Quantum trajectory sensing problem and its solution

The quantum trajectory sensing problem seeks quantum sensor states which enable the trajectories of incident particles to be distinguished using a single measurement. For an $n$-qubit sensor state to unambiguously discriminate a set of trajectories with a single projective measurement, all post-trajectory output states must be mutually orthogonal; therefore, the ${2}^{n}$ state coefficients must satisfy a system of constraints which is typically very large. Given that this system is generally challenging to solve directly, we introduce a group-theoretic framework which simplifies the criteria for sensor states and exponentially reduces the number of equations and variables involved when the trajectories obey certain symmetries. These simplified criteria yield general families of trajectory sensor states and provide bounds on the particle-sensor interaction strength required for perfect one-shot trajectory discrimination. Furthermore, we establish a link between trajectory sensing and quantum error correction, recognizing their common motivation to identify perturbations using projective measurements. Our sensor states in fact form quantum codes, and conversely, a number of familiar stabilizer codes (such as toric codes) also provide trajectory sensing capabilities. This connection enables noise-resilient trajectory sensing through the concatenation of sensor states with quantum error-correcting codes.