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

## TL;DR

This paper introduces a quantum algorithm for simulating quantum chemistry in arbitrary basis sets using qubitized quantum walks, exploiting sparsity and low-rank structures to significantly reduce resource requirements.

## Contribution

It extends quantum simulation techniques to arbitrary basis sets by leveraging sparsity and low-rank tensor factorizations, achieving improved scaling over previous methods.

## Key findings

- Achieves $ ilde{O}(N^{3/2} imes ext{lambda})$ T complexity for quantum chemistry simulation.
- Demonstrates a 700-fold reduction in surface code spacetime volume for FeMoco molecule.
- Provides circuits that outperform prior algorithms in resource efficiency.

## Abstract

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.

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Source: https://tomesphere.com/paper/1902.02134