A Geometric Approach to Quantum Distinguishers

This paper introduces a geometric framework for Q2 quantum distinguishers by combining the geometric approach to classical symmetric-key cryptanalysis with the generalized correlation extraction algorithm. Our main technical tool shows that one superposition query, followed by appropriate (unitary) change-of-basis operations, prepares a ``correlation state'' whose amplitudes are the entries of the geometric correlation matrix in the chosen basis. This yields a unified preparation-measurement template that recovers several known quantum distinguishers: (1) hidden structure detection via support constraints in Fourier-type bases (e.g., Simon, Bernstein-Vazirani, Deutsch-Jozsa), and (2) event probability deviation tests via amplitude estimation (covering standard quantizations of linear and differential distinguishers). We analyze when relevant distinguishing mass is diluted across many basis states, identify it as a cause of poor query efficiency in several recent distinguishers, and provide basis-specific mechanisms to concentrate the signal (phase-oracle row restriction in the Fourier setting; chosen-plaintext subset-state restriction in the quasidifferential setting) to restore quadratic advantage. We illustrate the framework on Fourier and quasidifferential instantiations and discuss obstacles for non-unitary integral bases.