Papers

We consider a class of redudancy-free Hamiltonians, which are those where the trace vanishes for any product of Hamiltonian terms in which at least one term appears an odd number of times. This includes Hamiltonians which are sums of products of Paulis with no relations between them. These Hamiltonians arise naturally in Hamiltonian Decoded Quantum Interferometry, which separates the hardness of preparing quantum states into two steps: a classical decoding step, and a step preparing the thermofield double state of such a redundancy-free Hamiltonian. We focus on the hardness of these Hamiltonians, both for intrinsic interest and to understand the complexity of the state preparation step in Hamiltonian DQI. We present several results showing that the problem becomes easy at quadratically lower temperature than for a general Hamiltonian, with the complexity at given inverse temperature $\beta$ depending on $\beta^2 d$, where $d$ is the degree of the anticommutation graph. We give an efficient classical algorithm for small $\beta^2 d$ to approximate the partition function, while for large $\beta^2 d$ we show that approximating the partition function is NP-hard. To this end, we introduce what we call an anticommutation glass, where frustration arises purely from anticommutation relations. We give a subexponential time quantum algorithm to prepare the thermofield state for small $\beta^2 d$, and present some results toward a polynomial time algorithm based on the Feiguin-Klich Hamiltonian. As a result of possible independent interest, we show that this Hamiltonian leads to a polynomial time algorithm for preparing thermofield double states of general Hamiltonians on an interaction graph of degree $d$ for $\beta\lesssim 1/d$. Finally, we show that estimating the ground state energy of redundancy-free Hamiltonians is QMA-complete.

Correlation functions are central to understanding the behavior of quantum systems, with their spectra giving direct access to quasiparticle and collective excitations. However, the long-time evolution required for sufficient spectral resolution is challenging to reach for both classical and quantum simulations. In this work, we introduce a principled approach for extending time series of quantum correlation functions and reconstructing their spectral densities, based on their connection with covariance sequences of stationary stochastic processes. This connection enables us to apply Burg's maximum-entropy principle to find the least-committal extension compatible with the measured values of the correlation function, which corresponds to an all-pole autoregressive model whose coefficients solve a set of Yule–Walker equations. The resulting extension is positive semi-definite by construction, guaranteeing the positivity of the spectrum and the automatic satisfaction of its total spectral weight sum rule. The approach applies to both scalar and matrix-valued correlation functions, including those with highly structured or continuous spectra. The generalization to the matrix case makes it possible to trade simulation time for additional measured correlation functions, enabling accurate time-series extension and spectral reconstruction from short-time data, even in the presence of noise.

Modern chess engines are silent experts: they play at a superhuman level, but do not offer explanations for their play. On the other hand, language models (LMs) can generate plausible-sounding explanations, but their weak playing strength limits the utility of their explanations. We introduce Queen, a 4B-parameter chess-language model that can explain its moves and plans while playing at the level of a typical Grandmaster. Our novel framework enables domain-specific reasoning through complementary components: an encoder-decoder architecture and an iterative distillation algorithm. This architecture integrates a silent expert chess encoder with an instruction-tuned LM through cross-attention, which we train via a question-answering curriculum to extract chess concepts from the encoder's representations. Building on this domain-adapted model, we iteratively improve its explanations with a natural-language analog of the Bellman update: the model analyzes the positions after its top candidate moves and consolidates them into an explanation of the current position, which is then distilled back into the model. Over seven iterations, our model gains over 900 Elo points (1782 to 2697), substantially surpassing all frontier models on both playing strength and puzzle accuracy, despite containing three orders of magnitude fewer parameters. Furthermore, LM-based evaluations show that our explanations are fluent and approach GPT-5.6-Sol (high) in coherence. The generality of our architecture and training procedure suggests a recipe for applying language models to domains where silent expert encoders are available, like games, robotics, and computer use.

