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

Towards large-scale quantum optimization solvers with few qubits