Quantum algorithms on noisy intermediate-scale quantum (NISQ) devices are expected to soon simulate quantum systems that are classically intractable. However, the non-negligible gate error present on NISQ devices impedes the implementation of many purely quantum algorithms, necessitating the use of hybrid quantum-classical algorithms. One such hybrid quantum-classical algorithm, is based upon quantum computed Hamiltonian moments ϕj Ĥn jϕ D E n ¼ 1,2,ÁÁÁ ð Þ , with respect to quantum state j ϕi. In this tutorial review, we will give a brief review of these quantum algorithms with focuses on the typical ways of computing Hamiltonian moments using quantum hardware and improving the accuracy of the estimated state energies based on the quantum computed moments. We also present a computation of the Hamiltonian moments of a four-site Heisenberg model and compute the energy and magnetization of the model utilizing the imaginary time evolution on current IBM-Q hardware. Finally, we discuss some possible developments and applications of Hamiltonian moment methods.

Colloidal nanoparticles with anisotropic interactions are promising building blocks for the fabrication of complex functional materials. A challenge in the self-assembly of colloidal particles is the rational design of geometry and chemistry to program the formation of a desired target structure. We report an inverse design procedure integrating Langevin dynamics simulations and evolutionary algorithms to engineer anisotropic patchy colloidal clusters to spontaneously assemble into a cubic diamond lattice possessing a complete photonic band gap. The combination of a tetrahedral cluster geometry and optimized placement of a single type of anisotropic interaction patch results in a colloidal building block predicted to assemble a cubic diamond lattice with more than 82% yield. This design represents an experimentally viable colloidal building block capable of high-fidelity assembly of a cubic diamond lattice.