Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2501.10314 · PRX Quantum

Fault-Tolerant Quantum Simulation of Generalized Hubbard Models

Andreas Juul Bay-Smidt, Frederik Ravn Klausen, Christoph Sünderhauf, Róbert Izsák, Gemma C. Solomon, Nick S. Blunt

Published Jul 30, 2025• 2 citations
arXivHTMLPDF

Abstract

Quantum simulations of strongly interacting fermionic systems, such as those described by the Hubbard model, are promising candidates for useful early fault-tolerant quantum computing applications. This paper presents tile Trotterization, a generalization of plaquette Trotterization, which uses a set of tiles to construct Trotter decompositions of arbitrary lattice Hubbard models. The tile Trotterization scheme also enables the simulation of more complex models, including the extended Hubbard model. We improve previous Hubbard-model commutator bounds, further provide tight commutator bounds for periodic extended Hubbard models, and demonstrate the use of tensor network methods for this task. We consider applications of tile Trotterization to simulate hexagonal lattice Hubbard models and compare the resource requirements of tile Trotterization for performing quantum phase estimation to a qubitization-based approach, demonstrating that tile Trotterization scales more efficiently with system size. These advancements significantly broaden the potential applications of early fault-tolerant quantum computers to models of practical interest in materials research and organic chemistry.

DOI
10.1103/gr4t-b1w5

Related papers

Early fault-tolerant simulations of the Hubbard model

Earl T. Campbell

Abstract Simulation of the Hubbard model is a leading candidate for the first useful applications of a fault-tolerant quantum computer. A recent study of quantum algorithms for early simulations of the Hubbard model [Kivlichan et al 2019 Quantum 4 296] found that the lowest resource costs were achieved by split-operator Trotterization combined with the fast-fermionic Fourier transform (FFFT) on an L × L lattice with length L = 2 k . On lattices with length L ≠ 2 k , Givens rotations can be used instead of the FFFT but lead to considerably higher resource costs. We present a new analytic approach to bounding the simulation error due to Trotterization that provides much tighter bounds for the split-operator FFFT method, leading to 16× improvement in error bounds. Furthermore, we introduce plaquette Trotterization that works on any size lattice and apply our improved error bound analysis to show competitive resource costs. We consider a phase estimation task and show plaquette Trotterization reduces the number of non-Clifford gates by a factor 5.5× to 9× (depending on the parameter regime) over the best previous estimates for 8 × 8 and 16 × 16 lattices and a much larger factor for other lattice sizes not of the form L = 2 k . In conclusion, we find there is a potentially useful application for fault-tolerant quantum computers using around one million Toffoli gates.

arXiv:2012.09238•Quantum Science and Technology•Oct 19, 2021•81 citations

Improved Fault-Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization

Ian Kivlichan, Craig Gidney, Dominic William Berry +9 more

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.

arXiv:1902.10673•Quantum•Jul 16, 2020•169 citations

Quantum simulation of nanographenes and Trotter error cancellation

Andreas Juul Bay-Smidt, Nina Glaser, Marcel D. Fabian +3 more

Fault-tolerant quantum computing is a promising tool for simulating molecules and materials, but frequently-considered applications require substantial resources, and the gap between hardware capabilities and requirements remains significant. We propose quantum simulation of nanographene $π$-systems as relevant and scalable problems to span the gap between early and large-scale fault-tolerant quantum computing. We examine the efficiency of Trotterized quantum simulation, present a detailed analysis of worst-case, average-case and energy eigenvalue Trotter errors, and show that these Trotter error estimates vary by orders of magnitude. Trotter eigenvalue errors are obtained from a novel tensor-network-based approach which allows spectral analysis of product formulas for systems beyond brute-force calculation. Notably, we observe a Trotter error cancellation phenomenon whereby the Trotter error for energy differences between low-lying eigenstates is significantly smaller than the Trotter error for absolute energies, resulting in approximately an order of magnitude circuit depth reduction for quantum phase estimation calculation of energy gaps. This is a significant result because for most chemical applications, only energy differences are of practical relevance. We estimate that calculation of energy gaps to chemical accuracy between the ground- and excited-states within the Pariser--Parr--Pople model for large 2D nanographenes (up to 140 spin orbitals) requires circuits with $< 3.2 \times 10^7$ Toffoli gates. This work shows that considering details of chemically-relevant applications and exploiting error cancellation can lead to substantial reductions in resource requirements.

quant-pharXiv:2605.00745•May 4, 2026•0 citations

Quantum simulation costs for Suzuki-Trotter decomposition of quantum many-body lattice models

