Superconducting microwave circuits form a versatile platform for storing andmanipulating quantum information. A major challenge to further scalability isto find approaches for connecting these systems over long distances and at highrates. One approach is to convert the quantum state of a microwave circuit tooptical photons that can be transmitted over kilometers at room temperaturewith little loss. Many proposals for electro-optic conversion between microwaveand optics use optical driving of a weak three-wave mixing nonlinearity toconvert the frequency of an excitation. Residual absorption of this opticalpump leads to heating, which is problematic at cryogenic temperatures. Here wepropose an alternative approach where a nonlinear superconducting circuit isdriven to interconvert between microwave-frequency andmillimeter-wave-frequency (300 GHz) photons. To understand the potential forquantum conversion between microwave and mm-wave photons, we consider thedriven four-wave mixing quantum dynamics of nonlinear circuits. In contrast tothe linear dynamics of the driven three-wave mixing converters, the proposedfour-wave mixing converter has nonlinear decoherence channels that lead to amore complex parameter space of couplings and pump powers that we map out. Weconsider physical realizations of such converter circuits by derivingtheoretically the upper bound on the maximum obtainable nonlinear couplingbetween any two modes in a lossless circuit, and synthesizing an optimalcircuit based on realistic materials that saturates this bound. Our proposedcircuit dissipates less than $10^{-9}$ times the energy of currentelectro-optic converters per qubit. Finally, we outline the quantum link budgetfor optical, microwave, and mm-wave connections, showing that our approach isviable for realizing interconnected quantum processors for intracity or quantumdatacenter environments.
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