encoder circuits built on the stabilizer formalism, the dominant contribution to circuit complexity comes from the use of controlled (CNOT) gates, making CNOT-count reduction a central circuit-design objective. Entanglement-assisted (EA) quantum QC-LDPC codes offer strong error-correction capabilities with structured parity-check matrices, but their practical use depends on efficient encoder circuits and the availability of pre-shared Bell pairs (ebits). In this paper, we adopt a prior entanglement-assisted QC-LDPC (EAQC) encoder construction. We formulate the encoder optimization as a search over GF(2) row operations that decompose the binary matrix derived from its CNOT sub-sequence. We solve this problem using a beam search algorithm guided by a Hamming-distance heuristic. For the tested EA quantum QC-LDPC code families, the proposed method achieves CNOT-count reductions of 7.3-34.0% relative to the baseline EAQC encoder. The optimized circuits also outperform the Patel-Markov-Hayes and greedy cost-minimization baselines, and are verified by stabilizer-tableau simulation. These results show that substantial encoder simplification is possible for structured EA QC-LDPC codes.

We prove a conjecture on the minimal number of generators of licci zero-dimensional ideals that are either monomial or have small Loewy colength.

Let $G$ be the group of power series $x+a_2x^2+a_3x^3+\cdots\in R[[x]]$ under substitution, where $R$ is a commutative ring with $1\neq 0$ of prime characteristic $p$. Given any $n\geq 1$, the subgroup $K_n=\{x+a_{n+1}x^{n+1}+a_{n+2}x^{n+2}+\cdots\,|\, a_i\in R\}$ is normal in $G$, and the quotient $G_n=G/K_n$ is the group of truncated polynomials over $R$ of degree $\leq n$ under substitution. In this paper, we compute the exponent of the image of $K_r$ in $G_n$, for all $r,n\geq 1$, indicating in every case a family of elements realizing this exponent.

The aim of this note is to present a simple strategy to answer negatively a decomposition question on tori posed by Colliot-Thélène and Sansuc in the article Principal Homogeneous Spaces under Flasque Tori: Applications of 1987. We then deduce a torus $T$ over $\mathbb{Q}$ and a prime number $p$ such that $T(\mathbb{Z}_p)\,T(\mathbb{Q})\not=T(\mathbb{Q}_p)$, where $T(\mathbb{Z}_p)$ denotes the maximal compact subgroup of $T(\mathbb{Q}_p)$.

Finite geometry (FG) codes combine the algebraic properties of classical block codes with the iterative belief propagation (BP) decoding ability of low-density parity-check~(LDPC) codes. However, exploiting both advantages in practice is hindered by the fact that the standard incidence matrix between $(\mu+1)$-flats and points is dense and contains many short cycles for any flat dimension $\mu\geq 1$. In this work, we propose to sparsify the decoding matrix based on pencil selection, formulated as a constant-dimension subspace packing problem and solved explicitly using lifted Gabidulin codes. For both affine and projective geometries, sparse parity-check matrices are constructed and verified for FG codes of lengths up to $1024$. Simulations on four FG codes show no visible error floor and around $0.5$~dB gain over corresponding 5G LDPC codes at a block error rate of $10^{-7}$.

We establish explicit subconvex bounds for central values of Rankin–Selberg $L$-functions $L(1/2,\pi\times\pi')$ associated with pairs of unitary cuspidal automorphic representations of $\mathrm{GL}_2$ over a number field. Building on the spectral reciprocity framework of Michel and Venkatesh, we develop a refined, fully explicit form of spectral reciprocity that allows for precise control of conductors and local test vectors. As a consequence, we obtain an explicit subconvex bound, which, even over $F=\mathbb{Q}$, improves all previously known results. We further apply these bounds to several arithmetic problems. These include effective equidistribution of CM suborbits on quaternionic Shimura varieties, quantitative equidistribution of totally geodesic submanifolds, a uniform quantitative form of dihedral quantum unique ergodicity over number fields, and an application to distinguishing cuspidal automorphic representations.

