We present a circuit design composed of a non-reciprocal device and Josephson junctions whose ground space is doubly degenerate and the ground states are approximate codewords of theGottesman-Kitaev-Preskill (GKP) code. We determine the low-energy dynamics of the circuit by working out the equivalence of this system to the problem of a single electron confined in a two-dimensional plane and under the effect of strong magnetic field and of a periodic potential. We find that the circuit is naturally protected against the common noise channels in superconducting circuits, such as charge and flux noise, implying that it can be used for passive quantum error correction. We also propose realistic design parameters for an experimental realization and we describe possible protocols to perform logical one- and two-qubit gates, state preparation and readout.

We theoretically investigate the simulation of moving cavities in a superconducting circuit setup. In particular, we consider a recently proposed experimental scenario where the phaseof the cavity field is used as a moving clock. By computing the error made when simulating the cavity trajectory with SQUIDs, we identify parameter regimes where the correspondence holds, and where time dilation, as well as corrections due to clock size and particle creation coefficients, are observable. These findings may serve as a guideline when performing experiments on simulation of moving cavities in superconducting circuits.

Stimulated by the recent implementation of a three-port Hall-effect microwave circulator of Mahoney et al. (MEA), we present model studies of the performance of this device. Our calculationsare based on the capacitive-coupling model of Viola and DiVincenzo (VD). Based on conductance data from a typical Hall-bar device obtained from a two-dimensional electron gas (2DEG) in a magnetic field, we numerically solve the coupled field-circuit equations to calculate the expected performance of the circulator, as determined by the S parameters of the device when coupled to 50Ω ports, as a function of frequency and magnetic field. Above magnetic fields of 1.5T, for which a typical 2DEG enters the quantum Hall regime (corresponding to a Landau-level filling fraction ν of 20), the Hall angle θH=tan−1σxy/σxx always remains close to 90∘, and the S parameters are close to the analytic predictions of VD for θH=π/2. As anticipated by VD, MEA find the device to have rather high (kΩ) impedance, and thus to be extremely mismatched to 50Ω, requiring the use of impedance matching. We incorporate the lumped matching circuits of MEA in our modeling and confirm that they can produce excellent circulation, although confined to a very small bandwidth. We predict that this bandwidth is significantly improved by working at lower magnetic field when the Landau index is high, e.g. ν=20, and the impedance mismatch is correspondingly less extreme. Our modeling also confirms the observation of MEA that parasitic port-to-port capacitance can produce very interesting countercirculation effects.