Product formulas for Trotter Suzuki simulation remain a practical route to Hamiltonian evolution on noisy intermediate scale quantum (NISQ) hardware, yet their accuracy hinges on three coupled choices: term grouping, product formula order, and timestep allocation. Toolchains such as Qiskit and Paulihedral lean on hand tuned heuristics, while the discrete nature of grouping and order makes naive gradient based optimization awkward. We describe a generate and evaluate loop: a conditional diffusion model proposes strategies, a physics informed neural network (PINN) supplies differentiable fidelity feedback, and a graph neural network (GNN) encodes commutator structure. Training spans a hybrid space (discrete grouping and order, continuous time steps); the closed loop uses REINFORCE and a Pareto tracker. On the transverse field Ising model (TFIM), under our primary comparison setup, the method reaches 85.6% of the fidelity of a fourth order Qiskit baseline (0.856) at roughly 21.8% of the circuit depth and 19.2% of the baseline CNOT count. Under an equal depth budget, fine tuning in the loop reached a best observed fidelity of 0.9994. Updated ablations show that, for a fixed training budget and default guidance knobs, module contributions depend on the training recipe and guidance hyperparameters CFG in particular needs to be tuned jointly with compute budget. Overall, the results suggest that "generative model and physics supervision" is a viable angle for NISQ oriented compilation, though where it wins still depends on the operating point.
Product formulas for Trotter Suzuki simulation remain a practical route to Hamiltonian evolution on noisy intermediate scale quantum (NISQ) hardware, yet their accuracy hinges on three coupled choices: term grouping, product formula order, and timestep allocation. Toolchains such as Qiskit and Paulihedral lean on hand tuned heuristics, while the discrete nature of grouping and order makes naive gradient based optimization awkward. We describe a generate and evaluate loop: a conditional diffusion model proposes strategies, a physics informed neural network (PINN) supplies differentiable fidelity feedback, and a graph neural network (GNN) encodes commutator structure. Training spans a hybrid space (discrete grouping and order, continuous time steps); the closed loop uses REINFORCE and a Pareto tracker. On the transverse field Ising model (TFIM), under our primary comparison setup, the method reaches 85.6% of the fidelity of a fourth order Qiskit baseline (0.856) at roughly 21.8% of the circuit depth and 19.2% of the baseline CNOT count. Under an equal depth budget, fine tuning in the loop reached a best observed fidelity of 0.9994. Updated ablations show that, for a fixed training budget and default guidance knobs, module contributions depend on the training recipe and guidance hyperparameters CFG in particular needs to be tuned jointly with compute budget. Overall, the results suggest that "generative model and physics supervision" is a viable angle for NISQ oriented compilation, though where it wins still depends on the operating point.
Hamiltonian simulation represents an important module in a large class of quantum algorithms and simulations such as quantum machine learning, quantum linear algebra methods, and modeling for physics, material science, and chemistry. One of the most prominent methods for realizing the time-evolution unitary is via the Trotter-Suzuki decomposition. However, there is a large class of possible decompositions for the infinitesimal time-evolution operator as the order in which the Hamiltonian terms are implemented is arbitrary. We introduce a perspective for generating a low-depth Trotter-Suzuki decomposition assuming the standard Clifford+rz gate set by adapting ideas from quantum error correction. We map a given Trotter-Suzuki decomposition to a constrained path on a graph which we deem the Pauli frame graph (PFG). Each node of the PFG represents the set of possible Hamiltonian terms currently available to be applied, Clifford operations represent a move from one node to another, and so the graph distance represents the gate cost of implementing the decomposition. The problem of finding the optimal decomposition is then equivalent to solving a problem similar to the traveling salesman. Although this is an NP-hard problem, we demonstrate the simplest heuristic, greedy search, and compare the resulting two-qubit gate count and circuit depth to more standard methods for a large class of scientifically relevant Hamiltonians, both fermionic and bosonic, found in chemical, vibrational, and condensed-matter problems which naturally scale. We find, in nearly every case we study, the resulting depth and two-qubit gate counts are less than those provided by standard methods, by as much as an order of magnitude. We also find the method is efficient and amenable to parallelization, making it scalable for problems of real interest.
