⚛️ quantum physics

Explicitly correlated ground states with a feedback-free variational hybrid algorithm

This paper proposes and experimentally demonstrates a feedback-free hybrid quantum-classical algorithm that uses a quantum processor to prepare a momentum-space matrix product state and classical post-processing to variationally optimize an explicitly correlated Jastrow-type correlator, achieving high-accuracy ground-state energy estimates for the 2D Fermi-Hubbard model with reduced sampling and gate requirements compared to other hybrid methods.

Original authors: Benjamin Corbett, Akimasa Miyake

Published 2026-10-14

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CC BY 4.0

Original authors: Benjamin Corbett, Akimasa Miyake

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of electrons, particles do not always behave like independent individuals moving through empty space. In certain materials, they become deeply entangled, their movements dictated by a fierce, immediate repulsion when they come too close to one another. This phenomenon, known as strong correlation, is the key to understanding how some materials conduct electricity, how they become magnetic, or how they might one day become superconductors. To predict these behaviors, scientists must solve a massive mathematical puzzle: calculating the exact energy state of a system where every electron is constantly reacting to every other electron. For decades, the most powerful classical computers have struggled with this task, often hitting a wall when the interactions become too intense or the system grows too large. The difficulty lies in a fundamental mismatch: the most efficient way to describe the long-range order of these electrons is in terms of their momentum, while the most efficient way to describe their short-range, violent repulsions is in terms of their physical position. Trying to combine these two descriptions on a classical computer has been like trying to solve a jigsaw puzzle where the pieces are cut from two different pictures.

A team of researchers at the University of New Mexico has found a way to bridge this divide using a quantum computer, not by asking the machine to solve the entire puzzle at once, but by using it to prepare a specific, high-quality starting point for a classical computer to finish. Their approach, detailed in a recent study, focuses on a theoretical model called the Fermi-Hubbard model, which describes electrons hopping between sites on a grid. The researchers used a quantum processor to generate a low-entanglement state of electrons in momentum space, a configuration that is difficult for classical computers to create efficiently. They then rotated this state into real space, effectively translating the description so that the short-range repulsions could be handled. Crucially, they did this without a back-and-forth loop where the computer and the classical algorithm constantly adjust each other. Instead, the quantum processor performed its task once, measured the results, and handed the data over to a classical computer. The classical computer then used this data to optimize a mathematical "correlator"—a tool that fine-tunes the description of the electron repulsions—entirely after the quantum experiment was over.

The experiment was conducted on a 32-qubit quantum computer, specifically the H2-2 model from Quantinuum. The team prepared a state representing a grid of 16 sites, half-filled with electrons, and measured the energy of the system. After filtering the data to remove outcomes that violated the known rules of particle conservation, they applied a calibration technique to correct for the small errors inherent in current quantum hardware. The result was a ground-state energy estimate that was remarkably close to the known exact value, with a relative error of just 3.5 percent. This level of accuracy was achieved using significantly fewer quantum measurements and fewer complex two-qubit gates than other leading methods that rely on repeated feedback loops or deep quantum circuits. The researchers demonstrated that by separating the quantum preparation from the classical optimization, they could reuse the same quantum data to test many different mathematical models of electron interaction without ever needing to run the quantum computer again.

The study also explored how this method might scale up to larger systems. By analyzing the data, the authors found that the number of measurements required to maintain a fixed level of precision grows exponentially with the size of the grid, but at a rate that is manageable for near-term quantum devices. They compared their approach to other hybrid algorithms, such as quantum-selected configuration interaction and quantum Krylov methods. While those methods can offer rigorous mathematical guarantees, they often require vastly more quantum resources—either millions of times more samples or thousands of times more complex gates—to reach the same level of accuracy on the small systems tested. The new method's ability to decouple the quantum measurement from the classical optimization allows for a flexibility that the other methods lack; researchers can swap out the mathematical model for the electron interactions after the data is collected, testing different theories against the same quantum experiment.

This work represents a proof of principle rather than a final solution to the problem of strongly correlated electrons. The researchers acknowledge that their current demonstration was limited to a small 4 by 4 grid, and that larger systems will require more advanced state preparation techniques. However, the success of the feedback-free hybrid algorithm suggests a viable path forward. By using the quantum processor strictly as a tool to generate a specific, hard-to-create state and then letting classical computers do the heavy lifting of optimization, the team has shown that quantum advantage in materials science might not require the massive, error-corrected machines of the distant future. Instead, it may be achievable today by carefully designing algorithms that respect the unique strengths of both quantum and classical hardware, turning a fundamental mismatch of bases into a complementary partnership.

Technical Summary: Explicitly Correlated Ground States with a Feedback-Free Variational Hybrid Algorithm

Problem Statement
Accurately describing strongly correlated electron systems, such as the 2D Fermi–Hubbard model, requires capturing dominant short-range correlations. Explicitly correlated methods achieve this by augmenting a reference state $|\phi\rangle$ with a diagonal correlator $C$ (e.g., a Jastrow or Gutzwiller factor). However, a fundamental basis mismatch arises: highly accurate reference states for these systems are often compact in momentum space (e.g., Matrix Product States or MPSs), while physically motivated diagonal correlators act naturally in real space. Classical algorithms struggle to bridge this gap; Variational Monte Carlo (VMC) requires efficient access to real-space amplitudes (which is $\#P$ -hard for momentum-space MPSs), while the transcorrelated method incorporates the correlator into the Hamiltonian but yields non-variational energies and requires complex biorthogonal calculations.

