Universal quantum circuits for quantum chemistry

Juan Miguel Arrazola, Olivia Di Matteo, Nicolás Quesada, Soran Jahangiri, Alain Delgado, and Nathan Killoran

Xanadu, Toronto, ON, M5G 2C8, Canada

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Universal gate sets for quantum computing have been known for decades, yet no universal gate set has been proposed for particle-conserving unitaries, which are the operations of interest in quantum chemistry. In this work, we show that controlled single-excitation gates in the form of Givens rotations are universal for particle-conserving unitaries. Single-excitation gates describe an arbitrary $U(2)$ rotation on the two-qubit subspace spanned by the states $|01\rangle, |10\rangle$, while leaving other states unchanged – a transformation that is analogous to a single-qubit rotation on a dual-rail qubit. The proof is constructive, so our result also provides an explicit method for compiling arbitrary particle-conserving unitaries. Additionally, we describe a method for using controlled single-excitation gates to prepare an arbitrary state of a fixed number of particles. We derive analytical gradient formulas for Givens rotations as well as decompositions into single-qubit and CNOT gates. Our results offer a unifying framework for quantum computational chemistry where every algorithm is a unique recipe built from the same universal ingredients: Givens rotations.

This work shows that a special type of gate, known as a controlled single-excitation gate, can be used to build any quantum circuit that preserves the number of particles in a fermionic system. These are the main transformations of interest in quantum chemistry. Controlled single-excitation gates are examples of Givens rotations, which therefore can be seen as the universal building blocks of quantum circuits for quantum chemistry.

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