Properties of higher-order Trotter formulas

Yikang Zhang, Thomas Barthel

Yiyuan Chen, Jonas Helsen, Māris Ozols

The Sachdev–Ye–Kitaev (SYK) model is a prominent model of strongly interacting fermions that serves as a toy model of quantum gravity and black hole physics. In this work, we study the Trotter error and gate complexity of the quantum simulation of the SYK model using Lie–Trotter–Suzuki formulas. Building on recent results by Chen and Brandão \cite{Anthony} — in particular their uniform smoothing technique for random matrix polynomials — we derive bounds on the first- and higher-order Trotter error of the SYK model, and subsequently find near-optimal gate complexities for simulating these models using Lie–Trotter–Suzuki formulas. For the k -local SYK model on n Majorana fermions, at time t , our gate complexity estimates for the first-order Lie–Trotter–Suzuki formula scales with O ~ ( n k + 5 2 t 2 ) for even k and O ~ ( n k + 3 t 2 ) for odd k , and the gate complexity of simulations using higher-order formulas scales with O ~ ( n k + 1 2 t ) for even k and O ~ ( n k + 1 t ) for odd k . Given that the SYK model has Θ ( n k ) terms, these estimates are close to optimal. These gate complexities can be further improved upon in the context of simulating the time evolution of an arbitrary fixed input state | ψ ⟩ , leading to a O ( n 2 ) -reduction in gate complexity for first-order formulas and O ( n ) -reduction for higher-order formulas.We also apply our techniques to the sparse SYK model, which is a simplified variant of the SYK model obtained by deleting all but a Θ ( n ) fraction of the terms in a uniformly i.i.d. manner. We find the average (over the random term removal) gate complexity for simulating this model using higher-order formulas scales with

Giuseppe Da Prato, Luciano Tubaro

Masuo Suzuki

Ibsal Assi, Michael Vogl, Meenu Kumari +1 more

We propose a variational alternative to the Trotter-Suzuki decomposition that provides greater control over errors while preserving the unitary structure of time evolution. The variational parameters in our ansatz are derived from a global action principle, where Euler-Lagrange equations govern their optimal dynamics. Unlike conventional wavefunction-based variational methods, our approach specifically targets the time evolution operation and this allows a single set of optimized parameters to be applied to any initial state for a fixed Hamiltonian avoiding costly optimization procedures. Our method outperforms the standard Trotter-Suzuki formulas, typically achieving higher accuracy than higher-order Suzuki schemes. This translates directly to quantum computing applications, where it enables the design of quantum circuits with fewer gates which reduces noise and improves precision. Although we focus on quantum dynamics, the method is broadly applicable to problems involving general time-evolution operators. Applied to various model Hamiltonians, our approach reduces errors by factors of 2 to 5 compared to Trotter-Suzuki decompositions, demonstrating its promise for accurate quantum simulation with improved efficiency. In certain cases, the variational ansatz achieves higher accuracy than more complex higher-order Suzuki formulas while reducing the gate count by nearly half within a single circuit layer. Furthermore, we derive approximate analytical expressions for the variational parameters up to cubic order in time, valid for generic Hamiltonians. These approximations enable long-time quantum simulations with improved accuracy over equivalent Suzuki decompositions, providing ready-to-use evolution formulas that match Suzuki's gate complexity while delivering better performance.

I. A. Assi, Michael Rudolf Vogl, Meenu Kumari +1 more

We propose a variational alternative to the Trotter-Suzuki decomposition that provides greater control over errors while preserving the unitary structure of time evolution. The variational parameters in our ansatz are derived from a global action principle, where Euler-Lagrange equations govern their optimal dynamics. Unlike conventional wavefunction-based variational methods, our approach specifically targets the time evolution operation, and this allows a single set of optimized parameters to be applied to any initial state for a fixed Hamiltonian avoiding costly optimization procedures. Our method outperforms the standard Trotter-Suzuki formulas, typically achieving higher accuracy than higher-order Suzuki schemes. This translates directly to quantum computing applications, where it enables the design of quantum circuits with fewer gates, which reduces noise and improves precision. Although we focus on quantum dynamics, the method is broadly applicable to problems involving general time-evolution operators. Applied to various model Hamiltonians, our approach reduces errors by factors of 2 to 5 compared to Trotter-Suzuki decompositions, demonstrating its promise for accurate quantum simulation with improved efficiency. In certain cases, the variational ansatz achieves higher accuracy than more complex higher-order Suzuki formulas while reducing the gate count by nearly half within a single circuit layer. Furthermore, we derive approximate analytical expressions for the variational parameters up to cubic order in time, valid for generic Hamiltonians. These approximations enable long-time quantum simulations with improved accuracy over equivalent Suzuki decompositions, providing ready-to-use evolution formulas that match Suzuki's gate complexity while delivering better performance.