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arXiv:2608.18650

(Almost) quadruply optimal unitary designs in 1D

Guoding Liu, Jonas Helsen

quant-phSubmitted Aug 19, 2026• Announced Aug 20, 2026• 0 citations
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Abstract

We construct $n$-qubit approximate unitary $k$-designs in 1D systems, achieving circuit depth $O(\log(n/\varepsilon) + k\log k)$ with relative error $\varepsilon$ and requiring $O(nk\log k)$ magic gates. This matches existing lower bounds $Ω(\log(n/\varepsilon) + k)$ for circuit depth, and $\widetildeΩ(nk)$ for the required number of $T$ gates, up to a $\log k$ factor, achieving simultaneous near-optimality in all parameters. Our construction is based on a combination and refinement of two existing results. We reduce the required magic block size for breaking Clifford symmetries in the magic-augmented circuit construction of Zhang et al. from $O(k\log k)$ to $O(\log k)$. We also improve the breakthrough construction of Chen et al. to construct a generating set of 1D local constant-depth circuits for the unitary group with a constant spectral gap, making $O(\log k)$-local random unitaries realizable in depth $O(k\log k)$. As a by-product, we provide a constant-size 1D-local generating set for the Clifford group, which we expect to be of independent interest. Combining the two results with the gluing lemma, we prove the final result.

Comments
50 pages, 2 figures. Comments are welcome!

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We construct ε\varepsilonε-approximate unitary kkk-designs on nnn qubits in circuit depth O(log⁡klog⁡log⁡nk/ε)O(\log k \log \log n k / \varepsilon)O(logkloglognk/ε). The depth is exponentially improved over all known results in all three parameters nnn, kkk, ε\varepsilonε. We further show that each dependence is optimal up to exponentially smaller factors. Our construction uses O~(nk)\tilde{O}(nk)O~(nk) ancilla qubits and O(nk){O}(nk)O(nk) bits of randomness, which are also optimal up to log⁡(nk)\log(n k)log(nk) factors. An alternative construction achieves a smaller ancilla count O~(n)\tilde{O}(n)O~(n) with circuit depth O(klog⁡log⁡nk/ε){O}(k \log \log nk/\varepsilon)O(kloglognk/ε). 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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We study the problem of constructing strong approximate unitary kkk-designs on DDD-dimensional grids (and more generally on Cartesian products of graphs), building on the work of Schuster et al. arXiv:2509.26310 which establishes strong unitary designs in 1D and in all-to-all connectivity. We provide two constructions. The first construction leverages the existing all-to-all connectivity result with general routing theory to provide flexible (but slightly suboptimal) strong kkk-designs in arbitrary connectivities. The second construction is more direct, requires no auxiliaries and has provably optimal depth (in the number of qubits nnn) for DDD-dimensional grids with constant dimension. Combining these techniques also allows us to construct strong pseudorandom unitaries on DDD-dimensional grids with provably optimal depth.