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Quantum simulation of the electronic structure problem is one of the most researched applications of quantum computing. The majority of quantum algorithms for this problem encode the wavefunction using N Gaussian orbitals, leading to Hamiltonians with O(N4) second-quantized terms. We avoid this overhead and extend methods to condensed phase materials by utilizing a dual form of the plane wave basis which diagonalizes the potential operator, leading to a Hamiltonian representation with O(N2) second-quantized terms. Using this representation, we can implement single Trotter steps of the Hamiltonians with linear gate depth on a planar lattice. Properties of the basis allow us to deploy Trotter- A nd Taylor-series-based simulations with respective circuit depths of O(N7/2) and Õ(N8/3) for fixed charge densities. Variational algorithms also require significantly fewer measurements in this basis, ameliorating a primary challenge of that approach. While our approach applies to the simulation of arbitrary electronic structure problems, the basis sets explored in this work will be most practical for treating periodic systems, such as crystalline materials, in the near term. We conclude with a proposal to simulate the uniform electron gas (jellium) using a low-depth variational ansatz realizable on near-term quantum devices. From these results, we identify simulations of low-density jellium as a promising first setting to explore quantum supremacy in electronic structure.

@article{Babbush2018, abstract = {Quantum simulation of the electronic structure problem is one of the most researched applications of quantum computing. The majority of quantum algorithms for this problem encode the wavefunction using N Gaussian orbitals, leading to Hamiltonians with O(N4) second-quantized terms. We avoid this overhead and extend methods to condensed phase materials by utilizing a dual form of the plane wave basis which diagonalizes the potential operator, leading to a Hamiltonian representation with O(N2) second-quantized terms. Using this representation, we can implement single Trotter steps of the Hamiltonians with linear gate depth on a planar lattice. Properties of the basis allow us to deploy Trotter- A nd Taylor-series-based simulations with respective circuit depths of O(N7/2) and {\~{O}}(N8/3) for fixed charge densities. Variational algorithms also require significantly fewer measurements in this basis, ameliorating a primary challenge of that approach. While our approach applies to the simulation of arbitrary electronic structure problems, the basis sets explored in this work will be most practical for treating periodic systems, such as crystalline materials, in the near term. We conclude with a proposal to simulate the uniform electron gas (jellium) using a low-depth variational ansatz realizable on near-term quantum devices. From these results, we identify simulations of low-density jellium as a promising first setting to explore quantum supremacy in electronic structure.}, author = {Babbush, Ryan and Wiebe, Nathan and McClean, Jarrod and McClain, James and Neven, Hartmut and Chan, Garnet Kin Lic}, doi = {10.1103/PhysRevX.8.011044}, issn = {21603308}, journal = {Physical Review X}, number = {1}, title = {{Low-Depth Quantum Simulation of Materials}}, volume = {8}, year = {2018} }

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