Matrix Representation of Arbitrarily Controlled Quantum Gates

Ajanta Das, Madhubrata Bhattacharya, Debabrata Datta

Manu Mitra

Pedro Juan Roig, Salvador Alcaraz, Katja Gilly +2 more

Quantum computing fundamentals are based on Hilbert spaces and linear algebra, which allows to describe quantum gates as matrix representations. In this paper, a matrix approach is taken in order to analyze basic quantum circuits, composed of basic quantum gates of 1, 2, and 3 qubits. The most common cases of all those gates are analyzed. One-qubit gates are expressed as 2x2 matrices, accounting for a rotation around a determined axis. Two-qubit gates are described as 4x4 matrices, standing for controlled gates. Three-quibit gates are defined as 8x8 matrices, extending the concept of controlled gates and rotations.

Qin Liu

Aim To establish the relationship between generalized controlled-NOT gates U(k,l)CN and U(l,k)CN.Methods Using some methods of operator theory.Results Generalized commutative gates C(k,l)n are introduced so as to exchange the kth quantum bit and lth quantum bit,a calculation formula is given.Conclusion A relationship between generalized controlled-NOT gates U(k,l)CN and U(l,k)CN is established,and the matrix representations of related gates are obtained.

Majid Mohammadi, Aliakbar Niknafs, Mohammad Eshghi

Zhiqiang Li

We introduce symbolic representation for quantum circuit,and show how n-qubit circuit can be described as a control qubit symbolic vector and a lower-order controlled gate,instead of the representation of higher-order unitary matrix.We extend the loopless Gray code binary generation algorithm and propose n elements k position binary generation algorithm.Besides,a fast quantum simulation algorithm(FQSA) is presented,it divides a qubit state vector into several operand groups,each group is multiplied by the same unitary matrix,and then the output can be given quickly.In contrast to any other state vector based algorithm,FQSA requires less memory,and has an optimal time complexity.The experimental result of QFT indicates that FQSA is more time-efficient and it can provide more qubits circuit simulation under the same condition.