High-impedance circuits for parity measurements of cat qubits

Encoding a qubit in the two degenerate steady states of an oscillator—which only exchanges pairs of photons with its environment—can exponentially suppress the bit-flip rate for large phase-space separations. The unsuppressed phase flips of these so-called "cat qubits" correspond to a change in the photon number parity of the oscillator, and they could be corrected using redundant encoding. In such a scheme, errors are detected via measurements of the joint parity between cat qubits, which could be implemented at the Hamiltonian level using effective parity-type couplings. We show that a parity-type Hamiltonian emerges from the conventional Josephson potential in the limit of high oscillator impedance. Here, the high impedance guarantees large fluctuations of the superconducting phase, which translates into large displacements in oscillator phase space. We present the design of a superconducting circuit that effectively realizes the parity-type Hamiltonian, as well as the status of its experimental implementation.