We study the query complexity of Boolean functions $f: \{0, 1\}^n \rightarrow \{0, 1\}$ in the noisy query model introduced by Feige, Raghavan, Peleg and Upfal [SICOMP 1994]. In this model, an algorithm can adaptively query the bits of an input vector, but each query result is independently flipped with constant probability $p \in (0, 1/2)$; repeated queries are allowed. The noisy query complexity $\mathsf{N}_p(f)$ of a function $f$ is defined as the minimum expected number of queries needed to compute $f(x)$ with error probability at most $1/3$, for the worst case input $x$. We prove a general lower bound on $\mathsf{N}_p(f)$ based on degree statistics of certain subgraphs of the Boolean hypercube. This is the first general lower bound beyond those implied by the simple observation that $\mathsf{N}_p(f)$ is lower bounded by the randomized query complexity. We show that this recovers (up to a constant factor) most previously known lower bounds on the noisy query complexity of Boolean functions, providing a unified framework for understanding these results and simplifying the proofs in several cases. Furthermore, this resolves in the affirmative an open problem of Gu, Li and Xu [COLT 2025] that $\mathsf{N}_p(f) = \Omega(\mathsf{I}(f) \log \mathsf{I}(f))$, where $\mathsf{I}(f)$ denotes the total influence of $f$. We also apply our general lower bound to obtain tight bounds on the noisy query complexity for several new functions.

A point of a scheme has an associated tangent cone, the spectrum of a standard graded algebra encoding the local singularity. Its homological complexity can be measured by its graded Betti table: a matrix that records a part of the structure of its graded, minimal free resolution over a polynomial ring. A natural question is whether the homological complexity of the tangent cones varies arbitrarily across a scheme. In this paper, we show that this is not the case for a quasi-excellent Noetherian scheme; over such schemes, only finitely many graded Betti tables can occur. More generally, we show that a coherent sheaf over a quasi-excellent Noetherian scheme admits finitely many graded Betti tables, and that the constancy loci for the graded Betti table are constructible. As an immediate consequence, regularity is bounded on a quasi-excellent Noetherian scheme.

We study isotropic random walks on complex projective space in the context of stochastic state-space exploration in open quantum systems. Using results from harmonic analysis on compact rank-one symmetric spaces, first we obtain explicit transition kernels and characterise the evolution of fidelity and geometric observables. Then, we establish the correspondence between the small-step limit and Fubini-Study Brownian motion, and also show that the long-time limit reproduces Haar-random state statistics. At the level of the density matrix, the ensemble evolution formally coincides with an isotropic depolarizing channel at all times. Our main results concern first-passage statistics to high-fidelity target states. In the Brownian limit, we derive a closed-form expression for the mean first-passage time and an explicit spectral representation of the first-passage-time distribution, with decay rates determined by the eigenvalues of the killed Fubini-Study Laplace-Beltrami operator. The resulting asymptotics reveal a strong dependence on target fidelity and Hilbert-space dimension, demonstrating the rapidly increasing difficulty of reaching high-fidelity targets through unbiased isotropic exploration. The model is purely geometric and does not incorporate microscopic Hamiltonians or Lindblad dynamics. Nevertheless, the results suggest that it offers a natural starting point for stochastic models of open quantum systems that incorporate Hamiltonian and dissipative effects, with potential applications to stochastic quantum state preparation and control.

In this paper, we obtain order estimates for the Kolmogorov widths of an intersection of a finite family of Besov classes $SB_{p_j,\theta_j}^{\overline{r}_j}(\mathbb{T}^d)$ with dominating mixed smoothness in a Besov space $B_{q,\sigma}^{\overline{l}}(\mathbb{T}^d)$ in the case $2<q, \, \sigma <\infty$ when the parameters satisfy certain conditions of general position.

