Better bounds on Grothendieck constants of finite orders

• URL: http://arxiv.org/abs/2409.03739v2
• Date: Fri, 20 Dec 2024 16:08:21 GMT
• Title: Better bounds on Grothendieck constants of finite orders
• Abstract summary: We exploit a recent Frank-Wolfe approach to provide good candidates for lower bounding some Grothendieck constants.<n>The complete proof relies on solving difficult binary quadratic optimisation problems.
• Score: 20.068273625719943
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Grothendieck constants $K_G(d)$ bound the advantage of $d$-dimensional strategies over $1$-dimensional ones in a specific optimisation task. They have applications ranging from approximation algorithms to quantum nonlocality. However, apart from $d=2$, their values are unknown. Here, we exploit a recent Frank-Wolfe approach to provide good candidates for lower bounding some of these constants. The complete proof relies on solving difficult binary quadratic optimisation problems. For $d\in\{3,4,5\}$, we construct specific rectangular instances that we can solve to certify better bounds than those previously known; by monotonicity, our lower bounds improve on the state of the art for $d\leqslant9$. For $d\in\{4,7,8\}$, we exploit elegant structures to build highly symmetric instances achieving even greater bounds; however, we can only solve them heuristically. We also recall the standard relation with violations of Bell inequalities and elaborate on it to interpret generalised Grothendieck constants $K_G(d\mapsto2)$ as the advantage of complex quantum mechanics over real quantum mechanics. Motivated by this connection, we also improve the bounds on $K_G(d\mapsto2)$.

