Analysis of selective photon addition in a bosonic mode with an ancillary qubit

Abstract

Quantum computation using bosonic modes is a promising alternative to discrete quantum computing with superconducting qubits. Bosonic modes utilize microwave cavities to encode and manipulate quantum information, leveraging the equally spaced energy levels of the cavity to create multi-photon states for robust encoding of quantum information. The dominant source of error in a bosonic mode is singlephoton loss, and finding error correction protocols for this error is an active area of research. Autonomous quantum error correction for a bosonic code was first applied experimentally by Gertler et. al. [J. M. Gertler et al., Nature 590, 243 (2021)], yet they did not reach the break-even point. An alternative protocol for recovering the cavity state after a photon loss is to apply a Selective Number-dependent Arbitrary-Phase Photon-Addition (SNAPPA) gate [M. Kudra et al., arXiv: 2212.12079 (2021)] followed by a quick qubit reset. SNAPPA was implemented experimentally by Kudra et al. yet no accurate theoretical model for this process exists. We derive an accurate and effective Hamiltonian for the SNAPPA gate, and simulate the dynamics to find close agreement with previous experimental results. The derivation highlights a general procedure for finding effective Hamiltonians where current procedures previously have failed.