Nathan M. Myers, Ryan Scott, Kwon Park +1 more

Quantum computers offer the potential to efficiently simulate the dynamics of quantum systems, a task whose difficulty scales exponentially with system size on classical devices. To assess the potential for near-term quantum computers to simulate many-body systems we develop a formalism to straightforwardly compute bounds on the number of Trotter steps needed to accurately simulate the time evolution of fermionic lattice models based on the first-order commutator scaling. We apply this formalism to two closely related many-body models prominent in condensed matter physics, the Hubbard and t-J models. We find that, while a naive comparison of the Trotter depth first seems to favor the Hubbard model, careful consideration of the model parameters and the allowable error for accurate simulation leads to a substantial advantage in favor of the t-J model. These results and formalism set the stage for significant improvements in quantum simulation costs.

arXiv:2302.04698•Physical Review Research•Jun 28, 2023•5 citations

Nearly tight Trotterization of interacting electrons

Su, Yuan, Hsin-Yuan Huang, Earl T. Campbell

We consider simulating quantum systems on digital quantum computers. We show that the performance of quantum simulation can be improved by simultaneously exploiting the commutativity of Hamiltonian, the sparsity of interactions, and the prior knowledge of initial state. We achieve this using Trotterization for a class of interacting electrons that encompasses various physical systems, including the plane-wave-basis electronic structure and the Fermi-Hubbard model. We estimate the simulation error by taking the transition amplitude of nested commutators of Hamiltonian terms within the η-electron manifold. We develop multiple techniques for bounding the transition amplitude and expectation of general fermionic operators, which may be of independent interest. We show that it suffices to use O(n^(5/3)/η^(2/3)+n^(4/3)η^(2/3)) gates to simulate electronic structure in the plane-wave basis with n spin orbitals and η electrons up to a negligible factor, improving the best previous result in second quantization while outperforming the first-quantized simulation when n=O(η²). We also obtain an improvement for simulating the Fermi-Hubbard model. We construct concrete examples for which our bounds are almost saturated, giving a nearly tight Trotterization of interacting electrons.

•Dec 16, 2020•46 citations

Nanostructure modelling with early fault tolerant quantum computers

Zhu Sun, Christian Binder, Balint Koczor +1 more

Semiconductor nanostructures are central to many developing technologies. Notably, double quantum dots are especially important for semiconductor spin-qubit architectures, quantum sensing applications, and quantum-dot solar cells. Accurate modelling is highly desirable but conventional methods can struggle when dynamics involve more than two interacting electrons. In this work, we present a quantum simulation framework capable of addressing multi-electron double quantum dots. We adopt an efficiently scaling 1$^\text{st}$ quantised representation of the system and develop algorithms based on both Trotterisation and Qubitisation. Incorporating insights from classical simulations enables us to produce resource estimates that are more realistic than those obtained from theoretical error bounds. Using a standard surface code model with physical noise at $10^{-3}$, our results indicate that the ground-state energy of four electrons in a double quantum dot can be estimated in approximately 22 hours using 226k physical qubits, or an eight-electron system in 3.3 days with 314k qubits (with runtimes falling dramatically when more qubits are available). We anticipate that incorporating recent advances in surface code architectures may reduce these costs significantly further. Our results suggest that early fault-tolerant quantum computers may become valuable tools for designing mature-era quantum technologies.

