Computational Advantage From The Quantum Superposition Of Multiple Temporal Orders Of Photonic Gates

Models for quantum computation with circuit connections subject to the quantum superposition principle have recently been proposed. In them, a control quantum system can coherently determine the order in which a target quantum system undergoes N gate operations. This process, known as the quantum N-switch, is a resource for several information-processing tasks. In particular, it provides a computational advantage-over fixed-gate-order quantum circuits-for phase-estimation problems involving N unknown unitary gates. However, the corresponding algorithm requires an experimentally unfeasible target-system dimension (super)exponential in N. Here, we introduce a promise problem for which the quantum N-switch gives an equivalent computational speedup with target-system dimension as small as 2 regardless of N. We use state-of-the-art multicore optical-fiber technology to experimentally demonstrate the quantum N-switch with N = 4 gates acting on a photonic-polarization qubit. This is the first observation of a quantum superposition of more than N = 2 temporal orders, demonstrating its usefulness for efficient phase estimation.