Quantum Gates Revisited: A Tensor Product Based Interpretation Model

Abstract: Quantum computers have been considered as powerful computing apparatus in the future. Various quantum gates and quantum circuits have been presented to solve classical computational problems using quantum mechanical systems. Quantum gates and quantum circuits can be expressed using the tensor product notation. However, the mathematical model of tensor products is usually limited to superposition of qubits. In this paper, we present a mathematical model to express complex quantum gates and quantum circuits. This mathematical model includes matrix operations such as matrix addition, matrix multiplication, direct sum, tensor product, and stride permutation. A quantum gate or a quantum circuit is expressed as a matrix formula, generally called, a tensor product formula. An interpretation model is also described to map operations of a tensor product formula to quantum operations. With this interpretation model, we are able to describe various quantum gates and quantum circuits succinctly and precisely and to develop a programming methodology for designing quantum algorithms effectively.