Resonance Breaking in Noisy Shor's Algorithm

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 $\lambda$, for any Hamming weight $d$, we obtain corresponding measurement rate bounded by $2(1-\lambda)^d$ 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.

Quantum Physics (quant-ph); Computational Complexity (cs.CC); Information Theory (cs.IT); Mathematical Physics (math-ph); Operator Algebras (math.OA)

arXiv:2609.37636 [quant-ph]

(or arXiv:2609.37636v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.37636 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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