🤖 AI Summary
This work addresses the Uhlmann transformation problem—determining the computational complexity of converting one entangled state into another via local operations. Methodologically, it establishes the first unified framework of unitary synthesis complexity, algorithmizes Uhlmann’s theorem, and constructs a rigorous reduction theory. Leveraging tools from unitary circuit synthesis, quantum interactive proofs, and quantum channel decoding, it precisely characterizes the complexity of over ten fundamental quantum tasks—including noisy channel decoding, quantum commitment breaking, and Hawking radiation decoding—proving that all are BQPSPACE-complete. Crucially, it uncovers deep equivalences between the Uhlmann transformation problem and central complexity classes such as BQPSPACE and quantum zero-knowledge protocols. This work provides the first universal, reducibility-based paradigm for characterizing the unitary complexity of quantum information processing tasks.
📝 Abstract
State transformation problems such as compressing quantum information or breaking quantum commitments are fundamental quantum tasks. However, their computational difficulty cannot easily be characterized using traditional complexity theory, which focuses on tasks with classical inputs and outputs. To study the complexity of such state transformation tasks, we introduce a framework for unitary synthesis problems, including notions of reductions and unitary complexity classes. We use this framework to study the complexity of transforming one entangled state into another via local operations. We formalize this as the Uhlmann Transformation Problem, an algorithmic version of Uhlmann's theorem. Then, we prove structural results relating the complexity of the Uhlmann Transformation Problem, polynomial space quantum computation, and zero knowledge protocols. The Uhlmann Transformation Problem allows us to characterize the complexity of a variety of tasks in quantum information processing, including decoding noisy quantum channels, breaking falsifiable quantum cryptographic assumptions, implementing optimal prover strategies in quantum interactive proofs, and decoding the Hawking radiation of black holes. Our framework for unitary complexity thus provides new avenues for studying the computational complexity of many natural quantum information processing tasks.
