arXiv subjects

Junhyung Lyle Kim

Publications and source records attributed to Junhyung Lyle Kim.

At least 19 recordsLinked to original sources

arXiv2026

The Quartic Gap Between Quantum and Dequantized Algorithms for Linear Algebra Is Tight

Quantum algorithms for linear algebra that access a matrix $A$ through quantum random access memory (QRAM) prepare quantum states proportional to the solutions of linear systems and related problems. Dequantized classical algorithms, when given sampling and query access to $A$, return sampling and query access to an approximate solution. Both kinds of algorithms use a number of queries that is polynomial in the normalized Frobenius norm $ρ=\|A\|_F/\|A\|$ and does not directly depend on the dimension. For linear systems at fixed condition number and accuracy, quantum algorithms use $\widetilde O(ρ)$ queries, whereas the best dequantized algorithms use $\widetilde O(ρ^4)$. We prove that this quartic gap is tight in the worst case, up to a logarithmic factor. For every constant $κ_0>1$, we construct dense linear systems with condition number at most $κ_0$ for which sampling from the solution within a sufficiently small constant total variation distance requires $Ω(ρ^4/\logρ)$ classical queries but only $Θ(ρ)$ quantum queries. The same separation holds, for errors below smaller thresholds, for positive-definite systems, least squares, ridge regression, matrix powers and square roots, imaginary-time and real-time Hamiltonian evolution, and sampling from the top principal component. Our results therefore show that many quantum algorithms for linear algebra, achieve a near-quartic speedup in the worst case, over classical algorithms with sampling and query access. All of these classical lower bounds follow from the same framework, which is based on a lower bound for sampling from distributions that depend on a fixed set of Fourier coefficients of an unknown Boolean function.

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arXiv2026

Hypergraph cycle enumeration for efficient inference of planted k-XOR with smooth trade-off

We introduce hypergraph cycle enumeration (HCE), a simple spectral framework for detecting and recovering planted signals in noisy hypergraph inference problems. We study HCE on one representative problem from this class, namely planted noisy k-XOR, also known as sparse Learning Parity with Noise (LPN). Both theoretically and empirically, HCE achieves comparable recovery behavior to Kikuchi method, matching Kikuchi hierarchy. We identify parameter regimes where HCE outperforms Kikuchi method, showing that neither algorithm is universally better. We also show a subquadratic quantum speedup for HCE by replacing one of the algorithm's steps with a quantum walk. Our classical and quantum algorithms do not refute the quartic quantum speedup of quantum Kikuchi method, which remains valid for the practically relevant regime of large size and high noise strength.

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arXiv2026

Faster Algorithms for Multimarginal Optimal Transport

We study constructive discrete multimarginal optimal transport (MOT) among $m$ distributions on $n$ points, where the cost tensor $C$ has $N=n^m$ entries. For additive accuracy $\varepsilon$, let $κ=\max\{1,(\max C-\min C)/\varepsilon\}$. Classically, we give two algorithms that return exactly feasible additive-$\varepsilon$ couplings. A deterministic box--simplex method with rounding runs in $O(m^2Nκ\log N)=\widetilde O(m^3Nκ)$ time, while an exact reduction to positive packing followed by rank-one completion runs in randomized time $\widetilde O(m^2Nκ)$. At fixed $κ$, both match the $Ω(N)$ cost of writing a dense coupling, up to factors in $m$ and logarithms. Quantumly, in the general entry-access model, tensor Sinkhorn followed by sparse recovery returns an exactly feasible additive-$\varepsilon$ coupling as a classical list of $\widetilde O(m^2nκ^2)$ atoms, using $\widetilde O(m^4\sqrt{Nn}\,κ^3)$ coherent cost queries without materializing the tensor. Finally, for fixed $m$ and constant normalized accuracy, we prove sparse-output lower bounds of $\widetildeΩ(N)$ randomized classical queries, even with unrestricted output size, and $\widetildeΩ(\sqrt{Nn})$ quantum queries for outputs with at most $n\operatorname{polylog}(n)$ atoms. Hence, for fixed $m$ and normalized accuracy between $(\log n)^{-O(1)}$ and a sufficiently small constant, explicit sparse MOT construction has query complexity $\widetildeΘ(N)$ classically and $\widetildeΘ(\sqrt{Nn})$ quantumly.

