Designs from magic-augmented Clifford circuits

• URL: http://arxiv.org/abs/2507.02828v1
• Date: Thu, 03 Jul 2025 17:41:03 GMT
• Title: Designs from magic-augmented Clifford circuits
• Authors: Yuzhen Zhang, Sagar Vijay, Yingfei Gu, Yimu Bao,
• Abstract summary: We prove that shallow Clifford circuits can generate approximate unitary and state $k$-designs with $epsilon$ relative error.<n>The required number of magic gates is parametrically reduced when considering $k$-designs with bounded additive error.
• Score: 1.5937802651366269
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $\epsilon$ relative error. The total circuit depth for these constructions on $N$ qubits is $O(\log (N/\epsilon)) +2^{O(k\log k)}$ in one dimension and $O(\log\log(N/\epsilon))+2^{O(k\log k)}$ 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.

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