arXiv subjects

David Gamarnik

Publications and source records attributed to David Gamarnik.

At least 19 recordsLinked to original sources

arXiv2026

Circuit complexity lower bounds for quantum spin glasses

A central question in quantum information theory is the circuit complexity of states arising from standard many-body models. We study this question for quantum $p$-spin glasses, random Hamiltonians whose interactions act on $p$-tuples of qubits through Pauli strings. Anschuetz, Gamarnik, and Kiani (arXiv:2404.07231) showed that the optimum energy is separated from the best energy achievable by product states. This leaves open whether shallow circuits can close the gap, since even depth-one circuits can generate entanglement. We show that the entanglement needed to close the product-state gap cannot be generated at shallow depth. When the average interaction degree grows with $n$, we prove that, for all sufficiently large fixed $p$, any circuit preparing an $n$-qubit state whose normalized energy is within a fixed positive constant of the optimum must have depth $\Omega_p(\log n)$. In the bounded-average-degree regime, we prove a fixed-depth obstruction: for every fixed $D$, a sufficiently large degree prefactor rules out depth-$D$ preparation of near-ground states. Both results hold uniformly over circuits with an arbitrary number of ancilla qubits. Our results give an obstruction in the spirit of the No Low-Energy Trivial States problem of Freedman and Hastings (arXiv:1301.1363), but for random quantum spin glasses rather than code-based Hamiltonians such as those of Anshu, Breuckmann, and Nirkhe (arXiv:2206.13228), whose ground states admit polynomial-size preparation circuits. This setting opens a probabilistic route to NLTS-like questions: we recast state-preparation lower bounds for random quantum Hamiltonians as uniform control of Gaussian processes indexed by shallow circuits.

quant-ph↗

arXiv2026

Price of Quality: Sufficient Conditions for Sparse Recovery using Mixed-Quality Data

We study sparse recovery when observations come from mixed-quality sources: a small collection of high-quality measurements with small noise variance and a larger collection of lower-quality measurements with higher variance. For this heterogeneous-noise setting, we establish sample-size conditions for information-theoretic and algorithmic recovery. On the information-theoretic side, we show that it is sufficient for $(n_1, n_2)$ to satisfy a linear trade-off defining the Price of Quality: the number of low-quality samples needed to replace one high-quality sample. In the agnostic setting, where the decoder is completely agnostic to the quality of the data, it is uniformly bounded, and in particular one high-quality sample is never worth more than two low-quality samples for this sufficient condition to hold. In the informed setting, where the decoder is informed of per-sample variances, the price of quality can grow arbitrarily large. On the algorithmic side, we analyze the LASSO in the agnostic setting and show that the recovery threshold matches the homogeneous-noise case and only depends on the average noise level, revealing a striking robustness of computational recovery to data heterogeneity. Together, these results give the first conditions for sparse recovery with mixed-quality data and expose a fundamental difference between how the information-theoretic and algorithmic thresholds adapt to changes in data quality.

stat.ML↗

arXiv2026

The stochastic block model has the overlap graph property for modularity

The overlap gap property (OGP) is a statement about the geometry of near-optimal solutions. Exhibiting OGP implies failure of a class of local algorithms; and has been observed to coincide with conjectured algorithmic limits in problems with statistical computational gap. We consider the Stochastic Block Model (SBM), where the graph has a planted partition with $k$ equal-size blocks which form the `communities', and where, for parameters $p>q$, vertices within the same community connect with probability $p$, while vertices in different communities connect with probability $q$, independently across pairs of vertices. Modularity--based clustering algorithms have become ubiquitous in applications. This article studies theoretical limits of local algorithms based on the modularity score on the SBM. We establish that modularity exhibits OGP on the SBM. This rules out a class of local algorithms based on modularity for recovery in the SBM, and shows slow mixing time for a related Markov Chain. Theoretically this is one of the few instances where OGP has been established for a `planted' model, as most such analyses to date consider the `null' model. As part of our analysis, we extend a result by Bickel and Chen 2009, who established that with high probability, the modularity optimal partition of SBM is $o(n)$ local moves away from the planted partition, where $n$ is the graph size. We show that, with high probability, any partition with modularity score sufficiently near the optimal value is close to the planted partition.

