Coherence as a Resource for Shor?s Algorithm

Shor's factoring algorithm provides a superpolynomial speedup over all known classical factoring algorithms. Here, we address the question of which quantum properties fuel this advantage. We investigate a sequential variant of Shor's algorithm with a fixed overall structure and identify the role of coherence for this algorithm quantitatively. We analyze this protocol in the framework of dynamical resource theories, which capture the resource character of operations that can create and detect coherence. This allows us to derive a lower and an upper bound on the success probability of the protocol, which depend on rigorously defined measures of coherence as a dynamical resource. We compare these bounds with the classical limit of the protocol and conclude that within the fixed structure that we consider, coherence is the quantum resource that determines its performance by bounding the success probability from below and above. Therefore, we shine new light on the fundamental role of coherence in quantum computation.

Automated summary

In this paper, we investigate a sequential variant of Shor's algorithm and quantitatively explore the role of coherence. By using the framework of dynamical resource theories, we determine the lower and upper bounds of the success probability of the protocol and find that coherence is the quantum resource that determines its performance within the fixed structure considered.