A Slice-Rank Drift Bound for Random Quantum \(k\)-SAT

Abstract:Random quantum satisfiability is a natural quantum analogue of random constraint satisfaction and a basic model for frustration-free local Hamiltonians. Despite extensive work on its satisfiable and unsatisfiable regimes, the quantitative location of the random quantum \(k\)-SAT threshold has remained poorly understood, with the best general upper bounds leaving a large gap to the known lower bounds. In this paper we prove a new upper bound on the satisfiability threshold of random quantum \(k\)-SAT. Our result improves the previously known asymptotic upper bound by a factor of order \(k\), giving a bound of order \(2^k/k\). The improvement is also significant at small values of \(k\); in particular, for random quantum \(3\)-SAT we obtain a substantially smaller explicit upper bound than the one previously available. The proof combines the geometric formulation of generic quantum satisfiability with a dimension-decay analysis of the full satisfying subspace. The key input is a multiplicative Shearer-type inequality for tensor-product subspaces, which quantifies how global dimension forces nontrivial local dimension on typical sets of qubits.

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Combinatorics (math.CO); Probability (math.PR)

arXiv:2607.23847 [quant-ph]

(or arXiv:2607.23847v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.23847 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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