A global view of quantum computation with noisy components

The operation of a quantum computer is considered as a general quantum operation on a mixed state on many qubits followed by a measurement. The general quantum operation is further represented as a Feynman-Vernon double path integral over the histories of the qubits and of an environment, and afterward tracing out the environment. The qubit histories are taken to be paths on the two-sphere $S^2$ as in Klauder's coherent-state path integral of spin, and the environment is assumed to consist of harmonic oscillators initially in thermal equilibrium, and linearly coupled to to qubit operators $\hat{S}_z$. The environment can then be integrated out to give a Feynman-Vernon influence action coupling the forward and backward history of each qubit. This representation allows to derive in a simple way an estimate that the total error of operation of a quantum computer scales linearly with the number of qubits and the time of operation, and allows to quantitatively discuss counter-arguments against quantum computing advanced by Kalai. This line of reasoning is applied to simple examples, and to Kitaev's toric code.