Strictly contractive quantum channels and physically realizable quantum computers

Abstract

We study the robustness of quantum computers under the influence of errors modeled by strictly contractive channels. A channel T is defined to be strictly contractive if, for any pair of density operators \ensuremath{\rho}, \ensuremath{\sigma} in its domain, $\ensuremath{\Vert}T\ensuremath{\rho}\ensuremath{-}T\ensuremath{\sigma}{\ensuremath{\Vert}}_{1}l~k\ensuremath{\Vert}\ensuremath{\rho}\ensuremath{-}\ensuremath{\sigma}{\ensuremath{\Vert}}_{1}$ for some $0l~kl1$ (here $\ensuremath{\Vert}\ensuremath{\cdot}{\ensuremath{\Vert}}_{1}$ denotes the trace norm). In other words, strictly contractive channels render the states of the computer less distinguishable in the sense of quantum detection theory. Starting from the premise that all experimental procedures can be carried out with finite precision, we argue that there exists a physically meaningful connection between strictly contractive channels and errors in physically realizable quantum computers. We show that, in the absence of error correction, sensitivity of quantum memories and computers to strictly contractive errors grows exponentially with storage time and computation time, respectively, and depends only on the constant k and the measurement precision. We prove that strict contractivity rules out the possibility of perfect error correction, and give an argument that approximate error correction, which covers previous work on fault-tolerant quantum computation as a special case, is possible.