math.PR↗

arXiv2026

Optimal Hardness of Online Algorithms for Large Common Induced Subgraphs

We study the problem of efficiently finding large common induced subgraphs of two independent Erd\H{o}s--R\'enyi random graphs $G_1, G_2 \sim \mathbb{G}(n,1/2)$. Recently, Chatterjee and Diaconis showed that the largest common induced subgraph of $G_1$ and $G_2$ has size $(4-o(1))\log_2 n$ with high probability. We first show that a simple greedy online algorithm finds a common induced subgraph of $G_1$ and $G_2$ of size $(2-o(1)) \log_2 n$ with high probability. Our main result shows that no online algorithm can find a common induced subgraph of $G_1$ and $G_2$ of size at least $(2+\varepsilon) \log_2 n$ with probability bounded away from $0$ as $n \to \infty$. Together, these results provide evidence that this problem exhibits a computation-to-optimization gap. To prove the impossibility result, we show that the solution space of the problem exhibits a version of the (multi) overlap gap property (OGP), and utilize an interpolation argument recently developed by Gamarnik, Kizilda\u{g}, and Warnke that connects OGP and online algorithms.

cs.DS↗

arXiv2026

The free energy limit of the SYK model at high temperature

The Sachdev-Ye-Kitaev (SYK) model is a disordered quantum mean-field model studied in condensed matter physics and the holographic theory of black holes. Its structural properties can be derived heuristically using a combination of the replica method and path integration techniques. Analyzing it mathematically rigorously, however, turned out to be notoriously difficult, even for basic questions such as computing the annealed free energy. In this paper we rigorously compute the free energy limit (annealed and quenched) for this model at high enough but constant temperature. Our results are in numerical agreement with the results derived by physics methods. Remarkably, though, our method of proof is novel and is different from the physics approach. It is based on (a) the theory of the component structure of sparse random graphs and (b) a variant of the cavity method, used widely in prior rigorous and heuristic treatments of classical spin glasses.

cond-mat.dis-nn↗

arXiv2026

Rigorous Asymptotics for First-Order Algorithms Through the Dynamical Cavity Method

Dynamical Mean Field Theory (DMFT) provides an asymptotic description of the dynamics of macroscopic observables in certain disordered systems. Originally pioneered in the context of spin glasses by Sompolinsky and Zippelius (1982), it has since been used to derive asymptotic dynamical equations for a wide range of models in physics, high-dimensional statistics and machine learning. One of the main tools used by physicists to obtain these equations is the dynamical cavity method, which has remained largely non-rigorous. In contrast, existing mathematical formalizations have relied on alternative approaches, including Gaussian conditioning, large deviations over paths, or Fourier analysis. In this work, we formalize the dynamical cavity method and use it to give a new proof of the DMFT equations for General First Order Methods, a broad class of dynamics encompassing algorithms such as Gradient Descent and Approximate Message Passing.

cond-mat.dis-nn↗

arXiv2026

Minimum Number of Monochromatic Subgraphs of a Random Graph

We consider the problem of minimizing the number of monochromatic subgraphs of a random graph, when each node of the host graph is assigned one of the two colors. Using a recently discovered contiguity between appearance of strictly balanced subgraphs $F$ in a random graph, and random hypergraphs where copies of $F$ are generated independently, we show that the minimum value converges to a limit, when the expected number of copies of $F$ is linear in the number of nodes $|V|$. Furthermore, using the connections with mean field spin glass models, we obtain an asymptotic expression for this limit as the normalized expected number of copies of $F$ and the size of $F$ diverge to infinity.

math.CO↗

arXiv2025

Theoretical Compression Bounds for Wide Multilayer Perceptrons

Pruning and quantization techniques have been broadly successful in reducing the number of parameters needed for large neural networks, yet theoretical justification for their empirical success falls short. We consider a randomized greedy compression algorithm for pruning and quantization post-training and use it to rigorously show the existence of pruned/quantized subnetworks of multilayer perceptrons (MLPs) with competitive performance. We further extend our results to structured pruning of MLPs and convolutional neural networks (CNNs), thus providing a unified analysis of pruning in wide networks. Our results are free of data assumptions, and showcase a tradeoff between compressibility and network width. The algorithm we consider bears some similarities with Optimal Brain Damage (OBD) and can be viewed as a post-training randomized version of it. The theoretical results we derive bridge the gap between theory and application for pruning/quantization, and provide a justification for the empirical success of compression in wide multilayer perceptrons.

