发表机构
Yau Mathematical Sciences Center and Department of Mathematics, Tsinghua University; Tsinghua University; Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（丘成桐数学科学中心; 清华大学; 北京雁栖湖应用数学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明任意小的一层去极化噪声会破坏Shor算法的共振现象，从而消除其指数级量子优势，并提出一个多项式时间的经典分解算法替代低频贡献。
AI 中文摘要
我们通过分析噪声量子电路中的共振破坏现象，论证了在任意小错误率的一层去极化噪声下，Shor 算法的指数级量子优势被破坏。首先，我们将 $n$ 量子比特字符串上的测量分布表示为 Pauli 路径积分中 $4^n$ 个波函数的叠加。在无噪声情况下，实现共振峰的比特串保证了分解大数时成功测量的恒定比率。其次，当量子电路中间层具有速率为 $\lambda$ 的独立去极化噪声时，对于任意汉明重量 $d$，我们得到高频项的对应测量率以 $2(1-\lambda)^d$ 为界，低频项以 $O(n^d)/2^{n/2}$ 为界。当 $n$ 和 $d$ 趋于无穷时，该比率趋近于零。共振破坏摧毁了指数级量子优势。此外，我们设计了一个多项式时间 $O(n^d)$ 的经典分解算法来替代低频贡献。
英文摘要
We argue that the exponential quantum advantage in Shor's algorithm is broken under one-layer depolarizing noise of arbitrary small error rate, by analyzing a resonance breaking phenomenon in noisy quantum circuits. First, we express the distribution of measurements on $n$-qubit strings as the superposition of the $4^n$ wave functions in the Pauli path integral. In the noiseless case, the bit-strings achieving resonant peaks guaranteed a constant rate of successful measurements to factor a large number. Secondly, when the middle layer of the quantum circuit has independent depolarizing noise of rate $λ$, for any Hamming weight $d$, we obtain corresponding measurement rate bounded by $2(1-λ)^d$ for higher frequency terms and by $O(n^d)/2^{n/2}$ for low frequency terms. The rate approaches to zero as $n$ and $d$ approaches to infinity. Resonance breaking destroys the exponential quantum advantage. Furthermore, we design a classical factoring algorithm in polynomial time $O(n^d)$ to substitute the low frequency contribution.
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