$\mathcal{O}(n)$ alternative to Quantum Fourier Transform with efficient neural net classical post-processing

TL;DR

This paper introduces a shallow, $ ext{O}(n)$-depth quantum-inspired circuit with neural network post-processing that can replace the traditional $ ext{O}(n^2)$ quantum Fourier transform in Shor's algorithm, maintaining essential information.

Contribution

The authors design and analyze shallow circuits that preserve key properties of the QFT, enabling classical neural network post-processing to effectively substitute for the quantum Fourier transform.

Findings

Shallow HP-$L$ circuits retain exponential Fisher information.

The $ ext{O}(n)$ HP-$1$ circuit can replace the $ ext{O}(n^2)$ QFT in Shor's algorithm.

Neural network post-processing efficiently reconstructs hidden subgroup information.

Abstract

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.