The Hubbard Model on Real Quantum Hardware: Doped, Frustrated, and Fully Interacting with Diagonal Hopping
To our knowledge, the first reported digital quantum-hardware simulation of the full interacting, doped, and frustrated 2D Hubbard model with diagonal hopping , run on IBM Phoenix.
Superconductors carry electrical current without resistance, enabling powerful magnets and more efficient electrical technologies. Yet superconductivity at higher temperatures in cuprates materials remains debated: competing physics of magnetism, electron pairing and charge patterns makes its microscopic origin difficult to establish.
The Hubbard model offers a simple framework to study such complex physics. The model describes electrons which hop between crystal lattice sites and repel each other when they occupy the same site. On a square lattice, adding electron vacancies, called hole doping, and diagonal hopping, which introduces “frustrated” magnetic interactions, creates a minimal setting for studying the competing physics relevant to cuprate superconductivity.
We focused on a Hubbard model on a frustrated LxL lattice containing a single electron vacancy. For a small size of L, this example can be solved exactly on a classical computer, providing an exact reference for testing a quantum algorithm. As the lattice grows, however, exact diagonalization becomes prohibitively expensive, while Quantum Monte Carlo and tensor-network methods face substantial difficulties in this regime.
These challenges make the 2D Hubbard model with frustration and doping an ideal target for quantum simulation and potential search for quantum advantage.
We used Haiqu AgenticOS to set up a research project and develop a new custom algorithm and circuit design for Krylov diagonalization and run it on IBM Phoenix quantum computer to estimate the system’s lowest energy.
Two recent experiments used gate-based quantum computers to study the 2D Fermi-Hubbard model. Google used 72 qubits on its Willow processor to simulate electron motion on grids as large as 6x6 [1]. Quantinuum used its Helios processor to measure signals associated with superconducting pairing in a hole-doped 6x6 grid [2]. In both studies, electrons moved only between directly adjacent sites. To our knowledge, our work is the first gate-based hardware experiment to also allow electrons to move diagonally between nearby sites. This added motion makes the model more representative of materials such as cuprates, while also making the quantum circuit more demanding to run.
What Does This Experiment Show?
The goal was to estimate the system’s lowest energy, a starting point for understanding how its electrons behave. Starting from a scientist’s algorithm idea, Haiqu AgenticOS helped develop the circuits, test alternative implementations, and diagnose hardware errors in an experiment to estimate the model’s lowest energy on a QPU.
The known classical answer provided a reference throughout this process. After optimization and error mitigation, the hardware estimate was consistent with the ideal algorithm’s prediction and uncertainty. The case shows how a research idea can be developed into an executable quantum experiment, with each design choice tested against simulations and hardware measurements.
How Does the Method Work?
The project began with a proposal to adapt Krylov quantum diagonalization (KQD) to the interacting electron model defined above. The KQD method finds the lowest-energy weighted combination of a small set of quantum states. Haiqu AgenticOS used a prepared reference state and two states obtained by simulating its evolution a short time forward and backward. Time evolution changes the relative phases of a state’s energy components, allowing a suitable combination to suppress higher-energy contributions.
To determine those weights, the quantum processor measures how the states overlap and the energy contributions involving each pair. Measuring each state’s energy separately is insufficient, because a combination also depends on interference between different states. These measurements supply the information a classical computer needs to find the lowest-energy combination within the chosen set.
The interference measurements use a helper qubit, called an ancilla, to distinguish two state-preparation paths. A conventional implementation makes the simulated evolution conditional on this qubit, adding many gates. Our construction instead controls simpler reversal operations interleaved with ordinary evolution, leaving the expensive evolution blocks uncontrolled.
Haiqu AgenticOS helped develop and validate this tailored construction, alongside the reference-state preparation and evolution circuits. Diagonal hopping prevents a simple reversal from applying directly to the whole model, so the construction handles the model’s components separately while retaining hopping and electron–electron repulsion.
On a smaller 2×3 validation system, the proposed approach reproduced the fully controlled calculation while reducing the CNOT count in the controlled block by a factor of approximately 4.6 (from 36,008 to 7,776 CNOTs).
What did Haiqu AgenticOS Do?
Haiqu AgenticOS helped turn the scientist’s KQD proposal into a tested hardware experiment, connecting literature review, algorithm development, circuit construction, and experimental feedback.