A key goal in the foundations of quantum mechanics is to identify operational constraints characterizing physical theories. Bell inequalities do so for classical probability theory, while Tsirelson's bound provides a first step for quantum mechanics. Here, we address the dual question: Can one certify that all correlations predicted by quantum theory are actually realizable? In this work, we construct a finite number of two-body correlations such that the only probabilistic theory that (1) realizes them, and (2) does so in a way that is stable under iterated teleportation, is quantum theory. For $n$ quantum systems of dimension $d$ each, Condition (1) can be verified using $\operatorname{poly}(d,n)$ measurement settings. Condition (2) may be understood as a hierarchy of tests, one for each number $N$ of teleportation steps. There is thus a sense in which finite-dimensional quantum theory can be self-tested. In particular, one can certify the existence of Bell inequality violations larger than any that have been directly observed.

Scarcity of labeled data limits development of deep learning biomarkers in oncology. We develop a two-stage AI model predicting pathological complete response (pCR) to neoadjuvant therapy in breast cancer. The first stage learns the transcriptome from histopathology using 8,742 patients across 32 cancer types, corroborated by pathologist review and spatial agreement with measured expression. This simplifies the second stage to predicting pCR from inferred expression and clinical variables. Developed using 1,080 patients (five cohorts) and evaluated in 1,412 patients (nine cohorts), the model achieves a pooled AUROC of 0.79 (95% CI, 0.73-0.85), discriminating responders within molecular subtypes. It outperforms histopathological biomarkers, remaining stable across intratumoral sampling and with minimal biopsy tissue. Ablations show transcriptome-wide inference improves discrimination over clinical variables alone or one-stage pathology models, and robustness by avoiding genomic assays' gene selection constraints. These results indicate that biologically informed compression may generalize to data-sparse applications in precision oncology.

We examine Higgsino dark matter (DM) as a potential explanation of the $248~\mathrm{keV}$ nuclear-recoil event of interest reported by the LUX-ZEPLIN (LZ) experiment, taking into account IceCube constraints on DM annihilation in the Sun. In the nearly pure Higgsino limit, two neutral Majorana states separated by a small mass splitting $\delta$ are connected by an off-diagonal coupling to the $Z$ boson, providing a natural realization of inelastic DM. Using a likelihood analysis, we identify 90% confidence regions compatible with the LZ event and consistent with the null results of IceCube searches, both with and without including a constraint from the absence of events in the LZ high-energy sideband. In our LZ analysis, we consider the DM distribution in the Standard Halo Model alone as well as possible enhancements of the high-speed tail induced by the Large Magellanic Cloud (LMC). We present four variants of bounds from IceCube on Higgsino dark matter, depending on various assumptions on cooling channels in the Sun, motivated by severe theoretical uncertainties about Higgsino-nucleon elastic scattering (unimportant for direct detection but crucial for IceCube bounds). Whereas the thermal Higgsino with a mass of $1.1~\mathrm{TeV}$ remains only marginally compatible with both the LZ event and the IceCube limits when the LMC-induced high-speed tail is included and both elastic cooling in the Sun and the LZ high-energy sideband constraint are neglected, higher mass Higgsinos are excellent candidates for the LZ event in light of IceCube constraints. Depending on the halo distribution, cooling interactions, and whether the LZ high-energy sideband is included, the compatible regions in the Higgsino parameter space $(m_\chi,\delta)$ extend from masses of $\sim2~\mathrm{TeV}$ to hundreds of TeV or higher, with mass splittings ranging approximately from $350$ to $525~\mathrm{keV}$.

