Experimentally feasible computational advantage from quantum superposition of gate orders

TL;DR

This paper demonstrates that quantum-controlled superpositions of gate orders can provide a provable computational advantage in quantum algorithms, especially for tasks involving only qubit gates, making experimental demonstration feasible.

Contribution

It introduces new tasks showing asymptotic quantum advantage with controlled gate orderings using only qubits, and analyzes their solutions within quantum circuit models.

Findings

Quantum-$n$-switch calls each gate once, causal algorithms call at least 2n-1 gates.

Fixed order algorithms require O(n log n) gates, more than the quantum-$n$-switch.

The tasks are suitable for experimental demonstration of quantum superposition advantages.

Abstract

In an ordinary quantum algorithm the gates are applied in a fixed order on the systems. The introduction of indefinite causal structures allows 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 (1) for which there is a provable computational advantage of a quantum-controlled ordering of gates in the asymptotic case and (2) 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.