The ability to produce and manipulate ultracold polar molecules has opened exciting avenues in quantum science. These molecules, cooled to within a millionth of a degree of absolute zero, exhibit exquisite control over both their motional and internal states. Standard techniques from ultracold atom physics, such as optical lattices and optical tweezers, are employed to confine these molecules into ordered arrays. Unlike atoms, molecules possess an intrinsic electric dipole moment, enabling the generation of controllable long-range dipole-dipole interactions (DDI) through electric and/or microwave fields. The interaction strength lies between those of highly magnetic atoms and Rydberg atoms, offering a unique regime for investigation. The vibrational and rotational degrees of freedom within the molecule result in a complex internal structure of long-lived states. This combination of properties allows for new experimental possibilities and applications in ultracold chemistry, few-body physics, precision measurement, quantum simulation, and quantum computation. This review focuses specifically on the use of ultracold molecules for quantum simulation and quantum computation, highlighting current experimental progress, challenges, and future directions.

Several earlier reviews have documented the remarkable progress in the field of ultracold molecules. These reviews chart the advancements in cooling, trapping, and manipulating these molecules, highlighting their potential applications in various areas of physics and chemistry. They often emphasize the challenges associated with achieving sufficient control over molecular properties to effectively realize the promised applications. This present review builds upon this existing body of work by specifically focusing on the latest developments in quantum simulation and quantum computation using ultracold molecules.

The review adopts a comprehensive approach by examining both theoretical and experimental aspects. The theoretical considerations involve a review of basic molecular structure and interactions, specifically focusing on the rigid-rotor Hamiltonian and its eigenstates. The impact of electric and microwave fields on creating static and oscillating dipole moments is detailed. The calculation of dipole-dipole interactions (DDI) between molecules is presented, highlighting the interaction's anisotropy. Quantum simulations with ultracold molecules are divided into two categories: those with molecules frozen in place by lattices or microtraps and those where molecules can move. The former leads to interacting spin models such as the dipolar XXZ model. Methods for engineering this model, including microwave dressing and Floquet engineering, are explained. The latter, allowing for molecular movement, requires suppression of reactive losses through techniques like the continuous quantum Zeno effect or microwave shielding. Various model Hamiltonians, including the t-J-V-W model and the Hubbard model, and the potential phases of matter they can describe, are discussed. Experimental challenges, such as anisotropic polarizability, light shifts, and low lattice filling, are addressed. The review also delves into measurements of spin dynamics, mentioning experiments using different molecular species such as ⁴⁰K⁸⁷Rb and ²³Na⁸⁷Rb and the utilization of quantum gas microscopes for site-resolved measurements. Quantum computation with molecules involves using optical tweezer arrays for trapping individual molecules, defining qubits in hyperfine or rotational states and the need for longer coherence times than gate operation times. Methods for achieving long coherence times, such as utilizing magnetically insensitive qubits and magic wavelength trapping are explained. The role of DDI in entangling molecules and implementing two-qubit gates, and the importance of methods for suppressing motional dephasing are further detailed. The review finishes by discussing avenues for scaling up the system and controlling collisions, mentioning techniques like microwave shielding and evaporative cooling leading to the first molecular Bose-Einstein condensates. The possibility of utilizing qudits and synthetic dimensions, along with approaches to enhance interaction strength using Rydberg atoms, is reviewed. Finally, the exploration of new molecular species, including polyatomic molecules, is discussed, emphasizing the potential applications of their unique properties in both quantum simulation and computation.

Significant progress has been made in controlling ultracold molecules, particularly in achieving long coherence times crucial for quantum computation. Magic wavelength trapping and advanced pulse sequences such as XY8 dynamical decoupling have extended coherence times dramatically. Deterministic entanglement of molecular pairs has been experimentally demonstrated, using dipolar spin-exchange interactions in CaF molecules, achieving Bell state fidelities of approximately 0.8. Quantum gas microscopy has allowed for site-resolved measurements of spin dynamics in two-dimensional lattices, directly revealing the anisotropy of DDI. Furthermore, the achievement of quantum degenerate molecular gases, including the first molecular Bose-Einstein condensates, via microwave shielding methods, opens new possibilities for high-density molecular samples in optical lattices. The exploration of synthetic dimensions by using the molecule's internal states also shows great promise for novel quantum simulations. Finally, the first observation of Rydberg blockade between an atom and a polar molecule indicates the feasibility of strong Rydberg-mediated interactions to engineer entanglement. These experiments showcase the growing maturity and power of quantum manipulation techniques for ultracold molecules.

The findings presented in this review address the central research question of leveraging the unique properties of ultracold molecules for quantum computation and simulation. The significant progress in coherence times, entanglement generation, and quantum degeneracy opens exciting prospects. The ability to engineer complex Hamiltonians and achieve high-fidelity gate operations pushes the boundaries of what is possible with traditional quantum systems. The development of quantum gas microscopy techniques provides unprecedented access to many-body correlations. The field is poised for rapid advancements, with ongoing efforts to enhance interaction strengths, explore more complex molecular systems, and scale up the experimental systems. These developments are not only advancing fundamental scientific understanding but also have the potential for technological breakthroughs in areas such as quantum computing and materials science.

Ultracold molecules offer an exciting platform for both quantum simulation and quantum computation. Recent advances in controlling their internal states, suppressing losses, and generating entanglement have established them as a powerful tool for exploring previously inaccessible aspects of quantum many-body physics. Future research directions include scaling up the number of molecules, improving gate fidelities, exploring novel molecular species, and harnessing the potential of Rydberg-mediated interactions. The versatility and richness of ultracold molecules suggest a bright future for this rapidly developing field.

While significant progress has been made, several limitations remain. Achieving high filling factors in optical lattices continues to be challenging. Suppression of reactive losses, particularly for itinerant molecular systems, requires further development of shielding techniques. Precise control over molecule-molecule separation in tweezer arrays remains crucial for scaling up quantum computations. The theoretical modeling of complex many-body systems is computationally demanding, highlighting the need for more sophisticated theoretical methods. Finally, the range of molecular species currently explored is limited, although the field is rapidly expanding.