We present FlowHMR, a framework for recovering physically plausible global 3D human motion from monocular video. Previous learning-based methods typically regress human motion directly from video and train the network with geometric supervision. However, recovering human motion from monocular video is inherently ambiguous in depth, and direct regression tends to collapse toward an averaged solution. Moreover, the recovered motions are not guaranteed to be physically plausible, so physics-based tracking of them often fails. To address these challenges, we formulate video motion capture as a video-conditioned motion generation problem and first pretrain a flow matching model for this task. Given an input video, the pretrained model generates diverse motion candidates, but not all of them are faithful to the video or physically trackable. We therefore post-train the model using Group Relative Policy Optimization (GRPO) with two rewards. A fidelity reward encourages consistency with the input video. A tracking reward favors motions that a physics-based controller can track successfully. Together, these rewards shift the model's output preference, so the post-trained model stays faithful to the input video while producing more physically plausible motion. We further introduce Wild-4K, a large and diverse dataset of about 4K internet videos, for evaluating human motion recovery in the wild. Qualitative and quantitative experiments on Wild-4K show that our method outperforms state-of-the-art methods in overall motion fidelity and achieves a physical tracking success rate of 82.47%, compared with 62.82% for the strongest baseline, GVHMR.

How much can the action of an unknown unitary dynamics on one family of states reveal about its action elsewhere? We develop a general framework for out-of-distribution quantum dynamics learning based on the first and second moments of the input distribution. These define a notion of the bias and expressivity of an ensemble, which together control generalization between arbitrary mixed-state training and testing distributions. Our framework shows that only studying the action of a unitary on random real states can learn its action on random complex states, that tensor products of unbiased pure-state ensembles preserve expressivity, and that single-qubit risks are independent of bias. It also recovers earlier locally scrambling results as loose special cases and yields tight product-to-Haar bounds. We conclude with a series of no-go results and limited generalization bounds for learning certain channels, which further serve to underscore the distinctive role of unitarity in enabling out-of-distribution generalization.

We examine the transfer of a pretrained SigLIP2 image encoder to seven class fire risk classification from aerial imagery. We introduce a reproducible partition of the public FireRisk training mirror and an implementation that records data provenance, preprocessing and model selection. Two initial runs compare a frozen encoder probe with full model adaptation. On the validation partition, full adaptation reaches 63.05% accuracy and 58.94% macro F1, compared with 55.95% and 50.19% for the probe. Both runs use one training seed and select their checkpoint on the same validation partition. These development results support further evaluation of SigLIP2 but do not establish performance on an independent test set or unseen regions. The accompanying code provides a common framework for repeated experiments and comparisons with additional visual encoders.

The configuration matrices of generalized complete intersection Calabi-Yau (gCICY) threefolds contain negative entries, so their defining line bundles are not nef and the Batyrev-Borisov construction does not apply directly. We study gCICY threefolds in $\mathbb{P}^1\times B$ with a $[3,-1]$ block, that is, cut out by sections of $O(3)\boxtimes L$ and $O(-1)\boxtimes(L+R)$, where $B$ is a smooth projective fourfold and $2L+R=-K_B$. Auxiliary variables turn the generalized section into a relative variation of GIT, whose opposite phase is the zero locus of a rank-3 bundle on $\mathbb{P}^2\times B$; for general data the two phases are isomorphic or related by $\int_B c_1(L)^4$ disjoint Atiyah flops. Over products of projective spaces there are exactly 18 Calabi-Yau threefold configurations with a $[3,-1]$ block. All of them except the non-Gorenstein case studied in the companion paper have Gorenstein toric nef phases. Thus, for every other configuration in this classification, mirror symmetry is reduced after variation of GIT to ordinary Batyrev-Borisov duality, which gives the mirror Hodge numbers of the generalized phase itself. We analyze four examples with a $[3,-1]$ block and one outside this class in detail. For each we construct an explicit mirror and compute its fundamental period and low-order Picard-Fuchs system; the indicial algebras at the maximally unipotent boundary points are isomorphic to the rational even cohomology rings of the A-model phases.

Jet vetoes are central to precision measurements of Higgs-boson properties at the Large Hadron Collider, where they are used to suppress backgrounds containing hard QCD radiation. We present the first prediction for the Higgs jet-vetoed cross section at the third logarithmic order in QCD, consistently combining the next-to-next-to-next-to-leading order calculation with state-of-the-art next-to-next-to-next-to-leading logarithmic resummation. At phenomenologically relevant veto scales ($p_T^{\rm v}\sim30$ GeV) and jet radii ($R=0.4$), the resummation increases the fixed-order prediction by about $3\%$, and their combination leads to a residual perturbative uncertainty of approximately $3\%$. Our results mark a milestone in the precise description of jet-vetoed Higgs-production at hadron colliders, reaching the level of precision relevant for the high-luminosity phase of the LHC.

