Quantum band structures skip per-material Hamiltonians: silicon's eight bands land within 1.78 meV of exact diagonalization

Quantum band-structure calculations can now take each k-point Hamiltonian directly from a pretrained universal tight-binding model instead of building one per material through density-functional theory plus Wannierization or hand fitting. In earlier quantum band-structure work the second-quantized Hamiltonian fed to the quantum solver was always constructed by hand for each material; Wannierization requires choosing disentanglement windows and initial projections and checking convergence, while empirical tight binding must be fitted to a target band structure. For silicon, 161 k-points along the Γ−X−W−K−Γ−L−U−W−L−K|U−X path were sampled, and the eight eigenvalues from the three-qubit variational quantum deflation run deviate from exact diagonalization of the same SlaKoNet Hamiltonian by 1.78 meV on average across all bands and k-points, with the largest deviation of 29.32 meV on the highest excited state after seven successive deflations.

The workflow feeds the neural Hamiltonian into a Qiskit variational quantum deflation pipeline that peels off each found eigenstate in turn, extracting a whole band set at one k-point. SlaKoNet learns the distance-dependent hopping and overlap integrals of the Slater-Koster form by automatic differentiation across 65 elements, then interpolates the learned parameters at runtime to produce H(k) and the overlap matrix S(k). Each 8×8 matrix is symmetrized and decomposed into Pauli strings on 3 qubits; the ansatz is EfficientSU2 at five repetitions with 36 trainable parameters, optimized by COBYLA on a noiseless statevector backend.

The same circuit and optimizer settings transfer without tuning to five conventional superconductors, holding errors at the millielectronvolt level. The elemental metals Al, Ta, V and Nb use two-qubit bases, while the rock-salt nitride ZrN uses a three-qubit basis, so qubit count follows the orbital basis size. Per-band mean absolute errors are 0.05, 0.02, 0.01 and 0.02 meV for Al, Ta, V and Nb, and 1.74 meV for ZrN, comparable to silicon's 1.78 meV.

On IBM Quantum hardware, the ground-state energy at one representative aluminum k-point was computed end to end. This run exercises only the ground state; resolving the deflated excited-state spectrum on a physical device was not attempted. On ibm_boston with 20480 shots per energy estimate, the error-mitigated estimate deviates by about 0.37 eV from the exact ground state of the same Hamiltonian, with a shot-noise standard error of about 0.06 eV per estimate.

The demonstration covers crystals within SlaKoNet's training domain of elements up to atomic number 65; the full silicon band comparison runs on a noiseless statevector simulator, and hardware execution covers one representative aluminum k-point ground state. The same workflow yields bands for Al, Ta, V, Nb and ZrN without retraining, with qubit count varying between 2 and 3 with the orbital basis. Phonon bands and a machine-learning-accelerated quantum dynamical mean-field theory are named as next directions.

Absolute electronic-structure accuracy is dominated by SlaKoNet's deviation from the TBmBJ reference, reported as 0.74 eV mean absolute bandgap error on a 52-material benchmark and about 2.3 eV on ultrawide-gap insulators. The released model includes neither spin-orbit coupling nor repulsive pair potentials, so heavy-element systems and total-energy quantities that rely on the repulsive part are out of reach. The deflation overlap penalty grows linearly with the number of found states, and the highest band shows errors about 30.3 times the ground state's. Hardware deviations of about 0.37 eV exceed chemical accuracy. In the correlated extension, the impurity solver is iterated perturbation theory, so quasiparticle weights are best read as correlation trends rather than converged values.