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arXiv:2606.30741

Theory and practice of Trotter product formulas for quantum chemistry

Pablo A. M. Casares, William Maxwell, Danial Motlagh, Hitarth Choubisa, Zy Niu, Ignacio Loaiza, Jonathan E. Mueller, Arne-Christian Voigt, Juan Miguel Arrazola, Stepan Fomichev

quant-phSubmitted Jun 29, 2026• Announced Jul 1, 2026• 0 citations
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Abstract

Trotter product formulas are a fundamental class of methods for Hamiltonian simulation, particularly attractive due to their low qubit requirements. However, they are often overlooked for use with fault-tolerant quantum algorithms, because of their perceived higher gate counts and the difficulty of estimating Trotter error. Here, we introduce Symmetry-Protected Randomized near-Integrable Trotter (SPRINT) formulas, a framework for building optimized product formulas for electronic structure Hamiltonians widely used in quantum chemistry. SPRINT integrates a generalization of classical near-integrability, randomization, symmetry protection, use of QROM, and other techniques into a thoroughly optimized methodology for Hamiltonian simulation. When applied to concrete simulation tasks, we find SPRINT leads to substantial reduction in gate count compared to previous approaches. Alongside SPRINT, we introduce and analyze a Generalized Rank Decomposition (GRADE) of electronic Hamiltonians that generalizes previous factorization methods. We apply these techniques to the task of simulating the X-ray absorption spectrum of Li$_4$Mn$_2$O, a candidate battery cathode material, leveraging recent advances in tight Trotter error estimation to carefully identify the best version of SPRINT for this problem. Using a Trotter error estimation tool developed in the PennyLane software platform, we show that SPRINT reduces the Toffoli gate cost by a factor of $4.5$ relative to the previous state of the art for this problem, with a gate cost only $\times 2.5$ higher than qubitization, while requiring a dramatic $\times 5.5$ fewer logical qubits. These results establish well-designed Trotter product formulas as an attractive Hamiltonian simulation method for industrially relevant problems in chemistry and materials science.

