arXiv subjects

David Gamarnik

Publications and source records attributed to David Gamarnik.

At least 37 records · Page 2Linked to original sources

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 $T_\mathrm{mix} = \exp[Ω(n^α)]$, we prove that quantum Gibbs samplers also have $T_\mathrm{mix} = \exp[Ω(n^α)]$. This applies to models like random $K$-SAT instances and spin glasses. For stabilizer Hamiltonians, we provide a concise proof of exponential lower bounds $T_\mathrm{mix} = \exp[Ω(n)]$ 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

Combinatorial NLTS From the Overlap Gap Property

In an important recent development, Anshu, Breuckmann, and Nirkhe [ABN22] resolved positively the so-called No Low-Energy Trivial State (NLTS) conjecture by Freedman and Hastings. The conjecture postulated the existence of linear-size local Hamiltonians on n qubit systems for which no near-ground state can be prepared by a shallow (sublogarithmic depth) circuit. The construction in [ABN22] is based on recently developed good quantum codes. Earlier results in this direction included the constructions of the so-called Combinatorial NLTS -- a weaker version of NLTS -- where a state is defined to have low energy if it violates at most a vanishing fraction of the Hamiltonian terms [AB22]. These constructions were also based on codes. In this paper we provide a "non-code" construction of a class of Hamiltonians satisfying the Combinatorial NLTS. The construction is inspired by one in [AB22], but our proof uses the complex solution space geometry of random K-SAT instead of properties of codes. Specifically, it is known that above a certain clause-to-variables density the set of satisfying assignments of random K-SAT exhibits an overlap gap property, which implies that it can be partitioned into exponentially many clusters each constituting at most an exponentially small fraction of the total set of satisfying solutions. We establish a certain robust version of this clustering property for the space of near-satisfying assignments and show that for our constructed Hamiltonians every combinatorial near-ground state induces a near-uniform distribution supported by this set. Standard arguments then are used to show that such distributions cannot be prepared by quantum circuits with depth o(log n). Since the clustering property is exhibited by many random structures, including proper coloring and maximum cut, we anticipate that our approach is extendable to these models as well.

quant-ph↗

arXiv2024

Sharp Thresholds Imply Circuit Lower Bounds: from random 2-SAT to Planted Clique

We show that sharp thresholds for Boolean functions directly imply average-case circuit lower bounds. More formally we show that any Boolean function exhibiting a sharp enough threshold at \emph{arbitrary} critical density cannot be computed by Boolean circuits of bounded depth and polynomial size. We also prove a partial converse: if a monotone graph invariant Boolean function does not have a sharp threshold then it can be computed on average by a Boolean circuit of bounded depth and polynomial size. Our general result also implies new average-case bounded depth circuit lower bounds in a variety of settings. (a) ($k$-cliques) For $k=Θ(n)$, we prove that any circuit of depth $d$ deciding the presence of a size $k$ clique in a random graph requires exponential-in-$n^{Θ(1/d)}$ size. (b)(random 2-SAT) We prove that any circuit of depth $d$ deciding the satisfiability of a random 2-SAT formula requires exponential-in-$n^{Θ(1/d)}$ size. To the best of our knowledge, this is the first bounded depth circuit lower bound for random $k$-SAT for any value of $k \geq 2.$ Our results also provide the first rigorous lower bound in agreement with a conjectured, but debated, "computational hardness" of random $k$-SAT around its satisfiability threshold. (c)(Statistical estimation -- planted $k$-clique) Over the recent years, multiple statistical estimation problems have also been proven to exhibit a "statistical" sharp threshold, called the All-or-Nothing (AoN) phenomenon. We show that AoN also implies circuit lower bounds for statistical problems. As a simple corollary of that, we prove that any circuit of depth $d$ that solves to information-theoretic optimality a "dense" variant of the celebrated planted $k$-clique problem requires exponential-in-$n^{Θ(1/d)}$ size.

cs.CC↗

arXiv2024

Cliques, Chromatic Number, and Independent Sets in the Semi-random Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. In this paper, we investigate the following three properties: containing a complete graph of order $k$, having the chromatic number at least $k$, and not having an independent set of size at least $k$.

