The product formula, commonly known as Trotter decomposition, is a central tool for digital quantum simulation, whose performance depends critically on how the Hamiltonian is partitioned into tractable blocks. Standard decompositions typically rely on direct commutativity among Hamiltonian terms in a chosen operator representation, which can lead to large residual errors and deep circuits for complex, practically relevant many-body quantum systems. We address this fundamental bottleneck by introducing a new decomposition principle that goes beyond commutativity, grouping Hamiltonian terms into local three-site clusters according to the underlying SU(2) symmetry of the local dynamics. We show that three-site generators fall into at most four SU(2)-symmetry classes, each admitting an effective two-qubit SU(4) representation with exact and efficient implementations. By reducing the number of clusters, this decomposition principle substantially suppresses commutator-induced errors and circuit overhead while preserving underlying physical structures that commutativity-based decompositions may violate. We demonstrate the proposed method on several physically relevant spin-lattice models, where the reduced cluster structure can even realise the second-order product formula without doubling the circuit depth, as would be required by conventional decompositions. Numerical simulations of a Kagome Heisenberg model with triangular spin-chirality interactions show that the proposed method reduces both state infidelity and average spin-chirality bias by more than three orders of magnitude compared with conventional decompositions, while using substantially fewer gates. These results establish local symmetry as a flexible and practical design principle for product-formula simulation, opening a route to more accurate and hardware-efficient simulations of broader classes of many-body systems.
The product formula, commonly known as Trotter decomposition, is a central tool for digital quantum simulation, whose performance depends critically on how the Hamiltonian is partitioned into tractable blocks. Standard decompositions typically rely on direct commutativity among Hamiltonian terms in a chosen operator representation, which can lead to large residual errors and deep circuits for complex, practically relevant many-body quantum systems. We address this fundamental bottleneck by introducing a new decomposition principle that goes beyond commutativity, grouping Hamiltonian terms into local three-site clusters according to the underlying SU(2) symmetry of the local dynamics. We show that three-site generators fall into at most four SU(2)-symmetry classes, each admitting an effective two-qubit SU(4) representation with exact and efficient implementations. By reducing the number of clusters, this decomposition principle substantially suppresses commutator-induced errors and circuit overhead while preserving underlying physical structures that commutativity-based decompositions may violate. We demonstrate the proposed method on several physically relevant spin-lattice models, where the reduced cluster structure can even realise the second-order product formula without doubling the circuit depth, as would be required by conventional decompositions. Numerical simulations of a Kagome Heisenberg model with triangular spin-chirality interactions show that the proposed method reduces both state infidelity and average spin-chirality bias by more than three orders of magnitude compared with conventional decompositions, while using substantially fewer gates. These results establish local symmetry as a flexible and practical design principle for product-formula simulation, opening a route to more accurate and hardware-efficient simulations of broader classes of many-body systems.
The Suzuki-Trotter decomposition, which digitalizes quantum time evolution, provides a promising framework for simulating quantum dynamics on quantum hardware and exploring quantum advantage over classical computation. However, conventional Trotter circuits require a large number of non-local gates, lowering their faithfulness to the ideal dynamics when implemented on current noisy quantum hardware. While most previous studies have focused on circuit optimization, we instead propose a new Trotter decomposition that is intrinsically circuit-efficient for simulating quantum dynamics on near-term devices. Our method substantially reduces both the residual error by Trotter decomposition and the number of CNOT operations compared to conventional Trotter decompositions by exploiting the symmetry of the target model to construct an effective Hamiltonian with fewer two-qubit gates. We demonstrate the noise robustness of the proposed approach through numerical simulations of a nine-site Heisenberg model under realistic noise, and further validate its experimental practicality on the IBM superconducting device, achieving a state fidelity exceeding 0.98 when combined with quantum error mitigation in the three-site case. The proposed circuit design is also compatible with existing circuit optimization techniques. Our results establish a practical route toward noise-resilient quantum simulation in many-body dynamics.
The Suzuki–Trotter decomposition, which digitalizes quantum time evolution, provides a promising framework for simulating quantum dynamics on quantum hardware and exploring quantum advantage over classical computation. However, conventional Trotter circuits require a large number of non-local gates, lowering their faithfulness to the ideal dynamics when implemented on current noisy quantum hardware. While most previous studies have focused on circuit optimization, we instead propose a new Trotter decomposition that is intrinsically circuit-efficient for simulating quantum dynamics on near-term devices. Our method substantially reduces both the residual error by Trotter decomposition and the number of CNOT operations compared to conventional Trotter decompositions by exploiting the symmetry of the target model to construct an effective Hamiltonian with fewer two-qubit gates. We demonstrate the noise robustness of the proposed approach through numerical simulations of a nine-site Heisenberg model under realistic noise and further validate its experimental practicality on the IBM superconducting device, achieving a state fidelity exceeding 0.98 when combined with quantum error mitigation in the three-site case. The proposed circuit design is also compatible with existing circuit optimization techniques. Our results establish a practical route toward noise-resilient quantum simulation in many-body dynamics.
Product formulas are a basic tool for simulating quantum many-body dynamics, but a fundamental degree of freedom in their application---the ordering of Hamiltonian terms---has remained largely unexplored. Here we systematically study such ordering effects, quantifying their impact on simulation accuracy and exploiting them to design more accurate product-formula simulations. We develop an efficiently computable, ordering-sensitive bound on the finite-step-size Trotter error for geometrically local Hamiltonians for arbitrarily high-order product formulas. For translation-invariant one-dimensional chains, our bounds are close to exact second-, fourth-, and sixth-order Trotter errors in small-system tests, while on large-scale systems they improve on previous higher-order analytical bounds by orders of magnitude. We further use the local structure of the error bounds and estimates to optimize ordering efficiently, reducing the long-chain problem to a minimum-mean-weight-cycle problem on a weighted de Bruijn graph. For random translation-invariant chains, sequential (staircase-like) orderings consistently outperform standard brickwall circuits; when circuit depth is restricted, the optimizer instead favors wave-like parallel-sequential orderings, which preserve much of the accuracy advantage over brickwall circuits. These results establish ordering as a useful degree of freedom for both understanding and improving product-formula simulation.
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.
Quantum simulation is a powerful tool for exploring complex quantum many-body systems such as condensed matter physics and gauge theories. Trotterization, which approximates the ideal time evolution operator by decomposing it into a sequence of local gate operations, is one of the most widely used quantum simulation algorithms. However, such Trotterized implementations generally fail to preserve the symmetries of the target Hamiltonian during compilation. As a result, they can drive quantum states out of symmetrically allowed subspaces, leading to unphysical dynamics and symmetry-violating algorithmic errors. In this work, we propose a symmetry-based Trotter error mitigation protocol using classical post-processing. By applying symmetry transformations to the initial state or interleaving them between discrete Trotter layers, and then averaging an ensemble of the resulting measurement outcomes via classical post-processing, our method systematically projects out the symmetry-violating components of the Trotter error while leaving the ideal dynamics unchanged. Importantly, this framework naturally accommodates non-local spatial symmetries and anti-unitary operations such as time reversal, which are difficult or impossible to implement directly with hardware-native quantum gates. We benchmark our protocol on the one-dimensional XY model and the one-dimensional Schwinger model. In the XY model, enforcing reflection symmetry suppresses the leading-order Trotter error, whereas in the Schwinger model, interleaving gauge transformations between Trotter layers enables gauge-twirling effectively to reduce unphysical violations of local Gauss's law. These results demonstrate that symmetry-based post-processing provides a depth-preserving route to substantially improving the fidelity of Trotterized quantum simulations on near-term devices.
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.