Quantum simulation is widely regarded as one of the most promising routes to genuine quantum advantage, yet most existing approaches to quantum chemistry are formulated in terms of closed-system, unitary dynamics and ground-state preparation within the Born–Oppenheimer approximation. In this review, we discuss a broader perspective motivated by the observation that naturally occurring quantum systems are rarely isolated and often reach physically relevant states only through relaxation, decoherence, and thermalization. We first examine what is and is not known about exponential quantum advantage in chemistry, emphasizing that coherent Hamiltonian simulation provides the clearest formal case for speed-up, while many open questions remain for realistic problems. We then discuss how dissipation might ideally be integrated into quantum chemistry on a fault-tolerant quantum computer, using recent proposals for chemically motivated dynamical simulation as a guiding vision. More generally, we highlight the practical appeal of this approach to enhancing the robustness of quantum algorithms.
Many-body quantum-optical systems, where a collection of emitters interacts through a common electromagnetic reservoir, exhibit rich out-of-equilibrium behavior and hold promise for applications in quantum technologies. However, exact numerical simulations of their dynamics quickly become unfeasible due to the exponential growth of the Hilbert space with system size. Semiclassical, phase-space approaches – such as the Truncated Wigner approximation (TWA) – provide computationally efficient alternatives by capturing leading-order quantum fluctuations. In this paper, we present a comprehensive overview of how to tackle problems in many-body quantum optics using phase-space methods. We derive the exact partial differential equation governing many-body dissipative evolution in any phase-space representation and discuss the approximations that yield the dissipative TWA proposed by Mink and Fleischhauer [SciPost Phys. 15, 233 (2023)]. We find that P and Q distributions are generally suboptimal for many-body quantum optics. Additionally, we extend the formalism to calculate multi-time correlation functions, thereby broadening the scope of phase-space simulations of open spin systems to include coherence and spectral properties, as well as directional correlations of collectively radiating emitters. These developments provide valuable tools for investigating exotic light sources driven by collective dissipation, driven-dissipative phase transitions, and a wealth of many-body phenomena arising in state-of-the-art experimental platforms.
Paraxial fluids of light are a promising platform for exploring collective phenomena in a highly tunable environment. These systems, which map the propagation of light through nonlinear media onto the wavefunction of effective 2D quantum fluids, offer a complementary approach to traditional platforms such as cold atomic gases or superfluid helium. In this review, we present a detailed overview of the theoretical framework underlying paraxial fluids of light, including the nonlinear Schrödinger equation (NLSE) and its mapping to the 2D+1 Gross-Pitaevskii equation (GPE). We explore the hydrodynamic formulation of these systems and we provide a comparative analysis of fluids of light and cold atomic gases, examining key parameters and figures of merit. We then review the recent experimental advances and the experimental platforms currently used to realize paraxial fluids of light, including hot atomic vapors, photorefractive crystals, and thermo-optic media. Additionally, we question the geometry of the system extending the analogy from 2D+1 to lower or higher dimensions. Looking forward, we outline the potential future directions for the field, including the use of laser cooled atoms as nonlinear media, the study of two-component mixtures, and the exploration of quantum effects beyond the mean-field approximation. These developments promise to deepen our understanding of quantum fluids and potentially contribute to advances in quantum technologies.