Literature and physics. It reviewed the research and assessed methods and fermion-to-qubit mappings suitable for the chosen Hubbard model.
Algorithm design. It translated the proposal into testable hypotheses, analytical checks, and a codebase validated against exact small-system calculations.
Circuit synthesis. It combined the reversal construction with a symmetry-informed basis change, shorter reference-state preparation, and routing that moves the ancilla along the chip to reduce communication overhead.
Hardware diagnosis. It investigated compilation and calibration problems and revised the implementation after noise-impacted initial hardware runs. It proposed and implemented an additional rotational-symmetry check. This enabled the C₂ correction described below.
The scientist directed the research, reviewed the claims, and made the final decisions. Tests and hardware results shaped the next steps: analytical checks ruled out a single global reversal, exact calculations validation exposed LLM-drifting errors, and test Phoenix runs prompted changes to the circuits and execution strategy.
How Did Haiqu AgenticOS Make the Circuit Run on IBM Phoenix?
Running the algorithm meant translating its evolution and measurement steps into physical gates on IBM Phoenix’s square grid of qubits. Long circuits accumulate errors that can obscure the interference KQD needs.
AgenticOS compared alternative implementations, refined the circuits, and used Haiqu Platform’s hardware-aware compilation to make them suitable for the device.
Representative circuit reductions
75.1% lower circuit depth: 2,210 → 550 gate layers.
47.5% fewer two-qubit CZ gates: 1,276 → 670.
The reversal construction reduced the cost of conditional evolution. Further changes addressed the remaining costs:
These changes worked together to shorten the circuits while preserving the quantities needed for the energy calculation.
Incorporating feedback from hardware runs
The feedback from initial hardware runs became part of the development process: AgenticOS analyzed discrepancies and used them to revise the experiment. A poorly performing qubit spoiled an initial test run on Phoenix QPU. Hence, later circuits were compiled through Haiqu Platform using updated device calibration data and application of dynamical decoupling on idle qubits helped preserve the ancilla’s interference signal.
The experiment also filtered measurement outcomes using two physical checks: the required electron count and the chosen mirror symmetry of the lattice. These checks improved the data, but they could not detect every error. Systematic errors in the gates could alter the interference while preserving both properties, biasing the energy estimate even in the accepted measurements.
AgenticOS helped with investigation on this remaining bias using calibration circuits whose expected results were known. This diagnosis prompted a further change: measure an additional lattice symmetry (180-degree rotation symmetry, called C₂) to help distinguish the desired signal from the remaining error contributions.
This separates measurements into two symmetry groups, even and odd. The prepared state contributes predominantly to the even group, while the diagnostic analysis suggested that the errors were more evenly distributed between them. Comparing the groups allowed their shared error contribution to be suppressed when reconstructing the energy estimate. This correction assumes that the errors populate both groups approximately equally and have similar effects on the measured quantities.
Results and Outlook
After optimization and symmetry-based error mitigation, the Phoenix experiment estimated the system’s energy at approximately −18.99t. We compared this with the exact classical solution and the ideal version of the same three-state KQD calculation. The latter includes the approximation error of the chosen algorithm, but no hardware noise.
Energy comparison
Exact ground-state energy: −19.13t
Ideal three-state KQD estimate: −19.00t
Corrected Phoenix estimate: −18.99t
Energies are expressed in units of t, the nearest-neighbor hopping strength.
The central hardware estimate is close to the ideal calculation. Its statistical uncertainty, however, corresponds to an energy interval from −19.48t to −18.57t.
Reaching this result required coordinated work across physics, circuit design, compilation, and error diagnosis. Under the scientist’s direction, AgenticOS helped develop the implementation and revise it in response to simulations and hardware measurements.
The experiment establishes initial feasibility of the circuit construction and measurement approach. It provides the basis to further test the approach on progressively larger lattices, where exact classical answers become increasingly difficult to obtain. Comparisons with approximate classical methods and internal consistency checks will become increasingly important as exact verification becomes less accessible.
Interested in running this kind of loop on your own problem? Book a demo with the Haiqu team.
Footnotes
[1] F. Alam et al., "Programmable digital quantum simulation of 2D Fermi-Hubbard dynamics using 72 superconducting qubits," arXiv:2510.26845 (2025).
[2] E. Granet et al., "Superconducting pairing correlations on a trapped-ion quantum computer," arXiv:2511.02125 (2025, v3 February 2026).