Unitary designs in nearly optimal depth
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908456627208192 |
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| author | Cui, Laura Schuster, Thomas Brandao, Fernando Huang, Hsin-Yuan |
| author_facet | Cui, Laura Schuster, Thomas Brandao, Fernando Huang, Hsin-Yuan |
| contents | We construct $\varepsilon$-approximate unitary $k$-designs on $n$ qubits in circuit depth $O(\log k \log \log n k / \varepsilon)$. The depth is exponentially improved over all known results in all three parameters $n$, $k$, $\varepsilon$. We further show that each dependence is optimal up to exponentially smaller factors. Our construction uses $\tilde{O}(nk)$ ancilla qubits and ${O}(nk)$ bits of randomness, which are also optimal up to $\log(n k)$ factors. An alternative construction achieves a smaller ancilla count $\tilde{O}(n)$ with circuit depth ${O}(k \log \log nk/\varepsilon)$. To achieve these efficient unitary designs, we introduce a highly-structured random unitary ensemble that leverages long-range two-qubit gates and low-depth implementations of random classical hash functions. We also develop a new analytical framework for bounding errors in quantum experiments involving many queries to random unitaries. As an illustration of this framework's versatility, we provide a succinct alternative proof of the existence of pseudorandom unitaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06216 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unitary designs in nearly optimal depth Cui, Laura Schuster, Thomas Brandao, Fernando Huang, Hsin-Yuan Quantum Physics Computational Complexity Information Theory Mathematical Physics We construct $\varepsilon$-approximate unitary $k$-designs on $n$ qubits in circuit depth $O(\log k \log \log n k / \varepsilon)$. The depth is exponentially improved over all known results in all three parameters $n$, $k$, $\varepsilon$. We further show that each dependence is optimal up to exponentially smaller factors. Our construction uses $\tilde{O}(nk)$ ancilla qubits and ${O}(nk)$ bits of randomness, which are also optimal up to $\log(n k)$ factors. An alternative construction achieves a smaller ancilla count $\tilde{O}(n)$ with circuit depth ${O}(k \log \log nk/\varepsilon)$. To achieve these efficient unitary designs, we introduce a highly-structured random unitary ensemble that leverages long-range two-qubit gates and low-depth implementations of random classical hash functions. We also develop a new analytical framework for bounding errors in quantum experiments involving many queries to random unitaries. As an illustration of this framework's versatility, we provide a succinct alternative proof of the existence of pseudorandom unitaries. |
| title | Unitary designs in nearly optimal depth |
| topic | Quantum Physics Computational Complexity Information Theory Mathematical Physics |
| url | https://arxiv.org/abs/2507.06216 |