Quantum encoder for fixed Hamming-weight subspaces

We present an exact n-qubit computational-basis amplitude encoder of real- or complex-valued data vectors of d=nk components into a subspace of fixed Hamming weight k. This represents a polynomial space compression. The circuit is optimal in that it expresses an arbitrary data vector using only $d-1$ (controlled) Reconfigurable Beam Splitter (RBS) gates and is constructed by an efficient classical algorithm that sequentially generates all bitstrings of weight k and identifies all gate parameters. An explicit compilation into CNOTs and single-qubit gates is presented, with the total CNOT-gate count of O(k\, d) provided in analytical form. In addition, we show how to load data in the binary basis by sequentially stacking encoders of different Hamming weights using O(d\,log(d)) CNOT gates. Moreover, using generalized RBS gates that mix states of different Hamming weights, we extend the construction to efficiently encode arbitrary sparse vectors. Finally, we perform an experimental proof-of-principle demonstration of our scheme on a commercial trapped-ion quantum computer. We successfully upload a q-Gaussian probability distribution in the non-log-concave regime with $n = 6$ and $k = 2$. We also showcase how the effect of hardware noise can be alleviated by quantum error mitigation. Our results constitute a versatile framework for quantum data compression with various potential applications in fields such as quantum chemistry, quantum machine learning, and constrained combinatorial optimizations.