Abstract: Quantum circuit complexity is a crucial metric across quantum information science, many-body physics, and high-energy physics, traditionally focused on closed systems. In this talk, we delve into the notion of embedded complexity in quantum circuits, characterizing interactions within larger systems. We introduce and analyze the complexity of projected states within subsystems, demonstrating that it is significantly influenced by the circuit volume—the aggregate of gates affecting both the subsystem and its complement. Our results indicate that ancillary qubits and measurements typically do not diminish the preparation cost of these projected states. We explore the implications of circuit volume for embedded complexity and present a spacetime conversion method that leverages random gate teleportation to optimize circuit volume. Additionally, we propose a streamlined shadow tomography protocol utilizing ancillary random states and Bell state measurements, providing valuable insights for practical implementations in quantum circuit design and analysis. Details are available in [arXiv: 2408.16602].
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