In this paper we develop a classical algorithm of complexity $O(2^n)$ to simulate parametrized quantum circuits (PQCs) of $n$ qubits. The algorithm is developed by finding $2$-sparse unitary matrices of order $2^n$ explicitly corresponding to any single-qubit and two-qubit control gates in an $n$-qubit system. Finally, we determine analytical expression of Hamiltonians for any such gate and consequently a local Hamiltonian decomposition of any PQC is obtained. All results are validated with numerical simulations.

翻译：本文提出了一种复杂度为$O(2^n)$的经典算法，用于模拟含$n$个量子比特的参数化量子电路（PQC）。该算法通过显式构造n量子比特系统中任意单量子比特门和两量子比特控制门对应的$2^n$阶2-稀疏酉矩阵来实现。最后，我们得到了任意此类门的哈密顿量解析表达式，并由此获得了任意参数化量子电路的局部哈密顿量分解。所有结果均通过数值模拟进行了验证。