Minimizing minor embedding energy: an application in quantum annealing

TL;DR

This paper investigates how to optimize the ferromagnetic coupling strength in minor embedding for quantum annealing, demonstrating that minimizing embedding energy enhances the probability of finding correct ground states.

Contribution

It introduces an analytical lower bound on the coupling strength that maximizes ground-state probability, improving upon previous bounds.

Findings

Ground-state probability is maximized when embedding energy is minimized.

A new, tighter lower bound on coupling strength is derived.

Experimental confirmation supports the theoretical results.

Abstract

A significant challenge in quantum annealing is to map a real-world problem onto a hardware graph of limited connectivity. If the maximum degree of the problem graph exceeds the maximum degree of the hardware graph, one employs minor embedding in which each logical qubit is mapped to a tree of physical qubits. Pairwise interactions between physical qubits in the tree are set to be ferromagnetic with some coupling strength $F<0$. Here we address the question of what value $F$ should take in order to maximise the probability that the annealer finds the correct ground-state of an Ising problem. The sum of $|F|$ for each logical qubit is defined as minor embedding energy. We confirm experimentally that the ground-state probability is maximised when the minor embedding energy is minimised, subject to the constraint that no domain walls appear in every tree of physical qubits associated with each embedded logical qubit. We further develop an analytical lower bound on $|F|$ which satisfies this constraint and show that it is a tighter bound than that previously derived by Choi (Quantum Inf. Proc. 7 193 (2008)).