Let $\Delta$ be the Neumann Laplacian on the interval $(0,\pi)$, and let $T>0$. An integration by parts formula is proven for the spectral fractional Laplacian, $(-\Delta)^\alpha$, for $\alpha \in (0,1)$. As an application, we prove well-posedness results for the structurally damped beam equation $$u_{tt}+\Delta^2 u+\rho (-\Delta)^\alpha u_t=0, x\in (0,\pi),t>0$$ with various boundary conditions including $$ u_x(0,t)=u_{xxx}(0,t)=0;\ u_x(\pi,t)=f(t),\ u_{xxx}(\pi,t)=0, $$ and $f\in L^2(0,T)$ and appropriate initial conditions. Viewing $f$ as a control, we prove null-controllability. Analagous results are proven for higher order controls, and for the Dirichlet Laplacian.

The cumulative distribution transform (CDT) is a quantile-based transport representation that exactly linearizes one-dimensional translations of positive densities. We study how this structure behaves under additive perturbations and how it can be exploited for shift recovery. Under a local nondegeneracy condition, we derive a first-order expansion showing that additive noise in physical space induces a nonlocal perturbation in CDT space through the primitive of the noise, weighted by the reciprocal density. This yields an explicit description of transform-domain sensitivity and shows, in particular, that perturbations are amplified in low-density regions. When the physical-space perturbation is modeled as a centered Gaussian random field, the induced first-order CDT perturbation is again Gaussian, with an explicit covariance kernel. We then use this structure to study recovery in CDT coordinates. In the known-template setting, the transport shift is obtained by projection onto the constant mode, giving an explicit estimator together with exactness in the noiseless case and a stability bound under perturbations. In the unknown-template setting, multiple observations permit joint recovery of the shifts and a common template up to the natural constant-mode gauge, leading to a simple de-shift–and–average procedure. We also consider a signed-signal analogue based on the signed cumulative distribution transform (SCDT), where shifts are estimated numerically by feature matching and unknown templates are recovered by alternating alignment and averaging. Numerical experiments validate the perturbation analysis and illustrate effective recovery for both density-valued and signed signals.

While long treated as empirical fits, Hill functions have been postulated to be the universal Hopfield barrier for sharpness of input-output responses by Martinez-Corral, Nam, DePace, and Gunawardena. A Hopfield barrier is a fundamental limit on how well biological systems can process information without expending energy. Their case rested on numerical findings for Hill coefficients $4$ and $6$. We give a precise formulation and proof of this: measuring sharpness by the supremum of the derivative in semi-log scale, any rational function $r(x)=(\alpha_0+\alpha_1 x+ \cdots +\alpha_n x^n)/(\beta_0 + \beta_1 x+ \cdots + \beta_n x^n)$ with real coefficients $0\leq \alpha_i\leq \beta_i$ has sharpness at most $n/4$, with equality if and only if $r$ is a Hill function with Hill coefficient $n$.

For primes $p\geq 5$, we determine those $q$ for which the projective special linear group $\psl{2}{q}$ over the finite field of order $q$ is $(2,p)$-generated – that is, there exist two elements of $\psl{2}{q}$ of orders $2$ and $p$, respectively, that generate $\psl{2}{q}$.

For linear regression with i.i.d. homoskedastic errors, optimally weighted second-order least squares (SLS) is the degree-two polynomial maximization method (PMM) estimating equation: both select the same optimal combination of $e$ and $e^2-\sigma^2$, share one influence function, and attain the slope variance $c_2g_2/N$. The identity is one of estimating equations and stops where the optimal combination depends on the design. Within the polynomial moment class, degree three holds an efficiency reserve: on four right-skewed laws the cubic direction adds between $30\%$ and $50\%$ asymptotic efficiency, and under symmetric platykurtic errors it is the only informative direction. The reserve belongs to the degree-three span, which an efficient generalized method of moments estimator also attains. We give the asymptotic law of the feasible plug-in estimator, a rule for choosing the degree, Lean 4 checks of the algebra, and simulation evidence.

