  Nonlinear boolean equation systems play an important role in a wide range of
applications. Grover's algorithm is one of the best-known quantum search
algorithms in solving the nonlinear boolean equation system on quantum
computers. In this paper, we propose three novel techniques to improve the
efficiency under Grover's algorithm framework. A W-cycle circuit construction
introduces a recursive idea to increase the solvable number of boolean
equations given a fixed number of qubits. Then, a greedy compression technique
is proposed to reduce the oracle circuit depth. Finally, a randomized Grover's
algorithm randomly chooses a subset of equations to form a random oracle every
iteration, which further reduces the circuit depth and the number of ancilla
qubits. Numerical results on boolean quadratic equations demonstrate the
efficiency of the proposed techniques.
