Symmetry-adapted encodings for qubit number reduction by point-group and other Boolean symmetries

Abstract

A symmetry-adapted fermion-to-spin mapping or encoding that is able to store information about the occupancy of the n spin-orbitals of a molecular system into a lower number of n − k qubits in a quantum computer (where the number of reduced qubits k ranges from 2 to 5 depending on the symmetry of the system) is introduced. This mapping reduces the computational cost of a quantum computing simulation and at the same time enforces symmetry constraints. These symmetry-adapted encodings (SAEs) can be explicitly seen as a block-diagonalization of the Jordan–Wigner qubit Hamiltonian, followed by an orthogonal projection. We provide the form of the Clifford tableau for a general class of fermion-to-qubit encodings, and then use it to construct the map that block-diagonalizes the Hamiltonian in the SAEs. The algorithm proposed does not require any further computations to obtain this map, which is derived directly from the character table of the molecular point group. An implementation of the algorithm is presented as an open-source Python package, QuantumSymmetry, a user guide and code examples. QuantumSymmetry uses open-source quantum chemistry software PySCF for Hartree–Fock calculations, and is compatible with quantum computing toolsets OpenFermion and Qiskit. QuantumSymmetry takes arbitrary user input such as the molecular geometry and atomic basis set to construct the qubit operators that correspond in the appropriate SAE to fermionic operators on the molecular system, such as the second-quantized electronic structure Hamiltonian. QuantumSymmetry is used to produce numerical examples of variational quantum algorithm simulations to find the ground state energy for a number of example molecules, for both Unitary Coupled Clusters with Singles and Doubles and Adaptive Derivative Assembled Pseudo-Trotter Variational Quantum Eigensolver ansätze. We show that, beyond the advantage given by the lower qubit count, the proposed encodings consistently result in shallower and less complex circuits with a reduced number of variational parameters that are able to reach convergence faster and without any loss of computed accuracy.