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中扮演着重要角色。格罗弗算法是在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。本文提出三种新技术以提高格罗弗算法框架下的效率。首先，W周期电路构造引入递归思想，在固定量子比特数条件下增加可求解布尔方程的数量。其次，提出贪婪压缩技术以减少预言机电路深度。最后，随机化格罗弗算法每次迭代从方程子集中随机选择构成随机预言机，进一步降低电路深度和辅助量子比特数。针对布尔二次方程的数值实验证明了所提技术的有效性。