Unitary quantum process tomography with unreliable pure input states

Quantum process tomography (QPT) methods aim at identifying a given quantum process. QPT is a major quantum information processing tool, since it allows one to characterize the actual behavior of quantum gates, which are the building blocks of quantum computers. The present paper focuses on the estimation of a unitary process. This class is of particular interest because quantum mechanics postulates that the evolution of any closed quantum system is described by a unitary transformation. The standard approach of QPT is to measure copies of a particular set of predetermined (generally pure) states after they have been modified by the process to be identified. The main problem with this setup is that preparing an input state and setting it precisely to a predetermined value is challenging and thus yields errors. These errors can be decomposed into a sum of centered errors (i.e., whose average on all the copies is zero) and systematic errors that are the same for all the copies. The latter is often the main source of error in QPT. The algorithm we introduce works for any input states that make QPT theoretically possible (i.e., unless there are several solutions due to, e.g., a lack of diversity in the input states). The fact that we do not require the input states to be precisely set to predetermined values means that we can use a trick to remove the issue of systematic errors by considering that some states' copies are unknown but measured before they go through the process to be identified. We achieve this by splitting the copies of each input state into several groups and measuring the copies of the kth group after they have successively been transferred through k instances of the process to be identified (each copy of each input state is used for only a single measurement). Using this approach, we can compute estimates of the measured states before and after they go through the process without using the knowledge we might have about the initial states. We test our algorithm with simulated data, and we assess its performance with a CNOT gate on a trapped -ions qubit quantum computer.