Approximating exponentials of commutators by optimized product formulas

TL;DR

This work develops optimized product formulas to approximate $e^{t^2 [A,B]}$ using exponentials of $A$ and $B$ with orders 4–6 achieved at substantially lower costs than prior approaches. By leveraging invariances and introducing counter-palindromic patterns, the authors define PCP and NCP schemes that reduce the number of order conditions and provide explicit coefficients up to order 6, with representative 4th–6th order schemes achieving significant efficiency gains. Numerical tests on Pauli matrices and random matrices demonstrate that the new schemes markedly outperform existing recurrences, particularly at higher orders (e.g., a 6th-order PCP scheme with 26 exponentials). Extensions to more general Lie polynomials, including $F = t(A+B) + t^2 R [A,B]$ and nested commutators such as $[A,[A,B]]$, as well as connections to the Zassenhaus formula, indicate broad applicability to quantum simulation, quantum control, and related areas. The results suggest that direct optimized formulas offer practical advantages over recursive constructions for achieving high-fidelity commutator evolutions.

Abstract

Trotter product formulas constitute a cornerstone quantum Hamiltonian simulation technique. However, the efficient implementation of Hamiltonian evolution of nested commutators remains an under explored area. In this work, we construct optimized product formulas of orders 3 to 6 approximating the exponential of a commutator of two arbitrary operators in terms of the exponentials of the operators involved. The new schemes require a reduced number of exponentials and thus provide more efficient approximations than other previously published alternatives. They can also be used as basic methods in recursive procedures to increase the order of approximation. We expect this research will improve the efficiency of quantum control protocols, as well as quantum algorithms such as the Zassenhaus-based product formula, Magnus operator-based time-dependent simulation, and product formula schemes with modified potentials.

Figures (5)

• Figure 1: Error committed in the approximation of $\mathrm{e}^{[A,B]}$ by different product formulas vs. the number of elementary exponentials when $A = -i \sigma_x$, $B = -i \sigma_z$ (left) and when $A$ and $B$ are $16 \times 16$ real matrices with random elements (right). The new methods of Table \ref{['table.3']} (in red) are compared with schemes designed in chen22epf (in gray). The improvement in efficiency of the new methods of order $4$, $5$ and $6$ is clearly visible, whereas the curves corresponding to the 3rd-order schemes $\mathcal{S}_3$ and $\mathcal{NCP}_6^{[3]}$ almost coincide.
• Figure 2: Error in the approximation of $\mathrm{e}^{[A,B]}$ obtained by the new product formulas proposed in this work when $A = -i \sigma_x$, $B = -i \sigma_z$ (left) and when $A$ and $B$ are $16 \times 16$ real matrices with random elements (right). Here we check the improvement of the optimized schemes of Table \ref{['table.4']} with respect to the product formulas of Table \ref{['table.3']}.
• Figure 3: Same as Figure \ref{['fig:effic']}, but now when approximating $\mathrm{e}^{t [A,B]}$ for a larger value $t=10$, with $A = -i \sigma_x$, $B = -i \sigma_z$. Notice the improved efficiency of the new 6th-order approximation (in red), especially with respect to previous schemes (in gray).
• Figure 4: Number of elementary exponentials (gates) required by each product formula to approximate $\mathrm{e}^{x^2 [-i \sigma_x, -i \sigma_z]}$ with error smaller than $10^{-4}$ (left) and $10^{-7}$ (right). This graph should be compared with Fig. 4 in chen22epf: $\mathcal{G}_5$ (involving 56 exponentials) are depicted there, whereas $\mathcal{G}_6$ (involving 98 exponentials) is obtained by applying the recursion of chen22epf to our 4th-order scheme $\mathcal{NCP}_{10}^{[4]}$. Again, the new approximations provide better results for all values of $x$ considered.
• Figure 5: Left: error vs time step for simulating $\mathrm{e}^{-i t (\sigma_x + \sigma_z)}$ using the triple-jump (Yoshida), Suzuki and symmetric Zassenhaus product formulas of order 4. All methods follow a 5th order power law for a single step. Right: Error vs cost for total simulation time $t = 1$.