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A SCHEMATIC DEFINITION OF QUANTUM POLYNOMIAL TIME COMPUTABILITY

Part of: Theory of computing Computability and recursion theory

Published online by Cambridge University Press:  08 September 2020

TOMOYUKI YAMAKAMI
TOMOYUKI YAMAKAMI*
Affiliation:
FACULTY OF ENGINEERING UNIVERSITY OF FUKUI 3-9-1 BUNKYO, FUKUI910-8507, JAPANE-mail: tomoyukiyamakami@gmail.com
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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 computer

Keywords

quantum computingquantum functionquantum Turing machinequantum circuitschematic definitiondescriptional complexitypolynomial-time computabilitynormal form theorem

MSC classification

Primary: 81P68: Quantum computation
Secondary: 68Q12: Quantum algorithms and complexity 68Q10: Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.) 03D75: Abstract and axiomatic computability and recursion theory
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 4 , December 2020 , pp. 1546 - 1587
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Copyright
© The Association for Symbolic Logic 2020

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