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Reducing the complexity of quantum algorithms to treat quantum chemistryproblems is essential to demonstrate an eventual quantum advantage ofNoisy-Intermediate Scale Quantum (NISQ) devices over their classicalcounterpart. Significant improvements have been made recently to simulate thetime-evolution operator $U(t) = e^{i\mathcal{\hat{H}}t}$ where$\mathcal{\hat{H}}$ is the electronic structure Hamiltonian, or to simulate$\mathcal{\hat{H}}$ directly (when written as a linear combination ofunitaries) by using block encoding or "qubitization" techniques. A fundamentalmeasure quantifying the practical implementation complexity of these quantumalgorithms is the so-called "1-norm" of the qubit-representation of theHamiltonian, which can be reduced by writing the Hamiltonian in factorized ortensor-hypercontracted forms for instance. In this work, we investigate theeffect of classical pre-optimization of the electronic structure Hamiltonianrepresentation, via single-particle basis transformation, on the 1-norm.Specifically, we employ several localization schemes and benchmark the 1-normof several systems of different sizes (number of atoms and active space sizes).We also derive a new formula for the 1-norm as a function of the electronicintegrals, and use this quantity as a cost function for an orbital-optimizationscheme that improves over localization schemes. This paper gives more insightsabout the importance of the 1-norm in quantum computing for quantum chemistry,and provides simple ways of decreasing its value to reduce the complexity ofquantum algorithms.

Orbital transformations to reduce the 1-norm of the electronic structure Hamiltonian for quantum computing applications