Quantum Circuit Synthesis Based On Orthogonal Matrices

Quantum computation is a new computing mode that follows the laws of quantum mechanics.It cannot only solve the mathematical problems that are difficult to compute by existing computers,but also greatly reduce the volume and energy consumption of hardware.The quantum circuit model,one of the most widely used quantum computation models,has achieved the evolution through quantum circuits which are composed of quantum logic gates to the initial state of the quantum.Therefore,it is critical to effectively prepare quantum state and generate an optimized quantum circuit for any quantum computation.In this paper,we made a research on the two problems with the selection of the rotation matrix and Householder matrix as transition matrices,and application of QR decomposition method.The main results are as follows:(1)Quantum circuit synthesis based on multiple rotation gates.According to the unitarity of quantum circuits,a multiple rotation gate with variable angle and a multiple rotation gate with fixed control are proposed,effective decomposition methods from these two types of gates to CNOT gates and single qubit gates as well as the optimization criteria for cascade between these two gates are given in this paper.For the quantum computation,QR decomposition method is adopted,the quantum circuit synthesis method based on multiple rotation gates is proposed,concrete synthesis and optimization processes are given,and the synthesis process is illustrated by taking 3 qubits computation as an example.After analysis,the quantum cost of the generated quantum circuit proves to be the lowest in the synthesis method using QR decomposition.(2)Quantum circuit synthesis based on multiple householder gates.According to the characteristics of multiplexer and the property of Householder matrix,a multiple Householder gate with variable dimension is proposed,and the physical realization of multiple Householder gate with m dimension in the 2n-pod quantum system is given for n qubits.Additionally,the difficulty in controlling the gate is proportional to its dimension.A synthesis method based on multiple Householder gates is proposed for quantum computation.The detailed synthesis process is given with an illustration on the the synthesis of any 4-qubits circuit.For n qubits computation,the computational complexity of the generated quantum circuit is between O(2n)and O(4n),and is inversely proportional to the dimension of the gate.Compared with the same kind of synthesis methods,quantum circuit synthesis method is prominent in advantages.Appropriate dimension values can be selected to construct the multiple Householder gate according to the actual situation.(3)Preparation of quantum states.According to multiple rotation gate and multiple Householder gate,we provided three methods for the preparation of quantum states,including a direct method for the quantum state based on multiple rotation gate,an indirect method for the quantum state based on multiple rotation gates and multiple rotation gates with fixed control,and an indirect preparation method for the quantum state based on multiple Householder gates.Procedures for generating quantum circuit in the three methods are respectively described in detail.The performance analysis and evaluation on the quantum circuits generating from those three methods indicate that the complexity of the two methods based on the multiple rotation gate is O(2n),the indirect preparation method is simpler and easier to implement,and the complexity of the method based on multiple Householder gates lies between 1 and O(2n),which is inversely proportional to the dimension of the gate.