Lower Bounds on Stabilizer Rank

Shir Peleg1, Amir Shpilka1, and Ben Lee Volk2

1Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel
2Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel

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The $\textit{stabilizer rank}$ 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 [7]. 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.

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Cited by

[1] Benjamin Lovitz and Vincent Steffan, "New techniques for bounding stabilizer rank", Quantum 6, 692 (2022).

[2] Hammam Qassim, Hakop Pashayan, and David Gosset, "Improved upper bounds on the stabilizer rank of magic states", Quantum 5, 606 (2021).

[3] Saeed Mehraban and Mehrdad Tahmasbi, Proceedings of the 56th Annual ACM Symposium on Theory of Computing 608 (2024) ISBN:9798400703836.

[4] Saeed Mehraban and Mehrdad Tahmasbi, "Quadratic Lower bounds on the Approximate Stabilizer Rank: A Probabilistic Approach", arXiv:2305.10277, (2023).

[5] Fulvio Gesmundo, "Geometry of Tensors: Open problems and research directions", arXiv:2304.10570, (2023).

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