⚛️ quantum physics

Computing band gaps of periodic materials via sample-based quantum diagonalization

This paper introduces a sample-based quantum diagonalization workflow that accurately predicts the band gaps of periodic materials like hafnium and zirconium dioxide by combining self-consistent lattice Hamiltonians with superconducting quantum processor sampling, outperforming traditional density functional theory and aligning with experimental results.

Original authors: Alan Duriez, Pamela C. Carvalho, Marco Antonio Barroca, Federico Zipoli, Ben Jaderberg, Rodrigo Neumann Barros Ferreira, Kunal Sharma, Antonio Mezzacapo, Benjamin Wunsch, Mathias Steiner

Published 2026-07-14

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Alan Duriez, Pamela C. Carvalho, Marco Antonio Barroca, Federico Zipoli, Ben Jaderberg, Rodrigo Neumann Barros Ferreira, Kunal Sharma, Antonio Mezzacapo, Benjamin Wunsch, Mathias Steiner

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

Imagine trying to predict how a material conducts electricity or blocks it, like guessing the height of a wall between two rooms. For decades, scientists have used a standard recipe called Density Functional Theory (DFT) to make these guesses. It's like using a map that assumes everyone walks in a straight line and ignores the fact that people bump into each other. This works fine for some materials, but when you get to tricky ones like hafnium dioxide (HfO2) and zirconium dioxide (ZrO2)—materials used in everything from computer chips to insulation—the "straight line" map fails. It misses the chaotic, bumpy interactions between electrons, leading to predictions that are too low compared to what happens in real life.

Enter a new, hybrid team-up: a mix of classical supercomputers and a brand-new type of quantum computer. The researchers didn't just throw a quantum computer at the problem; they built a special bridge. First, they used the old-school DFT to get a rough sketch of the material's electronic structure. Then, they translated that sketch into a "lattice Hamiltonian," which is like turning the material into a grid of tiny, interacting islands. On this grid, they added specific rules for how electrons repel each other (the "bumping" part) using parameters they calculated themselves.

To solve the puzzle of how these electrons behave on this grid, they used a method called Sample-Based Quantum Diagonalization (SQD). Think of the quantum computer not as a calculator that solves the whole equation at once, but as a super-fast sampler. It runs a quantum circuit (a specific set of instructions) millions of times to collect a massive list of "snapshots" or configurations of the electrons. In this study, they ran these circuits on a state-of-the-art superconducting quantum processor called ibm pittsburgh, which has 156 qubits. For HfO2, they used a circuit with 38 qubits, and for ZrO2, they used 46 qubits.

The magic happens when they take these quantum snapshots and feed them into a classical computer. The classical computer looks at this curated list of the most important electron arrangements and does the heavy math to find the lowest energy state. It's like the quantum computer is a scout finding the best paths through a forest, and the classical computer is the cartographer drawing the final map based on those paths.

The results? The team found that this quantum-classical combo predicted the "band gap" (the energy wall between occupied and empty electron states) for both HfO2 and ZrO2 with impressive accuracy. For ZrO2, the standard DFT methods failed completely, missing the experimental range entirely. The quantum method, however, landed right in the sweet spot of what independent lab experiments have measured. In fact, the difference between their quantum prediction and a highly accurate classical reference method (called HCI) was tiny: just 1.6 × 10⁻⁸ eV for HfO2 and 1.3 × 10⁻⁷ eV for ZrO2.

Importantly, the paper explicitly argues against the idea that the standard DFT approach is sufficient for these materials, noting that it underestimates the band gap because it ignores strong electron correlations. They also show that while adding just one type of correction (intra-site interactions, or "U") helps a little, it's the combination of both intra-site and inter-site interactions (the "V" part, representing how electrons on different atoms talk to each other) that is crucial. Their method outperformed not only DFT but also other advanced quantum-chemical benchmarks like CCSD(T).

While the results are a significant step forward, the authors are careful to frame this as a successful simulation on current, "pre-fault-tolerant" hardware. They didn't claim to have solved all material science problems; rather, they demonstrated that this specific workflow can produce useful, experimentally verifiable predictions for these two specific dielectrics. The method is fast and resource-efficient compared to other high-accuracy methods like GW approximation, suggesting that with better quantum processors in the future, this approach could be scaled up to tackle even more complex materials. For now, it proves that even with today's noisy quantum machines, we can start seeing the true shape of the electronic world.

Technical Summary: Computing Band Gaps of Periodic Materials via Sample-Based Quantum Diagonalization

Problem Statement
Predicting the electronic properties of periodic materials, particularly band gaps, remains a significant challenge in computational solid-state physics. Standard Density Functional Theory (DFT), while widely used, relies on mean-field approximations and the assumption of non-interacting electrons. These assumptions often fail to capture strong electronic correlations found in transition-metal oxides, leading to systematic underestimation of band gaps in wide-gap semiconductors and insulators. While methods like the Extended Hubbard (EH) model can incorporate correlation effects via intra-site ( $U$ ) and inter-site ( $V$ ) parameters, solving the resulting fermionic lattice Hamiltonians for large systems is computationally prohibitive for classical algorithms like Quantum Monte Carlo (QMC) or Density Matrix Renormalization Group (DMRG), which face limitations in applicability to fermionic systems or scalability to three dimensions.

