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 ${\cal O}(N^4)$ second-quantized terms. We avoid this overhead and extend methods to the condensed phase by utilizing a dual form of the plane wave basis which diagonalizes the potential operator, leading to a Hamiltonian representation with ${\cal O}(N^2)$ 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 and Taylor series based simulations with respective circuit depths of ${\cal O}(N^{7/2})$ and $\widetilde{\cal O}(N^{8/3})$ for fixed charge densities - both are large asymptotic improvements over all prior results. Variational algorithms also require significantly fewer measurements to find the mean energy in this basis, ameliorating a primary challenge of that approach. 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.
Highlights
The problem of electronic structure is to simulate the stationary properties of electrons interacting via Coulomb forces in an external potential
While Ref. [72] showed that the fermionic fast Fourier transform” (FFFT) could be realized with Oðlog NÞ depth using arbitrary two-qubit gates, in Appendix I, we extend the method of Ref. [71] to show that the FFFT can be implemented for three spatial dimensions using a planar lattice of qubits with OðNÞ depth
We have introduced efficient techniques that use the plane wave basis and its dual for quantum simulations of electronic structure
Summary
The problem of electronic structure is to simulate the stationary properties of electrons interacting via Coulomb forces in an external potential. The major challenge in developing low depth quantum algorithms for quantum chemistry is that electronic structure Hamiltonians often have as many as OðN4Þ terms, where N is the number of basis functions. This is problematic as many algorithms for time evolution and energy estimation have costs that scale explicitly with the number of terms. The basis is compact for uniform and near-uniform electron gasses (realized in simple metals as well as electrons in semiconductor wells), and there is well-developed infrastructure (e.g., pseudopotentials) to enable compact representations of atomistic materials [49,50] These methods are especially promising for extending the reach of quantum simulations into the domain of materials. In contrast to formally rigorous bounds, a tilde inside of a bound, e.g., Oð∼NÞ, indicates that the bound is obtained empirically
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