A Schematic Definition of Quantum Polynomial Time Computability
Tomoyuki Yamakami
TL;DR
The paper introduces a minimalist, inductive definition of quantum polynomial-time computability through a schematic class $Box^{\mathrm{QP}}_{1}$ built from a small, universal set of initial quantum functions and a few construction rules. It proves that $Box^{\mathrm{QP}}_{1}$ (and its norm-preserving variant) exactly characterizes $FBQP$, connecting unitary quantum computation with a Kleene-style normal form and offering a new lens on descriptional complexity and higher-type quantum functionals. Central to the result is a constructive reduction showing how any well-formed QTM can be simulated by a $\widehat{Box^{\mathrm{QP}}_{1}}$-function, together with a polynomial-time encoding/decoding scheme that yields high-probability outputs. The framework avoids the traditional well-formedness and uniformity constraints inherent in QTMs and circuits, facilitating potential applications to quantum programming languages and to the study of descriptional complexity of quantum languages and functionals. Overall, the work provides a compact, intuitive, and versatile description of quantum polynomial-time computability with broad theoretical and practical implications for programming, logic, and higher-type quantum computation.
Abstract
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.