In this paper we develop a classical algorithm of complexity $O(K \, 2^n)$ to simulate parametrized quantum circuits (PQCs) of $n$ qubits, where $K$ is the total number of one-qubit and two-qubit control gates. 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：本文提出一种复杂度为 $O(K \, 2^n)$ 的经典算法，用于模拟 $n$ 量子比特参数化量子电路，其中 $K$ 为单量子比特与双量子比特控制门的总数。该算法通过显式构造与 $n$ 量子比特系统中任意单量子比特及双量子比特控制门相对应的 $2$-稀疏酉矩阵（阶数为 $2^n$）而实现。最终，我们确定了任意此类门对应的哈密顿量解析表达式，进而获得了任意参数化量子电路的局域哈密顿量分解。所有结果均通过数值模拟验证。