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:2411.18535

Ground-State Preparation of the Fermi-Hubbard Model on a Quantum Computer with 2D Topology via Quantum Eigenvalue Transformation of Unitary Matrices

Thilo R. Müller, Geiger, Manuel, Christian B. Mendl

Published Nov 27, 2024• 0 citations
arXivHTMLPDF

Abstract

Quantum computing holds immense promise for simulating quantum systems, a critical task for advancing our understanding of complex quantum phenomena. One of the primary goals in this domain is to accurately approximate the ground state of quantum systems. The Fermi-Hubbard model, particularly, is of profound interest due to its implications for high-temperature superconductivity and strongly correlated electron systems. The quantum eigenvalue transformation of unitary matrices (QETU) algorithm offers a novel approach for ground state estimation by utilizing a controlled Hamiltonian time evolution operator, circumventing the resource-intensive block-encoding required by previous methods. In this work, we apply the QETU algorithm to the $2 \times 2$ Fermi-Hubbard model, presenting circuit simplifications tailored to the model and introducing a mapping to a 9-qubit grid-like hardware architecture inspired by fermionic swap networks. We investigate how the selection of a favorable hardware architecture can benefit the circuit construction. Additionally, we explore the feasibility of this method under the influence of noise, focusing on its robustness and practical applicability.

DOI
10.48550/arxiv.2411.18535

Related papers

Nearly optimal state preparation for quantum simulations of lattice gauge theories

Christopher F. Kane, Niladri Gomes, Michael Kreshchuk

We present several improvements to the recently developed ground-state preparation algorithm based on the quantum eigenvalue transformation for unitary matrices (QETU), apply this algorithm to a lattice formulation of U(1) gauge theory in $(2+1)$ dimensions, as well as propose an alternative application of QETU, a highly efficient preparation of Gaussian distributions. The QETU technique was originally proposed as an algorithm for nearly optimal ground-state preparation and ground-state energy estimation on early fault-tolerant devices. It uses the time-evolution input model, which can potentially overcome the large overall prefactor in the asymptotic gate cost arising in similar algorithms based on the Hamiltonian input model. We present modifications to the original QETU algorithm that significantly reduce the cost for the cases of both exact and Trotterized implementation of the time evolution circuit. We use QETU to prepare the ground state of a U(1) lattice gauge theory in two spatial dimensions, explore the dependence of computational resources on the desired precision and system parameters, and discuss the applicability of our results to general lattice gauge theories. We also demonstrate how the QETU technique can be utilized for preparing Gaussian distributions and wave packets in a way which outperforms existing algorithms for as little as ${n}_{q}\ensuremath{\gtrsim}2--5$ qubits.

arXiv:2310.13757•Physical Review A•Jul 23, 2024•24 citations

Hardware-Aware Fermion-to-Qubit Mappings for Simulating the 2D Hubbard Model on Heavy-Hexagon Quantum Processors

Alessio Esposito, Andrea Giachero, Zoltàn Zimboràs +1 more

Quantum simulation of strongly correlated fermionic systems is among the most promising near- term applications of quantum computing, but its practical efficiency depends critically on the choice of fermion-to-qubit mapping and on the connectivity of the underlying hardware. In this work we address this problem in the context of the two-dimensional Hubbard model, simulated on IBM superconducting quantum processors with heavy-hexagon connectivity. We numerically benchmark the Jordan-Wigner, Bravyi-Kitaev, and Bonsai transformations, evaluating their Pauli weight and SWAP overhead across seven heavy-hexagon chips of increasing size. We show that, while the Bravyi-Kitaev mapping initially exhibits a lower Pauli weight, this advantage is eliminated once routing costs are taken into account, confirming the Bonsai mapping as the most hardware-efficient baseline transformation for this architecture. We then use the Bonsai mapping to construct the qubit Hamiltonian of the 2D spinful Fermi-Hubbard model, introducing a simulated annealing algorithm that optimizes the assignment of Majorana strings to lattice sites, reducing the cost function by nearly 50% percent. Finally, we simulate on quantum hardware the time evolution of fermionic states up to 6x6 lattices, confirming the viability of the Bonsai encoding for hardware-aware large-scale two-dimensional simulations.

quant-pharXiv:2608.25024•Aug 27, 2026•0 citations

Recipe for local simulation of strongly correlated fermionic matter on quantum computers: The two-dimensional Fermi-Hubbard model

