Experimental Quantum Switching for Exponentially Superior Quantum Communication Complexity

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 $(0.696\pm 0.006)$ 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.

Figure 1

Schematic illustration of the exchange evaluation game. Alice and Bob receive inputs $(\mathit{x},f)$ and $(\mathit{y},g)$, respectively. There is another party, Charlie, who needs to compute the exchange evaluation function $EE(\mathit{x},f,\mathit{y},g)=f(\mathit{y})\oplus g(\mathit{x})$. Under the condition of one-way communication, for a causally ordered protocol, Alice and Bob can transmit information either from Alice to Bob, then Charlie (blue arrow) or from Bob to Alice, then Charlie (red arrow). Using a quantum switch, a control qubit in the state $|+{⟩}_{\mathrm{c}}=1/\sqrt{2}(|0{⟩}_{\mathrm{c}}+|1{⟩}_{\mathrm{c}})$ coherently controls the communication direction of the target system ${|0⟩}_{\mathrm{t}}$. Charlie computes the exchange evaluation function by measuring the control qubit. As a result, they solve the game with exponential advantage in the scaling of the required communication compared to any causally ordered protocol.

Figure 2

Experimental setup of the causally indefinite protocol. The pump light is emitted by a 780-nm continuous-wave laser diode (LD). After passing through a 945-nm short-pass filter (945SP) used to remove the residual long wavelengths from outside fluorescence, the pump light is focused into a 10 mm, periodically poled potassium titanyl phosphate (PPKTP) crystal to convert pump photons into pairs of orthogonally polarized photons at a wavelength of 1560 nm. The down-converted photons are separated by a polarizing beam splitter (PBS), and the residual pump photons are removed by two filters. One of the down-converted photons is sent to a circulator (Cir) and then enters the Sagnac loop via a beam splitter (BS). To herald the presence of this photon, the other one is directly detected by a high-quality superconducting nanowire single-photon detector (SNSPD1). In the Sagnac loop, Alice and Bob each possess a variable delay and a phase modulator (PM) to implement their unitary transformations. For each of the $n$ segments, either the long option or the short option is chosen, depending on whether the corresponding digit of $x$ in binary representation is 0 or 1. A fiber spool (FS) with a length of 7 km is used to store the pulse train. The interference results are monitored by SNSPD2 and SNSPD3. Abbreviations of other components: PC, polarization controller; FC, fiber coupler; DM, dichroic mirror; TDC, time-to-digital converter.

Figure 3

Error probability for the system size $n=12$. The value $k=12$ denotes the worst case of inputs $x=y=2^n-1$. The other values of $k\in \{0,1,\dots ,11\}$ correspond to Alice’s ($k+1$)-th bit being flipped from 1 to 0. The red dashed line shows the worst error probability including the uncertainty. The error bars indicate 1 standard deviation.

Figure 4

(a) Comparison of the transmitted information. The red points show our experimental results for different system sizes. The red dashed line is the best fit straight line for the four data points. The error bars are too small to be seen. For $n=12$, our experimental result clearly beats both classical protocols and quantum causally definite protocols. (b) The ratio $\gamma$ of the transmitted information $Q$ of our causally indefinite protocol, and the lower bound on the transmitted information $C$ for a given type causally ordered protocol.