Quantum embedding for complex systems

Jason D. Goodpaster, Taylor Barnes, Frederick R. Manby +1 more

Xiongfeng Ma, Zhenyu Du, Zi-Wen Liu

Abstract Quantum circuit complexity is a pivotal concept in quantum information, quantum many-body physics, and high-energy physics. While extensively studied for closed systems, the characterization and dynamics of circuit complexity are little understood for the situation where the system is embedded within a larger system, which encompasses measurement-assisted state preparation. To address this gap, we introduce the notion of embedded complexity, which accounts for both system extensions and measurements. We study the complexity of projected states in a subsystem after measuring its complement and find that in random circuits, the embedded complexity is lower-bounded by the circuit volume---the total number of gates affecting both the subsystem and its complement. This finding indicates that the total cost of preparing the projected state cannot be reduced by leveraging ancillary qubits and measurements in general. Our result underscores the operational meaning of circuit volume, as it characterizes the embedded complexity of the generated state. Specifically, for random circuits or Clifford circuits, we demonstrate a spacetime conversion that concentrates circuit volume onto a subsystem via a random gate teleportation approach. In scenarios of deep thermalization where the system interacts extensively with a larger system, our analysis suggests that the resulting projected states exhibit high complexity. Additionally, we introduce a shadow tomography protocol that employs only ancillary random states and Bell state measurements, circumventing the need to evolve the input state and thereby simplifying experimental controls.

Ayberk Demir Bakırcı

The increasing complexity of modern energy systems—driven by distributed renewable generation, dynamic grid behavior, and multi-scale interactions—necessitates computational models capable of capturing nonlinearities, stochastic variability, and high-dimensional optimization landscapes. Embedding-based quantum simulations offer a promising path forward by integrating quantum solvers into classical energy modeling frameworks, enabling more accurate analysis of localized subsystems while maintaining tractable large-scale system evaluations. This study investigates quantum embedding techniques applied to power flow analysis, renewable integration, grid stability assessment, and energy storage optimization. By partitioning complex energy networks into quantum-treated active regions embedded within classical architectures, the approach enhances the fidelity of system-level simulations while reducing computational overhead. The research highlights the potential of hybrid quantum–classical workflows to accelerate decision-making in smart grid design, improve forecasting under uncertainty, and support the development of sustainable engineering solutions. The results outline a roadmap for deploying quantum-embedded simulations in future energy infrastructures.

Max Nusspickel

Shreya Verma, Abhishek Mitra, Qiaohong Wang +8 more

One of the primary challenges in quantum chemistry is the accurate modeling of strong electron correlation. While multireference methods effectively capture such correlation, their steep scaling with system size prohibits their application to large molecules and extended materials. Quantum embedding offers a promising solution by partitioning complex systems into manageable subsystems. In this review, we highlight recent advances in multireference density matrix embedding and localized active space self-consistent field approaches for complex molecules and extended materials. We discuss both classical implementations and the emerging potential of these methods on quantum computers. By extending classical embedding concepts to the quantum landscape, these algorithms have the potential to expand the reach of multireference methods in quantum chemistry and materials.

Santosh Kumar Srivastava, Dinesh Khandelwal, Dhiraj Madan +3 more