Symmetry-Checking in Band Structure Calculations on a Noisy Quantum Computer

TL;DR

The paper tackles the challenge of identifying the symmetry of band crossings in solid-state systems on noisy quantum devices. It introduces a character-checking quantum circuit that compares symmetry characters of neighboring-state SALC bases to distinguish crossings from anticrossings, and demonstrates this approach on AA- and AB-stacked bilayer graphene. The authors correct SALC basis inconsistencies, test robustness against depolarizing noise, and validate the method on IBM’s Marrakesh hardware, achieving reliable identification of band-crossing points. This approach enables symmetry-resolved band-structure analysis on NISQ devices and can be extended to a wide range of materials and configurations.

Abstract

Band crossings in electronic band structures play an important role in determining the electronic, topological, and transport properties in solid-state systems, making them central to both condensed matter physics and materials science. The emergence of noisy intermediate-scale quantum (NISQ) processors has sparked great interest in developing quantum algorithms to compute band structure properties of materials. While significant research has been reported on computing ground state and excited state energy bands in the presence of noise that breaks the degeneracy, identifying the symmetry at crossing points using quantum computers is still an open question. In this work, we propose a method for identifying the symmetry of bands around crossings and anti-crossings in the band structure of bilayer graphene with two distinct configurations on a NISQ device. The method utilizes eigenstates at neighbouring $\mathbf{k}$ points on either side of the touching point to recover the local symmetry by implementing a character-checking quantum circuit that uses ancilla qubit measurements for a probabilistic test. We then evaluate the performance of our method under a depolarizing noise model, using four distinct matrix representations of symmetry operations to assess its robustness. Finally, we demonstrate the reliability of our method by correctly identifying the correct band crossings of AA-stacked bilayer graphene around $K$ point, using the character-checking circuit implemented on a noisy IBM quantum processor $ibm\_marrakesh$.

Figures (9)

• Figure 1: The atomic and band structures of AA and AB-stacked bilayer graphene. (a) and (b) are the atomic configurations of AA and AB-stacked bilayer graphene, respectively. (c) is the corresponding Brillouin zone. (d) illustrates the band structure of the AA-stacked configuration, while (e) represents the band structure of the AB-stacked configuration.
• Figure 2: The character check circuit and the corresponding numerical simulation results. (a) The character check circuit for checking if two quantum states $\ket{\psi_1}$ and $\ket{\psi_2}$ have the same character given the symmetry operation $S_1$ in the point group. The numerical simulations illustrating the band structures with band crossings for AA- and AB-stacked bilayer graphene are shown in (b) and (c), respectively.
• Figure 3: Depolarizing noise model simulation of AA-stacked bilayer graphene at the $\Gamma$ point using the character-checking circuit for the ground and first excited states. Symmetry operations yielding the same characters for the two states are shown in (a), while those yielding different characters are shown in (b).
• Figure 4: The band crossing verification of AA-stacked bilayer graphene using the character-checking circuit on $ibm\_marrakesh$. The band structures are shown with raw data in (a), and with corrected band crossings verified by the character-checking circuit in (b). The eigenvectors used for verification are labeled from $e1$ to $e12$.
• Figure 5: The chip geometry of the IBM quantum processor $ibm\_marrakesh$. The qubits and connections used in this work are highlighted in red.