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.

翻译：状态变换问题，例如压缩量子信息或破坏量子承诺，是基础性的量子任务。然而，使用关注于经典输入和输出的传统复杂性理论难以轻易表征这些任务的计算难度。为研究此类状态变换任务的复杂性，我们引入了一个酉综合问题的框架，包括归约和酉复杂性类等概念。我们利用该框架研究通过局域操作将一个纠缠态变换为另一个纠缠态的复杂性，并将其形式化为乌尔曼变换问题——乌尔曼定理的算法版本。随后，我们证明了关于乌尔曼变换问题复杂性、多项式空间量子计算与零知识协议之间关联的结构性结果。乌尔曼变换问题使我们能够表征量子信息处理中多种任务的复杂性，包括解码噪声量子信道、破坏可证伪的量子密码学假设、在量子交互式证明中实现最优证明者策略，以及解码黑洞的霍金辐射。因此，我们的酉复杂性框架为研究许多自然量子信息处理任务的计算复杂性提供了新途径。