Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917065054486528 |
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| author | Grewal, Sabee Iyer, Vishnu Kretschmer, William Liang, Daniel |
| author_facet | Grewal, Sabee Iyer, Vishnu Kretschmer, William Liang, Daniel |
| contents | We give a pair of algorithms that efficiently learn a quantum state prepared by Clifford gates and $O(\log n)$ non-Clifford gates. Specifically, for an $n$-qubit state $|ψ\rangle$ prepared with at most $t$ non-Clifford gates, our algorithms use $\mathsf{poly}(n,2^t,1/\varepsilon)$ time and copies of $|ψ\rangle$ to learn $|ψ\rangle$ to trace distance at most $\varepsilon$.
The first algorithm for this task is more efficient, but requires entangled measurements across two copies of $|ψ\rangle$. The second algorithm uses only single-copy measurements at the cost of polynomial factors in runtime and sample complexity. Our algorithms more generally learn any state with sufficiently large stabilizer dimension, where a quantum state has stabilizer dimension $k$ if it is stabilized by an abelian group of $2^k$ Pauli operators. We also develop an efficient property testing algorithm for stabilizer dimension, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_13409 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates Grewal, Sabee Iyer, Vishnu Kretschmer, William Liang, Daniel Quantum Physics Machine Learning We give a pair of algorithms that efficiently learn a quantum state prepared by Clifford gates and $O(\log n)$ non-Clifford gates. Specifically, for an $n$-qubit state $|ψ\rangle$ prepared with at most $t$ non-Clifford gates, our algorithms use $\mathsf{poly}(n,2^t,1/\varepsilon)$ time and copies of $|ψ\rangle$ to learn $|ψ\rangle$ to trace distance at most $\varepsilon$. The first algorithm for this task is more efficient, but requires entangled measurements across two copies of $|ψ\rangle$. The second algorithm uses only single-copy measurements at the cost of polynomial factors in runtime and sample complexity. Our algorithms more generally learn any state with sufficiently large stabilizer dimension, where a quantum state has stabilizer dimension $k$ if it is stabilized by an abelian group of $2^k$ Pauli operators. We also develop an efficient property testing algorithm for stabilizer dimension, which may be of independent interest. |
| title | Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates |
| topic | Quantum Physics Machine Learning |
| url | https://arxiv.org/abs/2305.13409 |