@inproceedings{9e0b69b8936c4189b6c43f86fd2041ad,

title = "Lower Bounds on Stabilizer Rank",

abstract = "The stabilizer rank of a quantum state ψ is the minimal r such that (Equation presented) for cj E C and stabilizer states φj. 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 Ω(n) on the stabilizer rank of such states, improving a previous lower bound of Ω(√n) of Bravyi, Smith and Smolin [5]. Further, we prove that for a sufficiently small constant δ, the stabilizer rank of any state which is δ-close to those states is Ω(√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 Fn2, 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.",

keywords = "Lower Bounds, Quantum Computation, Simulation of Quantum computers, Stabilizer rank",

author = "Shir Peleg and Volk, {Ben Lee} and Amir Shpilka",

note = "Publisher Copyright: {\textcopyright} Shir Peleg, Ben Lee Volk, and Amir Shpilka; licensed under Creative Commons License CC-BY 4.0; null ; Conference date: 31-01-2022 Through 03-02-2022",

year = "2022",

month = jan,

day = "1",

doi = "10.4230/LIPIcs.ITCS.2022.110",

language = "אנגלית",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

editor = "Mark Braverman",

booktitle = "13th Innovations in Theoretical Computer Science Conference, ITCS 2022",

}