Giancarlo Camilo

We present an exact $n$-qubit computational-basis amplitude encoder of real- or complex-valued data vectors of $d=\binom{n}{k}$ 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 $d-1$ (controlled) Reconfigurable Beam Splitter (RBS) gates and is constructed by an efficient classical algorithm that sequentially generates all bitstrings of weight $k$ and identifies the gates that superpose the corresponding states with the correct amplitudes. An explicit compilation into CNOTs and single-qubit gates is presented, with the total CNOT-gate count of $\mathcal{O}(k\, d)$ 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 $\mathcal{O}(d\,\log(d))$ CNOT gates. Moreover, using generalized RBS gates that mix states of different Hamming weights, we extend the construction to efficiently encode arbitrary sparse vectors. Experimentally, 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 showcase 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.

We introduce a variational quantum solver for combinatorial optimizations over $m=\mathcal{O}(n^k)$ binary variables using only $n$ qubits, with tunable $k>1$. The number of parameters and circuit depth display mild linear and sublinear scalings in $m$, respectively. Moreover, we analytically prove that the specific qubit-efficient encoding brings in a super-polynomial mitigation of barren plateaus as a built-in feature. This leads to unprecedented quantum-solver performances. For $m=7000$, numerical simulations produce solutions competitive in quality with state-of-the-art classical solvers. In turn, for $m=2000$, an experiment with $n=17$ trapped-ion qubits featured MaxCut approximation ratios estimated to be beyond the hardness threshold $0.941$. To our knowledge, this is the highest quality attained experimentally on such sizes. Our findings offer a novel heuristics for quantum-inspired solvers as well as a promising route towards solving commercially-relevant problems on near term quantum devices.