cs.LG↗

arXiv2025

Spin Glass Transitions Obstruct Decoded Quantum Interferometry

Quantum algorithms are believed to offer advantages in solving certain hard discrete optimization problems, yet identifying when such advantages persist in explicit distributions of problem instances remains a foundational challenge. Recently, a new quantum algorithm known as Decoded Quantum Interferometry (DQI) has been proposed to solve optimization problems by decoding a corresponding LDPC error-correcting code. Although DQI exhibits quantum advantage on certain structured problem instances, the possibility for advantage on random, unstructured problem instances is less well-understood. Here we prove that, assuming decoding threshold upper bounds satisfied by state-of-the-art decoders, DQI is asymptotically obstructed by a spin glass phase transition in random local combinatorial optimization problems. This phase transition is heralded by the onset of the overlap gap property (OGP), a topological fragmentation of the near-optimal solution space widely conjectured to exactly characterize the asymptotic performance of optimal efficient classical algorithms. Our results therefore indicate that DQI, applied on the best known efficient decoders, is unlikely to exhibit quantum advantage on unstructured problem instances. We support this result by proving that approximate message passing, a classical optimization algorithm, outperforms DQI on certain problem distributions.

quant-ph↗

arXiv2025

The Price of Sparsity: Sufficient Conditions for Sparse Recovery using Sparse and Sparsified Measurements

We consider the problem of support recovery for sparse binary signals from noisy linear measurements. For sparse Gaussian measurement matrices we identify sufficient conditions on the minimal sample size for maximum-likelihood recovery in the high-SNR regime $ds/p \to \infty$, where $p$ denotes the signal dimension, $s$ the number of non-zero components of the signal, and $d$ the expected number of non-zero components per row of measurement. Combined with known lower bounds, this yields an information-theoretic threshold of order $s\log(p/s) / \log(ds/p)$, making explicit the price of measurement sparsity. In particular, we highlight a regime where the sample-complexity loss from measurement sparsity is logarithmic while the computational gain is nearly linear. Second, we study recovery after sparsifying an originally dense Gaussian design: the observations are generated from the dense design, while estimation uses an independently sparsified design and a rescaled response. In the proportional regime $s=\alpha p$, $d=\psi p$, we prove that, for every fixed target error level $\delta$ and every slack $\varepsilon>0$, a sample size of order $p/\psi^2$ is sufficient for support recovery for arbitrarily small $\psi$.

stat.ML↗

arXiv2025

Finding a dense submatrix of a random matrix. Sharp bounds for online algorithms

We consider the problem of finding a dense submatrix of a matrix with i.i.d. Gaussian entries, where density is measured by average value. This problem arose from practical applications in biology and social sciences \cites{madeira-survey,shabalin2009finding} and is known to exhibit a computation-to-optimization gap between the optimal value and best values achievable by existing polynomial time algorithms. In this paper we consider the class of online algorithms, which includes the best known algorithm for this problem, and derive a tight approximation factor ${4\over 3\sqrt{2}}$ for this class. The result is established using a simple implementation of recently developed Branching-Overlap-Gap-Property \cite{huang2025tight}. We further extend our results to $(\mathbb R^n)^{\otimes p}$ tensors with i.i.d. Gaussian entries, for which the approximation factor is proven to be ${2\sqrt{p}/(1+p)}$.

math.PR↗

arXiv2025

Sequential Dynamics in Ising Spin Glasses

We present the first exact asymptotic characterization of sequential dynamics for a broad class of local update algorithms on the Sherrington-Kirkpatrick (SK) model with Ising spins. Focusing on dynamics implemented via systematic scan -- encompassing Glauber updates at any temperature -- we analyze the regime where the number of spin updates scales linearly with system size. Our main result provides a description of the spin-field trajectories as the unique solution to a system of integro-difference equations derived via Dynamical Mean Field Theory (DMFT) applied to a novel block approximation. This framework captures the time evolution of macroscopic observables such as energy and overlap, and is numerically tractable. Our equations serve as a discrete-spin sequential-update analogue of the celebrated Cugliandolo-Kurchan equations for spherical spin glasses, resolving a long-standing gap in the theory of Ising spin glass dynamics. Beyond their intrinsic theoretical interest, our results establish a foundation for analyzing a wide variety of asynchronous dynamics on the hypercube and offer new avenues for studying algorithmic limitations of local heuristics in disordered systems.

