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

• URL: http://arxiv.org/abs/2103.14753v3
• Date: Wed, 6 Oct 2021 10:33:41 GMT
• Title: Orbital transformations to reduce the 1-norm of the electronic structure Hamiltonian for quantum computing applications
• Authors: Emiel Koridon, Saad Yalouz, Bruno Senjean, Francesco Buda, Thomas E. O'Brien and Lucas Visscher
• Abstract summary: We investigate the effect of classical pre-optimization of the electronic structure Hamiltonian representation, via single-particle basis transformation, on the "1-norm" We derive a new 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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 (NISQ) devices over their classical counterpart. Significant improvements have been made recently to simulate the time-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 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 work, we investigate the effect of classical pre-optimization 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 new 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.

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