Local Hamiltonian decomposition of parametrized quantum circuits

A Parameterized Quantum Circuit (PQC) is a structured sequence of parametric quantum gates that enables the implementation of quantum dynamics in multi-qubit systems on quantum hardware. By optimizing gate parameters to minimize a cost function, PQCs are applied to tasks in quantum machine learning (QML), optimization, and simulation. However, selecting a PQC that provides a computational advantage over classical methods for a specific problem remains an open challenge, requiring a thorough examination to identify features that drive computational efficiency. In this work, we analyze PQCs constructed from parametric one-qubit and two-qubit control gates, decomposing them as products of exponentials of parameterized Hermitian matrices, referred to as local Hamiltonians of the PQC. This decomposition enables the derivation of analytical formulas for the probability amplitudes of the PQC's output state from a given input state. The local Hamiltonian operators are obtained by constructing explicit <span style="font-size:10.8333px">2ⁿ x 2ⁿ unitary matrices for single-qubit and two-qubit control gates in an

n-qubit PQC. This matrix representation also facilitates a two-cycle decomposition of permutation matrices of order 2ⁿ , associated with strings of X gates and CNOT gates. The algebraic and geometric properties of the parameterized dynamical Lie algebra defined by the local Hamiltonians offer valuable insights into the functionality and efficiency of the PQC in specific tasks.