Related papers

• Quantum Query Complexity of Finding a Tarski Fixed Point on a High-Dimensional Grid [31.45779401617503]
  Knaster-Tarski fixed-point theorem states that every monotone function over a complete lattice has a fixed point.<n>We prove an $(klog n)$ quantum query lower bound for finding a fixed point of a monotone function on $[n]k$, using the nonnegative spectral adversary method.
  arXiv  Detail & Related papers  (2026-09-03T13:06:12Z)
• Quantum Speedups for Stochastic Optimization with Heavy-Tailed Noise [49.730496294398726]
  We develop a novel quantum mean estimator for heavy-tailed random variables.<n>We derive stronger lower bounds for tail index>4/3$, showing that a nontrivial dependence on a dimension is unavoidable.<n>For a convex objective function, we propose a quantum projected gradient descent method.
  arXiv  Detail & Related papers  (2026-07-28T09:29:37Z)
• Optimal Lower Bounds for Hamiltonian Simulation [42.227880669333835]
  For Hamiltonian $H = sum_j h_j$, we proveally tight lower bounds on the gate and query complexities of simulating time evolution on a quantum computer.<n>Our hold for arbitrary term norms $|h_j|$, time $t$, and trace-distance error $$.
  arXiv  Detail & Related papers  (2026-07-22T07:41:32Z)
• Near-Optimal Regret in Adversarial Kernel Bandits [50.68324062892194]
  We study the adversarial kernel bandit problem, in which the loss at each round is induced by an arbitrary bounded element.<n>Our main result bounds the regret by $widetildeObig(sqrtT, d_*(),log|X|big)$, where $d_*()$ is a widely-adopted notion of effective dimension.
  arXiv  Detail & Related papers  (2026-05-26T06:10:24Z)
• Optimal Dimension-Free Sampling for Regularized Classification [56.72526267755301]
  We prove optimal sampling bounds achieving $(1pmvarepsilon)$-relative error for a broad class of Lipschitz continuous classification loss functions.<n>This includes important functions such as logistic and sigmoid loss, hinge loss, and ReLU loss, as prominent and popular representative examples.
  arXiv  Detail & Related papers  (2026-05-22T15:05:33Z)
• Optimal Quantum Logarithmic Trace Inequality [2.7074235008521246]
  We establish a sharp logarithmic trace inequality that strengthens recent bounds of Cheng et al.(arXiv:2507.07961)<n>We prove that $G_s$ is the optimal universal constant, in the sense that no smaller constant satisfies the inequality for all positive operators.<n>These sharper inequalities enhance key primitives in quantum information theory, including decoupling, convex-splitting, and covering lemmas.
  arXiv  Detail & Related papers  (2026-04-16T04:57:25Z)
• Hardness of High-Dimensional Linear Classification [58.29089693778071]
  We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem.<n>Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms.
  arXiv  Detail & Related papers  (2026-03-19T15:53:41Z)
• Tight inapproximability of max-LINSAT and implications for decoded quantum interferometry [0.0]
  We prove by a direct reduction from Hstad's theorem that no-time algorithm can exceed the randomassignment ratio $r/q$ by any constant.<n>This threshold coincides with the $ell/m to 0$ limit of the semicircle law governing decoded quantum interferometry.
  arXiv  Detail & Related papers  (2026-03-04T19:26:26Z)
• Instance-Dependent Regret Bounds for Learning Two-Player Zero-Sum Games with Bandit Feedback [60.610120215789976]
  We show that when a pure strategy Nash equilibrium exists, $c$ becomes zero, leading to an optimal instance-dependent regret bound. Our algorithm also enjoys last-iterate convergence and can identify the pure strategy Nash equilibrium with near-optimal sample.
  arXiv  Detail & Related papers  (2025-02-24T20:20:06Z)
• Efficient Quantum State Synthesis with One Query [0.0]
  We present a time analogue quantum algorithm making a single query (in superposition) to a classical oracle. We prove that every $n$-qubit state can be constructed to within 0.01 error by an $On/n)$-size circuit over an appropriate finite gate set.
  arXiv  Detail & Related papers  (2023-06-02T17:49:35Z)
• Pseudonorm Approachability and Applications to Regret Minimization [73.54127663296906]
  We convert high-dimensional $ell_infty$-approachability problems to low-dimensional pseudonorm approachability problems. We develop an algorithmic theory of pseudonorm approachability, analogous to previous work on approachability for $ell$ and other norms.
  arXiv  Detail & Related papers  (2023-02-03T03:19:14Z)
• A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization [53.044526424637866]
  In this paper we consider finding an approximate second-order stationary point (SOSP) that minimizes a twice different subject general non conic optimization. In particular, we propose a Newton-CG based-augmentedconjugate method for finding an approximate SOSP.
  arXiv  Detail & Related papers  (2023-01-10T20:43:29Z)
• Grothendieck bound in a single quantum system [0.0]
  Grothendieck's bound is used in the context of a single quantum system. The Grothendieck theorem is reformulated here in terms of arbitrary matrices.
  arXiv  Detail & Related papers  (2022-12-22T13:06:31Z)
• Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end [114.3957763744719]
  We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems. We show that the algorithm finds the optimal solution in time $O*(2(0.5-c)n)$ for an $n$-independent constant $c$. We also show that for a large fraction of random instances from the $k$-spin model and for any fully satisfiable or slightly frustrated $k$-CSP formula, statement (a) is the case.
  arXiv  Detail & Related papers  (2022-12-03T02:45:23Z)
• Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies [68.93512627479197]
  We study quantum Ordered Binary Decision Diagrams($OBDD$) model. We prove lower bounds and upper bounds for OBDD with arbitrary order of input variables. We extend hierarchy for read$k$-times Ordered Binary Decision Diagrams ($k$-OBDD$) of width.
  arXiv  Detail & Related papers  (2022-04-22T12:37:56Z)
• Improved upper bounds on the stabilizer rank of magic states [0.0]
  improvement is obtained by establishing a new upper bound on the stabilizer rank of $m$ copies of the magic state $|Trangle=sqrt2-1(|0rangle+eipi/4|1rangle)$ in the limit of large $m$. We obtain a strong simulation algorithm for circuits consisting of Clifford gates and $m$ instances of any (fixed) single-qubit $Z$-rotation gate with runtime $textpoly(n,m) 2m/2$.
  arXiv  Detail & Related papers  (2021-06-14T20:20:51Z)
• Private Stochastic Convex Optimization: Optimal Rates in $\ell_1$ Geometry [69.24618367447101]
  Up to logarithmic factors the optimal excess population loss of any $(varepsilon,delta)$-differently private is $sqrtlog(d)/n + sqrtd/varepsilon n.$ We show that when the loss functions satisfy additional smoothness assumptions, the excess loss is upper bounded (up to logarithmic factors) by $sqrtlog(d)/n + (log(d)/varepsilon n)2/3.
  arXiv  Detail & Related papers  (2021-03-02T06:53:44Z)
• Lower Bounds and Accelerated Algorithms for Bilevel Optimization [62.192297758346484]
  Bilevel optimization has recently attracted growing interests due to its wide applications in modern machine learning problems. We show that our results are more challenging than that of minimax applications.
  arXiv  Detail & Related papers  (2021-02-07T21:46:29Z)
• Epsilon-nets, unitary designs and random quantum circuits [0.11719282046304676]
  Epsilon-nets and approximate unitary $t$-designs are notions of unitary operations relevant for numerous applications in quantum information and quantum computing. We prove that for a fixed $d$ of the space, unitaries constituting $delta$-approx $t$-expanders form $epsilon$-nets for $tsimeqfracd5/2epsilon$ and $delta=left(fracepsilon3/2dright)d2$. We show that approximate tdesigns can be generated
  arXiv  Detail & Related papers  (2020-07-21T15:16:28Z)
• Second-Order Information in Non-Convex Stochastic Optimization: Power and Limitations [54.42518331209581]
  We find an algorithm which finds. epsilon$-approximate stationary point (with $|nabla F(x)|le epsilon$) using. $(epsilon,gamma)$surimate random random points. Our lower bounds here are novel even in the noiseless case.
  arXiv  Detail & Related papers  (2020-06-24T04:41:43Z)
• Quasi-polynomial time algorithms for free quantum games in bounded dimension [11.56707165033]
  We give a semidefinite program of size $exp(mathcalObig(T12(log2(AT)+log(Q)log(AT))/epsilon2big)) to compute additive $epsilon$-approximations on the values of two-player free games. We make a connection to the quantum separability problem and employ improved multipartite quantum de Finetti theorems with linear constraints.
  arXiv  Detail & Related papers  (2020-05-18T16:55:08Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.