Quadratization in discrete optimization and quantum mechanics

Abstract

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.

Contributors (in order of lines contributed on GitHub): Nike Dattani, Richard Tanburn, Andreas Soteriou, Nicholas Chancellor, Szilard Szalay, Elisabeth Rodriguez-Heck, Hou Tin Chau, Ka Wa Yip, Yudong Cao

References in corpus (10)

• Realizable Hamiltonians for Universal Adiabatic Quantum Computers
• On the construction of model Hamiltonians for adiabatic quantum computation and its application to finding low energy conformations of lattice protein models
• Non-perturbative k-body to two-body commuting conversion Hamiltonians and embedding problem instances into Ising spins
• Simulation of Many-Body Hamiltonians using Perturbation Theory with Bounded-Strength Interactions
• Prime factorization using quantum annealing and computational algebraic geometry
• Pegasus: The second connectivity graph for large-scale quantum annealing hardware
• A Direct Mapping of Max k-SAT and High Order Parity Checks to a Chimera Graph
• Stabilisers as a design tool for new forms of Lechner-Hauke-Zoller Annealer
• Embedding quadratization gadgets on Chimera and Pegasus graphs
• Tunable three-body coupler for superconducting flux qubits

Cited by in corpus (9)

• Unconstrained Binary Models of the Travelling Salesman Problem Variants for Quantum Optimization
• Autonomous Probabilistic Coprocessing with Petaflips per Second
• Pegasus: The second connectivity graph for large-scale quantum annealing hardware
• Computational Overhead of Locality Reduction in Binary Optimization Problems
• Practical designs for permutation symmetric problem Hamiltonians on hypercubes
• Using Quantum Annealers to Calculate Ground State Properties of Molecules
• Ising Machines for Diophantine Problems in Physics
• Decoding quantum error correction with Ising model hardware
• Efficient QUBO transformation for Higher Degree Pseudo Boolean Functions