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arXiv2026

Near-Optimal Quantum Lower Bounds for Convex Optimization via Fourier Rank

We establish a linear quantum query lower bound for high-accuracy convex optimization over an explicit family of $n$-dimensional ellipsoids. We focus on linear optimization with an explicitly given objective, where the feasible set is accessed through a membership oracle. We show that any algorithm that, for every unit linear objective, returns an exactly feasible point with additive objective error $Θ(n^{-1/2})$ requires $Ω(n)$ membership queries, where the ellipsoids are contained in the unit ball. The same lower bound can be shown to hold if the returned point is only required to be approximately feasible, within $Θ(n^{-3/2})$ distance from the feasible set. This resolves an open question posed by Chakrabarti, Childs, Li, and Wu~(\textit{Quantum}, 2020) and by van Apeldoorn, Gilyén, Gribling, and de Wolf~(\textit{Quantum}, 2020). Coupled with the upper bounds in these papers, the query complexity of high-accuracy convex optimization is characterized tightly up to logarithmic factors. The proof is built around a lower bound for determinant computation that is derived via a novel polynomial method based on Fourier-rank. In the continuous matrix phase-query model, computing the determinant of a real $n\times n$ matrix requires at least $n/2$ matrix-vector product queries. The construction also yields an $Ω(n)$ phase-query lower bound for estimating the minimum eigenvalue of a real symmetric $n\times n$ matrix to additive accuracy $Θ(n^{-2})$. These results extend the determinant and minimum-eigenvalue lower bounds of Childs, Hung, and Li~(ICALP 2021) from finite fields to the real-valued setting. Based on the same constructions, we also prove an $Ω(\min\{\sqrtκ,n\})$ gradient-query lower bound for constant-accuracy optimization of $1$-strongly convex and $κ$-smooth functions, which is optimal up to a logarithmic factor.

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arXiv2026

From Quantum Tunneling to Classical Optimization through Instantons

We use an instanton-based analysis to show that Path Integral Stochastic Gradient Descent (PISGD), a classical optimization algorithm instantiated from path integral Monte Carlo (PIMC), can inherit the quantum tunneling mechanism. Quantum tunneling is the phenomenon in which a quantum particle passes through a potential barrier that a classical trajectory cannot cross in the same way. This mechanism can contribute to the performance of quantum optimization algorithms, since a common bottleneck in optimization is that the solution becomes trapped in a local minimum, separated from better solutions by a potential barrier. Classical noise-driven optimizers, such as stochastic gradient descent (SGD), have escape times that scale exponentially with the potential barrier height. For quantum tunneling, the escape time is instead exponential in the bounce action, which equals twice the Agmon distance between the two basins. When the barrier is tall but thin, quantum tunneling can allow escaping local minima faster than such classical methods. Remarkably, this scaling is not exclusive to algorithms with genuine quantum dynamics. For PISGD applied to classical nonconvex global optimization, we adapt Kramers' escape-rate theory to path space and obtain the same exponential scaling as quantum tunneling, in terms of the bounce action rather than the barrier height. We confirm this scaling numerically by implementing PISGD for both separable and nonseparable potentials. That said, PISGD retains only incoherent tunneling, since coherence among multiple tunneling paths is absent. Our analysis therefore does not rule out a potential quantum advantage for quantum-tunneling-based algorithms over classical ones in general.

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arXiv2026

Quantum Simulation on Riemannian Manifolds

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.

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arXiv2026

Anytime Training with Schedule-Free Spectral Optimization

Standard neural network training relies on learning-rate schedules tied to a fixed horizon, leading to strong path dependence and costly re-tuning as data availability changes. Schedule-Free (SF) methods address this by removing explicit schedules, yet SF-AdamW, the current state-of-the-art anytime optimizer, consistently underperforms well-tuned AdamW baselines. We propose SF-NorMuon, a schedule-free spectral optimizer that closes this gap: with a single hyperparameter configuration, SF-NorMuon matches or exceeds tuned AdamW on 125M and 772M parameter language models across $1$--$8\times$ Chinchilla horizons, and trails horizon-aware, cosine-scheduled NorMuon by only $\sim 0.03$ nats, substantially narrowing the gap between anytime and scheduled spectral optimizers. On the theoretical side, we prove a stationarity guarantee for schedule-free spectral dynamics and identify weight decay at the fast iterate as essential for long-horizon stability. SF-NorMuon enables practitioners to obtain high-quality checkpoints at any point during training without committing to a horizon in advance. Fully matching horizon-aware spectral optimizers without a schedule remains an open problem, which we highlight as a direction for future work. By closing the performance gap with tuned AdamW baselines, SF-NorMuon makes horizon-free optimization more practical, taking a step towards truly open-ended, continual learning.

