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Funding Information: The authors thank S Pirandola, and S Gasparinetti for useful discussions, and thank the support from Project QMiCS (820505) of the EU Flagship on Quantum Technologies. M C, E Z C, R A, G F, S G, and Y O thank the support from FCT—Fundação para a Ciência e a Tecnologia (Portugal), namely through Projects UIDB/50008/2020 and UIDB/04540/2020, as well as from Project TheBlinQC supported by the EU H2020 QuantERA ERA-NET Cofund in Quantum Technologies and by FCT (QuantERA/0001/2017). M C acknowledges support from the DP-PMI and FCT through scholarship PD/BD/135186/2017. G F acknowledges support from FCT through scholarship SFRH/BD/145572/2019. T G-R, M R and M S acknowledge financial support from Basque Government QUANTEK Project from ELKARTEK program (KK-2021/00070), Spanish Ramón y Cajal Grant RYC-2020-030503-I, as well as from OpenSuperQ (820363) of the EU Flagship on Quantum Technologies, and the EU FET-Open Projects Quromorphic (828826) and EPIQUS (899368), and IQM Quantum Computers under the Project ‘Generating quantum algorithms and quantum processor optimization’. M R acknowledges support from UPV/EHU PhD Grant PIF21/289. M M and G C acknowledge funding from the European Research Council under Consolidator Grant No. 681311 (QUESS), and from the Academy of Finland through its Centers of Excellence Program (Project Nos. 312300 and 336810) and QEMP Project (319579). M R, K G F, and F D acknowledge support by the German Research Foundation via Germany’s Excellence Strategy (EXC-2111-390814868), the Elite Network of Bavaria through the program ExQM, the EU Flagship Project QMiCS (Grant No. 820505), the German Federal Ministry of Education and Research via the Projects QUARATE (Grant No. 13N15380) and QuaMToMe (Grant No. 16KISQ036), and the State of Bavaria via the Munich Quantum Valley and the Hightech Agenda Bayern Plus. | openaire: EC/H2020/820505/EU//QMiCS | openaire: EC/H2020/820363/EU//OpenSuperQ | openaire: EC/H2020/681311/EU//QUESS The field of propagating quantum microwaves is a relatively new area of research that is receiving increased attention due to its promising technological applications, both in communication and sensing. While formally similar to quantum optics, some key elements required by the aim of having a controllable quantum microwave interface are still on an early stage of development. Here, we argue where and why a fully operative toolbox for propagating quantum microwaves will be needed, pointing to novel directions of research along the way: from microwave quantum key distribution to quantum radar, bath-system learning, or direct dark matter detection. The article therefore functions both as a review of the state-of-the-art, and as an illustration of the wide reach of applications the future of quantum microwaves will open.

We describe a unified quantum approach for analyzing the scattering coefficients of superconducting microwave resonators with a variety of geometries. We also generalize the method to a chain of resonators with time delays, and reveal several transport properties similar to a photonic crystal. It is shown that both the quantum and classical analyses provide consistent results, and they together reveal different decay and decoherence mechanisms in a general microwave resonator. These results form a solid basis for understanding the scattering spectrums of networks of microwave resonators, and pave the way for applying superconducting microwave resonators in complex circuits.

We describe a unified classical approach for analyzing the scattering coefficients of superconducting microwave resonators with a variety of geometries. To fill the gap between experiment and theory, we also consider the influences of small circuit asymmetry and the finite length of the feedlines, and describe a procedure to correct them in typical measurement results. We show that, similar to the transmission coefficient of a hanger-type resonator, the reflection coefficient of a necklace- or bridge-type resonator does also contain a reference point which can be used to characterize the electrical properties of a microwave resonator in a single step. Our results provide a comprehensive understanding of superconducting microwave resonators from the design concepts to the characterization details.

