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#### Quantum Modeling of Traveling Wave Parametric Amplifiers for Superconducting Qubit Readout

Superconducting quantum computers are at the forefront of demonstrating quantum supremacy. Within such a quantum computer, tunable coupling among the coherent qubits allows for the implementation of quantum experiments and protocols. The qubit states from repeated experiments need to be measured to sample the resulting probability distribution. High readout fidelities require a reasonable signal-to-noise ratio, where the added noise due to amplification plays a crucial role. Thus, quantum-noise-limited traveling wave parametric amplifiers (TWPAs) are required as a first stage, with a cryogenic HEMT amplifier down the line. We describe quantum models for TWPAs, capable of predicting gain, losses, (quantum-)noise, and squeezing.