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arXiv2026

Quantum Speedups for Group Relaxations of Integer Linear Programs

Integer Linear Programs (ILPs) are a flexible and ubiquitous model for discrete optimization problems. Solving ILPs is \textsf{NP-Hard} yet of great practical importance. Super-quadratic quantum speedups for ILPs have been difficult to obtain because classical algorithms for many-constraint ILPs are global and exhaustive, whereas quantum frameworks that offer super-quadratic speedup exploit local structure of the objective and feasible set. We address this via quantum algorithms for Gomory's group relaxation. The group relaxation of an ILP is obtained by dropping nonnegativity on variables that are positive in the optimal solution of the linear programming (LP) relaxation, while retaining integrality of the decision variables. We present a competitive feasibility-preserving classical local-search algorithm for the group relaxation, and a corresponding quantum algorithm that, under reasonable technical conditions, achieves a super-quadratic speedup. When the group relaxation satisfies a nondegeneracy condition analogous to, but stronger than, LP non-degeneracy, our approach yields the optimal solution to the original ILP. Otherwise, the group relaxation tightens bounds on the optimal objective value of the ILP, and can improve downstream branch-and-cut by reducing the integrality gap; we numerically observe this on several practically relevant ILPs. To achieve these results, we derive efficiently constructible constraint-preserving mixers for the group relaxation with favorable spectral properties, which are of independent interest.

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arXiv2025

A Catalyst Framework for the Quantum Linear System Problem via the Proximal Point Algorithm

Solving systems of linear equations is a fundamental problem, but it can be computationally intensive for classical algorithms in high dimensions. Existing quantum algorithms can achieve exponential speedups for the quantum linear system problem (QLSP) in terms of the problem dimension, but the advantage is bottlenecked by condition number of the coefficient matrix. In this work, we propose a new quantum algorithm for QLSP inspired by the classical proximal point algorithm (PPA). Our proposed method can be viewed as a meta-algorithm that allows inverting a modified matrix via an existing \texttt{QLSP\_solver}, thereby directly approximating the solution vector instead of approximating the inverse of the coefficient matrix. By carefully choosing the step size $η$, the proposed algorithm can effectively precondition the linear system to mitigate the dependence on condition numbers that hindered the applicability of previous approaches. Importantly, this is the first iterative framework for QLSP where a tunable parameter $η$ and initialization $x_0$ allows controlling the trade-off between the runtime and approximation error.

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arXiv2025

End-to-end quantum algorithms for tensor problems

We present a comprehensive end-to-end quantum algorithm for tensor problems, including tensor PCA and planted kXOR, that achieves potential superquadratic quantum speedups over classical methods. We build upon prior works by Hastings~(\textit{Quantum}, 2020) and Schmidhuber~\textit{et al.}~(\textit{Phys.~Rev.~X.}, 2025), we address key limitations by introducing a native qubit-based encoding for the Kikuchi method, enabling explicit quantum circuit constructions and non-asymptotic resource estimation. Our approach substantially reduces constant overheads through a novel guiding state preparation technique as well as circuit optimizations, reducing the threshold for a quantum advantage. We further extend the algorithmic framework to support recovery in sparse tensor PCA and tensor completion, and generalize detection to asymmetric tensors, demonstrating that the quantum advantage persists in these broader settings. Detailed resource estimates show that 900 logical qubits, $\sim 10^{15}$ gates and $\sim 10^{12}$ gate depth suffice for a problem that classically requires $\sim 10^{23}$ FLOPs. The gate count and depth for the same problem without the improvements presented in this paper would be at least $10^{19}$ and $10^{18}$ respectively. These advances position tensor problems as a candidate for quantum advantage whose resource requirements benefit significantly from algorithmic and compilation improvements; the magnitude of the improvements suggest that further enhancements are possible, which would make the algorithm viable for upcoming fault-tolerant quantum hardware.