math.CO↗

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 $ε$-net of product states. Furthermore, we relate the maximal energy achieved over all states to a $p$-dependent constant $γ\left(p\right)$, 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 $\sqrt{γ\left(p\right)}E_{\text{product}}^\ast$. 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↗

arXiv2024

Product states optimize quantum $p$-spin models for large $p$

We consider the problem of estimating the maximal energy of quantum $p$-local spin glass random Hamiltonians, the quantum analogues of widely studied classical spin glass models. Denoting by $E^*(p)$ the (appropriately normalized) maximal energy in the limit of a large number of qubits $n$, we show that $E^*(p)$ approaches $\sqrt{2\log 6}$ as $p$ increases. 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. Our most notable and (arguably) surprising result proves the existence of near-maximal energy states which are product states, and thus not entangled. Specifically, we prove that with high probability as $n\to\infty$, for any $E<E^*(p)$ there exists a product state with energy $\geq E$ at sufficiently large constant $p$. Even more surprisingly, this remains true even when restricting to tensor products of Pauli eigenstates. Our approximations go beyond what is known from monogamy-of-entanglement style arguments -- the best of which, in this normalization, achieve approximation error growing with $n$. Our results not only challenge prevailing beliefs in physics that extremely low-temperature states of random local Hamiltonians should exhibit non-negligible entanglement, but they also imply that classical algorithms can be just as effective as quantum algorithms in optimizing Hamiltonians with large locality -- though performing such optimization is still likely a hard problem. Our results are robust with respect to the choice of the randomness (disorder) and apply to the case of sparse random Hamiltonian using Lindeberg's interpolation method. The proof of the main result is obtained by estimating the expected trace of the associated partition function, and then matching its asymptotics with the extremal energy of product states using the second moment method.

quant-ph↗

arXiv2024

Integrating High-Dimensional Functions Deterministically

We design a Quasi-Polynomial time deterministic approximation algorithm for computing the integral of a multi-dimensional separable function, supported by some underlying hyper-graph structure, appropriately defined. Equivalently, our integral is the partition function of a graphical model with continuous potentials. While randomized algorithms for high-dimensional integration are widely known, deterministic counterparts generally do not exist. We use the correlation decay method applied to the Riemann sum of the function to produce our algorithm. For our method to work, we require that the domain is bounded and the hyper-edge potentials are positive and bounded on the domain. We further assume that upper and lower bounds on the potentials separated by a multiplicative factor of $1 + O(1/Δ^2)$, where $Δ$ is the maximum degree of the graph. When $Δ= 3$, our method works provided the upper and lower bounds are separated by a factor of at most $1.0479$. To the best of our knowledge, our algorithm is the first deterministic algorithm for high-dimensional integration of a continuous function, apart from the case of trivial product form distributions.

cs.DS↗

arXiv2023

Computing the Volume of a Restricted Independent Set Polytope Deterministically

We construct a quasi-polynomial time deterministic approximation algorithm for computing the volume of an independent set polytope with restrictions. Randomized polynomial time approximation algorithms for computing the volume of a convex body have been known now for several decades, but the corresponding deterministic counterparts are not available, and our algorithm is the first of this kind. The class of polytopes for which our algorithm applies arises as linear programming relaxation of the independent set problem with the additional restriction that each variable takes value in the interval $[0,1-α]$ for some $α<1/2$. (We note that the $α\ge 1/2$ case is trivial). We use the correlation decay method for this problem applied to its appropriate and natural discretization. The method works provided $α> 1/2-O(1/Δ^2)$, where $Δ$ is the maximum degree of the graph. When $Δ=3$ (the sparsest non-trivial case), our method works provided $0.488<α<0.5$. Interestingly, the interpolation method, which is based on analyzing complex roots of the associated partition functions, fails even in the trivial case when the underlying graph is a singleton.

cs.DS↗

arXiv2023

Shattering in the Ising Pure $p$-Spin Model

We study the Ising pure $p$-spin model for large $p$. We investigate the landscape of the Hamiltonian of this model. We show that for any $γ>0$ and any large enough $p$, the model exhibits an intricate geometrical property known as the multi Overlap Gap Property above the energy value $γ\sqrt{2\ln 2}$. We then show that for any inverse temperature $\sqrt{\ln 2}<β<\sqrt{2\ln 2}$ and any large $p$, the model exhibits shattering: w.h.p. as $n\to\infty$, there exists exponentially many well-separated clusters such that (a) each cluster has exponentially small Gibbs mass, and (b) the clusters collectively contain all but a vanishing fraction of Gibbs mass. Moreover, these clusters consist of configurations with energy near $β$. Range of temperatures for which shattering occurs is within the replica symmetric region. To the best of our knowledge, this is the first shattering result regarding the Ising $p$-spin models. Our proof is elementary, and in particular based on simple applications of the first and the second moment methods.

