用浅层电路替代量子傅里叶变换，实现高效经典后处理
$\mathcal{O}(n)$ alternative to Quantum Fourier Transform with efficient neural net classical post-processing
- 设计基于哈达玛和受控相位门的浅层HP-1电路，保持移位不变性
- 数值实验表明其离散费雪信息指数增长，可替代传统QFT
- 适合关注量子算法优化与经典神经网络后处理的研究者
量子傅里叶变换（QFT）是隐藏子群问题（HSP）算法的关键组件，包括用于因式分解的肖尔算法。然而，QFT的电路深度对近期硬件构成挑战。为寻找更浅的替代方案，本文识别出QFT在解决HSP时依赖的两个特性：第一，QFT具有移位不变性，可消除随机全局移位；第二，测量结果中保留了关于隐藏子群生成元的信息，通过离散费雪信息进行量化。本文构建了一类使用哈达玛门和受控相位门的浅层电路——HP-L电路，并证明其保持移位不变性。数值分析显示，这些电路能保留指数增长的费雪信息。其中，复杂度为$/mathcal{O}(n)$的HP-1电路可在肖尔算法中替代$/mathcal{O}(n^2)$的QFT，且配合高效的神经网络实现经典后处理，已通过数值验证。
原文摘要 · Abstract (English)
The Quantum Fourier Transform (QFT) is required by hidden subgroup problem (HSP) algorithms, including Shor's algorithm for factoring. The circuit depth of the QFT remains challenging for near-term hardware. To find shallower alternatives we identify two properties that are exploited by the QFT to enable HSP. Firstly, the shift invariance of the QFT allows for the removal of a random overall shift. Secondly, the QFT retains information about the hidden subgroup generator accessible in the measurement outcomes. We quantify that information via the discrete Fisher information. We construct a family of shallow circuits using Hadamards and controlled-Phase gates, HP-$L$ circuits, that we prove preserve shift invariance. Numerical analysis shows these circuits retain exponentially growing Fisher information. The $\mathcal{O}(n)$ HP-$1$ can replace the $\mathcal{O}(n^2)$ QFT in Shor's algorithm, as demonstrated numerically, with an efficient neural network implementing classical post-processing.
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