Universal Entropy Ceiling in Finite-Depth Quantum Circuits: Numerical Evidence and a Modular-Forms Conjecture

We present large-scale numerical evidence for a universalentropy ceiling in finite-depth quantumcircuits. Using GPU-accelerated simulation of up to 106random quantum states across local dimensions d = 2 and d = 3, we demonstrate that the ratio of bipartite entanglement entropy to itstheoretical maximum saturates at a value strictly below unity that grows logarithmically with totalHilbert space dimension d n but never reaches 1. The ceilingdepends only on total dimension, not on local structure: configurationswith identical total dimension but different local factorisationssaturate at the same value. We connect this observation to the theoryof modular forms for SL2(Z), arguing that the weight-12 transition—wherethe Eisenstein subspace first fails to span the full space of modular forms—governs the onset of coupling-induced rigidity on the Bloch sphere. The ceilingis robust against non-standard inner products, extended state spaces, and modified coupling topologies. Counting independent coupling directionsbelow the weight-12 transition, we conjecture anupper bound of 211 = 2,048 logical qubits for any pairwise-coupled two-level architecture, falsifiableagainst current industry road-maps.