cond-mat.dis-nn↗

arXiv2025

Optimal Hardness of Online Algorithms for Large Independent Sets

We study the algorithmic problem of finding a large independent set in the Erd{\"o}s-R\'{e}nyi random graph $G(n,p)$. For constant $p$ and $b=1/(1-p)$, the largest independent set has size $2\log_b n$, while a simple greedy algorithm - revealing vertices sequentially and making decisions based only on previously seen vertices - finds an independent set of size $\log_b n$. In his seminal 1976 paper, Karp challenged to either improve this guarantee or establish its hardness. Decades later, this problem remains open - one of the most prominent algorithmic problems in the theory of random graphs. In this paper, we establish that a broad class of online algorithms fails to find an independent set of size $(1+\epsilon)\log_b n$ whp. This class includes Karp's algorithm as a special case, and extends it by allowing the algorithm to query exceptional edges, not yet "seen" by the algorithm. Our lower bound holds for $p\in [d/n,1-n^{-1/d}]$. In the dense regime (constant $p$), we also prove that our result is asymptotically tight with respect to the number of exceptional edges queried, by designing an online algorithm which beats the half-optimality threshold when the number of exceptional edges slightly exceeds our bound. Our result provides evidence for the algorithmic hardness of Karp's problem, by supporting the conjectured optimality of the greedy algorithm and establishing it within the class of online algorithms. Our proof relies on a refined analysis of the geometric structure of large independent sets, establishing a variant of the Overlap Gap Property (OGP). While OGP has predominantly served as a barrier to stable algorithms, online algorithms are inherently unstable, necessitating new ideas. Our proof refines the OGP framework by incorporating several new ideas (including temporal interpolation paths and stopping-times) that we expect to be useful for other online models.

cs.DS↗

arXiv2025

Turing in the shadows of Nobel and Abel: an algorithmic story behind two recent prizes

The 2021 Nobel Prize in physics was awarded to Giorgio Parisi ``for the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales,'' and the 2024 Abel Prize in mathematics was awarded to Michel Talagrand ``for his groundbreaking contributions to probability theory and functional analysis, with outstanding applications in mathematical physics and statistics.'' What remains largely absent in the popular descriptions of these prizes, however, is the profound contributions the works of both individuals have had to the field of \emph{algorithms and computation}. The ideas first developed by Parisi and his collaborators relying on remarkably precise physics intuition, and later confirmed by Talagrand and others by no less remarkable mathematical techniques, have revolutionized the way we think algorithmically about optimization problems involving randomness. This is true both in terms of the existence of fast algorithms for some optimization problems, but also in terms of our persistent failures of finding such algorithms for some other optimization problems. The goal of this article is to highlight these developments and explain how the ideas pioneered by Parisi and Talagrand have led to a remarkably precise characterization of which optimization problems admit fast algorithms, versus those which do not, and furthermore to explain why this characterization holds true.

math.PR↗

arXiv2024

Algorithmic Universality, Low-Degree Polynomials, and Max-Cut in Sparse Random Graphs

Universality, namely distributional invariance, is a well-known property for many random structures. For example, it is known to hold for a broad range of variational problems with random input. Much less is known about the algorithmic universality of specific methods for solving such variational problems. Namely, whether algorithms tuned to specific variational tasks produce the same asymptotic behavior across different input distributions with matching moments. In this paper, we establish algorithmic universality for a class of models, which includes spin glass models and constraint satisfaction problems on sparse graphs, provided that an algorithm can be coded as a low-degree polynomial (LDP). We illustrate this specifically for the case of the Max-Cut problem in sparse Erd\"os-R\'enyi graph $\mathbb{G}(n,d/n)$. We use the fact that the Approximate Message Passing (AMP) algorithm, which is an effective algorithm for finding near-ground states of the Sherrington-Kirkpatrick (SK) model, is well approximated by an LDP. We then establish our main universality result: the performance of the LDP based algorithms exhibiting a certain connectivity property, is the same in the mean-field (SK) and in the random graph $\mathbb{G}(n,d/n)$ setting, up to an appropriate rescaling. The main technical challenge we address in this paper is showing that the output of an LDP algorithm on $\mathbb{G}(n,d/n)$ is truly discrete, namely, that it is close to the set of points in the binary cube. This is achieved by establishing universality of coordinate-wise statistics of the LDP output across disorder ensembles, which implies that proximity to the cube transfers from the Gaussian to the sparse graph setting.

