Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.05644

Dimension Amplification for Tarski Fixed-Point Query Lower Bounds

Boyu Liu·Zihe Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that finding a fixed point of a monotone map on the nine-dimensional grid $[N]^9$ requires $Ω((\log N)^3)$ deterministic queries, even when the fixed point is unique and each query returns the entire function value. The proof gives a construction that raises the dimension from $d$ to $4d+1$, preserves uniqueness, and adds a logarithmic factor to the lower bound. Iteration gives $Ω((\log N)^{r+2})$ queries in dimension $(7\cdot4^r-1)/3$, for every fixed nonnegative integer $r$, with an implicit constant that may depend on $r$. Consequently, no finite logarithmic exponent bounds the query complexity in all fixed dimensions.

Explore related subjects

cs.CC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Boyu Liu, Zihe Wang. 2026-10-05. Dimension Amplification for Tarski Fixed-Point Query Lower Bounds. https://arxiv.org/abs/2610.05644

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Spectral Certificates and Non-commutative Sum-of-Squares Lower Bounds for Hamiltonians

A central question in quantum many-body physics is estimating the ground energy of a $k$-local Hamiltonian system. In this work, we present a spectral technique for certifying a lower bound on the ground energy of a random $n$-qubit Hamiltonian system defined as the sum of signed $k$-local Pauli operators. In particular, we prove that for any constant $\ell$, there exists an efficiently computable length $n^{O(\ell)}$ certificate that is always a lower bound on the ground energy with the promise that, with high probability over the random Hamiltonian distribution, the certificate value is an $\varepsilon$-good approximation of the true ground energy when the number of terms is sufficiently large. Second, we show by construction that this technique, while successful on average over random Hamiltonian systems, can fail to produce good certificates on worst-case instances. Our spectral technique for producing these certificates comes from extending classical results on $k$-XOR refutations to $k$-local Pauli Hamiltonians by crafting a quantum variant of the Kikuchi matrix for CSP refutations. To show the limitations of this technique, we prove non-commutative Sum-of-Squares lower bounds for worst-case signed $k$-local Pauli operators. More generally, we explore how the non-commutative Sum-of-Squares relaxation can be understood as augmenting the standard Sum-of-Squares relaxation with the commutation relations between the Pauli operators. We instantiate the resulting framework with a modification to prior quantum code-based NLTS Hamiltonians that yields stronger complexity guarantees for the low-energy space; our Hamiltonian family satisfies simultaneously (1) $Ω(\log n)$-circuit depth lower bounds for all low-energy states, (2) constant-gap NP-hardness to approximate the ground energy, and (3) a constant-gap non-commutative Sum-of-Squares integrality gap up to $Ω(n)$-levels.

cs.CC↗
arXiv2026

Exponential Quantum Advantage in Numbers-on-Forehead Communication

We give the first exponential quantum advantage in the general interactive three-party Numbers-on-Forehead (NOF) model for a decision problem. Previous separations hold only for restricted protocols like one-way communication for a relation. We construct an explicit partial Boolean function, the Interleaved Unitary Product problem, that requires only $O(\log n)$ NOF quantum communication but $\widetildeΩ(n^{1/32})$ randomized communication. This function builds on the two-party unitary product problem of Arunachalam, Girish, and Lifshitz (TQC 2024). The main technical obstacle is that discrepancy, the standard lower-bound method for NOF, also lower-bounds quantum communication. We instead develop a regularity-based argument for randomized NOF lower bounds, building on the approach of Kelley, Lovett, and Meka and adapting the regularity decomposition of Abboud, Fischer, Kelley, Lovett, and Meka (STOC 2024) to cylinder intersections. Combined with matrix-product estimates of Arunachalam, Girish, and Lifshitz, this yields our randomized lower bound.

cs.CC↗
arXiv2026

Fractal Gadgets for Neural Networks: The Complexity of the Narrow Regime

We study the verification problem for deep narrow ReLU neural networks: given a network of bounded width computing a piecewise-affine map on [0,1], does some input satisfy a prescribed output constraint? Classical NP-hardness proofs for ReLU verification use one neuron per Boolean variable and say nothing about networks of small constant width, while width-1 networks are easy to verify. We show that verification of ReLU networks is NP-complete at width 4 for arbitrary inputs in [0,1]. When inputs are restricted to a natural discrete encoding set, NP-completeness already holds at width 3. Together with polynomial-time decidability at width 1, this leaves open only width 2 on the encoding set, and widths 2 and 3 on [0,1]. The technical core is a fractal preprocessing gadget: a width-2 ReLU subnetwork whose iterate vanishes precisely near a finite Cantor-like subset of [0,1] with 2^n points. It reduces verification of a continuous function on [0,1] to verification on 2^n discrete points without increasing the width, and is the missing ingredient for width-bounded hardness reductions. The same construction yields further results at width 3 on the encoding set: the universal problem is coNP-complete, counting zeros is #P-complete, a majority variant is PP-complete, and approximating the minimum output within a constant gap inherited from Max-3Sat is NP-hard. The NP, coNP and inapproximability results lift to all of [0,1] at width 4; lifting counting and majority, and lifting at width 3, remain open.

cs.CC↗