The Use of Hilbert-Schmidt Decomposition for Implementing Quantum Gates

S. Balakrishnan, R. Sankaranarayanan

Dmitry Solenov

Yorick Hardy, W.‐H. Steeb

Quantum gates are represented by unitary operators on a Hilbert space. We consider the unitary operators on and describe the component operators on each Hilbert space in the product space. From these criteria we derive a decomposition for a specific class of unitary operators such that each operator in the product has Schmidt rank bounded by 2. This decomposition directly yields the XOR (controlled NOT) implementation of the SWAP operation.

Spandan Das, Supervisors Dr. Goutam Paul, Swarnendu Datta

In this report, we discuss a necessary and sufficient condition for the existence of Schmidt decomposition of a pure state in a multipartite Hilbert space. Schmidt decomposition exists for any pure state in a bipartite Hilbert space. It is extremely useful since it reduces the number of terms needed to represent a pure state using orthonormal bases of the respective parts. However, for 3 or more partite Hilbert spaces, Schmidt decomposition does not exist in general for an arbitrary pure state. In this thesis, we derive a condition that states whether a state in an n-partite Hilbert space is Schmidt-decomposable or not, i.e., whether we can represent it through the orthonormal bases of the respective parts using less number of terms that is required in general.

Juha J. Vartiainen, Ville Bergholm, O.-P. Saira +1 more

Luming Duan