cs.DS↗

arXiv2024

Slow Mixing of Quantum Gibbs Samplers

Preparing thermal (Gibbs) states is a common task in physics and computer science. Recent algorithms mimic cooling via system-bath coupling, where the cost is determined by mixing time, akin to classical Metropolis-like algorithms. However, few methods exist to demonstrate slow mixing in quantum systems, unlike the well-established classical tools for systems like the Ising model and constraint satisfaction problems. We present a quantum generalization of these tools through a generic bottleneck lemma that implies slow mixing in quantum systems. This lemma focuses on quantum measures of distance, analogous to the classical Hamming distance but rooted in uniquely quantum principles and quantified either through Bohr spectrum jumps or operator locality. Using our bottleneck lemma, we establish unconditional lower bounds on the mixing times of Gibbs samplers for several families of Hamiltonians at low temperatures. For classical Hamiltonians with mixing time lower bounds $T_\mathrm{mix} = \exp[\Omega(n^\alpha)]$, we prove that quantum Gibbs samplers also have $T_\mathrm{mix} = \exp[\Omega(n^\alpha)]$. This applies to models like random $K$-SAT instances and spin glasses. For stabilizer Hamiltonians, we provide a concise proof of exponential lower bounds $T_\mathrm{mix} = \exp[\Omega(n)]$ on mixing times of good $n$-qubit stabilizer codes at low constant temperature. Finally, we consider constant-degree classical Hamiltonians and show how to lift classical slow mixing results in the presence of a transverse field using Poisson Feynman-Kac techniques. We show generic results for models with linear free energy barriers, and we demonstrate that our techniques extend to models with sublinear free energy barriers by proving $T_\mathrm{mix} = \exp[n^{1/2-o(1)}]$ for the ferromagnetic 2D transverse field Ising model.

quant-ph↗

arXiv2024

Hardness of sampling solutions from the Symmetric Binary Perceptron

We show that two related classes of algorithms, stable algorithms and Boolean circuits with bounded depth, cannot produce an approximate sample from the uniform measure over the set of solutions to the symmetric binary perceptron model at any constraint-to-variable density. This result is in contrast to the question of finding \emph{a} solution to the same problem, where efficient (and stable) algorithms are known to succeed at sufficiently low density. This result suggests that the solutions found efficiently -- whenever this task is possible -- must be highly atypical, and therefore provides an example of a problem where search is efficiently possible but approximate sampling from the set of solutions is not, at least within these two classes of algorithms.

math.PR↗

arXiv2024

Bounds on the ground state energy of quantum $p$-spin Hamiltonians

We consider the problem of estimating the ground state energy of quantum $p$-local spin glass random Hamiltonians, the quantum analogues of widely studied classical spin glass models. Our main result shows that the maximum energy achievable by product states has a well-defined limit (for even $p$) as $n\to\infty$ and is $E_{\text{product}}^\ast=\sqrt{2 \log p}$ in the limit of large $p$. This value is interpreted as the maximal energy of a much simpler so-called Random Energy Model, widely studied in the setting of classical spin glasses. The proof of the limit existing follows from an extension of Fekete's Lemma after we demonstrate near super-additivity of the (normalized) quenched free energy. The proof of the value follows from a second moment method on the number of states achieving a given energy when restricting to an $\epsilon$-net of product states. Furthermore, we relate the maximal energy achieved over all states to a $p$-dependent constant $\gamma\left(p\right)$, which is defined by the degree of violation of a certain asymptotic independence ansatz over graph matchings. We show that the maximal energy achieved by all states $E^\ast\left(p\right)$ in the limit of large $n$ is at most $\sqrt{\gamma\left(p\right)}E_{\text{product}}^\ast$. We also prove using Lindeberg's interpolation method that the limiting $E^\ast\left(p\right)$ is robust with respect to the choice of the randomness and, for instance, also applies to the case of sparse random Hamiltonians. This robustness in the randomness extends to a wide range of random Hamiltonian models including SYK and random quantum max-cut.

quant-ph↗