# A Schematic Definition of Quantum Polynomial Time Computability

In the past four decades, the notion of quantum polynomial-time computability has been realized by the theoretical models of quantum Turing machines and quantum circuits. Here, we seek a 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 quantum functions. 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 other functions and to the theory of higher-type quantum functionals.

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