A Schematic Definition of Quantum Polynomial Time Computability

TL;DR

This paper introduces a simple, schematic model for quantum polynomial-time computability that aligns with existing models but avoids complex conditions, potentially advancing quantum complexity theory and programming language development.

Contribution

It provides the first schematic, inductive definition of quantum polynomial-time functions that matches standard computational models.

Findings

The schematic definition characterizes all polynomial-time quantum functions.

It simplifies the understanding of quantum computability models.

It opens new avenues for quantum complexity and language theories.

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.