In fault-tolerant block encodings of structured Hamiltonians, the support geometry can set how the non-Clifford cost of SELECT scales with the system size, however many operators act on each support. We construct permute-act-unpermute circuits: a CSWAP network fixed by this geometry brings a selected support to a fixed target register, a shared local circuit acts there, and the permutation is undone. The operators add only an $XZ$ mask on this register, at a cost independent of the system size, and one query can mix several support geometries. For two-site supports, we establish and attain the minimum CSWAP count within address-controlled networks of site transpositions. Compiled for Pauli terms, these circuits use $8\mathrm{S}N+o(N)$ $T$ gates with $O(\log N)$ permutation work qubits, where $N$ is the number of system qubits and $\mathrm{S}$ is determined by the support geometry, with $\mathrm{S}=1$ for nearest-neighbour interactions and $\mathrm{S}=3/2$ for all-to-all pairs; in both cases the correction is $O(\log N)$. We also introduce a bridged CSWAP, which remembers the logical value of its temporary AND across the target action and repairs it from the prepared labels, so that the inverse CSWAP undoes the exchange without recomputing the value. When the repair is Clifford, it halves the $T$ count of a matched forward-inverse CSWAP pair at the cost of one retained work qubit. For a Heisenberg ring, the complete block-encoding query reduces the $T$ count by a factor approaching $3$ as the system size grows, relative to flat and product-wise Pauli walks. For a five-orbital Anderson impurity model, which combines nearest-neighbour and all-to-all support geometries, the reduction is about $1.7$ at large bath sizes, relative to the cheapest of three comparators, each built from the methods of the references it follows.

Whether a mortality model needs a Makeham term, the constant representing background mortality, is commonly decided with a likelihood-ratio test. The constant cannot be negative, so testing its absence places it on a boundary, where the usual chi-squared calibration fails. Assuming independent Poisson death counts and standard regularity conditions, we show that when the constant is the only parameter on a boundary the statistic converges to an equal mixture of a point mass at zero and a chi-squared distribution with one degree of freedom: at the 5% level the critical value is 2.71, not 3.84. We state conditions under which the correction holds for Makeham models, verify them for Gompertz–Makeham and gamma-Gompertz–Makeham, and quantify the information about the constant, which fixes the local power, the smallest detectable term, and the loss of detectability as the age window narrows. When a gamma-frailty variance is estimated at a true value of zero, the limit is no longer a chi-squared mixture and the correction rejects too often. Monte Carlo experiments show the conventional cutoff rejecting at half the nominal level, a correctly calibrated test missing a term at its detection threshold in most samples, and excess rejections persisting as exposure grows.

Recent work established that, for independent erasures, storing $K$ logical qubits for $S$ time steps with error at most $\varepsilon$ requires a spacetime cost of $\Omega(S(K+\log(S/\varepsilon)))$. A matching construction was also given, but assumes ideal error correction. Here we show that the same scaling can be achieved with an explicit noisy error-correction circuit and efficient decoding, counting all state-preparation, gate, measurement, and wait locations, provided that the hardware supports long-range qubit connectivity and fast, reliable classical processing. Thus, reliability adds only a logarithmic overhead shared by all stored qubits. The construction works for sufficiently weak but otherwise general circuit noise, including faults during syndrome extraction and correlated faults, and for independent erasures it matches the best known lower bound up to constant factors. We attain this scaling using quantum Tanner codes, with each memory step consisting of one round of syndrome extraction followed by a fixed number of parallel classical decoder steps that reduce the residual error enough to keep later faults correctable, regardless of memory size, storage time, or target accuracy. Together, this gives exponentially small failure probability per step while keeping the cost of each memory step linear in the code size. The same analysis also extends to certain constant-depth logical Clifford operations.

