# Quadratization in discrete optimization and quantum mechanics

arXiv: https://arxiv.org/abs/1901.04405 · HTML: https://mlsanity.com/paper/1901.04405/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jan 2019.

## Built on by (most cited first, 5 of 5)

- [Application of QUBO solver using black-box optimization to structural design for resonance avoidance](https://mlsanity.com/paper/2204.04906/) (2022)
- [CCuantuMM: Cycle-Consistent Quantum-Hybrid Matching of Multiple Shapes](https://mlsanity.com/paper/2303.16202/) (2023)
- [Quantum-enhanced Computer Vision: Going Beyond Classical Algorithms](https://mlsanity.com/paper/2510.07317/) (2025)
- [Efficient QUBO transformation for Higher Degree Pseudo Boolean Functions](https://mlsanity.com/paper/2107.11695/) (2021)
- [Non-Negative Matrix Factorization Using Non-Von Neumann Computers](https://mlsanity.com/paper/2512.00675/) (2025)

## Abstract (arXiv)

A book about turning high-degree optimization problems into quadratic optimization problems that maintain the same global minimum (ground state). This book explores quadratizations for pseudo-Boolean optimization, perturbative gadgets used in QMA completeness theorems, and also non-perturbative k-local to 2-local transformations used for quantum mechanics, quantum annealing and universal adiabatic quantum computing. The book contains ~70 different Hamiltonian transformations, each of them on a separate page, where the cost (in number of auxiliary binary variables or auxiliary qubits, or number of sub-modular terms, or in graph connectivity, etc.), pros, cons, examples, and references are given. One can therefore look up a quadratization appropriate for the specific term(s) that need to be quadratized, much like using an integral table to look up the integral that needs to be done. This book is therefore useful for writing compilers to transform general optimization problems, into a form that quantum annealing or universal adiabatic quantum computing hardware requires; or for transforming quantum chemistry problems written in the Jordan-Wigner or Bravyi-Kitaev form, into a form where all multi-qubit interactions become 2-qubit pairwise interactions, without changing the desired ground state. Applications cited include computer vision problems (e.g. image de-noising, un-blurring, etc.), number theory (e.g. integer factoring), graph theory (e.g. Ramsey number determination), and quantum chemistry. The book is open source, and anyone can make modifications here: https://github.com/HPQC-LABS/Book_About_Quadratization.