Hamiltonian Simulation via Stochastic Zassenhaus Expansions

TL;DR

The paper introduces stochastic Zassenhaus expansions (SZEs), a family of ancilla-free quantum algorithms for Hamiltonian simulation that map nested Zassenhaus formulas to quantum gates and use randomized sampling to reduce circuit depth. SZEs fuse recursive Zassenhaus expansions with random-unitary sampling to approximate higher-order time evolutions as convex combinations of unitaries, preserving unitary dynamics while lowering gate counts. For a 10-qubit transverse-field Ising model, an 11th-order SZE achieves 42× fewer CNOTs than a standard 10th-order Suzuki-Trotter product formula, with empirical results showing substantial error reductions in favorable regimes. The analysis covers multi-variable error bounds across various Hamiltonian classes (nearest-neighbor, $j$-local, electronic-structure, power-law, and quasilocal) and identifies where SZEs provide the most benefit, notably in geometrically localized systems, offering a resource scaling bridge between product formulas and more advanced methods like quantum signal processing. Overall, SZEs present a practical route to high-order Hamiltonian simulation with favorable depth and gate-count trade-offs for a broad class of physically relevant systems.

Abstract

We introduce the stochastic Zassenhaus expansions (SZEs), a class of ancilla-free quantum algorithms for Hamiltonian simulation. These algorithms map nested Zassenhaus formulas onto quantum gates and then employ randomized sampling to minimize circuit depths. Unlike Suzuki-Trotter product formulas, which grow exponentially long with approximation order, the nested commutator structures of SZEs enable high-order formulas for many systems of interest. For a 10-qubit transverse-field Ising model, we construct an 11th-order SZE with 42x fewer CNOTs than the standard 10th-order product formula. Further, we empirically demonstrate regimes where SZEs reduce simulation errors by many orders of magnitude compared to leading algorithms.

Figures (3)

• Figure 1: Average number of CNOT gates vs. approximation order $p$ for a tranverse-field Ising model with $n=10$ qubits, before applying compilation algorithms. The stochastic Zassenhaus expansions (SZE) reduce gate counts compared to both the standard Suzuki-Trotter product formulas (PF) and the minimal product formulas morales_greatly_2022. For the same $k$-th order nested Zassenhaus formulas, we compare stochastic schemes to order $p=2k-1$ and $p=2k+1$, showing that the extra stochastic terms negligibly increase gate costs.
• Figure 2: Trace distance error vs. time $t$ for a transverse-field Ising model with $n=10$ qubits and an equal superposition starting state, $\ket{+}^{\otimes n}$. The points are empirically calculated trace distances, and the lines are power-law fits to the smallest $5$ time steps. The labels show each algorithms' asymptotic scaling in time.
• Figure 3: Trace distance error vs. system size $n$ for the transverse-field Ising model. The simulation begins with the equal superposition starting state of $\ket{+}^{\otimes n}$ and evolves for a fixed time step of $t=0.03$. The points are empirically calculated trace distances, and the lines are power-law fits to the largest 5 system sizes. The labels show each algorithms' predicted scaling with respect to system size.