[Paper Review] Abelian Hypergroups and Quantum Computation

Juan Bermejo-Vega, Kevin C. Zatloukal|arXiv (Cornell University)|Sep 18, 2015

Computability, Logic, AI Algorithms98 references3 citations

TL;DR

This paper introduces an abelian hypergroup stabilizer formalism to generalize the Gottesman-Knill theorem and develop a provably efficient quantum algorithm for finding hidden subhypergroups in nilpotent abelian hypergroups. By connecting the hidden normal subgroup problem (HNSP) in nonabelian groups to the hidden subhypergroup problem (HSHP) via conjugacy classes, the authors achieve a new quantum algorithm for HNSP on nilpotent groups, demonstrating classical simulability and adaptive Fourier sampling techniques for non-unitary stabilizers.

ABSTRACT

Motivated by a connection, described here for the first time, between the hidden normal subgroup problem (HNSP) and abelian hypergroups (algebraic objects that model collisions of physical particles), we develop a stabilizer formalism using abelian hypergroups and an associated classical simulation theorem (a la Gottesman-Knill). Using these tools, we develop the first provably efficient quantum algorithm for finding hidden subhypergroups of nilpotent abelian hypergroups and, via the aforementioned connection, a new, hypergroup-based algorithm for the HNSP on nilpotent groups. We also give efficient methods for manipulating non-unitary, non-monomial stabilizers and an adaptive Fourier sampling technique of general interest.

Motivation & Objective

• To understand the quantum algorithm for the hidden normal subgroup problem (HNSP) in nonabelian groups by generalizing the abelian group stabilizer formalism beyond groups.
• To establish a formal connection between the HNSP and abelian hypergroups, showing that the structure of conjugacy classes in nonabelian groups forms an abelian hypergroup.
• To develop a stabilizer formalism for abelian hypergroups that enables classical simulation and efficient quantum algorithms, extending the Gottesman-Knill framework.
• To provide a new quantum algorithm for HNSP on nilpotent groups via the hypergroup formalism, with provable efficiency and adaptive Fourier sampling.
• To extend tools for non-unitary, non-monomial stabilizers and demonstrate classical simulability of related quantum circuits.

Proposed method

• The authors define abelian hypergroups as generalizations of groups where collisions produce sets of outcomes, modeling particle interactions.
• They establish that the conjugacy classes of a nonabelian group form an abelian hypergroup under multiplication, capturing the structure of normal subgroups.
• A stabilizer formalism is constructed for abelian hypergroups, generalizing the Gottesman-Knill theorem to allow classical simulation of quantum circuits over these structures.
• The method includes an adaptive Fourier sampling technique tailored for non-unitary and non-monomial stabilizer states.
• The formalism is applied to nilpotent abelian hypergroups, yielding an efficient quantum algorithm for the hidden subhypergroup problem (HSHP).
• Key equations include character orthogonality and probability bounds for measuring trivial irreps, used to analyze success probability.

Experimental results

Research questions

• RQ1 Can the hidden normal subgroup problem (HNSP) in nonabelian groups be reformulated as a hidden subhypergroup problem (HSHP) in abelian hypergroups?
• RQ2 Does a stabilizer formalism for abelian hypergroups exist that enables classical simulation of quantum circuits, generalizing the Gottesman-Knill theorem?
• RQ3 Can efficient quantum algorithms be constructed for HSHP in nilpotent abelian hypergroups using this formalism?
• RQ4 What is the success probability of measuring the trivial irrep in the Fourier sampling of hypergroup states, and how does it relate to subgroup structure?
• RQ5 Which classes of non-nilpotent, super-solvable groups admit efficient quantum algorithms via this hypergroup approach?

Key findings

• The paper establishes a formal connection between the HNSP in nonabelian groups and the HSHP in abelian hypergroups, showing that conjugacy classes form an abelian hypergroup under multiplication.
• An abelian hypergroup stabilizer formalism is developed, generalizing the Gottesman-Knill theorem to allow efficient classical simulation of quantum circuits over abelian hypergroups.
• An efficient quantum algorithm is constructed for finding hidden subhypergroups in nilpotent abelian hypergroups, with success probability bounded away from zero by a constant (e.g., at most 2/3 for dihedral groups).
• The algorithm achieves high success probability for dihedral groups, with the probability of measuring the trivial irrep bounded by 2/3 when n is odd and 1/2 when n is even, both constants independent of n.
• The method includes an adaptive Fourier sampling technique that enables efficient measurement of non-unitary, non-monomial stabilizer states, extending the scope of simulation techniques.
• The algorithm fails on some super-solvable groups (e.g., the affine group over Z_p), showing the method is not universally applicable, but succeeds on others like dihedral groups, highlighting a non-trivial class of solvable groups for which HNSP is efficiently solvable.