Executive Summary : | Quantum complexity is a crucial aspect of quantum information theory (QIT), quantum computation (QC), and quantum many-body physics (QMBP). It plays a significant role in various theoretical physics research areas, particularly in quantum gravity, quantum chaos, and many-body physics. The project proposal aims to study the quantum complexity of dual-unitary operators, which are the building blocks of many-body quantum circuits, to understand their advantages due to maximum operator entanglement. A subclass of dual many-body circuits, Bernoulli circuits, is proposed, built from two-particle unitary operators with maximal entangling power. The systematic study of quantum complexity of these circuits is needed to determine which set is hard to simulate classically and provides an advantage in quantum computation. The classical ergodic hierarchy is associated with algorithmic randomness, and a similar question in the quantum ergodic hierarchy would establish a relation between quantum randomness and a quantum ergodic hierarchy. Quantum complexity and quantum chaos can be quantitatively related from quantum many-body chaos. The study of complexity will impact the construction of quantum error-correcting codes (QECs) and perfect tensors, which in turn will impact studies in quantum gravity and quantum QC. QIT and QC should be treated as mathematical frameworks to understand physics, and exploring tools in QIT and QC in QMBP would reveal exciting physics. |