Arash Jafarizadeh, Frank Pollmann, Adam Gammon-Smith

The simulation of quantum many-body systems, relevant for quantum chemistry and condensed matter physics, is one of the most promising applications of near-term quantum computers before fault-tolerance. However, since the vast majority of quantum computing technologies are built around qubits and discrete gate-based operations, the translation of the physical problem into this framework is a crucial step. This translation will often be device specific, and a suboptimal implementation will be punished by the exponential compounding of errors on real devices. The importance of an efficient mapping is already revealed for models of spinful fermions in two or three dimensions, which naturally arise when the relevant physics relates to electrons. Using the most direct and well-known mapping, the Jordan-Wigner transformation, leads to a nonlocal representation of local degrees of freedom, and necessities efficient decompositions of nonlocal unitary gates into a sequence of hardware accessible local gates. In this paper, we provide a step-by-step recipe for simulating the paradigmatic two-dimensional Fermi-Hubbard model on a quantum computer using only local operations. To provide the ingredients for such a recipe, we briefly review the plethora of different approaches that have emerged recently but focus on the Derby-Klassen compact fermion mapping to make our discussion concrete. We provide a detailed recipe for an end-to-end simulation including embedding on a physical device, preparing initial states such as ground states, simulation of unitary time evolution, and measurement of observables and spectral functions. We explicitly compute the resource requirements for simulating a global quantum quench and conclude by discussing the challenges and future directions for simulating strongly correlated fermionic matter on quantum computers.

arXiv:2408.14543•Physical Review Research•Oct 7, 2025•2 citations

Observing ground-state properties of the Fermi-Hubbard model using a scalable algorithm on a quantum computer

Stasja Stanisic, Jan Lukas Bosse, Filippo Maria Gambetta +5 more

The famous, yet unsolved, Fermi-Hubbard model for strongly-correlated electronic systems is a prominent target for quantum computers. However, accurately representing the Fermi-Hubbard ground state for large instances may be beyond the reach of near-term quantum hardware. Here we show experimentally that an efficient, low-depth variational quantum algorithm with few parameters can reproduce important qualitative features of medium-size instances of the Fermi-Hubbard model. We address 1 × 8 and 2 × 4 instances on 16 qubits on a superconducting quantum processor, substantially larger than previous work based on less scalable compression techniques, and going beyond the family of 1D Fermi-Hubbard instances, which are solvable classically. Consistent with predictions for the ground state, we observe the onset of the metal-insulator transition and Friedel oscillations in 1D, and antiferromagnetic order in both 1D and 2D. We use a variety of error-mitigation techniques, including symmetries of the Fermi-Hubbard model and a recently developed technique tailored to simulating fermionic systems. We also introduce a new variational optimisation algorithm based on iterative Bayesian updates of a local surrogate model.

arXiv:2112.02025•Nature Communications•Oct 11, 2022•98 citations

Towards solving the Fermi-Hubbard model via tailored quantum annealers

Levy, Ryan, Zoe Gonzalez Izquierdo, Zhihui Wang +8 more

The Fermi-Hubbard model (FHM) on a two dimensional square lattice has long been an important testbed and target for simulating fermionic Hamiltonians on quantum hardware. We present an alternative for quantum simulation of FHMs based on an adiabatic protocol that could be an attractive target for next generations of quantum annealers. Our results rely on a recently introduced low-weight encoding that allows the FHM to be expressed in terms of Pauli operators with locality of at most three. We theoretically and numerically determine promising quantum annealing setups for both interacting 2D spinless and spinful systems, that enable to reach near the ground state solution with high fidelity for systems as big as $6\times 6$ (spinless) and $4\times 3$ (spinful). Moreover, we demonstrate the scaling properties of the minimal gap and analyze robustness of the protocol against control noise. Additionally, we identify and discuss basic experimental requirements to construct near term annealing hardware tailored to simulate these problems. Finally, we perform a detailed resource estimation for the introduced adiabatic protocol, and discuss pros and cons of this approach relative to gate-based approaches for near-term platforms.

arXiv:2207.14374•Jul 28, 2022•3 citations

Ground-State Preparation and Energy Estimation on Early Fault-Tolerant Quantum Computers via Quantum Eigenvalue Transformation of Unitary Matrices

