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

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.