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The advent of quantum-enhanced sensing methods in axion dark matter searches holds the promise of surpassing the quantum-limited scan rate by an order of magnitude or more. Superconducting circuits enable the generation of squeezed states in axion haloscopes and the direct detection of microwave photons. But the strong magnetic fields required in axion haloscopes complicate the use of superconducting circuits, necessitating their placement tens of centimeters from the cavity and connection via a lossy transmission line. We show analytically and demonstrate experimentally that a cavity can inherit the quantum amplification properties of a remote superconducting circuit through lossy transmission-line modes, without incurring the associated loss. The superconducting circuit can be operated in three distinct modes: as a noiseless mixer that converts cavity photons to a different frequency, as a phase-preserving quantum-limited amplifier, and as a quantum non-demolition (QND) amplifier of a single quadrature. We demonstrate that the QND amplifier yields a tenfold increase in scan rate relative to the conventional haloscope strategy employing a matched quantum-limited amplifier at the cavity output, which we benchmark using the noiseless mixer operation. Beyond providing an in-situ quantum-limited benchmark, a noiseless mixer could be combined with a fixed-frequency photon counter to enable direct photon detection in a frequency-tunable haloscope cavity.