Methodology
The authors propose a hybrid quantum-classical workflow to simulate electronic states and predict band gaps for periodic materials. The methodology consists of three primary stages:

Lattice Hamiltonian Construction:
- The electronic structure is first calculated using DFT (specifically DFT+U+V) to obtain ground-state densities and self-consistent Hubbard parameters ( $U$ and $V$ ).
- These DFT results are projected onto a localized atomic orbital basis set to construct a tight-binding Hamiltonian.
- This projection yields a general lattice Hamiltonian (Equation 1) defined by hopping parameters ( $t_{pq}$ ) and self-consistent intra-site ( $U_p$ ) and inter-site ( $V_{pq}$ ) interaction parameters. The system is mapped such that each atomic orbital corresponds to a single site in the lattice.
- The Hamiltonian is expressed in reciprocal space but investigated specifically at the $\Gamma$ -point.
Quantum Sampling and Diagonalization:
- The lattice Hamiltonian is mapped onto a generalized electronic Hamiltonian suitable for quantum sampling, effectively treating the periodic material as a molecular system for the purpose of the algorithm.
- The Sample-Based Quantum Diagonalization (SQD) method is employed. This involves:
  - Using a Local Unitary Cluster Jastrow (LUCJ) ansatz to approximate the ground state wavefunction. The ansatz parameters are initialized using amplitudes from a classical Coupled-Cluster Singles and Doubles (CCSD) calculation.
  - Executing the parametrized quantum circuit on a superconducting quantum processor (IBM's ibm pittsburgh with a Heron r3 QPU) to sample electronic configurations (bitstrings).
  - Constructing a reduced configuration subspace from these samples and classically diagonalizing the projected Hamiltonian within this subspace to obtain ground state energies.
- An extended version, Ext-SQD, is utilized to improve accuracy. This algorithm removes low-probability configurations from the initial SQD set and applies excitation operators to generate an extended configuration set, allowing for refined energy calculations with smaller configuration spaces.
Band Gap Calculation:
- The direct band gap at the $\Gamma$ -point is derived from the ground state energies of the Hamiltonian in three symmetry sectors: $N_e$ (valence electrons), $N_e - 1$ , and $N_e + 1$ .
- The band gap ( $E_g$ ) is calculated using the formula: $E_g = E[N_e - 1] + E[N_e + 1] - 2E[N_e]$ .

Key Contributions

Generalized Lattice Representation: The paper introduces a general framework for representing the electronic structure of periodic materials as a lattice Hamiltonian with material-specific, self-consistently derived interaction parameters, bridging the gap between DFT and quantum simulation.
Application of SQD to Solids: The authors successfully adapt the SQD method, previously applied to molecular structures, to periodic materials by mapping the Extended Hubbard model to a form compatible with quantum-chemical sampling techniques.
Hardware Demonstration: The workflow is executed on a state-of-the-art, pre-fault-tolerant superconducting quantum processor (156 qubits) to simulate hafnium dioxide (HfO $_2$ ) and zirconium dioxide (ZrO $_2$ ).

Results

Accuracy vs. Classical Benchmarks: The Ext-SQD method demonstrates high accuracy, with ground state energy errors relative to Full Configuration Interaction (HCI) benchmarks being on the order of $10^{-8}$ eV for HfO $_2$ and $10^{-7}$ eV for ZrO $_2$ . The method outperforms standard CCSD and CCSD(T) approaches in the tested configuration spaces.
Band Gap Predictions:
- For HfO $_2$ , the quantum-computed band gap falls within the range of experimental values reported in literature and outperforms all levels of DFT simulation (including DFT+U+V) and GW approximation results presented for comparison.
- For ZrO $_2$ , DFT simulations at all levels fail to reach the experimental band gap range. In contrast, the quantum-computed band gaps successfully achieve the experimental range.
Role of Interactions: The study highlights that while intra-site interactions ( $U$ ) cause noticeable shifts in DFT, the inter-site interactions ( $V$ ) play a dominant role in the lattice representation for accurately capturing the band gap.
Efficiency: The quantum sampling procedure is noted to be relatively fast, producing results comparable to the computationally expensive GW approximation with fewer resources.

Significance
The paper claims that this work demonstrates the utility of hybrid quantum-classical workflows on pre-fault-tolerant quantum computers for applied materials science. The primary significance lies in the ability of these methods to produce useful, experimentally verifiable property predictions (specifically band gaps) that surpass standard mean-field approaches (DFT) and align with laboratory experiments. The authors suggest that this approach effectively samples electronic correlations that are neglected in mean-field theories. While the current computational bottleneck remains the classical calculation of configuration interaction, the results indicate that as quantum processor quality improves, these workflows will offer a scalable path to simulating complex periodic materials with high accuracy.

Technical Summary: Computing Band Gaps of Periodic Materials via Sample-Based Quantum Diagonalization

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