A Remarkable Application of Zassenhaus Formula to Strongly Correlated Electron Systems

TL;DR

This work tackles the challenge of efficiently simulating quantum systems by decomposing the exponential of a sum of non-commuting operators. By exploiting a no-mixed adjoint property, the authors derive a closed-form Zassenhaus expansion: $e^{\hat{X}+\hat{Y}}=e^{\hat{X}}\exp\big(\sum_{n\ge 1}\frac{(-1)^{n-1}}{n!}ad^{n-1}_{\hat{X}}\hat{Y}\big)$, which becomes exact in a finite product for operators arising in mono- and di-excitations of electronic states. They apply this to a Unitary Frozen Pair CC (UfpCC) framework, particularly a 2D-block variant, showing that the excitation operators satisfy the no-mixed adjoint property and thus admit an exact circuit decomposition into Givens gates with a simple parameter map. For the general case of $N$ electron pairs, they derive a matrix-based expression using $M$ to yield an explicit, finite product of exponentials, with a gate count bounded by $\frac{N(N+1)}{2}$, providing a practical, Trotter-free route to quantum simulations of strongly correlated systems. Overall, the paper connects Lie-algebraic decompositions with quantum circuit design, offering a scalable, exact approach for preparing disentangled UCC states on NISQ devices and motivating further pair-based ansätze in quantum chemistry and beyond.

Abstract

We show that the Zassenhaus decomposition for the exponential of the sum of two non-commuting operators, simplifies drastically when these operators satisfy a simple condition, called the no-mixed adjoint property. An important application to a Unitary Coupled Cluster method for strongly correlated electron systems is presented. This ansatz requires no Trotterization and is exact on a quantum computer with a finite number of Givens gate equals to the number of free parameters. The formulas obtained in this work also shed light on why and when optimization after Trotterization gives exact solutions in disentangled forms of unitary coupled cluster.

Figures (3)

• Figure 1: Two possible abstract circuits for a cluster operator, made of two mono- and one di-excitation for the LiH molecule in a minimal basis set. The $8$ qubits represents the following electron pairs in $6$ APSG-optimized orbitals, as advocated in Cassam_hal-05210807, for the 2D-block geminal method: $\hat{S}^{+}_{1}\ket{0}$, $\hat{S}^{+}_{2}\ket{0}$, $\hat{S}^{+}_{12}\ket{0}$, $\hat{S}^{+}_{3}\ket{0}$, $\hat{S}^{+}_{4}\ket{0}$, $\hat{S}^{+}_{34}\ket{0}$, $\hat{S}^{+}_{5}\ket{0}$, $\hat{S}^{+}_{6}\ket{0}$. The last two qubits correspond to the lithium doubly occupied $p_x$ and $p_y$ orbitals, whose populations are optimized only in the reference wave function through a closed-shell pair unitary operator (not represented). They are not affected by the broken-pair cluster operator within the 2D-block ansatz, and the associated open-shell qubit $\hat{S}^{+}_{56}\ket{0}$ is not necessary. The left-hand side circuit makes use of two and four-qubit Givens gates, while the right-hand side one uses a conditional two-qubit Givens gate in place of the four-qubit Givens gate. Note that, this implies algebraic dependencies among the angle parameters, supposed small. The circuit drawings are exported from the IBM Quantum Qiskit packages and according to their convention, in practice, the sign of the gate angles should be reversed with respect to those appearing in the main text.
• Figure 2: Comparison of the transpiled circuits of Fig.1 on Heron Torino IBM quantum computer. Transpilations have been performed at level 3 of optimization for both circuits. The upper circuit corresponds to the four-qubit Givens gate and has a shorter depth than the lower one, corresponding to the conditional two-qubit Givens gate. However, the total count of two-qubit CZ basic gates is larger in the former than in the latter ($44$ against $39$). The circuit drawings are exported from the IBM Quantum Qiskit packages.
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