  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.
