Symmetry boost of the fidelity of Shor factoring

Abstract

In Shor's algorithm quantum subroutines occur with the structure $\mathcal{F}\mathcal{U}{\mathcal{F}}^{\ensuremath{-}1}$, where $\mathcal{F}$ is a unitary transform and $\mathcal{U}$ is performing a quantum computation. Examples are quantum adders and subunits of quantum modulo adders. In this paper we show, both analytically and numerically, that if, in analogy to spin echoes, $\mathcal{F}$ and ${\mathcal{F}}^{\ensuremath{-}1}$ can be implemented symmetrically when executing Shor's algorithm on actual, imperfect quantum hardware, such that $\mathcal{F}$ and ${\mathcal{F}}^{\ensuremath{-}1}$ have the same hardware errors, a symmetry boost in the fidelity of the combined $\mathcal{F}\mathcal{U}{\mathcal{F}}^{\ensuremath{-}1}$ quantum operation results when compared to the case in which the errors in $\mathcal{F}$ and ${\mathcal{F}}^{\ensuremath{-}1}$ are independently random. Running the complete gate-by-gate implemented Shor algorithm, we show that the symmetry-induced fidelity boost can be as large as a factor 4. While most of our analytical and numerical results concern the case of over- and under-rotation of controlled rotation gates, in the numerically accessible case of Shor's algorithm with a small number of qubits, we show explicitly that the symmetry boost is robust with respect to more general types of errors. While, expectedly, additional error types reduce the symmetry boost, we show explicitly, by implementing general off-diagonal $\mathrm{SU}(N)$ errors ($N=2,4,8$), that the boost factor scales like a Lorentzian in $\ensuremath{\delta}/\ensuremath{\sigma}$, where $\ensuremath{\sigma}$ and $\ensuremath{\delta}$ are the error strengths of the diagonal over- and underrotation errors and the off-diagonal $\mathrm{SU}(N)$ errors, respectively. The Lorentzian shape also shows that, while the boost factor may become small with increasing $\ensuremath{\delta}$, it declines slowly (essentially like a power law) and is never completely erased. We also investigate the effect of diagonal nonunitary errors, which, in analogy to unitary errors, reduce but never erase the symmetry boost. Going beyond the case of small quantum processors, we present analytical scaling results that show that the symmetry boost persists in the practically interesting case of a large number of qubits. We illustrate this result explicitly for the case of Shor factoring of the semiprime RSA-1024, where, analytically, focusing on over- and underrotation errors, we obtain a boost factor of about 10. In addition, we provide a proof of the fidelity product formula, including its range of applicability.