Computational Advantage from a Quantum Superposition of Qubit Gate Orders

In an ordinary quantum algorithm the gates are applied in a fixed order on the systems. The introduction of indefinite causal structures allows us to relax this constraint and control the order of the gates with an additional quantum state. It is known that this quantum-controlled ordering of gates can reduce the query complexity in deciding a property of black-box unitaries with respect to the best algorithm in which the gates are applied in a fixed order. However, all tasks explicitly found so far require unitaries that either act on unbounded dimensional quantum systems in the asymptotic limit (the limiting case of a large number of black-box gates) or act on qubits, but then involve only a few unitaries. Here we introduce tasks (i) for which there is a provable computational advantage of a quantum-controlled ordering of gates in the asymptotic case and (ii) that require only qubit gates and are therefore suitable to demonstrate this advantage experimentally. We study their solutions with the quantum n-switch and within the quantum circuit model and find that while the n-switch requires to call each gate only once, a causal algorithm has to call at least 2n-1 gates. Furthermore, the best known solution with a fixed gate ordering calls O[n log_{2}(n)] gates.