所有酉算子都具有恒定深度的量子电路
All Unitaries Have Constant Depth Quantum Circuits
- Columbia University（哥伦比亚大学）
- University of Washington（华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明所有n量子比特酉算子可用多项式深度电路近似，并在允许无界扇出门时实现恒定深度，通过连接酉合成问题与局部可解码码和私有信息检索。
AI中文摘要：
众所周知，每个 $n$ 量子比特的酉算子都可以通过使用单量子比特和双量子比特门的 $2^{O(n)}$ 深度量子电路来实现。即使允许无限数量的辅助量子比特，指数深度对于一般酉算子是否必要的问题一直是开放的。我们表明，也许令人惊讶的是，所有酉算子都可以通过深度为 $\poly(n,\log 1/\epsilon)$ 的单量子比特和双量子比特门电路近似到算子范数 $\epsilon$，并使用 $2^{O(n)}$ 个辅助量子比特。换句话说，每个 $n$ 量子比特的酉算子都可以并行化到多项式深度。此外，如果我们允许无界扇出门，这些电路可以进一步减少到恒定深度。我们的构造利用了Aaronson和Kuperberg的酉合成问题与复杂性理论和密码学中的局部可解码码和私有信息检索之间的新颖关系。
英文摘要：
It is well-known that every $n$-qubit unitary can be implemented by a $2^{O(n)}$-depth quantum circuit using single- and two-qubit gates. It has been open whether exponential depth is *necessary* for general unitaries, even when allowing an unlimited number of ancilla qubits. Here we show, perhaps surprisingly, that all unitaries can be implemented exactly by a circuit of one- and two-qubit gates of depth $\mathsf{poly}(n)$ with $2^{O(n)}$ ancilla qubits. In other words, every $n$-qubit unitary can be parallelized to polynomial depth. In fact, our depth bound is *linear* in $n$, which is the best possible, and an exponential improvement on the previous best bound of $2^{n/2}$ due to Rosenthal [TQC 2022, Quantum 2026]. Moreover, if we allow unbounded fan-out gates, these circuits can be further reduced to *constant* depth. Our construction takes advantage of a novel relationship connecting the unitary synthesis problem of Aaronson and Kuperberg to locally-decodable codes and private information retrieval from complexity theory and cryptography, and has a natural interpretation in bosonic quantum computation.