math.PR↗

arXiv2023

Barriers for the performance of graph neural networks (GNN) in discrete random structures. A comment on~\cite{schuetz2022combinatorial},\cite{angelini2023modern},\cite{schuetz2023reply}

Recently graph neural network (GNN) based algorithms were proposed to solve a variety of combinatorial optimization problems, including Maximum Cut problem, Maximum Independent Set problem and similar other problems~\cite{schuetz2022combinatorial},\cite{schuetz2022graph}. The publication~\cite{schuetz2022combinatorial} stirred a debate whether GNN based method was adequately benchmarked against best prior methods. In particular, critical commentaries~\cite{angelini2023modern} and~\cite{boettcher2023inability} point out that simple greedy algorithm performs better than GNN in the setting of random graphs, and in fact stronger algorithmic performance can be reached with more sophisticated methods. A response from the authors~\cite{schuetz2023reply} pointed out that GNN performance can be improved further by tuning up the parameters better. We do not intend to discuss the merits of arguments and counter-arguments in~\cite{schuetz2022combinatorial},\cite{angelini2023modern},\cite{boettcher2023inability},\cite{schuetz2023reply}. Rather in this note we establish a fundamental limitation for running GNN on random graphs considered in these references, for a broad range of choices of GNN architecture. These limitations arise from the presence of the Overlap Gap Property (OGP) phase transition, which is a barrier for many algorithms, both classical and quantum. As we demonstrate in this paper, it is also a barrier to GNN due to its local structure. We note that at the same time known algorithms ranging from simple greedy algorithms to more sophisticated algorithms based on message passing, provide best results for these problems \emph{up to} the OGP phase transition. This leaves very little space for GNN to outperform the known algorithms, and based on this we side with the conclusions made in~\cite{angelini2023modern} and~\cite{boettcher2023inability}.

cs.LG↗

arXiv2023

Geometric Barriers for Stable and Online Algorithms for Discrepancy Minimization

For many computational problems involving randomness, intricate geometric features of the solution space have been used to rigorously rule out powerful classes of algorithms. This is often accomplished through the lens of the multi Overlap Gap Property ($m$-OGP), a rigorous barrier against algorithms exhibiting input stability. In this paper, we focus on the algorithmic tractability of two models: (i) discrepancy minimization, and (ii) the symmetric binary perceptron (\texttt{SBP}), a random constraint satisfaction problem as well as a toy model of a single-layer neural network. Our first focus is on the limits of online algorithms. By establishing and leveraging a novel geometrical barrier, we obtain sharp hardness guarantees against online algorithms for both the \texttt{SBP} and discrepancy minimization. Our results match the best known algorithmic guarantees, up to constant factors. Our second focus is on efficiently finding a constant discrepancy solution, given a random matrix $\mathcal{M}\in\mathbb{R}^{M\times n}$. In a smooth setting, where the entries of $\mathcal{M}$ are i.i.d. standard normal, we establish the presence of $m$-OGP for $n=Θ(M\log M)$. Consequently, we rule out the class of stable algorithms at this value. These results give the first rigorous evidence towards a conjecture of Altschuler and Niles-Weed~\cite[Conjecture~1]{altschuler2021discrepancy}. Our methods use the intricate geometry of the solution space to prove tight hardness results for online algorithms. The barrier we establish is a novel variant of the $m$-OGP. Furthermore, it regards $m$-tuples of solutions with respect to correlated instances, with growing values of $m$, $m=ω(1)$. Importantly, our results rule out online algorithms succeeding even with an exponentially small probability.

cs.CC↗

arXiv2022

Densest Subgraphs of a Dense Erdös-Rényi Graph. Asymptotics, Landscape and Universality

