Symmetry protected subspaces in quantum simulations

Abstract

This work demonstrates an efficient algorithm which leverages transitive closure on graphs to identify symmetry protected subspaces of quantum unitary operators in the σz basis. The algorithm’s time complexity is linear to the size of the subspace. This subspace constrains states in quantum time evolution to a smaller subspace. If this subspace is small enough, a classical computer can complete the quantum simulation task; however, if this subspace has exponential scaling and is thus infeasible to calculate, a quantum computer is required to complete the same simulation task. In this regime, the symmetry protected subspace is leveraged to reduce quantum computation errors with post-selection. To do this, a probabilistic algorithm is presented which can determine if a measured state is within the subspace in polynomial time. Both algorithms are benchmarked on three quantum simulations: the Heisenberg XXX Hamiltonian, T2 Quantum Cellular Automata (QCA), and F4 QCA. QCA simulations are implemented as update rules which define quantum time evolution. Finally, the paper concludes by implementing post-selection on a quantum circuit emulator with noise with the three aforementioned simulations.