Quantum Fourier transform (QFT) is a key ingredient of many quantum algorithms. In typical applications such as phase estimation, a considerable number of ancilla qubits and gates areused to form a Hilbert space large enough for high-precision results. Qubit recycling reduces the number of ancilla qubits to one, but it is only applicable to semi-classical QFT and requires repeated measurements and feedforward within the coherence time of the qubits. In this work, we explore a novel approach based on resonators that forms a high-dimensional Hilbert space for the realization of QFT. By employing the perfect state-transfer method, we map an unknown multi-qubit state to a single resonator, and obtain the QFT state in the second oscillator through cross-Kerr interaction and projective measurement. A quantitive analysis shows that our method allows for high-dimensional and fully-quantum QFT employing the state-of-the-art superconducting quantum circuits. This paves the way for implementing various QFT related quantum algorithms.

as well as the generation of phononic and photonic quantum states [3-10]."]Electromechanical systems realize this optomechanical interaction in the microwave regime. In this context, capacitive coupling arrangements demonstrated interaction rates of up to 280 Hz [11]. Complementary, early proposals [12-15] and experiments [16,17] suggest that inductive coupling schemes are tunable and have the potential to reach the vacuum strong-coupling regime. Here, we follow the latter approach by integrating a partly suspended superconducting quantum interference device (SQUID) into a microwave resonator. The mechanical displacement translates into a time varying flux in the SQUID loop, thereby providing an inductive electromechanical coupling. We demonstrate a sideband-resolved electromechanical system with a tunable vacuum coupling rate of up to 1.62 kHz, realizing sub-aN Hz-1/2 force sensitivities. Moreover, we study the frequency splitting of the microwave resonator for large mechanical amplitudes confirming the large coupling. The presented inductive coupling scheme shows the high potential of SQUID-based electromechanics for targeting the full wealth of the intrinsically nonlinear optomechanics Hamiltonian.

We present a hybrid system consisting of a superconducting coplanar waveguide resonator coupled to a nanomechanical string and a transmon qubit acting as nonlinear circuit element.We perform spectroscopy for both the transmon qubit and the nanomechanical string. Measuring the ac-Stark shift on the transmon qubit as well as the electromechanically induced absorption on the string allows us to determine the average photon number in the microwave resonator in both the low and high power regimes. In this way, we measure photon numbers that are up to nine orders of magnitude apart. We find a quantitative agreement between the calibration of photon numbers in the microwave resonator using the two methods. Our experiments demonstrate the successful combination of superconducting circuit quantum electrodynamics and nano-electromechanics on a single chip.

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 fastreadout 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 Z-fidelity of 82%. We also find that this enhancement is not limited by fundamental constraints.

The concept of parity describes the inversion symmetry of a system and is of fundamental relevance in the standard model, quantum information processing, and field theory. In quantumelectrodynamics, parity is conserved and selection rules (SRs) appear when matter is probed with electromagnetic radiation. However, typically large field gradients are required to engineer the parity of the light-matter interaction operator for natural atoms. In this work, we instead irradiate a specifically designed superconducting artificial atom with spatially shaped microwave fields to select the interaction parity in situ. In this way, we observe dipole and quadrupole SRs for single state transitions and induce transparency via longitudinal coupling. Furthermore, we engineer an artificial potassium-like atom with adjustable wave function parity originating from an artificial orbital momentum provided by a resonator. Our work advances light-matter interaction to a new level with promising application perspectives in simulations of chemical compounds, quantum state engineering, and relativistic physics.

Josephson parametric amplifiers (JPA) have become key devices in quantum science and technology with superconducting circuits. In particular, they can be utilized as quantum-limitedamplifiers or as a source of squeezed microwave fields. Here, we report on the detailed measurements of five flux-driven JPAs, three of them exhibiting a hysteretic dependence of the resonant frequency versus the applied magnetic flux. We model the measured characteristics by numerical simulations based on the two-dimensional potential landscape of the dc superconducting quantum interference devices (dc-SQUID), which provide the JPA nonlinearity, for a finite screening parameter βL>0 and demonstrate excellent agreement between the numerical results and the experimental data. Furthermore, we study the nondegenerate response of different JPAs and accurately describe the experimental results with our theory.

We present a systematic analysis of the internal losses of superconducting coplanar waveguide microwave resonators based on niobium thin films on silicon substrates. At millikelvintemperatures and low power, we find that the characteristic saturation power of two-level state (TLS) losses shows a pronounced temperature dependence. Furthermore, TLS losses can also be introduced by Nb/Al interfaces in the center conductor, when the interfaces are not positioned at current nodes of the resonator. In addition, we confirm that TLS losses can be reduced by proper surface treatment. For resonators including Al, quasiparticle losses become relevant above \SI{200}{\milli\kelvin}. Finally, we investigate how losses generated by eddy currents in the conductive material on the backside of the substrate can be minimized by using thick enough substrates or metals with high conductivity on the substrate backside.

We investigate the coupling of an ensemble of phosphorus donors in an isotopically purified 28Si host lattice interacting with a superconducting coplanar waveguide resonator. The microwavetransmission spectrum of the resonator shows a normal mode splitting characteristic for high cooperativity. The evaluated collective coupling strength geff is of the same order as the loss rate of the spin system γ, indicating the onset of strong coupling. We develop a statistical model to describe the influence of temperature on the coupling strength from 50mK to 3.5K and find a scaling of the coupling strength with the square root of the number of thermally polarized spins.

A superconducting qubit coupled to an open transmission line represents an implementation of the spin-boson model with a broadband environment. We show that this environment can beengineered by introducing partial reflectors into the transmission line, allowing to shape the spectral function, J({\omega}), of the spin-boson model. The spectral function can be accessed by measuring the resonance fluorescence of the qubit, which provides information on both the engineered environment and the coupling between qubit and transmission line. The spectral function of a transmission line without partial reflectors is found to be Ohmic over a wide frequency range, whereas a peaked spectral density is found for the shaped environment. Our work lays the ground for future quantum simulations of other, more involved, impurity models with superconducting circuits.

We propose a hybrid system with quantum mechanical three-body interactions between photons, phonons, and qubit excitations. These interactions take place in a circuit quantum electrodynamicalarchitecture with a superconducting microwave resonator coupled to a transmon qubit whose shunt capacitance is free to mechanically oscillate. We show that this system design features a three-mode polariton–mechanical mode and a nonlinear transmon–mechanical mode interaction in the strong coupling regime. Together with the strong resonator–transmon interaction, these properties provide intriguing opportunities for manipulations of this hybrid quantum system. We show, in particular, the feasibility of cooling the mechanical motion down to its ground state and preparing various nonclassical states including mechanical Fock and cat states and hybrid tripartite entangled states.