quant-pharXiv:2606.06442•Jun 5, 2026•0 citations
Quantum simulations of strongly interacting fermionic systems, such as those described by the Hubbard model, are promising candidates for useful early fault-tolerant quantum computing applications. This paper presents tile Trotterization, a generalization of plaquette Trotterization, which uses a set of tiles to construct Trotter decompositions of arbitrary lattice Hubbard models. The tile Trotterization scheme also enables the simulation of more complex models, including the extended Hubbard model. We improve previous Hubbard-model commutator bounds, further provide tight commutator bounds for periodic extended Hubbard models, and demonstrate the use of tensor network methods for this task. We consider applications of tile Trotterization to simulate hexagonal lattice Hubbard models and compare the resource requirements of tile Trotterization for performing quantum phase estimation to a qubitization-based approach, demonstrating that tile Trotterization scales more efficiently with system size. These advancements significantly broaden the potential applications of early fault-tolerant quantum computers to models of practical interest in materials research and organic chemistry.
Abstract Simulation of the Hubbard model is a leading candidate for the first useful applications of a fault-tolerant quantum computer. A recent study of quantum algorithms for early simulations of the Hubbard model [Kivlichan et al 2019 Quantum 4 296] found that the lowest resource costs were achieved by split-operator Trotterization combined with the fast-fermionic Fourier transform (FFFT) on an L × L lattice with length L = 2 k . On lattices with length L ≠ 2 k , Givens rotations can be used instead of the FFFT but lead to considerably higher resource costs. We present a new analytic approach to bounding the simulation error due to Trotterization that provides much tighter bounds for the split-operator FFFT method, leading to 16× improvement in error bounds. Furthermore, we introduce plaquette Trotterization that works on any size lattice and apply our improved error bound analysis to show competitive resource costs. We consider a phase estimation task and show plaquette Trotterization reduces the number of non-Clifford gates by a factor 5.5× to 9× (depending on the parameter regime) over the best previous estimates for 8 × 8 and 16 × 16 lattices and a much larger factor for other lattice sizes not of the form L = 2 k . In conclusion, we find there is a potentially useful application for fault-tolerant quantum computers using around one million Toffoli gates.
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.
Fault-tolerant quantum computing is a promising tool for simulating molecules and materials, but frequently-considered applications require substantial resources, and the gap between hardware capabilities and requirements remains significant. We propose quantum simulation of nanographene πππ-systems as relevant and scalable problems to span the gap between early and large-scale fault-tolerant quantum computing. We examine the efficiency of Trotterized quantum simulation, present a detailed analysis of worst-case, average-case and energy eigenvalue Trotter errors, and show that these Trotter error estimates vary by orders of magnitude. Trotter eigenvalue errors are obtained from a novel tensor-network-based approach which allows spectral analysis of product formulas for systems beyond brute-force calculation. Notably, we observe a Trotter error cancellation phenomenon whereby the Trotter error for energy differences between low-lying eigenstates is significantly smaller than the Trotter error for absolute energies, resulting in approximately an order of magnitude circuit depth reduction for quantum phase estimation calculation of energy gaps. This is a significant result because for most chemical applications, only energy differences are of practical relevance. We estimate that calculation of energy gaps to chemical accuracy between the ground- and excited-states within the Pariser--Parr--Pople model for large 2D nanographenes (up to 140 spin orbitals) requires circuits with <3.2×107< 3.2 \times 10^7<3.2×107 Toffoli gates. This work shows that considering details of chemically-relevant applications and exploiting error cancellation can lead to substantial reductions in resource requirements.
Quantum computers offer the potential to efficiently simulate the dynamics of quantum systems, a task whose difficulty scales exponentially with system size on classical devices. To assess the potential for near-term quantum computers to simulate many-body systems we develop a formalism to straightforwardly compute bounds on the number of Trotter steps needed to accurately simulate the time evolution of fermionic lattice models based on the first-order commutator scaling. We apply this formalism to two closely related many-body models prominent in condensed matter physics, the Hubbard and t-J models. We find that, while a naive comparison of the Trotter depth first seems to favor the Hubbard model, careful consideration of the model parameters and the allowable error for accurate simulation leads to a substantial advantage in favor of the t-J model. These results and formalism set the stage for significant improvements in quantum simulation costs.
We consider simulating quantum systems on digital quantum computers. We show that the performance of quantum simulation can be improved by simultaneously exploiting the commutativity of Hamiltonian, the sparsity of interactions, and the prior knowledge of initial state. We achieve this using Trotterization for a class of interacting electrons that encompasses various physical systems, including the plane-wave-basis electronic structure and the Fermi-Hubbard model. We estimate the simulation error by taking the transition amplitude of nested commutators of Hamiltonian terms within the η-electron manifold. We develop multiple techniques for bounding the transition amplitude and expectation of general fermionic operators, which may be of independent interest. We show that it suffices to use O(n^(5/3)/η^(2/3)+n^(4/3)η^(2/3)) gates to simulate electronic structure in the plane-wave basis with n spin orbitals and η electrons up to a negligible factor, improving the best previous result in second quantization while outperforming the first-quantized simulation when n=O(η²). We also obtain an improvement for simulating the Fermi-Hubbard model. We construct concrete examples for which our bounds are almost saturated, giving a nearly tight Trotterization of interacting electrons.
Semiconductor nanostructures are central to many developing technologies. Notably, double quantum dots are especially important for semiconductor spin-qubit architectures, quantum sensing applications, and quantum-dot solar cells. Accurate modelling is highly desirable but conventional methods can struggle when dynamics involve more than two interacting electrons. In this work, we present a quantum simulation framework capable of addressing multi-electron double quantum dots. We adopt an efficiently scaling 1st^\text{st}st quantised representation of the system and develop algorithms based on both Trotterisation and Qubitisation. Incorporating insights from classical simulations enables us to produce resource estimates that are more realistic than those obtained from theoretical error bounds. Using a standard surface code model with physical noise at 10−310^{-3}10−3, our results indicate that the ground-state energy of four electrons in a double quantum dot can be estimated in approximately 22 hours using 226k physical qubits, or an eight-electron system in 3.3 days with 314k qubits (with runtimes falling dramatically when more qubits are available). We anticipate that incorporating recent advances in surface code architectures may reduce these costs significantly further. Our results suggest that early fault-tolerant quantum computers may become valuable tools for designing mature-era quantum technologies.