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arXiv2025

Mechanisms for Quantum Advantage in Global Optimization of Nonconvex Functions

We present new theoretical mechanisms for quantum speedup in the global optimization of nonconvex functions, expanding the scope of quantum advantage beyond traditional tunneling-based explanations. As our main building-block, we demonstrate a rigorous correspondence between the spectral properties of Schrödinger operators and the mixing times of classical Langevin diffusion. This correspondence motivates a mechanism for separation on functions with unique global minimum: while quantum algorithms operate on the original potential, classical diffusions correspond to a Schrödinger operators with a WKB potential having nearly degenerate global minima. We formalize these ideas by proving that a real-space adiabatic quantum algorithm (RsAA) achieves provably polynomial-time optimization for broad families of nonconvex functions. First, for block-separable functions, we show that RsAA maintains polynomial runtime while known off-the-shelf algorithms require exponential time and structure-aware algorithms exhibit arbitrarily large polynomial runtimes. These results leverage novel non-asymptotic results in semiclassical analysis. Second, we use recent advances in the theory of intrinsic hypercontractivity to demonstrate polynomial runtimes for RsAA on appropriately perturbed strongly convex functions that lack global structure, while off-the-shelf algorithms remain exponentially bottlenecked. In contrast to prior works based on quantum tunneling, these separations do not depend on the geometry of barriers between local minima. Our theoretical claims about classical algorithm runtimes are supported by rigorous analysis and comprehensive numerical benchmarking. These findings establish a rigorous theoretical foundation for quantum advantage in continuous optimization and open new research directions connecting quantum algorithms, stochastic processes, and semiclassical analysis.

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arXiv2025

On Speedups for Convex Optimization via Quantum Dynamics

We explore the potential for quantum speedups in convex optimization using discrete simulations of the Quantum Hamiltonian Descent (QHD) framework, as proposed by Leng et al., and establish the first rigorous query complexity bounds. We develop enhanced analyses for quantum simulation of Schrödinger operators with black-box potential via the pseudo-spectral method, providing explicit resource estimates independent of wavefunction assumptions. These bounds are applied to assess the complexity of optimization through QHD. Our findings pertain to unconstrained convex optimization in $d$ dimensions. In continuous time, we demonstrate that QHD, with suitable parameters, can achieve arbitrarily fast convergence rates. The optimization speed limit arises solely from the discretization of the dynamics, mirroring a property of the classical dynamics underlying QHD. Considering this cost, we show that a $G$-Lipschitz convex function can be optimized to an error of $ε$ with $\widetilde{\mathcal{O}}(d^{1.5}G^2 R^2/ε^2)$ queries. Moreover, under reasonable assumptions on the complexity of Hamiltonian simulation, $\widetildeΩ(d/ε^2)$ queries are necessary. Thus, QHD does not offer a speedup over classical zeroth order methods with exact oracles. However, we demonstrate that the QHD algorithm tolerates $\widetilde{\mathcal{O}}(ε^3/d^{1.5}G^2 R^2)$ noise in function evaluation. We show that QHD offers a super-quadratic query advantage over all known classical algorithms tolerating this level of evaluation noise in the high-dimension regime. Additionally, we design a quantum algorithm for stochastic convex optimization that provides a super-quadratic speedup over all known classical algorithms in the high-dimension regime. To our knowledge, these results represent the first rigorous quantum speedups for convex optimization achieved through a dynamical algorithm.

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arXiv2025

A simple analysis of a quantum-inspired algorithm for solving low-rank linear systems

We describe and analyze a simple algorithm for sampling from the solution $\mathbf{x}^* := \mathbf{A}^+\mathbf{b}$ to a linear system $\mathbf{A}\mathbf{x} = \mathbf{b}$. We assume access to a sampler which allows us to draw indices proportional to the squared row/column-norms of $\mathbf{A}$. Our algorithm produces a compressed representation of some vector $\mathbf{x}$ for which $\|\mathbf{x}^* - \mathbf{x}\| < \varepsilon \|\mathbf{x}^* \|$ in $\widetilde{O}(κ_{\mathsf{F}}^4 κ^2 / \varepsilon^2)$ time, where $κ_{\mathsf{F}} := \|\mathbf{A}\|_{\mathsf{F}}\|\mathbf{A}^{+}\|$ and $κ:= \|\mathbf{A}\|\|\mathbf{A}^{+}\|$. The representation of $\mathbf{x}$ allows us to query entries of $\mathbf{x}$ in $\widetilde{O}(κ_{\mathsf{F}}^2)$ time and sample proportional to the square entries of $\mathbf{x}$ in $\widetilde{O}(κ_{\mathsf{F}}^4 κ^6)$ time, assuming access to a sampler which allows us to draw indices proportional to the squared entries of any given row of $\mathbf{A}$. Our analysis, which is elementary, non-asymptotic, and fully self-contained, simplifies and clarifies several past analyses from literature including [Gilyén, Song, and Tang; 2022, 2023] and [Shao and Montanaro; 2022].

