Trotterization is a technique that allows one to approximate a time evolution of a Hamiltonian by repeatedly evolving the individual terms of the Hamiltonian one-at-a-time for small time durations. Bounds on the error of this approximation exist; however, they are typically loose and moreover, it is known that the true error can be greatly influenced by the order in which the terms of the Hamiltonian are evolved. In this work, we consider various ordering strategies that exploit the commutation structure of the Hamiltonian, in addition to a few other baseline ordering strategies. These commutation-based strategies involve dividing the terms of the Hamiltonian into groups where all the terms within each group commute with one another. These groupings can be obtained by using graph coloring techniques on what we call the "commutation graph" of the Hamiltonian. We prove various results regarding the structure and properties of such commutation graphs for certain classes of Hamiltonians. We also empirically calculate the (true) Trotter error using these ordering strategies on various 1D and 2D Heisenberg-style systems.
Trotterization is a technique that allows one to approximate a time evolution of a Hamiltonian by repeatedly evolving the individual terms of the Hamiltonian one-at-a-time for small time durations. Bounds on the error of this approximation exist; however, they are typically loose and moreover, it is known that the true error can be greatly influenced by the order in which the terms of the Hamiltonian are evolved. In this work, we consider various ordering strategies that exploit the commutation structure of the Hamiltonian, in addition to a few other baseline ordering strategies. These commutation-based strategies involve dividing the terms of the Hamiltonian into groups where all the terms within each group commute with one another. These groupings can be obtained by using graph coloring techniques on what we call the "commutation graph" of the Hamiltonian. We prove various results regarding the structure and properties of such commutation graphs for certain classes of Hamiltonians. We also empirically calculate the (true) Trotter error using these ordering strategies on various 1D and 2D Heisenberg-style systems.
Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter number. However, this increases the length of the quantum circuits required, which may be impractical. It is therefore desirable to find methods of reducing the Trotter error through alternate means. The Trotter error is dependent on the order in which individual term unitaries are applied. Due to the factorial growth in the number of possible orderings with respect to the number of terms, finding an optimal strategy for ordering Trotter sequences is difficult. In this paper, we propose three ordering strategies, and assess their impact on the Trotter error incurred. Initially, we exhaustively examine the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate how the optimal ordering scheme depends on the compatibility graph of the Hamiltonian, and show how it varies with increasing bond length. We then use 44 molecular Hamiltonians to evaluate two strategies based on coloring their incompatibility graphs, while considering the properties of the obtained colorings. We find that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative to this, the difference between ordering schemes can be substantial, being approximately on the order of millihartrees. The coloring-based ordering schemes are reasonably promising—particularly for systems involving heavy atoms—however further work is required to increase dependence on the magnitude of terms. Finally, we consider ordering strategies based on the norm of the Trotter error operator, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian.
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.
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.
Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter number. However, this increases the length of the quantum circuits required, which may be impractical. It is therefore desirable to find methods of reducing the Trotter error through alternate means. The Trotter error is dependent on the order in which individual term unitaries are applied. Due to the factorial growth in the number of possible orderings with respect to the number of terms, finding an optimal strategy for ordering Trotter sequences is difficult. In this paper, we propose three ordering strategies, and assess their impact on the Trotter error incurred. Initially, we exhaustively examine the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate how the optimal ordering scheme depends on the compatibility graph of the Hamiltonian, and show how it varies with increasing bond length. We then use 44 molecular Hamiltonians to evaluate two strategies based on coloring their incompatibility graphs, while considering the properties of the obtained colorings. We find that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative to this, the difference between ordering schemes can be substantial, being approximately on the order of millihartrees. The coloring-based ordering schemes are reasonably promising—particularly for systems involving heavy atoms—however further work is required to increase dependence on the magnitude of terms. Finally, we consider ordering strategies based on the norm of the Trotter error operator, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian.
Quantum algorithms for quantum chemistry and other many-body fermionic systems work by expressing the Hamiltonian in a basis of qubits and fragmenting the Hamiltonian into a sum of products of Pauli operators whose exponentials are easily encoded on a quantum device. Applying the product of exponentials, known as Trotterization, leads to an error associated with the non-commutativity of operators. This error can lead to breaking the symmetries of the Hamiltonian because the fragments are not symmetry conserving in general. Nonetheless, many algorithms for time evolution rely on Trotterization, including time evolution and quantum phase estimation. We show that we can express the Hamiltonian in terms of Hermitian excitation operators which map to sums of commuting Pauli strings for any encoding and conserve symmetries corresponding to Abelian groups of symmetry operators. Symmetries corresponding to non-Abelian groups, on the other hand, are not fully conserved by Trotterized Hermitian excitation operators, so we developed ``operator kirigami'' to cut the sum of non-commuting operators by orthogonal projection and to fold terms together using unitary rotations. We tested pools of operators for small molecules and basis sets, and found that electron number and spin symmetry conserving pools led to greater errors that decreased for larger molecules and were negated with second-order Trotterization. Our work shows the potential for testing quantum computing algorithms on classical computers by adapting tools used in electronic structure theory with conserved symmetries.
Solving the electronic structure problem via unitary evolution of the electronic Hamiltonian is one of the promising applications of digital quantum computers. One of the practical strategies to implement the unitary evolution is via Trotterization, where a sequence of short-time evolutions of fast-forwardable (i.e. efficiently diagonalizable) Hamiltonian fragments is used. Given multiple choices of possible Hamiltonian decompositions to fast-forwardable fragments, the accuracy of the Hamiltonian evolution depends on the choice of the fragments. We assess efficiency of multiple Hamiltonian partitioning techniques using fermionic and qubit algebras for the Trotterization. Use of symmetries of the electronic Hamiltonian and its fragments significantly reduces the Trotter error. This reduction makes fermionic-based partitioning Trotter errors lower compared to those in qubit-based techniques. However, from the simulation-cost standpoint, fermionic methods tend to introduce quantum circuits with a greater number of T-gates at each Trotter step and thus are more computationally expensive compared to their qubit counterparts.