For a commutative ring $R$ with unity, its algebraic $K$-theory space $K(R)$ may be obtained by group-completing the symmetric monoidal category of finitely generated free $R$-modules under direct sum. A natural question is what happens when one group-completes with respect to the tensor product structure instead. In this note, we give a direct proof of the folklore theorem that the resulting group-completion is the rationalization of $K(R)$, up to $\pi_0$. We also discuss how a similar group-completion would give the $p$-perfection and, more generally, the localization of $K(R)$ at any non-trivial multiplicatively closed subset $S \subseteq \mathbb{Z}_{> 0}$. The localization statement can be recovered from a localization theorem of May. We give a plus-construction proof without using the full machinery of multiplicative infinite loop space theory.

For many trees $T$, the Ramsey number of $T$, denoted by ${\mathcal R}(T)$, is determined by the sizes of the partition classes in its unique bipartition. In 1976, Burr proved that when $T$ has partition classes of size $t_1$ and $t_2$ with $t_1 \le t_2$, the Ramsey number is at least $\max(2t_2-1,2t_1+t_2-1)$, and conjectured that this is tight. While counterexamples have been found for some pairs $(t_1, t_2)$, a main focus of research on this problem has been determining ratios $t_2/t_1$ or bounds on the maximum degree of $T$ for which Burr's bound is either exactly or asymptotically tight. We essentially resolve these questions for lopsided trees. Specifically, we show that (a) there are counterexamples whenever $t_2 \ge 2t_1$, with the order of magnitude of the difference between the largest Ramsey numbers and Burr's bound being $\max \left( t_1^2/t_2, \sqrt{t_1} \right)$, and (b) for $t_2 \ge 500 t_1$, Burr's bound is tight when $\Delta(T) \le t_2 - t_1$, but is off by at least $C \log t_2$ (even when $t_2 \ge 2 t_1$) when $\Delta(T) \gtrsim t_2 - t_1$. In particular, this shows that Burr's bound need not hold for $t$-vertex trees $T$ with $\Delta(T) \approx t/3$.

We prove the Harish-Chandra isomorphism theorem for supersymmetric spaces, describing the polynomial algebra of eigenvalues of invariant differential operators. The polynomials obtained satisfy novel invariance conditions, which remain somewhat mysterious. We also prove the Harish-Chandra isomorphism for ghost distributions, which satisfy a `square root' of the invariance conditions coming from invariant differential operators. All proofs are algebraic, and rely on a rank-one reduction argument and the Chevalley restriction theorem.

Several recent works have addressed the problem of deconvolution under a linear model, where the goal is to estimate a completely unknown $G_0$ from a vector of noisy observations $\boldsymbol{Y} = X\boldsymbol{\beta} + \boldsymbol{\epsilon}$, assuming the coefficients $\beta_j$ are i.i.d. unobserved realizations from $G_0$. Assuming $G_0$ has a density $g_0$, we study theoretically a Bayesian nonparametric method proposed in Weinstein et al. (2025) that postulates a Pólya tree prior $\Pi$ on $g_0$ and bases a deconvolution estimate on the posterior distribution $\Pi(\cdot|\boldsymbol{Y})$. Our main result asserts that under the true model (fixed and unknown $g_0$), and under a suitable condition on the minimum eigenvalue of $X^\top X$, the posterior $\Pi(\cdot|\boldsymbol{Y})$ concentrates around $g_0$ in sup-norm. The analysis presented builds on and extends results from Castillo (2017), where posterior consistency of Pólya trees was proved for density estimation, the simpler problem of estimating $g_0$ when observing the coefficients $\beta_j$ directly.

We propose a low-complexity Chase decoder for optical interconnects that formulates test pattern selection as a generalized maximum coverage problem. For concatenated RS-BCH and oFEC codes, our decoder achieves the standard Chase decoding performance with 25% and 61.3% fewer test patterns, respectively.

In this note, we derive a large-deviation-type scaling limit for a discretization of the Martingale Representation Theorem. Somewhat surprisingly, and to the best of our knowledge, this result has not been previously obtained in the literature.

