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 Shors 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.
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