Lower Bounds on Stabilizer Rank
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...
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10.02.2022
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Shpilka, Amir Ben Lee Volk Peleg, Shir |
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| Copyright | 2022. This work is published under http://creativecommons.org/licenses/by-nc-nd/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| DOI | 10.48550/arxiv.2106.03214 |
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| Snippet | The stabilizer rank of a quantum state \(\psi\) is the minimal \(r\) such that \(\left| \psi \right \rangle = \sum_{j=1}^r c_j \left|\varphi_j \right\rangle\)... |
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| SubjectTerms | Boolean algebra Boolean functions Complexity theory Lower bounds Mathematical analysis Polynomials Quadratic equations Qubits (quantum computing) Subspaces Tensors |
| Title | Lower Bounds on Stabilizer Rank |
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