Implementation of Homological Quantum Rotor Codes

Protecting quantum information is most often though of as encoding logical qubits into a larger number of physical qubits. Although physical systems in the lab, more often than not, are described by larger, possibly infinite, Hilbert spaces. A widely know example are quantum modes or harmonic oscillators. Taking into account the full Hilbert space when designing error correcting codes is a promising route to get the most out of a given hardware. Recently we introduced homological quantum rotor codes as a class of quantum codes using quantum rotors to encode both infinite as well as finite dimensional logical quantum systems. We show how to construct and analyze homological quantum rotor codes via their connection with the homology of integer chain complexes and the role torsion plays in encoding finite dimensional logical systems. We also show how they are connected to superconducting qubits known as zero-pi qubits and therefore allow to design new circuits for implementing such protected superconducting qubits. Another way of implementing these rotor codes is to embed them into quantum modes. We also present how it is possible to obtain a very general class of multimode bosonic codes, nicknamed tiger codes, from quantum rotor codes. Many properties of the rotor codes translate to the properties of tiger codes. This give a general framework for multimode bosonic codes which are not concatenation of single (or a few) mode bosonic code with a discrete variable code.