Unitary designs in nearly optimal depth

• URL: http://arxiv.org/abs/2507.06216v2
• Date: Sat, 19 Jul 2025 04:11:58 GMT
• Title: Unitary designs in nearly optimal depth
• Authors: Laura Cui, Thomas Schuster, Fernando Brandao, Hsin-Yuan Huang,
• Abstract summary: We construct unitary designs on qubits in circuit depth $O(log k log log n k / varepsilon)$.<n>The depth is exponentially improved over all known results in all three parameters $n$, $k$, $varepsilon$.<n>We also develop a new analytical framework for bounding errors in quantum experiments involving many queries to random unitaries.
• Score: 40.28216388589026
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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