Improved bounds on stabilizer extent and Clifford rank

Abstract:We prove that every pure state of stabilizer rank at most $k$ has stabilizer extent at most $2^{O(\sqrt{k\log(k+1)})}$, and establish the analogous bound for the squared Clifford coefficient norm of Clifford-rank-$k$ operators. This implies stabilizer fidelity at least $2^{-O(\sqrt{k\log(k+1)})}$, resolving the quantitative conjecture of (Mehraban-Tamasbi, STOC, 2025), and proves an $\Omega(n^2/\log n)$ 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 $\Omega(n^2/\log n)$ 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 $O((\log n)^2/\log\log n)$, respectively, a $\log n$ improvement over prior work (Kalra-Sinha, Quantum, 2026). We also obtain tomography algorithms for states of stabilizer rank at most $k$, with $poly(n)2^{O(\sqrt{k\log(k+1)})}$ time and copy complexity, a nearly square-root improvement in the exponent over the best-known algorithm.

arXiv:2610.06819 [quant-ph]

(or arXiv:2610.06819v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.06819 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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