Quantum phenomena in materials underpin a range of emerging technologies, including spin-based quantum technologies, efficient energy transport materials and ultra-narrow bandwidth lasers.^1,2,3 Emergent behaviour such as quantum magnetism, superconductivity and superradiance⁴ arise from the complex interplay between electronic and structural properties; electronic features including strong electron correlation, spin-orbit coupling and reduced dimensionality can lead to phenomena such as unconventional superconductivity and room-temperature spin coherences, whilst structural factors such as crystal symmetry, doping concentrations and Moiré twist patterns are pivotal in shaping these quantum characteristics.^5,6 Computational quantum materials discovery requires both highly advanced theoretical models of the electronic structure and high-throughput approaches for identifying stable crystal structures and predicting their properties.^3,7

Strongly correlated electrons, ubiquitous in quantum materials, challenge conventional density functional theory (DFT). Quantum embedding methods, such as Density Matrix Embedding Theory (DMET) and Quantum Defect Embedding Theory (QDET), are powerful tools for describing strongly correlated electronic states in materials. QDET solves an effective Hamiltonian for a strongly-correlated subset of DFT orbitals using full configuration interaction, parameterized via a Green's function approach.⁸ DMET, however, maps the solid-state problem onto a self-consistent quantum impurity coupled to a mean-field bath, with the impurity solved by high-level methods.⁹ The application of these advanced techniques is rapidly growing, from analysing superconducting cuprates to describing quantum spin defects in semiconductors.^8,9

Model Hamiltonians, such as the multi-band Hubbard model, are increasingly used to describe the low-energy physics of quantum materials.¹⁰ While the constrained random phase approximation is the traditional choice for parametrising these models,¹¹ the newly developed moment-conserved RPA may offer superior accuracy by conserving instantaneous two-point correlation functions.^12,13 Powerful numerical techniques like Determinant Quantum Monte Carlo have recently been pioneered for solving the model Hamiltonian and predicting quantum phenomena such as pairing susceptibilities.¹⁴

Such theoretical methods are also essential for computational discovery of spin defects in semiconductors, a promising platform for room-temperature qubits.^3,15 Advanced theoretical treatments are essential to predict defect electronic, magnetic, and optical properties, incorporating effects like spin-orbit and spin-phonon coupling which determine spin coherence and optical manipulation characteristics. The current state-of-the-art combines DFT studies of semiconductor bulk properties with ab initio treatments of the defect; quantum embedding methods are emerging as a promising alternative.^16,17

Given the immense diversity of materials, high-throughput screening is a cornerstone of modern materials discovery. DFT, particularly with state-of-the-art approximations like r2SCAN+rVV10, remains the workhorse for reliably determining material structures; such calculations often offer critical insight into both a systems stability and electronic structure.^7,18,19,20 Machine learning (ML) is transforming materials discovery by slashing the computational cost of such calculations, allowing a wider exploration of composition space.^21,22

Computational quantum materials modelling is advancing rapidly, however reconciling methods treating strongly correlated electrons with computational workflows employed in modern materials discovery remains relatively unexploited. The synergy of advanced theory, high-performance computing and ML has the potential to drive breakthroughs in quantum materials discovery and accelerate development of emerging technologies, from novel qubit platforms to room-temperature superconductors.