arXiv subjects

Fernando G. Jeronimo

Publications and source records attributed to Fernando G. Jeronimo.

2 recordsLinked to original sources

arXiv2026

Stable Regularity Lemmas: Efficient Algorithms and Essentially Tight Littlestone Bounds

In this paper, we determine the precise asymptotics of the number of parts of stable regularity equipartitions in terms of the Littlestone dimension: every graph $G$ of Littlestone dimension $\operatorname{Lit}(G)\leq\ell$ has a regular equipartition into excellent sets with $(1 + o_{ε\to 0,\ell}(1))\cdotε^{-\ell-1}$ parts and in the other direction, for every $\ell\in\mathbb{N}_+$, there is an infinite family of graphs, all of Littlestone dimension $\ell$, whose equipartitions into good sets must have size at least $(1 + o_{ε\to 0,\ell}(1))\cdotε^{-\ell-1}$. Dropping the equitability condition, we determine the asymptotics of non-equitable partitions up to a multiplicative $\log(1/ε)$: every graph $G$ with $\operatorname{Lit}(G)\leq\ell$ has a regular partition into excellent sets with $(1 + o_{ε\to 0,\ell}(1))\cdotε^{-\ell}\cdot\ln(1/ε)$ parts and in the other direction, for every $\ell\in\mathbb{N}_+$, there is an infinite family of graphs, all of Littlestone dimension $\ell$, whose partitions into good sets must have size at least $(1 + o_{ε\to 0,\ell}(1))\cdotε^{-\ell}$. We also show that such partition can be obtained algorithmically efficiently in an approximation scheme fashion: replacing the $o_{ε\to 0,\ell}(1)$ term above by a constant $c > 0$, we obtain randomized $O_{c,ε,\ell}(n\cdot\log(n))$-time algorithms for partitions/equipartitions into good sets, a deterministic $O_{c,ε,\ell}(n^2)$-time algorithm for partitions into good sets, a deterministic $O_{c,ε,\ell}(n^6)$-time algorithm for equipartitions into good sets, a deterministic $O_{c,\ell,ε}(1)\cdot n^{O(\ell\cdot 2^{2\cdot\ell+4})}$-time algorithm for partitions/equipartitions into excellent sets, and $O_{c,ε,\ell}(\log(n+1))$-space algorithms for partitions/equipartitions into good/excellent sets.

math.CO↗

arXiv2024

Dimension Independent Disentanglers from Unentanglement and Applications

Quantum entanglement is a key enabling ingredient in diverse applications. However, the presence of unwanted adversarial entanglement also poses challenges in many applications. In this paper, we explore methods to "break" quantum entanglement. Specifically, we construct a dimension-independent k-partite disentangler (like) channel from bipartite unentangled input. We show: For every $d,\ell\ge k$, there is an efficient channel $Λ: \mathbb{C}^{d\ell} \otimes \mathbb{C}^{d\ell} \to \mathbb{C}^{dk}$ such that for every bipartite separable state $ρ_1\otimes ρ_2$, the output $Λ(ρ_1\otimesρ_2)$ is close to a k-partite separable state. Concretely, for some distribution $μ$ on states from $\mathbb{C}^d$, $$ \left\|Λ(ρ_1 \otimes ρ_2) - \int | ψ\rangle \langle ψ|^{\otimes k} dμ(ψ)\right\|_1 \le \tilde O \left(\left(\frac{k^{3}}{\ell}\right)^{1/4}\right). $$ Moreover, $Λ(| ψ\rangle \langle ψ|^{\otimes \ell}\otimes | ψ\rangle \langle ψ|^{\otimes \ell}) = | ψ\rangle \langle ψ|^{\otimes k}$. Without the bipartite unentanglement assumption, the above bound is conjectured to be impossible. Leveraging our disentanglers, we show that unentangled quantum proofs of almost general real amplitudes capture NEXP, greatly relaxing the nonnegative amplitudes assumption in the recent work of QMA^+(2)=NEXP. Specifically, our findings show that to capture NEXP, it suffices to have unentangled proofs of the form $| ψ\rangle = \sqrt{a} | ψ_+ \rangle + \sqrt{1-a} | ψ_- \rangle$ where $| ψ_+ \rangle$ has non-negative amplitudes, $| ψ_- \rangle$ only has negative amplitudes and $| a-(1-a) | \ge 1/poly(n)$ with $a \in [0,1]$. Additionally, we present a protocol achieving an almost largest possible gap before obtaining QMA^R(k)=NEXP$, namely, a 1/poly(n) additive improvement to the gap results in this equality.

quant-ph↗