  A topological quantum computer should allow intrinsically fault-tolerant
quantum computation, but there remains uncertainty about how such a computer
can be implemented. It is known that topological quantum computation can be
implemented with limited quasiparticle braiding capabilities, in fact using
only a single mobile quasiparticle, if the system can be properly initialized
by measurements. It is also known that measurements alone suffice without any
braiding, provided that the measurement devices can be dynamically created and
modified. We study a model in which both measurement and braiding capabilities
are limited. Given the ability to pull nontrivial Fibonacci anyon pairs from
the vacuum with a certain success probability, we show how to simulate
universal quantum computation by braiding one quasiparticle and with only one
measurement, to read out the result. The difficulty lies in initializing the
system. We give a systematic construction of a family of braid sequences that
initialize to arbitrary accuracy nontrivial composite anyons. Instead of using
the Solovay-Kitaev theorem, the sequences are based on a quantum algorithm for
convergent search.
