[Paper Review] Minimizing polynomial functions on quantum computers

Raouf Dridi, Hedayat Alghassi+1 more|arXiv (Cornell University)|Mar 19, 2019

Quantum Computing Algorithms and Architecture16 references4 citations

TL;DR

This paper proposes a novel integration of computational algebraic geometry with adiabatic quantum computing (AQC), demonstrating how Groebner bases and polynomial system solving can be used to model and analyze quantum annealing processes. It presents concrete quantum-ready algorithms for binary optimization, factoring, and compiling using Ising Hamiltonians, and introduces a new method to compute Groebner bases on quantum computers, linking them to non-degenerate critical points in AQC via Morse theory.

ABSTRACT

This expository paper reviews some of the recent uses of computational algebraic geometry in classical and quantum optimization. The paper assumes an elementary background in algebraic geometry and adiabatic quantum computing (AQC), and concentrates on presenting concrete examples (with Python codes tested on a quantum computer) of applying algebraic geometry constructs: solving binary optimization, factoring, and compiling. Reversing the direction, we also briefly describe a novel use of quantum computers to compute Groebner bases for toric ideals. We also show how Groebner bases play a role in studying AQC at a fundamental level within a Morse theory framework. We close by placing our work in perspective, by situating this leg of the journey, as part of a marvelous intellectual expedition that began with our ancients over 4000 years ago.

Motivation & Objective

• To bridge algebraic geometry and adiabatic quantum computing through practical, computable methods.
• To demonstrate how polynomial optimization problems can be encoded and solved on quantum annealers using algebraic geometry constructs.
• To introduce a new quantum algorithm for computing Groebner bases of toric ideals.
• To establish a theoretical link between Groebner bases and the non-degenerate critical points in AQC Hamiltonians.
• To provide accessible, code-tested examples of quantum algorithms grounded in algebraic geometry.

Proposed method

• The paper uses the duality between algebraic varieties and coordinate rings to translate polynomial systems into quantum-mechanical Ising Hamiltonians.
• It applies Groebner basis computation to triangulate polynomial systems, enabling efficient solution via quantum annealing.
• The method maps anti-crossings in AQC to saddle points of a polynomial function f(s, λ), where s is the adiabatic path parameter and λ is a coupling parameter.
• The critical points of f are found by computing the variety of the ideal generated by ∂f/∂s and ∂f/∂λ, and non-degeneracy is verified via the Hessian determinant.
• The saturation of this ideal with the Hessian determinant isolates non-degenerate critical points, which are essential for avoiding level crossings in AQC.
• The approach is validated through Python implementations tested on D-Wave quantum processors, demonstrating feasibility on real hardware.

Experimental results

Research questions

• RQ1 How can algebraic geometry be systematically applied to model and solve binary optimization problems on quantum annealers?
• RQ2 Can Groebner bases be computed efficiently on quantum computers, and what is their role in quantum algorithm design?
• RQ3 How do non-degenerate critical points in the adiabatic evolution Hamiltonian relate to the structure of polynomial systems and quantum tunneling?
• RQ4 What is the role of Morse theory in characterizing anti-crossings in AQC using polynomial functions?
• RQ5 Can the fundamental dynamics of AQC be analyzed through the lens of algebraic geometry, particularly via ideals and their varieties?

Key findings

• The paper successfully encodes binary optimization problems into Ising Hamiltonians using algebraic geometry, enabling quantum annealing solutions.
• A Groebner basis computation was used to triangulate a system of polynomial equations, reducing the problem to a solvable form on a quantum processor.
• The method demonstrated that anti-crossings in AQC correspond to saddle points of a polynomial function f(s, λ), which are identified via the vanishing of the gradient and non-vanishing Hessian determinant.
• The authors show that non-degenerate critical points in AQC can be detected using Groebner bases and ideal saturation, providing a computational criterion for avoiding level crossings.
• Theoretical analysis confirms that the Hessian determinant's non-vanishing at a critical point ensures non-degeneracy, which is essential for adiabatic evolution.
• The paper provides working Python code for Groebner basis computation and quantum annealing mapping, tested on real D-Wave hardware, validating the approach in practice.