Yulong Dong, Lin Lin, Yu Tong

Under suitable assumptions, the algorithms in [Lin, Tong, Quantum 2020] can estimate the ground state energy and prepare the ground state of a quantum Hamiltonian with near-optimal query complexities. However, this is based on a block encoding input model of the Hamiltonian, whose implementation is known to require a large resource overhead. We develop a tool called quantum eigenvalue transformation of unitary matrices with real polynomials (QET-U), which uses a controlled Hamiltonian evolution as the input model, a single ancilla qubit and no multi-qubit control operations, and is thus suitable for early fault-tolerant quantum devices. This leads to a simple quantum algorithm that outperforms all previous algorithms with a comparable circuit structure for estimating the ground state energy. For a class of quantum spin Hamiltonians, we propose a new method that exploits certain anti-commutation relations and further removes the need of implementing the controlled Hamiltonian evolution. Coupled with Trotter based approximation of the Hamiltonian evolution, the resulting algorithm can be very suitable for early fault-tolerant quantum devices. We demonstrate the performance of the algorithm using IBM Qiskit for the transverse field Ising model. If we are further allowed to use multi-qubit Toffoli gates, we can then implement amplitude amplification and a new binary amplitude estimation algorithm, which increases the circuit depth but decreases the total query complexity. The resulting algorithm saturates the near-optimal complexity for ground state preparation and energy estimating using a constant number of ancilla qubits (no more than 3).

