Abstract structure of unitary oracles for quantum algorithms
William Zeng (University of Oxford), Jamie Vicary (University of, Singapore, University of Oxford)
TL;DR
This paper provides an abstract algebraic framework for understanding quantum oracles using dagger-Frobenius algebras, leading to a new algorithm for identifying group homomorphisms and insights into signal-flow networks.
Contribution
It introduces a novel algebraic structure for quantum oracles and applies it to develop a deterministic group homomorphism identification algorithm.
Findings
New algebraic characterization of quantum oracles
Development of a deterministic group homomorphism algorithm
Application to categorical theory of signal-flow networks
Abstract
We show that a pair of complementary dagger-Frobenius algebras, equipped with a self-conjugate comonoid homomorphism onto one of the algebras, produce a nontrivial unitary morphism on the product of the algebras. This gives an abstract understanding of the structure of an oracle in a quantum computation, and we apply this understanding to develop a new algorithm for the deterministic identification of group homomorphisms into abelian groups. We also discuss an application to the categorical theory of signal-flow networks.
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