Band theory on a quantum computer

Band theory is a successful approach to studying the electronic structure of periodic systems under a single-electron approximation. Typically, band theory considers a Hamiltonian in reciprocal space, with hopping parameters as functions of k, a vector in the Brillouin zone. We demonstrate a technique to perform band theory on a quantum computer. We also demonstrate an alternative which considers a Hamiltonian in real space, with constant hopping parameters and with k input directly into the quantum circuit. We show this alternative adds Θ(n) qubits, Θ(n²) gate operations, and a factor of Θ(n) to algorithm complexity, where n is the maximum binary resolution of k. We validate our approach with results obtained from IBMQ cloud computers. Finally, we suggest how our approach may be adapted to consider correlations between multiple electrons.