发表机构
Technical University of Munich; Munich Center for Quantum Science and Technology; National University of Singapore; Centre for Quantum Technologies（慕尼黑工业大学; 慕尼黑量子科学与技术中心; 新加坡国立大学; 量子技术中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种并行经典算法，利用退极化噪声将一维浅层量子电路分割为对数宽度独立片段，实现高效采样，并证明一维噪声恒定深度电路无量子优势，其行为可由随机AC^0电路模拟。
AI 中文摘要
我们考虑由作用于排列在一条线上的 n 个量子比特上的 d 层最近邻双量子比特门组成的量子电路，其中每个量子比特在每一层之前以恒定概率独立地发生退极化。我们描述了一种随机并行算法，该算法从任何此类电路的输出分布中采样，总变差误差为 δ，并行运行时间为 2^{O(d)} log log(n/δ)，并需要 n 2^{O(d)} 次基本实数算术运算。在没有噪声的情况下，相同方法给出并行运行时间为 O(log n) 的精确采样器。对于恒定深度，我们进一步表明，噪声量子电路的输入/输出行为可由随机 AC^0 电路（即具有无界扇入 AND 和 OR 门以及 NOT 门的多项式大小和恒定深度的布尔电路）在恒定误差内重现。因此，在一维中由噪声恒定深度量子电路解决的每个关系问题，也以基本相同的成功概率由随机 AC^0 电路解决。这排除了一维噪声电路中的无条件量子优势，该优势已在二维和三维噪声浅层电路中确立，其中相同门集上的多项式大小经典电路需要深度 Ω(log n / log log n)。我们的算法利用了退极化噪声将一维电路切割成对数宽度的独立片段这一事实，每个片段都可以并行精确采样。
英文摘要
We consider quantum circuits consisting of $d$ layers of nearest-neighbor two-qubit gates acting on $n$ qubits arranged on a line, where every qubit is independently depolarized with a constant probability before each layer. We describe a randomized parallel algorithm which samples from the output distribution of any such circuit to within total variation error $δ$, with parallel runtime $2^{O(d)}\log\log(n/δ)$ and $n 2^{O(d)}$ elementary real-arithmetic operations. Without noise, the same approach gives an exact sampler with parallel runtime $O(\log n)$. For constant depth, we further show that the input/output behavior of the noisy quantum circuit is reproduced up to a constant error by a randomized $\mathsf{AC}^0$-circuit, that is, a Boolean circuit of polynomial size and constant depth with unbounded fan-in AND and OR gates and NOT gates. Consequently, every relation problem solved by a noisy constant-depth quantum circuit in one dimension is also solved, with essentially the same success probability, by a randomized $\mathsf{AC}^0$-circuit. This rules out, for noisy circuits in one dimension, the unconditional quantum advantage established for noisy shallow circuits in two and three dimensions, where polynomial-size classical circuits over the same gate set require depth $Ω(\log n/\log\log n)$. Our algorithm exploits the fact that depolarizing noise cuts a one-dimensional circuit into independent pieces of logarithmic width, each of which can be sampled exactly in parallel.
