  In the past four decades, the notion of quantum polynomial-time computability
has been mathematically modeled by quantum Turing machines as well as quantum
circuits. This paper seeks the third model, which is a quantum analogue of the
schematic (inductive or constructive) definition of (primitive) recursive
functions. For quantum functions mapping finite-dimensional Hilbert spaces to
themselves, we present such a schematic definition, composed of a small set of
initial quantum functions and a few construction rules that dictate how to
build a new quantum function from the existing ones. We prove that our
schematic definition precisely characterizes all functions that can be
computable with high success probabilities on well-formed quantum Turing
machines in polynomial time, or equivalently uniform families of
polynomial-size quantum circuits. Our new, schematic definition is quite simple
and intuitive and, more importantly, it avoids the cumbersome introduction of
the well-formedness condition imposed on a quantum Turing machine model as well
as of the uniformity condition necessary for a quantum circuit model. Our new
approach can further open a door to the descriptional complexity of quantum
functions, to the theory of higher-type quantum functionals, to the development
of new first-order theories for quantum computing, and to the designing of
programming languages for real-life quantum computers.
