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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.02828 |
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| _version_ | 1866914373700681728 |
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| author | Zhang, Yuzhen Vijay, Sagar Gu, Yingfei Bao, Yimu |
| author_facet | Zhang, Yuzhen Vijay, Sagar Gu, Yingfei Bao, Yimu |
| contents | We introduce magic-augmented Clifford circuits -- architectures in which Clifford circuits are preceded and/or followed by constant-depth circuits of non-Clifford (``magic") gates -- as a resource-efficient way to realize approximate $k$-designs, with reduced circuit depth and usage of magic. We prove that shallow Clifford circuits, when augmented with constant-depth circuits of magic gates, can generate approximate unitary and state $k$-designs with $ε$ relative error. The total circuit depth for these constructions on $N$ qubits is $O(\log (N/ε)) +2^{O(k\log k)}$ in one dimension and $O(\log\log(N/ε))+2^{O(k\log k)}$ in all-to-all circuits using ancillas, which improves upon previous results for small $k \geq 4$. Furthermore, our construction of relative-error state $k$-designs only involves states with strictly local magic. The required number of magic gates is parametrically reduced when considering $k$-designs with bounded additive error. As an example, we show that shallow Clifford circuits followed by $O(k^2)$ single-qubit magic gates, independent of system size, can generate an additive-error state $k$-design. We develop a classical statistical mechanics description of our random circuit architectures, which provides a quantitative understanding of the required depth and number of magic gates for additive-error state $k$-designs. We also prove no-go theorems for various architectures to generate designs with bounded relative error. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02828 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Designs from magic-augmented Clifford circuits Zhang, Yuzhen Vijay, Sagar Gu, Yingfei Bao, Yimu Quantum Physics Statistical Mechanics Strongly Correlated Electrons Information Theory High Energy Physics - Theory We introduce magic-augmented Clifford circuits -- architectures in which Clifford circuits are preceded and/or followed by constant-depth circuits of non-Clifford (``magic") gates -- as a resource-efficient way to realize approximate $k$-designs, with reduced circuit depth and usage of magic. We prove that shallow Clifford circuits, when augmented with constant-depth circuits of magic gates, can generate approximate unitary and state $k$-designs with $ε$ relative error. The total circuit depth for these constructions on $N$ qubits is $O(\log (N/ε)) +2^{O(k\log k)}$ in one dimension and $O(\log\log(N/ε))+2^{O(k\log k)}$ in all-to-all circuits using ancillas, which improves upon previous results for small $k \geq 4$. Furthermore, our construction of relative-error state $k$-designs only involves states with strictly local magic. The required number of magic gates is parametrically reduced when considering $k$-designs with bounded additive error. As an example, we show that shallow Clifford circuits followed by $O(k^2)$ single-qubit magic gates, independent of system size, can generate an additive-error state $k$-design. We develop a classical statistical mechanics description of our random circuit architectures, which provides a quantitative understanding of the required depth and number of magic gates for additive-error state $k$-designs. We also prove no-go theorems for various architectures to generate designs with bounded relative error. |
| title | Designs from magic-augmented Clifford circuits |
| topic | Quantum Physics Statistical Mechanics Strongly Correlated Electrons Information Theory High Energy Physics - Theory |
| url | https://arxiv.org/abs/2507.02828 |