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arXiv:2506.20483 · Physical Review Applied

Symmetry checking in band-structure calculations on a noisy quantum computer

Shaobo Zhang, Akib Karim, Harry M. Quiney, Muhammad Usman

Published Dec 2, 2025• 1 citation
arXivHTMLPDF

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 $ibm\_marrakesh$.

DOI
10.1103/vpyb-ynmz

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Graphene, a material of extraordinary interest in condensed matter physics due to its unique electronic properties, presents a fundamental challenge: the precise calculation of its band structure. This Master’s Thesis addresses this problem by proposing the use of one of the most widely used algorithms in quantum computing. To achieve this, a tight-binding model is used that describes the electron hopping between the atoms of the single-layer hexagonal lattice that defines graphene. This model is extended to include not only the interactions of each atom in the lattice with adjacent atoms (first neighbors), but also with the next closest atoms (second neighbors), allowing for the capture of more subtle physical effects. The resulting hamiltonian is then prepared for execution in the quantum simulators of IBM’s Qiskit platform. The results under ideal conditions demonstrate that the algorithm reproduces analytical solutions with high fidelity, capturing essential features such as Dirac points, where valence and conduction bands intersect, causing electrons to behave as massless particles, and electron-hole symmetry breaking, which reflects the difference in the energetic behavior of electrons and their absences (holes). However, when introducing a noise model that emulates current quantum processors (inherently noisy, intermediate-scale devices, the socalled NISQ era), a systematic deviation is observed that underscores the limiting impact of the hardware. The study concludes that VQE is a conceptually viable and powerful tool for materials simulation, but its practical application to achieve quantum advantage will critically depend on the development of advanced error mitigation techniques.
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Abstract Monitored quantum circuits offer great perspectives for exploring the interplay of quantum information and complex quantum dynamics. These systems exhibit entanglement and purification phase transitions, along with various symmetry-enforced and ordered non-equilibrium phases. The central question is whether these phases can persist in real-world noisy devices. We study the fate of the symmetry-enforced absorbing state and charge-sharpening transitions in the presence of noise, and establish that noise results in coherent and incoherent symmetry-breaking effects. The coherent effects blur the distinction between phases, turning sharp transitions into crossovers, but states far from phase boundaries largely retain their essential character. We find, corrective feedback and postselected measurements can mitigate noise, stabilizing the absorbing and charge-sharp phases. Hence, if challenges like postselection are addressed, errors do not prevent the observation of symmetry-enforced phases in noisy intermediate-scale quantum (NISQ) devices. Additionally, we propose a symmetry-based method to characterize noise and gate fidelity.
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Abstract Protocols for quantum measurement are an essential part of quantum computing. Measurements are no longer confined to the final step of computation but are increasingly embedded within quantum circuits as integral components of noise-resilient algorithms. However, each observable typically requires a distinct measurement basis, often demanding a different circuit configuration. As the number of such configurations typically grows with the number of qubits, measurements constitute a major bottleneck. Focusing on electronic structure calculations in crystalline systems, we propose a measurement protocol that restricts the required measurement configurations to an absolute minimum of just three, independent of the number of qubits. This makes it one of the few known protocols that do not scale with qubit number. In particular, we derive the measurement protocol from the symmetries of tight-binding (TB) Hamiltonians and implement it within the Orthogonal-Ansatz variational quantum eigensolver algorithm. We demonstrate its performance on three systems, namely a two-dimensional CuO 2 square lattice (3 qubits), bilayer graphene with hexagonal (Honeycomb) lattice (4 qubits) and three-dimensional diamond lattice (10 qubits). Beyond TB systems, the protocol can be extended to enable efficient initial state preparation for many-body Hamiltonians, such as multi-orbital Hubbard models in a momentum space.
This work addresses the critical challenge of excited-state preparation for semiconductor band structure calculations. We introduce a variational adiabatic eigensolver (VAE) protocol that combines adiabatic evolution with variational optimization to prepare high-fidelity eigenstates on noisy intermediate-scale quantum (NISQ) devices. Applying a momentum-space truncation, we accurately compute the electronic band structure of silicon -- an idealized infinite periodic system -- using only a modest number of qubits. Our approach employs multi-qubit parameterized circuits and a phase-based loss function, overcoming limitations of conventional methods. These limitations include the circuit-construction difficulty in traditional adiabatic approaches and the reduced accuracy of variational quantum eigensolvers for excited states. Through rigorous numerical simulation and experimental implementation on a superconducting quantum processor, we successfully prepare silicon's valence-band and conduction-band eigenstates. Single-shot readout yields state fidelities exceeding 96%, and the measured energy expectations agree with theoretical band energies within 0.5 eV. Further refinement via single-frequency oscillation fitting reduces the energy deviation to below 0.01 eV. This framework provides a robust and practical pathway for precisely determining electronic structures in quantum materials.