Experimentally Demonstrating Indefinite Causal Order Algorithms to Solve the Generalized Deutsch's Problem

Deutsch's algorithm is the first quantum algorithm to demonstrate an advantage over classical algorithms. Here, Deutsch's problem is generalized to $n$ functions and a quantum algorithm with an indefinite causal order is proposed to solve this problem. The algorithm not only reduces the number of queries to the black box by half compared to the classical algorithm, but also significantly decreases the complexity of the quantum circuit and the number of required quantum gates compared to the generalized Deutsch's algorithm. The algorithm is experimentally demonstrated in a stable Sagnac loop interferometer with a common path, which overcomes the obstacles of both phase instability and low fidelity of the Mach–Zehnder interferometer. The experimental results show both ultrahigh and robust success probabilities $\approx 99.7\%$. This study opens a path toward solving practical problems with indefinite cause-order quantum circuits.