Debugging Quantum Processes Using Monitoring Measurements

Since observation on a quantum system may cause the system state collapse, it is usually hard to find a way to monitor a quantum process, which is a quantum system that continuously evolves. We propose a protocol that can debug a quantum process by monitoring, but not disturb the evolution of the system. This protocol consists of an error detector and a debugging strategy. The detector is a projection operator that is orthogonal to the anticipated system state at a sequence of time points, and the strategy is used to specify these time points. As an example, we show how to debug the computational process of quantum search using this protocol. By applying the Skolem--Mahler--Lech theorem in algebraic number theory, we find an algorithm to construct all of the debugging protocols for quantum processes of time independent Hamiltonians.