Given a finite-range, non-commuting Hamiltonian on a 1D chain of finite spins, we prove a modified logarithmic Sobolev inequality with constant $\Omega(1/\log n)$ for the heat-bath dynamics, a quasi-local Gibbs sampler, and for the regularised heat-bath dynamics. This implies rapid mixing of both samplers towards the Gibbs state in relative entropy and trace norm, with mixing times of order $\mathcal{O}(\log n\cdot \log(n/\epsilon))$. The interactions may be spatially inhomogeneous. The proof combines weak quasi-factorisation of relative entropy, decay-of-correlation estimates, a uniform conditional local gap, and a buffered comparison between conditional entropy and quadratic energy. The resulting bounded entropy defect is removed using the global gap. To the best of our knowledge, this is the first proof of MLSI in the non-commuting regime.

Realizing practical quantum computers requires quantum error correction, but the most widely implemented approach, the planar surface code, demands a substantial physical qubit overhead. Here, we demonstrate that partitioning quantum processors into manageable, flat modules with non-local inter-module connections provides a natural solution. By wiring sparse, static boundary connections between Euclidean planar modules, we construct modular hyperbolic surface and color codes, based on new families of semi-hyperbolic codes. Using modular syndrome extraction circuits and efficient neural network and matching decoders, our circuit-level simulations show that this modular memory design can reduce the physical qubit overhead by tenfold or more compared to the surface code, even under elevated inter-module seam error rates. Projecting to larger system sizes, we find modular codes with encoding rate $k/n=1/16$ and distance $d\geq 22$, exceeding both the rate and distance of the $[[288,12,18]]$ two-gross bivariate bicycle code while using mostly nearest-neighbor gates within planar modules, and implying an over $30\times$ overhead reduction relative to surface codes. We develop fault-tolerant walking circuits for implementing logic via code automorphisms, which we co-design with low-weight, highly symmetrical logical bases and a modular extractor system that is $\approx 4.5\times$ smaller than the code itself. Combining these methods, our modular memory can be used as dense storage in a universal fault-tolerant architecture, unlocking the efficiency of high-rate codes without sacrificing the fabrication advantages of planar modules.

This paper develops a risk-aware guidance law for an attacker to intercept a stationary target in the presence of multiple moving defenders while respecting the control input limits. We represent defender threats via engagement zones (EZs) (regions of state-space from which attacker interception is feasible) and employ the dynamic maneuvering cue (DMC) (minimum heading correction to exit EZs) as a deterministic measure of geometric risk. Safety is enforced through a prescribed bound on the DMC to enable controlled EZ penetration subject to a specified risk tolerance. For multiple defenders, the corresponding safety constraints are combined through a smooth-minimum aggregation. The proposed guidance strategy drives the attacker toward the target when the engagement remains safe and transitions to an avoidance maneuver as the prescribed safety boundary is approached. The resulting online optimization-free guidance law directly bounds the applied lateral acceleration and initiates a bounded preemptive maneuver before corrective action is required. Closed-loop analysis establishes preservation of the prescribed risk constraints under pointwise bounded-input feasibility and finite-time target interception under a sustained target-accessibility condition. Numerical studies validate the proposed strategy and illustrate the tradeoff between allowable geometric risk and interception performance.

In this article, we present the results relating different rationality properties with $\mathbb{A}^1$-homotopy theoretic invariants.

The dynamics of cells, organisms, and fluids are often modeled as probability distributions evolving over time. Reconstructing and extrapolating this evolution from unpaired snapshots requires assumptions about the underlying process. Wasserstein gradient flows are a common choice, but they cannot describe conservative or periodic dynamics. Lagrangian mechanics in Wasserstein space covers both, but existing methods for learning it are simulation-based: they run a numerical solver at every training step, which makes training expensive. We propose Double-Stitch, a simulation-free method that learns these mechanics by penalizing the residual of the equation of motion along a learned population path. We derive this equation from a Clebsch variational principle that does not require gradient velocities, and show that the residual vanishes exactly when the equation holds. We test Double-Stitch on synthetic, single-cell and ocean vortex datasets and find that it matches or outperforms gradient-flow methods and simulation-based WLM on most tasks, while training $4$-$14$ times faster than WLM. We provide a JAX implementation of Double-Stitch at https://github.com/BasisResearch/stitching.

Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductivity, Cooper pairing at finite-momentum driven by spin imbalance, remains difficult to study numerically in two dimensions. The fermion sign problem in the spin-imbalanced attractive Hubbard model severely limits quantum Monte Carlo access to the low temperatures where pairing correlations are most pronounced. We show that the Holstein model, in which the effective attraction is retarded by phonon dynamics, strongly suppresses sign fluctuations under spin imbalance, enabling unbiased determinant quantum Monte Carlo simulations deep in this regime. With increasing magnetization, zero-momentum pairing evolves into finite-momentum FFLO pairing, evidenced by oscillating real-space pair correlations and a shift of the pairing susceptibility peak to finite $\vec{Q}$. The finite-momentum susceptibility substantially exceeds its zero-momentum value. At high density $n \sim 0.875$, near magnetization $m \approx 1-n$ where the majority-spin density approaches half filling, FFLO pairing coexists with checkerboard charge-density-wave order at short range, while staggered longitudinal spin correlations emerge as a magnetic halo at larger distances. These spatial textures provide experimentally accessible signatures of finite-momentum pairing and motivate cold-atom realizations of FFLO physics.

We study exact recovery for non-Abelian topological surface codes based on the quantum double $D(G)$ of any finite group $G$. We determine an orthogonal error basis, that is comprised of group multiplication and irreducible representation operators. Whenever the errors are supported on closed loops on the lattice or dual lattice, there is a redundancy in the basis, due to stabilizers. To remedy this, we introduce a gauge-fixing that results in a complete and orthogonal error basis on the code space. Assuming as input predetermined, neutral correctable error clusters, we construct charge and flux transfer circuits that move the errors onto ancillas that then get projected out. This is an exact and deterministic recovery protocol, applicable to surface codes for any finite, in particular non-Abelian, group $G$.

The Redfield equation with time-dependent coefficients (or TCL2 equation) gives one of the simplest perturbative descriptions of open quantum system dynamics derived from a microscopic theory. Its status as an approximation is nevertheless subtle: while perturbation theory fixes the reduced dynamics order-by-order in the system-environment coupling, solving the Redfield equation yields a particular resummation that need not preserve complete positivity. Here we compare Redfield evolution with an exponentiated (Magnus) resummation of the same second-order perturbative map, which is completely positive and trace preserving (CPTP) for a broad class of physical conditions. We also construct a simple CPTP resolvent resummation, illustrating the non-uniqueness of resummations of the same perturbative map. The Redfield and Magnus prescriptions agree to second order but differ through time-ordering corrections at higher orders. We exhibit a weak-coupling criterion which guarantees that Redfield evolution, even when non-Markovian, remains close to a CPTP map. We illustrate the criterion in two examples, focusing on finite-dimensional (qubit) systems. Our results clarify when, and why, Redfield evolution can remain reliable even when it is highly non-Lindbladian.

LLM-guided evolutionary methods, such as AlphaEvolve, have emerged as powerful approaches for challenging computational optimization problems, such as circle packing. However, prior work typically optimizes performance gain over a fixed number of iterations. We argue that practical optimization should maximize gain per unit cost. To this end, we propose FrugalEvo, a cost-aware evolutionary framework where a stronger, higher-cost LLM explores solution strategies, and a cheaper LLM implements them and iteratively refines the resulting code. We also design a cache-efficient evolution process, where our harness and prompts maximize the sharing of prefixes across different evolution steps, to improve cache reuse. To measure solution quality throughout a fixed cost budget, we introduce Budget-Aware Area Under the Curve (BA-AUC), defined as the area under the best-so-far evaluation score curve over cumulative LLM cost, up to the budget. Across 10 mathematical and systems optimization tasks, FrugalEvo matches or surpasses state-of-the-art baselines, including OpenEvolve, ShinkaEvolve, AdaEvolve, and EvoX, in final solution quality and achieves higher BA-AUC on 9 tasks. It also achieves higher average performance than these baselines on 10 algorithmic optimization tasks from ALE-Bench-Lite. Notably, on circle packing, FrugalEvo achieves new state-of-the-art performance with GPT-5.6 Terra and Luna for only 1.68 USD and with GLM-5.3 and its Flash variant for only 0.55 USD, matching or surpassing all baselines, including multi-agent methods such as CORAL and SwarmResearch, which cost approximately 50 USD on average.

