Keeping quantum data quantum slashes the samples needed

When you want to purify or clone a quantum state, the standard approach measures it first and reconstructs — but that destroys coherence. Skipping measurement and processing quantum data directly requires only O(1/ε) copies to hit precision ε, versus Ω(d/ε) for any measure-first protocol — an exponential saving in the system dimension d. The result establishes a sharp, proven boundary between quantum and classical inference strategies, with direct implications for quantum computing tasks like density matrix exponentiation.