We consider the problem of estimating the edge density of densest $K$-node subgraphs of an Erdös-Rényi graph $\mathbb{G}(n,1/2)$. The problem is well-understood in the regime $K=Θ(\log n)$ and in the regime $K=Θ(n)$. In the former case it can be reduced to the problem of estimating the size of largest cliques, and its extensions. In the latter case the full answer is known up to the order $n^{3\over 2}$ using sophisticated methods from the theory of spin glasses. The intermediate case $K=n^α, α\in (0,1)$ however is not well studied and this is our focus. We establish that that in this regime the density (that is the maximum number of edges supported by any $K$-node subgraph) is ${1\over 4}K^2+{1+o(1)\over 2}K^{3\over 2}\sqrt{\log (n/K)}$, w.h.p. as $n\to\infty$, and provide more refined asymptotics under the $o(\cdot)$, for various ranges of $α$. This extends earlier similar results where this asymptotics was confirmed only when $α$ is a small constant. We extend our results to the case of ''weighted'' graphs, when the weights have either Gaussian or arbitrary sub-Gaussian distributions. The proofs are based on the second moment method combined with concentration bounds, the Borell-TIS inequality for the Gaussian case and the Talagrand's inequality for the case of distributions with bounded support (including the $\mathbb{G}(n,1/2)$ case). The case of general distribution is treated using a novel symmetrized version of the Lindeberg argument, which reduces the general case to the Gaussian case. Finally, using the results above we conduct the landscape analysis of the related Hidden Clique Problem, and establish that it exhibits an overlap gap property when the size of the clique is $O(n^{2\over 3})$, confirming a hypothesis stated in a previous related work.

math.PR↗

arXiv2022

Disordered Systems Insights on Computational Hardness

In this review article, we discuss connections between the physics of disordered systems, phase transitions in inference problems, and computational hardness. We introduce two models representing the behavior of glassy systems, the spiked tensor model and the generalized linear model. We discuss the random (non-planted) versions of these problems as prototypical optimization problems, as well as the planted versions (with a hidden solution) as prototypical problems in statistical inference and learning. Based on ideas from physics, many of these problems have transitions where they are believed to jump from easy (solvable in polynomial time) to hard (requiring exponential time). We discuss several emerging ideas in theoretical computer science and statistics that provide rigorous evidence for hardness by proving that large classes of algorithms fail in the conjectured hard regime. This includes the overlap gap property, a particular mathematization of clustering or dynamical symmetry-breaking, which can be used to show that many algorithms that are local or robust to changes in their input fail. We also discuss the sum-of-squares hierarchy, which places bounds on proofs or algorithms that use low-degree polynomials such as standard spectral methods and semidefinite relaxations, including the Sherrington-Kirkpatrick model. Throughout the manuscript, we present connections to the physics of disordered systems and associated replica symmetry breaking properties.

cond-mat.dis-nn↗

arXiv2022

Performance and limitations of the QAOA at constant levels on large sparse hypergraphs and spin glass models

The Quantum Approximate Optimization Algorithm (QAOA) is a general purpose quantum algorithm designed for combinatorial optimization. We analyze its expected performance and prove concentration properties at any constant level (number of layers) on ensembles of random combinatorial optimization problems in the infinite size limit. These ensembles include mixed spin models and Max-$q$-XORSAT on sparse random hypergraphs. Our analysis can be understood via a saddle-point approximation of a sum-over-paths integral. This is made rigorous by proving a generalization of the multinomial theorem, which is a technical result of independent interest. We then show that the performance of the QAOA at constant levels for the pure $q$-spin model matches asymptotically the ones for Max-$q$-XORSAT on random sparse Erdős-Rényi hypergraphs and every large-girth regular hypergraph. Through this correspondence, we establish that the average-case value produced by the QAOA at constant levels is bounded away from optimality for pure $q$-spin models when $q\ge 4$ and is even. This limitation gives a hardness of approximation result for quantum algorithms in a new regime where the whole graph is seen.

quant-ph↗

arXiv2022

Algorithms and Barriers in the Symmetric Binary Perceptron Model

The symmetric binary perceptron ($\texttt{SBP}$) exhibits a dramatic statistical-to-computational gap: the densities at which known efficient algorithms find solutions are far below the threshold for the existence of solutions. Furthermore, the $\texttt{SBP}$ exhibits a striking structural property: at all positive constraint densities almost all of its solutions are 'totally frozen' singletons separated by large Hamming distance \cite{perkins2021frozen,abbe2021proof}. This suggests that finding a solution to the $\texttt{SBP}$ may be computationally intractable. At the same time, the $\texttt{SBP}$ does admit polynomial-time search algorithms at low enough densities. A conjectural explanation for this conundrum was put forth in \cite{baldassi2020clustering}: efficient algorithms succeed in the face of freezing by finding exponentially rare clusters of large size. However, it was discovered recently that such rare large clusters exist at all subcritical densities, even at those well above the limits of known efficient algorithms \cite{abbe2021binary}. Thus the driver of the statistical-to-computational gap exhibited by this model remains a mystery. In this paper, we conduct a different landscape analysis to explain the algorithmic tractability of this problem. We show that at high enough densities the $\texttt{SBP}$ exhibits the multi Overlap Gap Property ($m-$OGP), an intricate geometrical property known to be a rigorous barrier for large classes of algorithms. Our analysis shows that the $m-$OGP threshold (a) is well below the satisfiability threshold; and (b) matches the best known algorithmic threshold up to logarithmic factors as $m\to\infty$. We then prove that the $m-$OGP rules out the class of stable algorithms for the $\texttt{SBP}$ above this threshold. We conjecture that the $m \to \infty$ limit of the $m$-OGP threshold marks the algorithmic threshold for the problem.