While the ultimate limits on the rate of quantum communication over noisy channels are well-established, very few efficient coding schemes achieving these limits are known. This contrasts sharply with transmitting classical data over classical channels, where polar codes and low-density parity-check codes both efficiently achieve the classical capacity of any noisy channel. In this work we construct an efficient entanglement generation scheme which achieves the coherent information for a broad class of channels. For degradable channels in the class, which includes truncated versions of the pure loss Bosonic channel, the resulting scheme achieves the quantum capacity. The construction makes use of polar codes for two kinds of classical-input, quantum-output (CQ) channels: those with commuting outputs and those whose outputs are heralded mixtures of pure states. The former case is amenable to decoding by classical polar decoding algorithms, the latter by the belief propagation with quantum messages (BPQM) algorithm. To our knowledge, this construction is the first explicit and efficiently decodable protocol that achieves the energy-constrained quantum capacity of the pure-loss Bosonic channel.

We investigate algorithms for the quantum simulation of the Schrödinger equation on a Riemannian manifold, where the kinetic operator is defined by the Laplace–Beltrami operator corresponding to the metric. Our first algorithms are based on a global spectral method based on the identification of an efficient transform to the eigenbasis of the Laplace–Beltrami operator. We use this method to provide explicit, efficient, quantum simulation algorithms for the Riemannian Schrödinger equation on tori and spheres with their standard metrics, simplices with the Wright–Fisher metric, truncated positive orthants and their invertible affine images with the log-barrier Hessian metric, and $\ell_p$ balls with a metric induced by the Duffy map. Our second algorithm is based on a coherent simulation of local spectral methods on multiple charts, and is in principle applicable to any compact manifold. We first analyze this algorithm in the continuum and derive conditions under which a polynomial spectral cutoff suffices. We also provide a discretization analysis of a polynomial spectral cutoff for tensor-products of constant-dimensional manifolds. Finally, we consider applications of these methods to optimization and physical simulation. For optimization, we provide results including a generalization and convergence analysis of Quantum Hamiltonian Descent for geodesically convex functions that leads to explicit algorithms on the sphere and simplex, and a Riemannian generalization of the Real-Space Adiabatic Algorithm. For physical simulation, we show that our algorithms can simulate certain spatially discretized field theories, including a variant of the nonlinear sigma model.

Product formulas are a basic tool for simulating quantum many-body dynamics, but a fundamental degree of freedom in their application—the ordering of Hamiltonian terms—has remained largely unexplored. Here we systematically study such ordering effects, quantifying their impact on simulation accuracy and exploiting them to design more accurate product-formula simulations. We develop an efficiently computable, ordering-sensitive bound on the finite-step-size Trotter error for geometrically local Hamiltonians for arbitrarily high-order product formulas. For translation-invariant one-dimensional chains, our bounds are close to exact second-, fourth-, and sixth-order Trotter errors in small-system tests, while on large-scale systems they improve on previous higher-order analytical bounds by orders of magnitude. We further use the local structure of the error bounds and estimates to optimize ordering efficiently, reducing the long-chain problem to a minimum-mean-weight-cycle problem on a weighted de Bruijn graph. For random translation-invariant chains, sequential (staircase-like) orderings consistently outperform standard brickwall circuits; when circuit depth is restricted, the optimizer instead favors wave-like parallel-sequential orderings, which preserve much of the accuracy advantage over brickwall circuits. These results establish ordering as a useful degree of freedom for both understanding and improving product-formula simulation.

