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Shor's factoring algorithm provides a super-polynomial speed-up over allknown classical factoring algorithms. Here, we address the question of whichquantum properties fuel this advantage. We investigate a sequential variant ofShor's algorithm with a fixed overall structure and identify the role ofcoherence for this algorithm quantitatively. We analyze this protocol in theframework of dynamical resource theories, which capture the resource characterof operations that can create and detect coherence. This allows us to derive alower and an upper bound on the success probability of the protocol, whichdepend on rigorously defined measures of coherence as a dynamical resource. Wecompare these bounds with the classical limit of the protocol and conclude thatwithin the fixed structure that we consider, coherence is the quantum resourcethat determines its performance by bounding the success probability from belowand above. Therefore, we shine new light on the fundamental role of coherencein quantum computation.

On the Role of Coherence in Shor's Algorithm