Comments
21 pages main tex, 51 pages in total, 17 figures

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Trotter product formulas are a leading approach for Hamiltonian simulation on quantum computers, yet their practical performance has remained difficult to assess due to the challenge of accurately estimating the Trotter error. In this work, we develop new theoretical results, algorithms, and software tools that advance the state-of-the-art in Trotter error estimation by orders of magnitude in both scale and accuracy. On the theoretical side, we prove that in the asymptotic limit the error of a product formula depends on the diagonal elements of the Baker-Campbell-Hausdorff (BCH) error operator in the eigenbasis of the Hamiltonian, rather than its full spectral norm -- yielding an improved scaling for Hamiltonian simulation using product formulas. On the algorithmic side, we introduce a compact representation of the BCH expansion that reduces the number of commutators from O(n3)\mathcal{O}(n^3)O(n3) to O(n)\mathcal{O}(n)O(n) for second-order, and from O(n5)\mathcal{O}(n^5)O(n5) to O(n2)\mathcal{O}(n^2)O(n2) for fourth-order formulas on nnn fragments, complemented by an importance sampling scheme to further reduce the computational cost. We provide implementations of these techniques in software and demonstrate their power on two applications: (i) X-ray absorption spectroscopy of an electronic Hamiltonian (Li4_44​Mn2_22​O) at up to 56 qubits using tensor networks; and (ii) vibronic dynamics of naphthalene at over 100 qubits using ML-MCTDH, where we find that naive analytical bounds overestimate the required number of Trotter steps by nearly five orders of magnitude. Our framework enables, for the first time, the accurate estimation of Trotter error at practically relevant system sizes, providing a foundation for fair algorithmic comparisons and rational design of product formulas.
Product formulas, also known as Trotter formulas, are among the most widely used and practical methods for simulating quantum systems on quantum computers. Here we introduce two new classes of randomized product formulas for simulating Hamiltonians with separated energy scales, H=A+αBH=A+αBH=A+αB, where ααα is small. In the standard access model, where one can implement exponentials of AAA and BBB separately, our randomized formulas achieve O(α2)\mathcal O(α^2)O(α2) error scaling at the cost of only doubling the gate depth of the corresponding deterministic formula. We further prove an Ω(α)Ω(α)Ω(α) lower bound for deterministic product formulas. In a stronger access model, allowing exponentials of A+αBℓA+αB_\ellA+αBℓ​ for B=∑ℓBℓB = \sum_{\ell}B_\ellB=∑ℓ​Bℓ​, our randomized formula, based on Trotter Heuristic Resource Improved Formulas for Time-dynamics (THRIFT)~[J. L. Bosse et al., Nat. Commun. 16, 2673 (2025)], achieves O(α3)\mathcal O(α^3)O(α3) error scaling with only constant-factor expected gate overhead. We also establish an Ω(α2)Ω(α^2)Ω(α2) lower bound for deterministic product formulas in this access model. Numerical simulations confirm gate-count reductions for simulating physically motivated systems.
With quantum computers on the horizon, it is essential to understand what problems they can be used to solve. One potential application is simulating quantum systems, in particular, systems with many interacting electrons. However, there is a disparity between the computational resources required to study these problems and the capabilities of modern experimental quantum devices. In this thesis, we make progress toward closing this gap by reducing the cost of quantum simulation of quantum chemistry on a number of fronts. In the first half, we develop three improved quantum simulation algorithms by approximating time evolution under the Hamiltonian using a truncated Taylor series rather than the Trotter-Suzuki decomposition. We do this first for the molecular electronic structure Hamiltonian in two representations, obtaining asymptotic improvements in terms of both the number of orbitals and the required precision. We also apply the truncated Taylor series paradigm to quantum simulation in real space, where we consider simulation of general wave functions discretized on a uniform grid. In the second half, by working with Hamiltonians tailored specifically to quantum algorithms, we demonstrate that Trotter steps of the electronic structure Hamiltonian can be simulated in exactly N depth and with only linear connectivity, where N is the number of spin-orbitals. This resolves the problem of non-locality in fermion-to-qubit mappings, and has wide-ranging applications for variational and phase estimation-based quantum simulation of quantum chemistry. We further study this algorithm on an error-corrected quantum computer using the surface code. Benchmarking it alongside other product formula approaches, we show that quantum simulation of scientifically important instances of the electronic structure problem that are classically intractable can be performed with only a few hundred thousand physical qubits and fewer than one billion costly fault-tolerant gates. Finally, we study the effect of using randomized Hamiltonians in iterative phase estimation, and how this allows us to reduce the costs of simulation as well as to exploit knowledge of the ground state in how we design our simulation algorithm.
Quantum phase estimation requires simulating the evolution of the Hamiltonian, for which product formulas are attractive due to their smaller qubit cost and ease of implementation. However, the estimation of the error incurred by product formulas is usually pessimistic and task-agnostic, which poses problems for assessing their performance in practice for problems of interest. In this work, we study the error of product formulas for the specific task of quantum energy estimation. To this end, we employ the theory of Trotter error with a Magnus-based expansion of the effectively simulated Hamiltonian. The result is a generalization of previous energy estimation error analysis of gapped eigenstates to arbitrary order product formulas. As an application, we discover a 9-term second-order product formula with an energy estimation error that is quadratically better than Trotter-Suzuki. Furthermore, by leveraging recent work on low-energy dynamics of product formulas, we provide tighter bounds for energy estimation error in the low-energy subspace. We show that for Hamiltonians with some locality and positivity properties, the cost can achieve up to a quadratic asymptotic speedup in terms of the target error.
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.
Recent work has deployed linear combinations of unitaries techniques to reduce the cost of fault-tolerant quantum simulations of correlated electron models. Here, we show that one can sometimes improve upon those results with optimized implementations of Trotter-Suzuki-based product formulas. We show that low-order Trotter methods perform surprisingly well when used with phase estimation to compute relative precision quantities (e.g. energies per unit cell), as is often the goal for condensed-phase systems. In this context, simulations of the Hubbard and plane-wave electronic structure models with N < 10 5 fermionic modes can be performed with roughly O ( 1 ) and O ( N 2 ) T complexities. We perform numerics revealing tradeoffs between the error and gate complexity of a Trotter step; e.g., we show that split-operator techniques have less Trotter error than popular alternatives. By compiling to surface code fault-tolerant gates and assuming error rates of one part per thousand, we show that one can error-correct quantum simulations of interesting, classically intractable instances with a few hundred thousand physical qubits.