Aggregating quantum networks

Quantum networking allows the transmission of information in ways unavailable in the classical world. The information encoded in a single qubit can now be split and transmitted in a coherent way over different routes. This aggregation allows information to be transmitted in a fault tolerant way between different parts of the quantum network (or the future internet), even when that is not achievable with a single path approach. It is a quantum phenomenon not available in conventional telecommunication networks either. We show how the multiplexing of independent quantum channels allows a distributed form of quantum error correction to protect the transmission of quantum information between nodes or users of a quantum network. Combined with spatial-temporal single-photon multiplexing, we observe a significant drop in network resources required to transmit that quantum signal, even when only two channels are involved. This work goes far beyond the concepts of channel capacities and shows how quantum networking may operate in the future.

Figure 1

(a) Logic circuit used for the encoding of the (3,1,2) QRS code with qutrits; (b) gives the corresponding circuit using qubits while (c) illustrates how the Toffoli gate can be implemented for multiplexed photons using only a single photon-photon cnot gate. This is potentially an important resource savings [50, 51]. Here Pol.${}_1$, Pol.${}_2$ and ${\mathrm{TB}}_2$ stand for the polarization DOFs of photon 1 and photon 2 and time-bin DOF of photon 2 respectively. OS is an optical switch that splits photon 2 into two other modes (long and short). A cnot between photon 1 and the long component of photon 2 is performed followed by another OS that recombines the modes.

Figure 2

(a) Three quantum channels connecting two parties (Alice and Bob) with different number of qudits and transmission probabilities required to reach the threshold success probability ${\overline{P}}_S$ for the 7-qudits QRS code. (b) The same threshold probability can be reached with the aggregating network approach in which Alice and Bob are now connected by two channels having less transmission probability than the blue (dashed line) and red (solid line) channels or less qudits per channel than the blue and black channels (dotted line).

Figure 3

(a) Photon transmission probabilities required to reach ${\overline{P}}_S$ for different QRS codes. The dotted curves correspond to the quantum multiplexing 211-qudit QRS case ($k=106)$, in which each photon is carrying 8 qubits, where all qudits are traveling in the same channel (black), and 50 (orange), 80 (yellow), and 100 (purple) qudits are traveling in the channel having transmission probability $p_{t_2}.$ The solid (dashed) curves refer to the no-multiplexing (multiplexing) case for the 43-qudit QRS code ($k=22)$. The black curves are the cases when all qudits are traveling in the same channel. The red, yellow, and green curves are, respectively, the cases when 5, 10, and 20 qudits are traveling in the channel with transmission probability $p_{t_2}.$ In (b) we show a zoom of (a) that highlights the advantage of using the aggregation network approach for a range of values of the transmission probabilities of the 43-qudit QRS code. The blue thick solid line is when $p_{t_1}=p_{t_2}.$

Figure 4

Pictorial representation of the second aggregate networks scenario in which multiplexed photons (dots outside the boxes) are encoding different qudits (boxes) for (a) the 7-qudit QRS code and for (b) the 11-qudit QRS code. The empty dots are the photons traveling in the higher quality channel. Each qudit is encoded with the qubits (dots inside the boxes) linked to the corresponding photon.

Figure 5

In (a) [(b)], the black curve (dashed line) is the transmission probability of the 7 (11)-qudit QRS code required to reach the threshold success probability when all photons are carrying 3 (2) qubits each. The red curve (solid curve) gives the nominal values of the transmission probabilities required to reach the threshold success probability of two channels for the code represented in Fig. 4 and 4.

Figure 6

Logic circuit to create a qutrit using (a) two polarized photons and (b) one multiplexed photon.