The collective excitation spectra of quantum magnets provide a direct test of quantum computers as tools for studying strongly correlated matter. Hardware noise limits the circuit depths available for these calculations, leaving the question of how much magnetic information can survive circuit compression. Using IonQ's 36 qubit Forte Enterprise processor, we simulate the magnetic excitation spectra and spin dynamics of a 36 site Heisenberg XXZ chain describing the quantum magnet Cs2CoCl4 in a transverse field using quantum circuits containing just 420 two qubit gates. The workflow combines an ancilla free local kick protocol with classical tensor network compression, encoding both ground state preparation and time evolution. The reconstructed dynamical structure factors capture the evolution from a broad two spinon continuum toward a dispersive branch with magnon character near the expected critical field. We observe an interesting field driven sign reversal of the nearest neighbor spin correlations which shows a complementary signature of this magnetic evolution. Our error mitigation methods bring the reconstructed total spectra into close agreement with density matrix renormalization group calculations, with Pearson correlation and structural similarity above 0.9 and normalized mean squared error below 0.15. We also perform comparisons with experimental inelastic neutron scattering measurements to connect the recovered excitations to the material's magnetic response. Our results demonstrate that circuit compression and error mitigation allow noisy quantum processors to recover both collective excitations and local magnetic correlations providing a quantitative benchmark for quantum spectroscopy and a basis for extending these methods to more complex magnetic systems.

Quantum Max-Cut is a well-studied local Hamiltonian problem for which the classical complexity of the bipartite case remains open. We give a polynomial-time classical randomized algorithm for the spin-$\tfrac12$ antiferromagnetic Heisenberg model, equivalently Quantum Max-Cut, on dense balanced bipartite expanders. This class includes $G(n,n,p)$ with high probability for every fixed $p>0$. The algorithm estimates the ground energy and any ground state edge correlation to additive error $\varepsilon$ in time polynomial in $n$ and $1/\varepsilon$. We construct a Markov chain on perfect matchings of the complete bipartite graph, each matching representing a product of singlets, that converges to the ground state, and use the expansion of the graph to show that polynomially many steps and samples of this chain suffice.

Linear combination of Hamiltonian simulation (LCHS) provides an asymptotically optimal framework for the quantum simulation of linear non-unitary dynamics. The LCHS approach approximately implements the Peano–Baker propagator through an integral representation determined by a kernel function. We establish an LCHS approximation theorem for a broadened class of admissible kernel functions, defined by natural complex-analytic conditions. In the standard time-independent block-encoding access model, we explicitly construct, for each target error $\varepsilon$, the unique kernel function in this class minimizing the query-cost functional and show that the minimum of this functional has the asymptotic expansion $\tfrac{2\mathrm{e}}{\pi}\big(\log\frac{1}{\varepsilon}-\log\log\frac{1}{\varepsilon}+o(1)\big)$ as ${\varepsilon\downarrow0}$. The provably optimal leading coefficient $2\mathrm{e}/\pi$ improves by a factor of approximately 2.1 on the coefficient obtained empirically by Low and Somma through numerical minimization over their family of optimally scaling kernel functions. We construct a spectrally convergent quadrature with explicit bounds on operator error and linear combination of unitaries (LCU) subnormalization, proving that this asymptotic improvement persists after discretization into a finite linear combination of unitaries suitable for implementation. Our results are based on Hardy space theory and convex analysis.

We prove sharp Hardy-type inequalities in Log-weighted Sobolev spaces. Due to the non-attainability of the sharp constant, we improve these inequalities by adding suitable remainder terms consisting of optimal weights and sharp constants. Consequently, this gives an infinite series improvement in Log-weighted Sobolev spaces. Next, by using these improved Hardy-type inequalities, we also prove an optimal Leray-Trudinger-type improvement. Furthermore, we derive interpolation between Hardy-type inequalities and Leray-Trudinger-type inequalities.