Funding Information: The authors thank R. Assouly, R. Dassonneville, and B. Huard for useful discussions. All authors acknowledge support from QMiCS (Grant No. 820505) of the EU Flagship on Quantum Technologies. T.G.-R. and M.S. acknowledge financial support from the Basque Government QUANTEK project under the ELKARTEK program (KK-2021/00070), the Spanish Ramón y Cajal Grant No. RYC-2020-030503-I, project Grant No. PID2021-125823NA-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” and “ERDF Invest in your Future”, OpenSuperQ (820363) of the EU Flagship on Quantum Technologies, the EU FET-Open projects Quromorphic (828826) and EPIQUS (899368), and IQM Quantum Computers under the project “Generating quantum algorithms and quantum processor optimization”. M.C. and Y.O. thank the support from FCT – Fundação para a Ciência e a Tecnologia (Portugal), namely through project UIDB/04540/2020, as well as from project TheBlinQC supported by the EU H2020 QuantERA ERA-NET Cofund in QuantumTechnologies and by FCT (QuantERA/0001/2017). M.C. acknowledges support from the DP-PMI and FCT through scholarship PD/BD/135186/2017. M.R., F.F., F.D., and K.F. acknowledge support from the German Research Foundation via Germany’s Excellence Strategy (EXC-2111-390814868), Elite Network of Bavaria through program ExQM, and the German Federal Ministry of Education and Research via project QUARATE (Grant No. 13N15380) and project QuaMToMe (Grant No. 16KISQ036). This research is part of the Munich Quantum Valley, which is supported by the Bavarian state government with funds from the Hightech Agenda Bayern Plus. The research of V.S. is supported by the Basque Government through the BERC 2022–2025 program and by the Ministry of Science, Innovation, and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718. M.M. acknowledges funding from the European Research Council under Consolidator Grant No. 681311 (QUESS), from the Jane and Aatos Erkko Foundation and the Technology Industries of Finland Centennial Foundation through their Future Makers program, and from the Academy of Finland through its Centers of Excellence Program (Projects No. 312300 and No. 336810). | openaire: EC/H2020/820505/EU//QMiCS | openaire: EC/H2020/681311/EU//QUESS Microwave technology plays a central role in current wireless communications, including mobile communication and local area networks. The microwave range shows relevant advantages with respect to other frequencies in open-air transmission, such as low absorption losses and low-energy consumption, and in addition, it is the natural working frequency in superconducting quantum technologies. Entanglement distribution between separate parties is at the core of secure quantum communications. Therefore, understanding its limitations in realistic open-air settings, especially in the rather unexplored microwave regime, is crucial for transforming microwave quantum communications into a mainstream technology. Here, we investigate the feasibility of an open-air entanglement distribution scheme with microwave two-mode squeezed states. First, we study the reach of direct entanglement transmission in open air, obtaining a maximum distance of approximately 500 m with parameters feasible for state-of-the-art experiments. Subsequently, we adapt entanglement distillation and entanglement swapping protocols to microwave technology in order to reduce the environment-induced entanglement degradation. The employed entanglement distillation helps to increase quantum correlations in the short-distance low-squeezing regime by up to 46%, and the reach of entanglement increases by 14% with entanglement swapping. Importantly, we compute the fidelity of a continuous-variable quantum teleportation protocol using open-air-distributed entanglement as a resource. Finally, we adapt this machinery to explore the limitations of quantum communication between satellites, where the impact of thermal noise is substantially reduced and diffraction losses are dominant.

Description, Experimental microwave teleportation brings quantum local area networks within reach by exceeding the no-cloning limit., The field of quantum communication promises to provide efficient and unconditionally secure ways to exchange information, particularly, in the form of quantum states. Meanwhile, recent breakthroughs in quantum computation with superconducting circuits trigger a demand for quantum communication channels between spatially separated superconducting processors operating at microwave frequencies. In pursuit of this goal, we demonstrate the unconditional quantum teleportation of propagating coherent microwave states by exploiting two-mode squeezing and analog feedforward over a macroscopic distance of d = 0.42 m. We achieve a teleportation fidelity of F = 0.689 ± 0.004, exceeding the asymptotic no-cloning threshold. Thus, the quantum nature of the teleported states is preserved, opening the avenue toward unconditional security in microwave quantum communication.

Superconducting 3D microwave cavities offer state-of-the-art coherence times and a well controlled environment for superconducting qubits. In order to realize at the same time fast readout and long-lived quantum information storage, one can couple the qubit both to a low-quality readout and a high-quality storage cavity. However, such systems are bulky compared to their less coherent 2D counterparts. A more compact and scalable approach is achieved by making use of the multimode structure of a 3D cavity. In our work, we investigate such a device where a transmon qubit is capacitively coupled to two modes of a single 3D cavity. The external coupling is engineered so that the memory mode has an about 100 times larger quality factor than the readout mode. Using an all-microwave second-order protocol, we realize a lifetime enhancement of the stored state over the qubit lifetime by a factor of $6$ with a fidelity of approximately $80\%$ determined via quantum process tomography. We also find that this enhancement is not limited by fundamental constraints., 5 pages, 4 figures