发表机构
Stanford University; Google Quantum AI（斯坦福大学; 谷歌量子AI）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究\(n\)个量子比特上酉群\(\mathsf{SU}(2^n)\)随机游走谱隙，给出随机泡利旋转和砌砖随机酉电路的恒定谱隙下界，且与\(n\)无关，还证明了克利福德酉算子的类似结果。
AI 中文摘要
我们证明了在\(n\)个量子比特上的酉群\(\mathsf{SU}(2^n)\)上的以下随机游走的谱隙的恒定下界。（i）随机泡利旋转：随机均匀地选择一个\(n\)量子比特泡利算子\(P\)和一个角度\(\theta\in\mathbb{R}/2\pi\mathbb{Z}\)，并应用\(e^{\mathrm i\theta P}\)。（ii）砌砖随机酉电路：从\(\mathsf{SU}(4)\)中独立地随机均匀选择\(n - 1\)个酉算子\(U_i\)，并在两个量子比特\(2j - 1, 2j\)上应用\(U_{2j - 1}\)，然后在两个量子比特\(2j, 2j + 1\)上应用\(U_{2j}\)。重要的是，谱隙与\(n\)无关，并且均匀地适用于\(\mathsf{SU}(2^n)\)的所有有限维酉表示，包括那些出现在酉\(t\)-设计中的表示。我们还证明了关于克利福德酉算子的类似恒定间隙结果，这对于我们关于砌砖随机酉电路的结果是必不可少的。
英文摘要
We prove constant lower bounds for the spectral gap of the following random walks on unitary groups $\mathsf{SU}(2^n)$ on $n$ qubits. (i) Random Pauli Rotation: choose an $n$-qubit Pauli operator $P$ and an angle $θ\in [0,2π)$, both uniformly at random, and apply $e^{\mathrm i θP}$. (ii) Brickwork Random Unitary Circuit: choose $n-1$ unitaries $U_{i}$ uniformly at random from $\mathsf{SU}(4)$ independently, and apply $U_{2j-1}$ on two qubits $2j-1, 2j$ and then $U_{2j}$ on two qubits $2j, 2j+1$. Importantly, the spectral gap lower bounds are independent of $n$ and apply for all finite dimensional unitary representations of $\mathsf{SU}(2^n)$ uniformly, including those that appear in unitary $t$-designs. We also prove analogous constant-gap results for orthogonal and Clifford groups; the latter is indispensable for our result on the Brickwork Random Unitary Circuit.
Comments32 pages. 1 figrue. Julia and Mathematica code (v2) analogous results on orthogonal groups, simplified calculation using Gelfand pairs, and unitary complexity growth spelled out
