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 co

ndition.