A Bottom-up Approach to Constructing Symmetric Variational Quantum Circuits

In the age of noisy quantum processors, the exploitation of quantum symmetries can be quite beneficial in the efficient preparation of trial states, an important part of the variational quantum eigensolver algorithm. The benefits include building quantum circuits which are more compact, with lesser number of paramaters, and more robust to noise, than their non-symmetric counterparts. Leveraging on ideas from representation theory we show how to construct symmetric quantum circuits. Similar ideas have been previously used in the field of tensor networks to construct symmetric tensor networks. We focus on the specific case of particle number conservation, that is systems with U(1) symmetry. Based on the representation theory of U(1), we show how to derive the particle-conserving exchange gates, which are commonly used in constructing hardware-efficient quantum circuits for fermionic systems, like in quantum chemistry, material science, and condensed-matter physics. We tested the effectiveness of our circuits with the Heisenberg XXZ model.