Measuring Out-of-Time-Order Correlators with Projected Revivability

The out-of-time-order correlator (OTOC) has emerged as an important indicator for quantum chaos. However, it remains a major challenge in many systems to measure the OTOC due to the required implementation of time-reversal in the system. We propose a protocol to measure the OTOC using an auxillary quantum bit. Our approach is focused on systems that contain an island of revivability---a subspace within which the systems' dynamics exhibit quasi-periodic revivals. This allows approximate time-reversal in this subspace, and enables a qubit-based measurement of the OTOC. We provide an approach to find the island of revivability by showing that, in the classical limit, this subspace corresponds to an island of stability, i.e., a subspace within which the system is periodic. We demonstrate the effectiveness of this concept in a simple system of two coupled oscillators, and show that it is able to probe the non-revivable region nearby, enabling observation of OTOC growth consistent with quantum chaos.