Two-Qubit Unitary Quantum Process Tomography by Multiple-Delay Output Measurements for One Unknown Input Pure State Value

Abstract

Quantum process tomography (QPT) methods aim at identifying a given quantum process. QPT is a major quantum information processing tool, since it especially 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 two-qubit 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. Usual QPT methods can be hard to implement in practice because they require the user to prepare copies of a set of known quantum states. In contrast, the proposed methods only require the user to be able to provide copies of a single unknown input state which can be estimated along with the process. However, we need to be able to make measurements of the state of the system after 5 different time delays ∆ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> , 2∆ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> , 3∆ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> , 4∆ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> , 5∆ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> (one measurement per copy). We detail a least square approach which exploits tools used in the aerospace community. A slower but more precise maximum likelihood method is also introduced. It uses a simple parametrization of unitary matrices.