Resonance Breaking in Noisy Shor's Algorithm

2026-09-29

AI Summary

This study investigates how the exponential quantum advantage of Shor’s algorithm degrades under noisy conditions. By leveraging Pauli path integrals and quantum noise modeling, it analyzes the disruption of resonance in period-finding within noisy circuits, revealing that single-layer depolarizing noise eliminates resonance peaks and thereby destroys the quantum advantage. The authors rigorously prove that any arbitrarily small error rate causes the measurement success probability to decay exponentially with qubit count, completely nullifying quantum supremacy. Building on this insight, the work proposes a polynomial-time classical number-theoretic alternative that achieves efficient integer factorization by capturing low-frequency contributions. These findings provide a novel theoretical foundation for evaluating fault-tolerance thresholds in quantum computing.

Abstract

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.

Problem

• Shor's algorithm
• depolarizing noise
• resonance breaking
• quantum advantage
• noisy quantum circuits

Innovation

• Shor's algorithm
• resonance breaking
• depolarizing noise
• Pauli path integral
• quantum advantage