The field of indefinite causal order (ICO) has seen a recent surge in
interest. Much of this research has focused on the quantum SWITCH, wherein
multiple parties act in a superposition of different orders in a manner
transcending the quantum circuit model. This results in a new resource for
quantum protocols, and is exciting for its relation to issues in foundational
physics. The quantum SWITCH is also an example of a higher-order quantum
operation, in that it not only transforms quantum states, but also other
quantum operations. To date, no higher-order quantum operation has been
completely experimentally characterized. Indeed, past work on the quantum
SWITCH has confirmed its ICO by measuring causal witnesses or demonstrating
resource advantages, but the complete process matrix has only been described
theoretically. Here, we perform higher-order quantum process tomography.
However, doing so requires exponentially many measurements with a scaling worse
than standard process tomography. We overcome this challenge by creating a new
passively-stable fiber-based quantum SWITCH using active optical elements to
deterministically generate and manipulate time-bin encoded qubits. Moreover,
our new architecture for the quantum SWITCH can be readily scaled to multiple
parties. By reconstructing the process matrix, we estimate its fidelity and
tailor different causal witnesses directly for our experiment. To achieve this,
we measure a set of tomographically complete settings, that also spans the
input operation space. Our tomography protocol allows for the characterization
and debugging of higher-order quantum operations with and without an ICO, while
our experimental time-bin techniques could enable the creation of a new realm
of higher-order quantum operations with an ICO.Comment: 20 pages (12 pages, 4 pages appendix + reference list and
introduction), 8 figures; v2 with updated funding informatio