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.

翻译：非线性布尔方程组在众多应用领域中扮演着重要角色。Grover算法是量子计算机上求解非线性布尔方程组最著名的量子搜索算法之一。本文提出了三种新技术，以提升Grover算法框架下的求解效率。W循环电路构造引入递归思想，在给定量子比特数固定的条件下，增加了可求解布尔方程的数量。其次，提出一种贪婪压缩技术以降低预言机电路深度。最后，随机化Grover算法在每次迭代中随机选择方程子集来构建随机预言机，从而进一步减少了电路深度和辅助量子比特的数量。在布尔二次方程上的数值实验结果验证了所提技术的有效性。