  We investigate models of nonlinear quantum computation based on deterministic
positive trace-preserving (PTP) channels and evolution equations. The models
are defined in any finite Hilbert space, but the main results are for dimension
$N \! = \! 2$. For every normalizable linear or nonlinear positive map $\phi$
on bounded linear operators $X$, there is an associated normalized PTP channel
$ \phi(X) / {\rm tr}[\phi(X)]$. Normalized PTP channels include unitary mean
field theories, such as the Gross-Pitaevskii equation for interacting bosons,
as well as models of linear and nonlinear dissipation. They classify into 4
types, yielding 3 distinct forms of nonlinearity whose computational power we
explore. In the qubit case these channels support Bloch ball torsion and other
distortions studied previously, where it has been shown that such nonlinearity
can be used to increase the separation between a pair of close qubit states,
suggesting an exponential speedup for state discrimination. Building on this
idea, we argue that this operation can be made robust to noise by using
dissipation to induce a bifurcation to a novel phase where a pair of attracting
fixed points create an intrinsically fault-tolerant nonlinear state
discriminator.
