# Temperature scaling law for quantum annealing optimizers

## TL;DR

Quantum annealers operating at fixed finite temperatures face fundamental limitations in scalability, requiring temperature to decrease with problem size, following a specific scaling law for effective optimization.

## Contribution

We derive a temperature scaling law for quantum annealers, showing the necessity of temperature reduction with problem size for scalable optimization.

## Key findings

- Temperature must decrease at least logarithmically with problem size.
- Experimental and simulation results support the derived scaling law.
- Finite temperature limits the scalability of quantum annealing optimizers.

## Abstract

Physical implementations of quantum annealing unavoidably operate at finite temperatures. We point to a fundamental limitation of fixed finite temperature quantum annealers that prevents them from functioning as competitive scalable optimizers and show that to serve as optimizers annealer temperatures must be appropriately scaled down with problem size. We derive a temperature scaling law dictating that temperature must drop at the very least in a logarithmic manner but also possibly as a power law with problem size. We corroborate our results by experiment and simulations and discuss the implications of these to practical annealers.