Compiling time-evolution operators of the form U(t)=e−iHt into hardware-native gate sequences is a central bottleneck for digital quantum simulation on noisy intermediate-scale quantum (NISQ) devices. Generic transpilation treats U(t) as an arbitrary unitary, discarding the structure of Hamiltonian dynamics and producing circuits whose depth exceeds hardware coherence limits. We introduce a structure-aware compilation framework that treats product-formula decompositions as synthesis primitives rather than simulation approximations. The method combines (i) native placement of Hamiltonian terms onto the hardware coupling map, (ii) adaptive selection of Trotter blocks via a greedy discretization procedure, and (iii) variational refinement using a Trotter-initialized ansatz. Across Heisenberg, Ising, and XY models with n=3--8 qubits, the compiled circuits achieve fidelities F>0.996 with approximately linear scaling in the number of entangling gates, while generic synthesis produces circuits that are orders of magnitude deeper. On IBM Torino hardware, we observe a regime in which shorter approximate circuits outperform deeper exact decompositions: a 27-CX circuit achieves higher hardware fidelity (Fhw=0.987) than a 187-CX exact circuit. These results demonstrate that, in the NISQ regime, structure-aware approximate compilation can outperform exact structure-agnostic synthesis, providing a practical pathway for executing Hamiltonian dynamics without requiring pulse-level control.
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.
Trotterization is a standard approach for simulating quantum time evolution on quantum computers, where the Hamiltonian is split into local terms and each term is applied in sequence. The order of these terms affects the fidelity of the simulation when they do not commute, so the choice of ordering directly impacts the accuracy of the simulation. We study this problem for one-dimensional XXZ Heisenberg Hamiltonians using a structured set of 24 candidate orderings derived from colorings of the Hamiltonian's commutation graph and their group permutations. Finding the best candidate for large systems becomes prohibitive because fidelity evaluation is computationally expensive. In this work, we train a transformer encoder on smaller systems to predict the best candidate ordering for larger systems directly from Hamiltonian and Trotter-configuration features, without computing candidate fidelities at inference time. The model is trained on in-range systems of 3 to 14 qubits with 15-qubit systems held out for validation. Experimental results show that the model reaches a mean test fidelity gap of 0.00115 relative to the best of the 24 candidates on out-of-range systems of 16 to 20 qubits. A training-size sweep further shows that generalization emerges once training includes systems up to L=8 qubits, with validation at L=9, and the gap continues to decrease as the training range grows. To our knowledge, this is the first application of a learned model to Trotter ordering, and it motivates future work on AI-guided Trotter ordering with generalization across Hamiltonian families and system types.
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.
Trotterization approximates quantum time evolution by sequentially applying Hamiltonian terms. Because noncommuting terms introduce ordering-dependent errors, selecting an optimal term ordering is a combinatorial problem over a factorial search space. Prior approaches rely on fixed heuristics or on selection among predefined structured orderings, both of which require simulating candidates before choosing among them. For 1D and 2D Heisenberg-style Hamiltonians, we instead learn the ordering directly with a physics-aware ranking transformer that assigns a scalar score to each Hamiltonian term and predicts an ordering by sorting these scores. Physical structure enters through commutator-biased attention and an anticommutation-weighted ranking loss, and the models are trained on simulated-annealing (SA) reference orderings. We train separate models for first- and second-order Trotterization, each jointly on chains up to 14 qubits and lattices up to 12 qubits, and evaluate them on unseen chains with 16-20 qubits and lattices with 16 and 20 qubits. Predicted orderings are evaluated by the median gap in simulation fidelity to the SA reference. At first order, the model reaches a pooled gap below 10^-4 on chains, 0.0144 and 0.0088 on triangular lattices, and 0.0948 and 0.0823 on rectangular lattices at 16 and 20 qubits; at second order, the gaps are 0.0114, 0.0364 and 0.0244, and 0.1348 and 0.1239. The prediction exceeds the SA reference on 34% of first-order and 8% of second-order chain instances and can match its fidelity even when the two sequences differ because commuting terms may be rearranged without changing the Trotter unitary. The learned models generalize to larger systems across all three geometries, performing best on chains and triangular lattices, and produce an ordering in one forward pass without candidate enumeration, simulated annealing, or fidelity evaluation.