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解量子承诺，是基本的量子任务。然而，其计算难度难以用侧重于经典输入输出任务的传统复杂性理论直接刻画。为研究此类状态变换任务的复杂性，我们提出了一个幺正合成问题的框架，包括归约概念和幺正复杂性类。我们利用此框架研究通过局域操作将一种纠缠态转换为另一种纠缠态的复杂性，并将其形式化为乌尔曼变换问题——即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和零知识协议之间结构性的关系。乌尔曼变换问题使我们能够刻画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、破解可证伪量子密码假设、实现量子交互证明中最优证明者策略，以及解码黑洞霍金辐射。因此，我们的幺正复杂性框架为研究众多自然量子信息处理任务的计算复杂性提供了新途径。