  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 integralof
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 histories of
the qubits. This representation allows to derive in a simple way estimates that
the total error of operation of a quantum computer without error correction
scales linearly with the number of qubits and the time of operation. It also
allows to discuss Kitaev's toric code interacting with an environment in the
same manner.
