Unitary Complexity and the Uhlmann Transformation Problem

TL;DR

This paper introduces a framework for analyzing the complexity of quantum state transformation tasks, focusing on the Uhlmann Transformation Problem, and relates it to various fundamental quantum information processing tasks.

Contribution

It formalizes the Uhlmann Transformation Problem within a new unitary complexity framework, connecting it to quantum computation, cryptography, and black hole information decoding.

Findings

Relates the Uhlmann Transformation Problem to polynomial space quantum computation.

Links the problem to zero knowledge protocols.

Characterizes complexity of tasks like decoding quantum channels and black hole radiation.

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.