Finding exponential separation between quantum and classical information tasks is like striking gold in quantum information research. Such an advantage is believed to hold for quantum computing but is proven for quantum communication complexity. Recently, a novel quantum resource called the quantum switch-which creates a coherent superposition of the causal order of events, known as quantum causality-has been harnessed theoretically in a new protocol providing provable exponential separation. We experimentally demonstrate such an advantage by realizing a superposition of communication directions for a two-party distributed computation. Our photonic demonstration employs d-dimensional quantum systems, qudits, up to d = 2 13 dimensions and demonstrates a communication complexity advantage, requiring less than (0.696 ± 0.006) times the communication of any causally ordered protocol. These results elucidate the crucial role of the coherence of communication direction in achieving the exponential separation for the one-way processing task, and open a new path for experimentally exploring the fundamentals and applications of advanced features of indefinite causal structures.Computation by separated parties with minimal communication is the focus of communication complexity, which has applications to distributed computing, very-large-scale integration, streaming algorithms, and more [1]. For quantum information, communication complexity is especially exciting as exponential quantum-classical gaps can be proven [2][3][4][5][6][7][8][9]. By contrast, exponential quantum-classical gaps for computation tasks such as factorization [10] depend on the best-known classical algorithm, and thus are strongly believed but not rigorously proven. Experimentally, quantum communication complexity has been studied in proof of principle for the quantum fingerprinting protocol [11][12][13] and beyond [1,14,15].The quantum switch provides a new communication complexity tool that leads to another instance of exponential quantum advantage [16]. The quantum switch is a device where a control qubit determines the order in which two transformations are performed on a target system [17,18]. When the control is in a superposition of logical states, the order of the operations is causally indefinite; i.e., there is a superposition of the ordering of target operations. The quantum switch has broad relevance in the context of quantum causality [19] including applications to studies of quantum gravity [19][20][21][22], communication complexity [16, * These authors contributed equally to this work.2 23], witnessing causality [17,[24][25][26][27][28] and deciding whether a given indefinite causal order is physical [29,30]. In quantum computing, the quantum switch can reduce the query complexity for some tasks compared to causally ordered protocols [18,31] -this advantage has been demonstrated for single-qubit control and single-qubit target circuits [32]. In quantum communication, the quantum switch enhances the communication rate beyond the limits of ...
