Symmetry boost of the fidelity of Shor factoring

Frequently, subroutines in quantum computers have the structure $\mathcal{F}\mathcal{U}\mathcal{F}^{-1}$, where $\mathcal{F}$ is some unitary transform and $\mathcal{U}$ is performing a quantum computation. In this paper we suggest that if, in analogy to spin echoes, $\mathcal{F}$ and $\mathcal{F}^{-1}$ can be implemented symmetrically such that $\mathcal{F}$ and $\mathcal{F}^{-1}$ have the same hardware errors, a symmetry boost in the fidelity of the combined $\mathcal{F}\mathcal{U}\mathcal{F}^{-1}$ quantum operation results. Running the complete gate--by--gate implemented Shor algorithm, we show that the fidelity boost can be as large as a factor 10. Corroborating and extending our numerical results, we present analytical scaling calculations that show that a symmetry boost persists in the practically interesting case of a large number of qubits. Our analytical calculations predict a minimum boost factor of about 3, valid for all qubit numbers, which includes the boost factor 10 observed in our low-qubit-number simulations. While we find and document this symmetry boost here in the case of Shor's algorithm, we suggest that other quantum algorithms might profit from similar symmetry-based performance boosts whenever $\mathcal{F}\mathcal{U}\mathcal{F}^{-1}$ sub-units of the corresponding quantum algorithm can be identified.