Local Hamiltonian decomposition and classical simulation of parametrized quantum circuits

TL;DR

The paper tackles the efficient classical simulation of parametrized quantum circuits by introducing a local Hamiltonian decomposition that represents single-qubit and two-qubit gates as exponentials of local Hermitian operators. Leveraging a complete quantum binary tree ${\texttt{QT}}_n$, it reveals that the corresponding unitaries are 2-sparse and derives explicit Hamiltonians $H$ with $C_U = e^{- m i H}$ for two-qubit control gates, as well as for strings of single-qubit gates. It then shows that any PQC unitary can be written as a product of such local evolutions, yielding a decomposition $U(\boldsymbol{\theta}) = \prod_{k=1}^K U_k(\boldsymbol{\theta}_k) = \prod_{k=1}^K e^{- m i \sum_p \lambda_{pk}(\boldsymbol{\theta}_k) H_{pk}(\boldsymbol{\theta}_k)}$, which enables a classical algorithm to compute probability amplitudes in $O(K 2^n)$ time. Numerical experiments on up to 4-qubit circuits validate the theory with very small errors (≈ $10^{-15}$) and illustrate practical applicability to quantum machine learning tasks such as PQC design and screening. Overall, the work provides a scalable, analytically grounded framework for local-Hamiltonian representations of PQCs and efficient classical simulation of their amplitudes.

Abstract

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.

Figures (8)

• Figure 1: A single layer parametric quantum circuit and its equivalent quantum circuit with separated two-qubit control gates and string of single-qubit control gates
• Figure 2: $\texttt{QT}_3$
• Figure 3: Formation of the matrix $\widehat{U}^{(1)}$
• Figure 4: Formation of the matrix $\widehat{U}$, the block in the green rectangle. The small blocks are terminal nodes of ${\texttt{QT}}_n,$ and the red cross on alternatives of them represents that $C_U$ does not act on the basis elements inside it.
• Figure 5: (a) and (c): Errors given by $\|U(\theta) - e^{-\iota H(\theta)}\|_F$ vs $\theta$ for two-qubit control gates $i-j CR_X(\theta)$ for $100$ values of $\theta\in [-\pi, \pi]$ in a $4$-qubit system corresponding to $i<j$ and $i>j$ respectively. (b) and (d): Violin plots for the Error vs gate corresponding to (a) and (b) respectively.