Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization

Dominic W. Berry1, Craig Gidney2, Mario Motta3, Jarrod R. McClean2, and Ryan Babbush2

1Department of Physics and Astronomy, Macquarie University, Sydney, NSW 2109, Australia
2Google Research, Venice, CA 90291, United States
3Division of Chemistry, California Institute of Technology, Pasadena, CA 91125, United States

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Recent work has dramatically reduced the gate complexity required to quantum simulate chemistry by using linear combinations of unitaries based methods to exploit structure in the plane wave basis Coulomb operator. Here, we show that one can achieve similar scaling even for arbitrary basis sets (which can be hundreds of times more compact than plane waves) by using qubitized quantum walks in a fashion that takes advantage of structure in the Coulomb operator, either by directly exploiting sparseness, or via a low rank tensor factorization. We provide circuits for several variants of our algorithm (which all improve over the scaling of prior methods) including one with $\widetilde{\cal O}(N^{3/2} \lambda)$ T complexity, where $N$ is number of orbitals and $\lambda$ is the 1-norm of the chemistry Hamiltonian. We deploy our algorithms to simulate the FeMoco molecule (relevant to Nitrogen fixation) and obtain circuits requiring about seven hundred times less surface code spacetime volume than prior quantum algorithms for this system, despite us using a larger and more accurate active space.

Simulation of quantum chemistry is one of the most important potential applications of quantum computers, because it could be used to design new molecules for a wide range of applications. For example, it could be used to gain understanding of biological Nitrogen fixation by simulation of the FeMoco molecule. We show how to greatly accelerate the simulation of quantum chemistry by taking advantage of the structure of the system, together with several other new advances in quantum algorithms. Our most efficient approach takes advantage of the fact that the description of the energy has many terms that are close to zero and can be ignored. Together with a more efficient way of inputting information about the nonzero terms into the quantum algorithm, for the example of FeMoco we achieve a speedup of about a factor of 700.

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