This work proposes a numerically efficient implementation of sparse parallel vectors within the open-source finite element library deal.II. The main algorithmic ingredient is the matrix-free evaluation of the Hamiltonian operator by cell-wise quadrature. Based on an a-priori chosen support for each vector, we develop algorithms and data structures to perform (i) matrix-free sparse matrix multivector products (SpMM), (ii) the projection of an operator onto a sparse sub-space (inner products), and (iii) post-multiplication of a sparse multivector with a square matrix. The node-level performance is analyzed using a roofline model. Our matrix-free implementation of finite element operators with sparse multivectors achieves a performance of 157 GFlop/s on an Intel Cascade Lake processor with 20 cores. Strong and weak scaling results are reported for a representative benchmark problem using quadratic and quartic finite element base}, author = {Davydov, Denis and Kronbichler, Martin}, doi = {10.1145/3399736}, faupublication = {yes}, journal = {ACM Transactions on Parallel Computing}, pages = {1-30}, peerreviewed = {Yes}, title = {{Algorithms} and {Data} {Structures} for {Matrix}-{Free} {Finite} {Element} {Operators} with {MPI}-{Parallel} {Sparse} {Multi}-{Vectors}}, volume = {7}, year = {2020} }