cs.CC↗

arXiv2022

Hardness of Random Optimization Problems for Boolean Circuits, Low-Degree Polynomials, and Langevin Dynamics

We consider the problem of finding nearly optimal solutions of optimization problems with random objective functions. Two concrete problems we consider are (a) optimizing the Hamiltonian of a spherical or Ising $p$-spin glass model, and (b) finding a large independent set in a sparse Erdős-Rényi graph. The following families of algorithms are considered: (a) low-degree polynomials of the input; (b) low-depth Boolean circuits; (c) the Langevin dynamics algorithm. We show that these families of algorithms fail to produce nearly optimal solutions with high probability. For the case of Boolean circuits, our results improve the state-of-the-art bounds known in circuit complexity theory (although we consider the search problem as opposed to the decision problem). Our proof uses the fact that these models are known to exhibit a variant of the overlap gap property (OGP) of near-optimal solutions. Specifically, for both models, every two solutions whose objectives are above a certain threshold are either close or far from each other. The crux of our proof is that the classes of algorithms we consider exhibit a form of stability. We show by an interpolation argument that stable algorithms cannot overcome the OGP barrier. The stability of Langevin dynamics is an immediate consequence of the well-posedness of stochastic differential equations. The stability of low-degree polynomials and Boolean circuits is established using tools from Gaussian and Boolean analysis -- namely hypercontractivity and total influence, as well as a novel lower bound for random walks avoiding certain subsets. In the case of Boolean circuits, the result also makes use of Linal-Mansour-Nisan's classical theorem. Our techniques apply more broadly to low influence functions and may apply more generally.

cs.CC↗

arXiv2022

Circuit Lower Bounds for the p-Spin Optimization Problem

We consider the problem of finding a near ground state of a $p$-spin model with Rademacher couplings by means of a low-depth circuit. As a direct extension of the authors' recent work [Gamarnik, Jagannath, Wein 2020], we establish that any poly-size $n$-output circuit that produces a spin assignment with objective value within a certain constant factor of optimality, must have depth at least $\log n/(2\log\log n)$ as $n$ grows. This is stronger than the known state of the art bounds of the form $Ω(\log n/(k(n)\log\log n))$ for similar combinatorial optimization problems, where $k(n)$ depends on the optimality value. For example, for the largest clique problem $k(n)$ corresponds to the square of the size of the clique [Rossman 2010]. At the same time our results are not quite comparable since in our case the circuits are required to produce a solution itself rather than solving the associated decision problem. As in our earlier work, the approach is based on the overlap gap property (OGP) exhibited by random $p$-spin models, but the derivation of the circuit lower bound relies further on standard facts from Fourier analysis on the Boolean cube, in particular the Linial-Mansour-Nisan Theorem. To the best of our knowledge, this is the first instance when methods from spin glass theory have ramifications for circuit complexity.

cs.CC↗

arXiv2021

The Overlap Gap Property: a Geometric Barrier to Optimizing over Random Structures

The problem of optimizing over random structures emerges in many areas of science and engineering, ranging from statistical physics to machine learning and artificial intelligence. For many such structures finding optimal solutions by means of fast algorithms is not known and often is believed not possible. At the same time the formal hardness of these problems in form of say complexity-theoretic $NP$-hardness is lacking. In this introductory article a new approach for algorithmic intractability in random structures is described, which is based on the topological disconnectivity property of the set of pair-wise distances of near optimal solutions, called the Overlap Gap Property. The article demonstrates how this property a) emerges in most models known to exhibit an apparent algorithmic hardness b) is consistent with the hardness/tractability phase transition for many models analyzed to the day, and importantly c) allows to mathematically rigorously rule out large classes of algorithms as potential contenders, in particular the algorithms exhibiting the input stability (insensitivity).

cs.CC↗