Abstract
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 the Gottesman-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.
Highlights
Building a quantum computer in a physical system is a formidably challenging task because of the inherent fragility of physical quantum bits
The key idea behind quantum error correction (QEC) [1,2] is to use logical qubits that can be protected against certain likely errors, extending the lifetime of the encoded quantum information and allowing for fault-tolerant quantum computation [3,4]
The QEC codes that have been most successful in enhancing the lifetime of quantum information have been built from continuous variable (CV) systems [11,12,13], such as a single microwave cavity mode
Summary
Building a quantum computer in a physical system is a formidably challenging task because of the inherent fragility of physical quantum bits (qubits). III, we show how this Hamiltonian can be derived from the low-energy description of a single electron in a high magnetic field and a periodic potential The dynamics of this system is equivalent to that of a gyrator connected to two Josephson junctions, but the solid-state jargon more reveals the intimate relation to Hofstadter’s butterfly [34,37]: The GKP states are obtained at a specific point in the butterfly. IV, we study the effect of an additional parabolic confinement potential, which in the circuit model consists of the addition of inductances in parallel to the Josephson junctions For this setting, the ground space of the resulting Hamiltonian is twofold degenerate up to an exponentially small gap, and the eigenstates of the system resemble superpositions of normalizable, approximate GKP code words [16]. VIII, we summarize our results and give an outlook on further work
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