Researchers have developed a new method to quantify the complexity of state preparation in dense quantum $p$-spin Hamiltonians, operating on $n$ qubits. This approach extends beyond limitations based solely on quantum circuit light cones, introducing "effective profile complexity" derived from the metric entropy of Pauli profiles. These profiles record the expectations of all Pauli operators supported on exactly $p$ qubits. This advancement allows for a more precise characterization of the computational difficulty in achieving certain quantum states.

The study reveals that classes of states with uniformly bounded quadratic effective profile complexity remain separated from the ground-state energy by a positive multiple of $\sqrt{n}$ for sufficiently large fixed $p$. At subquadratic effective profile complexity, the class cannot outperform a suitable benchmark class at leading order, with product states providing a universal benchmark. The proof of these results combines an adaptation of a nonsymmetric quantum de Finetti theorem with Gaussian process entropy bounds.

Applying this framework, it is shown that attaining near-ground-state energy requires $\Omega(n^2/\log n)$ one- and two-qubit gates, even with arbitrary discardable ancillas. The work also obtains depth-width tradeoffs, entanglement-depth and matrix product state bond-dimension lower bounds. Furthermore, obstructions are identified for both orientations at every fixed level of Parham's magic hierarchy, with a total circuit width of $O(n)$. For the first-level reverse magic class, where a shallow circuit is followed by an unrestricted Clifford circuit, the bounds also allow arbitrarily many clean ancillas at fixed shallow-circuit depth. A sharper benchmark shows that Clifford+$T$ circuits with $o(n)$ $T$-gates have no leading-order energy advantage over product stabilizer states, even with unrestricted Clifford operations and arbitrary discardable ancillas.