Methodology
The authors propose a hybrid quantum-classical algorithm that decouples the quantum state preparation from the variational optimization of the correlator, eliminating the need for a quantum-classical feedback loop. The workflow proceeds as follows:

Classical Pre-computation: A transcorrelated Density Matrix Renormalization Group (DMRG) calculation generates a low-bond-dimension momentum-space MPS $|\phi\rangle$ . This state is optimized self-consistently using a Gutzwiller-like similarity transformation ( $C_{tc} = e^{JD}$ ), yielding a non-variational reference energy.
Quantum Execution: A quantum processor prepares the momentum-space MPS $|\phi\rangle$ . It then applies a fermionic Fourier transformation ( $U_{FT}$ ) to rotate the state into real space.
Measurement: The rotated state is measured in bases fixed solely by the Hamiltonian terms (diagonal occupation basis and hopping bases). Crucially, these measurement circuits are independent of the parameters $\theta$ of the subsequent variational correlator $C_\theta = \exp(\sum \theta_r D_r)$ .
Classical Post-processing: The raw measurement outcomes are calibrated using a quadratic program to reproduce known expectation values from the classical momentum-space MPS (e.g., $\langle D \rangle$ , $\langle T \rangle$ ). These calibrated samples are then used to evaluate the variational energy functional:
$E_{ec}(\theta) = \frac{\langle \phi | C_\theta H C_\theta | \phi \rangle}{\langle \phi | C_\theta^2 | \phi \rangle}$
The parameters $\theta$ are optimized entirely classically to minimize this energy. Because the measurement bases do not depend on $\theta$ , the same dataset can be reused for any choice of correlator form or parameters.

Key Contributions

Feedback-Free Variational Optimization: Unlike standard Variational Quantum Eigensolvers (VQE) that require iterative circuit updates, this method fixes all quantum circuits before execution. The variational optimization occurs entirely in classical post-processing, allowing for the exploration of arbitrary real-space diagonal correlators (including high-parameter forms) without additional quantum runs.
Basis Conflict Resolution: The algorithm leverages the quantum processor to perform the computationally expensive rotation from momentum space to real space, enabling the use of compact momentum-space MPS references with real-space correlators.
Calibration Protocol: A novel calibration scheme reweights measurement samples to match low-order moments of the reference MPS, significantly reducing statistical errors and bias without requiring a full reconstruction of the wavefunction.

Results
The algorithm was demonstrated on a 4 $\times$ 4 half-filled Fermi–Hubbard model ( $U/t=4$ ) using Quantinuum's H2-2 trapped-ion quantum computer (32 qubits).

Hardware Performance: Using a bond-dimension-4 MPS, the raw postselected energy was $-11.49 \pm 0.49$ . After calibration, the energy improved to $-13.14 \pm 0.07$ , achieving a relative error of $3.5 \pm 0.5\%$ compared to the exact ground state.
Sample Efficiency: Compared to Quantum-Selected Configuration Interaction (QSCI), the proposed method required significantly fewer samples (approx. $3.6 \times 10^4$ vs. $\sim 2 \times 10^8$ ) to reach comparable accuracy.
Gate Efficiency: Compared to Quantum Krylov Subspace Diagonalization (QKSD), the method required fewer two-qubit gates (464 for the rotation circuit vs. 544 for QKSD to match the same accuracy, and 696 vs. 22,304 for higher accuracy targets).
Scaling: The study suggests that for half-filled systems, the sampling cost scales as $\exp(cN)$ with $c < 0.1$ , though this remains a hypothesis requiring further verification.

Significance and Claims
The authors position this work as a "proof of principle" rather than a demonstration of full quantum advantage. The significance lies in demonstrating a viable pathway to combine the strengths of classical tensor network methods (compact momentum-space references) with quantum hardware (exact real-space rotation) without the overhead of feedback loops.

The paper claims that this approach offers a distinct advantage over other feedback-free hybrid algorithms (like QSCI and QKSD) by requiring fewer resources (samples and gates) for the specific benchmark tested. It highlights that while the energy estimate is variational, the observables derived from the single state $C|\phi\rangle$ are more accurate than those from biorthogonal transcorrelated methods. The authors note that the method is particularly well-suited for regimes where auxiliary-field Quantum Monte Carlo suffers from sign problems (e.g., away from half-filling or with dilute systems), provided that small-bond-dimension MPSs remain effective reference states. The work concludes that while current hardware limitations restrict the system size, the feedback-free nature and resource efficiency make it a promising candidate for near-term quantum simulation of strongly correlated materials.

Technical Summary: Explicitly Correlated Ground States with a Feedback-Free Variational Hybrid Algorithm

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