Improved bounds on stabilizer extent and Clifford rank

We prove that every pure state of stabilizer rank at most k has stabilizer extent at most 2 O(sqrt k (k+1)), and establish the analogous bound for the squared Clifford coefficient norm of Clifford-rank- k operators. This implies stabilizer fidelity at least 2-O(sqrt k (k+1)), resolving the quantitative conjecture of (Mehraban-Tamasbi, STOC, 2025), and proves an Ω(n 2/ n) lower bound for the approximate stabilizer rank of tensor powers of any non-stabilizer qubit state. The latter result generalizes the best-known lower bound for tensor powers of T-states (Mehraban-Tamasbi, STOC, 2024) to arbitrary non-stabilizer qubit states, including magic states. As a consequence of the Clifford rank--norm inequality, we obtain an Ω(n 2/ n) lower bound for exact representations of n-bit AND function by quadratic phases, improving the previous best-known linear bound. Further consequences rule out pseudorandom state and unitary ensembles with approximate stabilizer and Clifford rank O((n) 2/ n), respectively, a n improvement over prior work (Kalra-Sinha, Quantum, 2026). We also obtain tomography algorithms for states of stabilizer rank at most k, with poly(n)2 O(sqrt k (k+1)) time and copy complexity, a nearly square-root improvement in the exponent over the best-known algorithm.