Lower Bounds on Stabilizer Rank - Peleg, Shir; Shpilka, Amir; Ben Lee Volk (arXiv.org, 2022)
The stabilizer rank of a quantum state \(\psi\) is the minimal \(r\) such that \(\left| \psi \right \rangle = \sum_{j=1}^r c_j \left|\varphi_j \right\rangle\) for \(c_j \in \mathbb{C}\) and stabilizer states \(\varphi_j\). The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the \(n\)-th tensor power of single-qubit magic states. We prove a lower bound of \(\Omega(n)\) on the stabilizer rank of such states, improving a previous lower bound of \(\Omega(\sqrt{n})\) of Bravyi, Smith and Smolin (arXiv:1506.01396). Further, we prove that for a sufficiently small constant \(\delta\), the stabilizer rank of any state which is \(\delta\)-close to those states is \(\Omega(\sqrt{n}/\log n)\). This is the first non-trivial lower bound for approximate stabilizer rank. Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of \(\mathbb{F}_2^n\), and we use tools from analysis of boolean functions and complexity theory. The proof of the first result involves a careful analysis of directional derivatives of quadratic polynomials, whereas the proof of the second result uses Razborov-Smolensky low degree polynomial approximations and correlation bounds against the majority function.