Orbital transformations to reduce the 1-norm of the electronic structure Hamiltonian for quantum computing applications
Emiel Koridon, Saad Yalouz, Bruno Senjean, Francesco Buda, Thomas E. O’Brien, Lucas Visscher
Abstract
Reducing the complexity of quantum algorithms to treat quantum chemistry problems is essential to demonstrate an eventual quantum advantage of noisy-intermediate scale quantum devices over their classical counterpart. Significant improvements have been made recently to simulate the time-evolution operator $U(t)={e}^{i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{H}}t}$, where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{H}}$ is the electronic structure Hamiltonian, or to simulate $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{H}}$ directly (when written as a linear combination of unitaries) by using block encoding or qubitization techniques. A fundamental measure quantifying the practical implementation complexity of these quantum algorithms is the so-called 1-norm of the qubit representation of the Hamiltonian, which can be reduced by writing the Hamiltonian in factorized or tensor-hypercontracted forms, for instance. In this paper, we investigate the effect of classical preoptimization of the electronic structure Hamiltonian representation, via single-particle basis transformation, on the 1-norm. Specifically, we employ several localization schemes and benchmark the 1-norm of several systems of different sizes (number of atoms and active space sizes). We also derive a formula for the 1-norm as a function of the electronic integrals and use this quantity as a cost function for an orbital-optimization scheme that improves over localization schemes. This paper gives more insights about the importance of the 1-norm in quantum computing for quantum chemistry and provides simple ways of decreasing its value to reduce the complexity of quantum algorithms.