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 $d=(\genfrac{}{}{0}{}{n}{k})$ components into a subspace of fixed Hamming weight $k$. This represents a polynomial space compression of degree $k$. The circuit is optimal in that it expresses an arbitrary data vector using only $d\ensuremath{-}1$ (controlled) reconfigurable beam splitter (RBS) gates. It is constructed by an efficient classical algorithm that sequentially generates all bitstrings of weight $k$ and identifies the gates that superpose the corresponding states with the correct amplitudes. An explicit compilation into $\mathrm{CNOT}\mathrm{s}$ and single-qubit gates is presented, with the total $\mathrm{CNOT}$-gate count of $\mathcal{O}(kd)$ 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 $\mathcal{O}(d\mathrm{log}(d))\phantom{\rule{0.2em}{0ex}}\mathrm{CNOT}$ gates. Moreover, using generalized RBS gates that mix states of different Hamming weights, we extend the construction to efficiently encode arbitrary sparse vectors. We perform a 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 illustrate how the effect of hardware noise can be alleviated by quantum error mitigation. Numerically, we show how our encoder can improve the performance of variational quantum algorithms for problems that include particle-preserving symmetries. 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.