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Pure qubit states admit a geometric representation on the extended complex plane via stereographic projection, allowing their discrete-time evolution to be described through conformal maps. We identify fractional linear conformal maps as a unifying framework encompassing unitary dynamics, non-unitary linear dynamics, and non-unitary nonlinear dynamics, classified by their action on the Hilbert space. Within this framework, we harness the non-linear chaotic maps (as a part of fractional non-linear conformal maps)- specifically those acting on the Julia set to engineer a ”quantum microscope.” This device leverages the extreme sensitivity of chaotic dynamics to exponentially amplify the distinguishability of nonorthogonal states. We quantify the microscope’s ”magnification power” through temporal Leggett-Garg correlations, providing a device-independent benchmark for self-testing. By linking the characteristic waiting times of these maps to state discrimination costs, this work combines chaos theory with quantum correlations to push the limits of precision in quantum metrology. |