Quantum computation of a quasiparticle band structure with the quantum-selected configuration interaction

TL;DR

This work tackles the challenge of computing quasiparticle band structures for strongly correlated systems on noisy quantum devices. It introduces a hybrid quantum-classical scheme that replaces full VQE ground-state optimization with quantum-selected configuration interaction QSCI to form a compact ground-state representation, which is then used by quantum subspace expansion QSE to obtain quasiparticle bands. Matrix elements for QSE are computed from the compact ground-state representation, enabling classical diagonalization of the subspace and scalable performance as system size grows. Demonstrated on silicon using a $16$-qubit IBM processor, the resulting quasiparticle bands agree with exact diagonalization, showcasing a practical route toward accurate simulations of correlated materials on near-term quantum hardware.

Abstract

Quasiparticle band structures are fundamental for understanding strongly correlated electron systems. While solving these structures accurately on classical computers is challenging, quantum computing offers a promising alternative. Specifically, the quantum subspace expansion (QSE) method, combined with the variational quantum eigensolver (VQE), provides a quantum algorithm for calculating quasiparticle band structures. However, optimizing the variational parameters in VQE becomes increasingly difficult as the system size grows, due to device noise, statistical noise, and the barren plateau problem. To address these challenges, we propose a hybrid approach that combines QSE with the quantum-selected configuration interaction (QSCI) method for calculating quasiparticle band structures. QSCI may leverage the VQE ansatz as an input state but, unlike the standard VQE, it does not require full optimization of the variational parameters, making it more scalable for larger quantum systems. Based on this approach, we demonstrate the quantum computation of the quasiparticle band structure of a silicon using 16 qubits on an IBM quantum processor.

Figures (6)

• Figure 1: (Color online) Schematic description of our approach to calculate the quasiparticle band structure in this paper. It is based on QSCI for the ground-state calculation, combined with QSE to find quasiparticle bands.
• Figure 2: (Color online) Structure of the quantum circuit used as VQE ansatz in this study. The HF state, $\ket{10101\cdots 010}$, is generated by applying $X$ gates to the initialized state $\ket{\psi_0}=\ket{0}^{\otimes 16}$ of 16 qubits. Note that we adopt an unusual ordering in the mapping of spin orbitals to qubits as described in the main text. $R_y(\theta)$ represents the rotation about the $y$-axis. $d$ is the number of repeated layers, or depth. $16(d+1)$ variational parameters of $\theta_{0\mathchar'- 15}^{(i)}$ ($i=1,\cdots, d+1$) are optimized during VQE calculations. A partially optimized state is used as the input state for QSCI. We take $d=3$ in the numerical study.
• Figure 3: (Color online) Configuration of the 127 qubits on the IBM Quantum processor $\mathsf{ibm\_kyoto}$. The 16 qubits used in this study are highlighted. See the main text for details.
• Figure 4: (Color online) Quasiparticle band structure of the Si crystal obtained by our QSE-QSCI approach with 16 qubits of the quantum device $\mathsf{ibm\_kyoto}$. 50 most frequent configurations are used in the QSCI calculation. For comparison, we also show the result based on FCI for the ground-state calculation combined with QSE.
• Figure 5: (Color online) QSCI results for the ground state of the Si crystal, shown with the VQE optimization history at the $L$ point on the band path. The QSCI results are obtained by the noiseless simulation or the quantum device with the sampling outcomes (a) before or (b) after the post-selection. In the noiseless simulation, each of the VQE states are used as the input for QSCI with the number of electron configurations varied ($R = 10, 30, 50$). The quantum-device result of QSCI is shown for the 400th iteration step. The energy obtained by each method is plotted as the difference from the FCI energy in Hartree. The energy obtained by the HF calculation is also shown for reference.