cs.DS↗

arXiv2025

Fast Convex Optimization with Quantum Gradient Methods

We study quantum algorithms based on quantum (sub)gradient estimation using noisy function evaluation oracles, and demonstrate the first dimension-independent query complexities (up to poly-logarithmic factors) for zeroth-order convex optimization in both smooth and nonsmooth settings. Interestingly, only using noisy function evaluation oracles, we match the first-order query complexities of classical gradient descent, thereby exhibiting exponential separation between quantum and classical zeroth-order optimization. We then generalize these algorithms to work in non-Euclidean settings by using quantum (sub)gradient estimation to instantiate mirror descent and its variants, including dual averaging and mirror prox. By leveraging a connection between semidefinite programming and eigenvalue optimization, we use our quantum mirror descent method to give a new quantum algorithm for solving semidefinite programs, linear programs, and zero-sum games. We identify a parameter regime in which our zero-sum games algorithm is faster than any existing classical or quantum approach.

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arXiv2025

Solving Hidden Monotone Variational Inequalities with Surrogate Losses

Deep learning has proven to be effective in a wide variety of loss minimization problems. However, many applications of interest, like minimizing projected Bellman error and min-max optimization, cannot be modelled as minimizing a scalar loss function but instead correspond to solving a variational inequality (VI) problem. This difference in setting has caused many practical challenges as naive gradient-based approaches from supervised learning tend to diverge and cycle in the VI case. In this work, we propose a principled surrogate-based approach compatible with deep learning to solve VIs. We show that our surrogate-based approach has three main benefits: (1) under assumptions that are realistic in practice (when hidden monotone structure is present, interpolation, and sufficient optimization of the surrogates), it guarantees convergence, (2) it provides a unifying perspective of existing methods, and (3) is amenable to existing deep learning optimizers like ADAM. Experimentally, we demonstrate our surrogate-based approach is effective in min-max optimization and minimizing projected Bellman error. Furthermore, in the deep reinforcement learning case, we propose a novel variant of TD(0) which is more compute and sample efficient.

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arXiv2024

On the Error-Propagation of Inexact Hotelling's Deflation for Principal Component Analysis

Principal Component Analysis (PCA) aims to find subspaces spanned by the so-called principal components that best represent the variance in the dataset. The deflation method is a popular meta-algorithm that sequentially finds individual principal components, starting from the most important ones and working towards the less important ones. However, as deflation proceeds, numerical errors from the imprecise estimation of principal components propagate due to its sequential nature. This paper mathematically characterizes the error propagation of the inexact Hotelling's deflation method. We consider two scenarios: $i)$ when the sub-routine for finding the leading eigenvector is abstract and can represent various algorithms; and $ii)$ when power iteration is used as the sub-routine. In the latter case, the additional directional information from power iteration allows us to obtain a tighter error bound than the sub-routine agnostic case. For both scenarios, we explicitly characterize how the errors progress and affect subsequent principal component estimations.

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arXiv2024

Adaptive Federated Learning with Auto-Tuned Clients

Federated learning (FL) is a distributed machine learning framework where the global model of a central server is trained via multiple collaborative steps by participating clients without sharing their data. While being a flexible framework, where the distribution of local data, participation rate, and computing power of each client can greatly vary, such flexibility gives rise to many new challenges, especially in the hyperparameter tuning on the client side. We propose $Δ$-SGD, a simple step size rule for SGD that enables each client to use its own step size by adapting to the local smoothness of the function each client is optimizing. We provide theoretical and empirical results where the benefit of the client adaptivity is shown in various FL scenarios.

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arXiv2024

When is Momentum Extragradient Optimal? A Polynomial-Based Analysis

The extragradient method has gained popularity due to its robust convergence properties for differentiable games. Unlike single-objective optimization, game dynamics involve complex interactions reflected by the eigenvalues of the game vector field's Jacobian scattered across the complex plane. This complexity can cause the simple gradient method to diverge, even for bilinear games, while the extragradient method achieves convergence. Building on the recently proven accelerated convergence of the momentum extragradient method for bilinear games \citep{azizian2020accelerating}, we use a polynomial-based analysis to identify three distinct scenarios where this method exhibits further accelerated convergence. These scenarios encompass situations where the eigenvalues reside on the (positive) real line, lie on the real line alongside complex conjugates, or exist solely as complex conjugates. Furthermore, we derive the hyperparameters for each scenario that achieve the fastest convergence rate.

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