# Optimisation

Physics-inspired optimisation leverages physical systems' properties, particularly in optics and photonics, to tackle a wide array of complex optimisation problems, offering new possibilities beyond the capabilities of traditional computing methods. Physics-inspired optimisation encompasses a broad spectrum of solutions for various optimisation problems, including linear and nonlinear tasks in binary, integer, real, or complex variables, both with and without constraints. This extensive applicability enables its use across diverse sectors such as social sciences, finance, telecommunications, and biological and chemical industries.

Nonlinear optimisation problems often approximated through Quadratic Programming (QP), aim to minimise quadratic functions subject to linear constraints. This approach is useful in contexts like least squares regression and support vector machine training. Such problems often align with the objective functions of 2-local spin Hamiltonians in physical systems. These physical systems can emulate the optimisation process by mapping variables to spins and the objective function to a Hamiltonian.

Optimisation in physical systems can occur in quasi-equilibrium or non-equilibrium regimes. Quantum annealing utilises adiabatic evolution from a simple initial Hamiltonian to a final one encoding the objective function. However, a shrinking spectral gap can make this process less efficient for larger systems or complex Hamiltonians.

Conversely, non-equilibrium systems like lasers and photonic or polaritonic condensates, being non-Hermitian, aim to minimise losses en route to coherence. Loss minimisation involves bosonic stimulation and the coherence of operations at the threshold, leading to an emergent state that minimises losses and effectively represents the objective spin Hamiltonian.

Once overlooked due to connectivity and evolution time issues in classical architectures, Hopfield networks have regained interest, particularly in photonic neural networks. These networks can rapidly converge to optimal solutions owing to their fast operational timescale and the capability to carry multiple signals through a single optical waveguide.

Real-life optimisation problems, often challenging for classical computers, can be reformulated into finding the ground state of spin Hamiltonians. Ising and XY models are commonly employed, with the Ising model being particularly pertinent for discrete combinatorial optimisation problems. The minimisation of these Hamiltonians can be represented in mathematical terms, with the Ising model focusing on binary variables and the XY model on continuous vectors.

Complexity theory classifies these problems into different categories based on the computational effort required. Problems like the Ising model are NP-hard, indicating their computational difficulty. Despite this, universal spin Hamiltonians exist, capable of reproducing all classical spin models.

Recent research in physical systems for optimisation has focused on creating 'hard' instances for spin Hamiltonians connected to phase transitions in statistical physics. Understanding the nature of these hard instances allows one to benchmark and evaluate the performance of classical and quantum simulators and algorithms.