A unified framework for quantum random walk algorithms on general graphs

We propose a unified framework for quantum walk algorithms on general graphs, which introduces the concept of unitary labelling into the quantum walk algorithm. In our framework, by assigning the incoming or outgoing arcs of the vertices with distinct labels, unitary properties of the quantum walks can be reserved. For a non-regular graph, auxiliary arcs are added to satisfy the constraint of unitary labelling. For non-unitary quantum walks, under the same framework, we provide a solution by intermediate measurement. This solution performs the Hadamard operator on auxiliary qubits and makes measurement after each step of the walk. Though the unitary constraint can be dissatisfied by applying such a solution, we show that the properties of the quantum walks are still reserved. With this intermediate measurement, the labelling constraint can be alleviated, and the walks on unitary graphs can exhibit the same probability distribution as the unitary quantum walks. Some simulation results over general graphs are given to justify our design.