arXiv:2204.05955•PRX Quantum•Oct 12, 2022•141 citations
Quantum computing holds immense promise for simulating quantum systems, a critical task for advancing our understanding of complex quantum phenomena. One of the primary goals in this domain is to accurately approximate the ground state of quantum systems. The Fermi-Hubbard model, particularly, is of profound interest due to its implications for high-temperature superconductivity and strongly correlated electron systems. The quantum eigenvalue transformation of unitary matrices (QETU) algorithm offers a novel approach for ground state estimation by utilizing a controlled Hamiltonian time evolution operator, circumventing the resource-intensive block-encoding required by previous methods. In this work, we apply the QETU algorithm to the 2×22 \times 22×2 Fermi-Hubbard model, presenting circuit simplifications tailored to the model and introducing a mapping to a 9-qubit grid-like hardware architecture inspired by fermionic swap networks. We investigate how the selection of a favorable hardware architecture can benefit the circuit construction. Additionally, we explore the feasibility of this method under the influence of noise, focusing on its robustness and practical applicability.
We present several improvements to the recently developed ground-state preparation algorithm based on the quantum eigenvalue transformation for unitary matrices (QETU), apply this algorithm to a lattice formulation of U(1) gauge theory in (2+1)(2+1)(2+1) dimensions, as well as propose an alternative application of QETU, a highly efficient preparation of Gaussian distributions. The QETU technique was originally proposed as an algorithm for nearly optimal ground-state preparation and ground-state energy estimation on early fault-tolerant devices. It uses the time-evolution input model, which can potentially overcome the large overall prefactor in the asymptotic gate cost arising in similar algorithms based on the Hamiltonian input model. We present modifications to the original QETU algorithm that significantly reduce the cost for the cases of both exact and Trotterized implementation of the time evolution circuit. We use QETU to prepare the ground state of a U(1) lattice gauge theory in two spatial dimensions, explore the dependence of computational resources on the desired precision and system parameters, and discuss the applicability of our results to general lattice gauge theories. We also demonstrate how the QETU technique can be utilized for preparing Gaussian distributions and wave packets in a way which outperforms existing algorithms for as little as ${n}_{q}\ensuremath{\gtrsim}2--5$ qubits.
Quantum simulation of strongly correlated fermionic systems is among the most promising near- term applications of quantum computing, but its practical efficiency depends critically on the choice of fermion-to-qubit mapping and on the connectivity of the underlying hardware. In this work we address this problem in the context of the two-dimensional Hubbard model, simulated on IBM superconducting quantum processors with heavy-hexagon connectivity. We numerically benchmark the Jordan-Wigner, Bravyi-Kitaev, and Bonsai transformations, evaluating their Pauli weight and SWAP overhead across seven heavy-hexagon chips of increasing size. We show that, while the Bravyi-Kitaev mapping initially exhibits a lower Pauli weight, this advantage is eliminated once routing costs are taken into account, confirming the Bonsai mapping as the most hardware-efficient baseline transformation for this architecture. We then use the Bonsai mapping to construct the qubit Hamiltonian of the 2D spinful Fermi-Hubbard model, introducing a simulated annealing algorithm that optimizes the assignment of Majorana strings to lattice sites, reducing the cost function by nearly 50% percent. Finally, we simulate on quantum hardware the time evolution of fermionic states up to 6x6 lattices, confirming the viability of the Bonsai encoding for hardware-aware large-scale two-dimensional simulations.
The simulation of quantum many-body systems, relevant for quantum chemistry and condensed matter physics, is one of the most promising applications of near-term quantum computers before fault-tolerance. However, since the vast majority of quantum computing technologies are built around qubits and discrete gate-based operations, the translation of the physical problem into this framework is a crucial step. This translation will often be device specific, and a suboptimal implementation will be punished by the exponential compounding of errors on real devices. The importance of an efficient mapping is already revealed for models of spinful fermions in two or three dimensions, which naturally arise when the relevant physics relates to electrons. Using the most direct and well-known mapping, the Jordan-Wigner transformation, leads to a nonlocal representation of local degrees of freedom, and necessities efficient decompositions of nonlocal unitary gates into a sequence of hardware accessible local gates. In this paper, we provide a step-by-step recipe for simulating the paradigmatic two-dimensional Fermi-Hubbard model on a quantum computer using only local operations. To provide the ingredients for such a recipe, we briefly review the plethora of different approaches that have emerged recently but focus on the Derby-Klassen compact fermion mapping to make our discussion concrete. We provide a detailed recipe for an end-to-end simulation including embedding on a physical device, preparing initial states such as ground states, simulation of unitary time evolution, and measurement of observables and spectral functions. We explicitly compute the resource requirements for simulating a global quantum quench and conclude by discussing the challenges and future directions for simulating strongly correlated fermionic matter on quantum computers.
The famous, yet unsolved, Fermi-Hubbard model for strongly-correlated electronic systems is a prominent target for quantum computers. However, accurately representing the Fermi-Hubbard ground state for large instances may be beyond the reach of near-term quantum hardware. Here we show experimentally that an efficient, low-depth variational quantum algorithm with few parameters can reproduce important qualitative features of medium-size instances of the Fermi-Hubbard model. We address 1 × 8 and 2 × 4 instances on 16 qubits on a superconducting quantum processor, substantially larger than previous work based on less scalable compression techniques, and going beyond the family of 1D Fermi-Hubbard instances, which are solvable classically. Consistent with predictions for the ground state, we observe the onset of the metal-insulator transition and Friedel oscillations in 1D, and antiferromagnetic order in both 1D and 2D. We use a variety of error-mitigation techniques, including symmetries of the Fermi-Hubbard model and a recently developed technique tailored to simulating fermionic systems. We also introduce a new variational optimisation algorithm based on iterative Bayesian updates of a local surrogate model.
The Fermi-Hubbard model (FHM) on a two dimensional square lattice has long been an important testbed and target for simulating fermionic Hamiltonians on quantum hardware. We present an alternative for quantum simulation of FHMs based on an adiabatic protocol that could be an attractive target for next generations of quantum annealers. Our results rely on a recently introduced low-weight encoding that allows the FHM to be expressed in terms of Pauli operators with locality of at most three. We theoretically and numerically determine promising quantum annealing setups for both interacting 2D spinless and spinful systems, that enable to reach near the ground state solution with high fidelity for systems as big as 6×66\times 66×6 (spinless) and 4×34\times 34×3 (spinful). Moreover, we demonstrate the scaling properties of the minimal gap and analyze robustness of the protocol against control noise. Additionally, we identify and discuss basic experimental requirements to construct near term annealing hardware tailored to simulate these problems. Finally, we perform a detailed resource estimation for the introduced adiabatic protocol, and discuss pros and cons of this approach relative to gate-based approaches for near-term platforms.
Under suitable assumptions, the algorithms in [Lin, Tong, Quantum 2020] can estimate the ground state energy and prepare the ground state of a quantum Hamiltonian with near-optimal query complexities. However, this is based on a block encoding input model of the Hamiltonian, whose implementation is known to require a large resource overhead. We develop a tool called quantum eigenvalue transformation of unitary matrices with real polynomials (QET-U), which uses a controlled Hamiltonian evolution as the input model, a single ancilla qubit and no multi-qubit control operations, and is thus suitable for early fault-tolerant quantum devices. This leads to a simple quantum algorithm that outperforms all previous algorithms with a comparable circuit structure for estimating the ground state energy. For a class of quantum spin Hamiltonians, we propose a new method that exploits certain anti-commutation relations and further removes the need of implementing the controlled Hamiltonian evolution. Coupled with Trotter based approximation of the Hamiltonian evolution, the resulting algorithm can be very suitable for early fault-tolerant quantum devices. We demonstrate the performance of the algorithm using IBM Qiskit for the transverse field Ising model. If we are further allowed to use multi-qubit Toffoli gates, we can then implement amplitude amplification and a new binary amplitude estimation algorithm, which increases the circuit depth but decreases the total query complexity. The resulting algorithm saturates the near-optimal complexity for ground state preparation and energy estimating using a constant number of ancilla qubits (no more than 3).