Protected superconducting qubit encoded in the topological phase of the transverse Ising model

The 1D transverse field Ising model (TFIM) is a well-studied spin model which exhibits a topological phase transition when the coupling energy overcomes the on-site energy. Experimentally reaching the topological regime with atomic or photonic systems is challenging due to the prohibitively large coupling strength required. In this work, we present a superconducting circuit that can implement the topological phase of the TFIM. The circuit is composed of a chain of inductively coupled transmon qubits whose coupling strength and on-site energy can be tuned by biasing the couplers with an external flux. We show how the quasi-degenerate ground states of the chain can be used to encode a logical qubit which is exponentially protected against charge noise. We estimate T1 as a function of the qubit coupling strength accounting for both ohmic and 1/f charge and flux noise. When the protection of the chain against charge noise is such that the flux noise becomes the dominant relaxation channel, we find an optimal T1 in the millisecond range.