Let $\lambda = \lambda(\beta,\sigma,a,b)$ denote the top Lyapunov exponent for the linearization along trajectories of the noisy damped non-linear oscillator $\ddot{x}+\beta \dot{x} + ax+bx^3 = \sigma \dot{W}_t$, where $a$, $b$ and $\beta$ are all positive and $\sigma \neq 0$. In 2004 Arnold, Imkeller and Sri Namachchivaya stated without proof that $\lambda(\varepsilon^2 \beta,\varepsilon \sigma,a,b) \sim \overline{\lambda} \varepsilon^{2/3}$ as $\varepsilon \to 0$ with $\overline{\lambda} > 0$. This paper contains a proof of this assertion.

In this paper, we apply the I-method to establish global well-posedness for the fractional nonlinear Schrödinger equation with initial data u_0 \in H^s(R^d) for s <\alpha/2, i.e., below the energy threshold. Moreover, for radial initial data, we combine a modiffed Morawetz estimate-recovered via Balakrishnan's formula-with the I-method to obtain improved results. In the same spirit, we employ the "upside-down" I-method to derive polynomial-in-time growth bounds for the higher-order Sobolev norm. The main difffculty stems from the fact that Strichartz estimates for the fractional Schrödinger equation has a loss of derivatives, and the problem is always L^2-supercritical, thereby requiring more delicate analysis.

We study stochastic interacting particle systems whose interaction structure is described by dense weighted directed graphs converging to a graphon. In the thermodynamic limit, we prove a law of large numbers for the empirical measure process and derive a deterministic nonlinear master equation describing the macroscopic evolution. The limiting equation retains the heterogeneous interaction structure of the microscopic system through the limiting graphon, allowing for spatially non-homogeneous behaviors such as localized or community-type interactions.

Combinatorial memory is a class of memory in which information is encoded in the set of paths through a structured mesh. In this work, we introduce a systematic encoding framework, referred to as the Color-Rule-Function (CRF) approach, for representing information in combinatorial memory. The method consists of four key steps: selecting a sequence of paths in the mesh, assigning values (e.g., colors) to each cell, defining a set of rules based on the values encountered along each path, and constructing a Boolean function that determines the state of each path. . The coding procedure is illustrated by several examples. The design space scales of the CRF scale fundamentally faster compared to conventional memory. This apparent advantage arises from the use of rule-based and functional representations but is accompanied by increased hardware complexity. A possible hardware realization of the CRF framework is discussed. Importantly, the hardware overhead can be substantially reduced through the use of customized modules. The examples of the customized design are described in the text. The combination of CRF coding with customized module design may lead to a practical advantage in data storage density. According to the estimates, the data storage density may exceed Exabit per centimeter squared. A key problem that requires further investigation is related to the minimum Hamming distance between an arbitrary target bit sequence and the closest sequence realizable within the CRF framework under fixed hardware constraints.

The current study presents a three-dimensional linear stability analysis of particle-laden Couette-Poiseuille flow suspended in a Newtonian fluid between two parallel plates, with the lower plate coated by a porous medium. The influence of suspended particles is examined using a two-domain formulation in which particles are confined to the fluid layer and do not penetrate the porous substrate. The particle-laden suspension is modeled using the dusty-gas framework, while the flow within the porous layer is described by the volume-averaged Navier-Stokes (VANS) equations. In particle-laden flows over impermeable walls, particle inertia may either stabilize or destabilize the flow depending on the governing parameters. In contrast, the presence of a porous layer introduces an additional permeability-dependent destabilizing mechanism that fundamentally modifies these classical trends. Consequently, particle loading can reduce the critical Reynolds number at sufficiently high permeability, even in parameter regimes where particles stabilize the corresponding rigid-wall flow. The coupled formulation also introduces additional disturbance branches associated with fluid-particle coupling near the permeable interface. Although these modes remain stable across the parameter space investigated, they modify the eigenspectrum and influence the dominant instability by altering coupling pathways. Furthermore, unlike impermeable-wall Couette-Poiseuille flow, where increasing the Couette component generally stabilizes the flow, the porous-wall configuration exhibits a monotonic decrease in the critical Reynolds number over the range examined. These results demonstrate that porous boundaries can fundamentally alter established stability behavior in particle-laden shear flows through permeability-dependent coupling between the suspension and the porous substrate.