Unitary quantum process tomography with unreliable pure input states

Abstract: Quantum process tomography (QPT) methods aim at identifying a given quantum process. 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 QTP 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 centred errors (i.e. whose average on all the copies is zero) and systematic errors that are the same on all the copies, the latter is often the main source of error in QPT. The algorithm we introduce in the current paper works for any input states that make QPT theoretically possible. 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 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 $k$-th group after they have successively been transferred through $k$ instances of the process to be identified (each copy of each input state is only measured once). Using this trick we can compute estimates of the measured states before and after they go through the process without using the knowledge we might have on the initial states. We test our algorithm both on simulated data and experimentally to identify a CNOT gate on a trapped-ions qubit quantum computer.