Polynomial-time tolerant testing stabilizer states
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916477871849472 |
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| author | Arunachalam, Srinivasan Dutt, Arkopal |
| author_facet | Arunachalam, Srinivasan Dutt, Arkopal |
| contents | We consider the following task: suppose an algorithm is given copies of an unknown $n$-qubit quantum state $|ψ\rangle$ promised $(i)$ $|ψ\rangle$ is $\varepsilon_1$-close to a stabilizer state in fidelity or $(ii)$ $|ψ\rangle$ is $\varepsilon_2$-far from all stabilizer states, decide which is the case. We show that for every $\varepsilon_1>0$ and $\varepsilon_2\leq \varepsilon_1^C$, there is a $\textsf{poly}(1/\varepsilon_1)$-sample and $n\cdot \textsf{poly}(1/\varepsilon_1)$-time algorithm that decides which is the case (where $C>1$ is a universal constant). Our proof includes a new definition of Gowers norm for quantum states, an inverse theorem for the Gowers-$3$ norm of quantum states and new bounds on stabilizer covering for structured subsets of Paulis using results in additive combinatorics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06289 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomial-time tolerant testing stabilizer states Arunachalam, Srinivasan Dutt, Arkopal Quantum Physics Computational Complexity Data Structures and Algorithms We consider the following task: suppose an algorithm is given copies of an unknown $n$-qubit quantum state $|ψ\rangle$ promised $(i)$ $|ψ\rangle$ is $\varepsilon_1$-close to a stabilizer state in fidelity or $(ii)$ $|ψ\rangle$ is $\varepsilon_2$-far from all stabilizer states, decide which is the case. We show that for every $\varepsilon_1>0$ and $\varepsilon_2\leq \varepsilon_1^C$, there is a $\textsf{poly}(1/\varepsilon_1)$-sample and $n\cdot \textsf{poly}(1/\varepsilon_1)$-time algorithm that decides which is the case (where $C>1$ is a universal constant). Our proof includes a new definition of Gowers norm for quantum states, an inverse theorem for the Gowers-$3$ norm of quantum states and new bounds on stabilizer covering for structured subsets of Paulis using results in additive combinatorics. |
| title | Polynomial-time tolerant testing stabilizer states |
| topic | Quantum Physics Computational Complexity Data Structures and Algorithms |
| url | https://arxiv.org/abs/2408.06289 |