Quantum link bootstrapping using a RuleSet-based communication protocol

Establishing end-to-end quantum connections requires quantified link characteristics, and operations need to coordinate decision making between nodes across a network. We introduce the RuleSet-based communication protocol for supporting quantum operations over distant nodes to minimize classical packet transmissions for guaranteeing synchronicity. RuleSets are distributed to nodes along a path at connection setup time, and hold lists of operations that need to be performed in real time. We simulate the RuleSet-based quantum link bootstrapping protocol, which consists of recurrent purifications and link-level tomography, to quantify the quantum link fidelity and its throughput. Our Markov-chain Monte Carlo simulation includes various error sources, such as the memory error, gate error, and channel error, modeled on currently available hardware. We found that when two quantum nodes, each with 100 memory qubits capable of emitting photons ideally to the optical fiber, are physically connected with a 10-km MeetInTheMiddle link, the Recurrent Single selection–Single error purification (RSs-Sp) protocol is capable of bringing up the fidelity from an average input $F_r=0.675$ to around $F_r=0.865$ with a generation rate of 1106 Bell pairs per second, as determined by simulated tomography. The system gets noisier with longer channels, in which case errors may develop faster than the purification gain. In such a situation, a stronger purification method, such as the double selection-based purification, shows an advantage for improving the fidelity. The knowledge acquired from bootstrapping can later be distributed to nodes within the same network, and used for other purposes such as route selection.

Figure 1

The Quantum Internet. Different networks are operated under different technologies. That includes the physical link architecture, data link protocol, routing protocol, and any other necessary technology layered as in the OSI reference model [5]. Connecting those networks forms the Quantum Internet.

Figure 2

Examples of entanglement swapping breakdowns without multihop purification (each link may require purification). (a) Perform entanglement swapping from left to right. This keeps the information flow consistent, but requires many steps. (b) Node B and Node D perform entanglement swapping first, and Node C connects the end nodes afterwards. This has one less step than the previous one. (c) All middle nodes perform entanglement swapping simultaneously. This requires only one step, but it may work poorly with low quality links. (d) Ad hoc. Entanglement swapping is performed as soon as resources are available. The number of steps varies from execution to execution.

Figure 3

The MeetInTheMiddle model. In the real world, the position of the BSA could be limited for various reasons, such as geographical constraints.

Figure 4

The SenderReceiver model.

Figure 5

Node A generating and distributing RuleSets. In this example, there are five nodes in total, which requires a total of five distinct RuleSets.

Figure 6

Bias in knowledge regarding the state of resources. Node C receives the success or failure acknowledgment later than Node A and B.

Figure 7

An example structure of a RuleSet. The resource allocated in the second Rule has been already passed through the first.

Figure 8

BSA node architecture. Classical communications, shown as dashed arrows, will be sent through the classical channel via NICs.

Figure 9

Impact of the number of measurements on the reconstructed fidelity error over the MeetInTheMiddle link. $F_r$ is the reconstructed fidelity and $F_a$ is the actual fidelity.

Figure 10

Change in the accuracy of the reconstructed fidelity relative to the actual fidelity.

Figure 11

Single selection–Single error (X) purification (Ss-Sp). Consumes a single Bell pair $|{\mathrm{\Psi }}_{C,D}^{+}\rangle$ to detect the X error on $|{\mathrm{\Psi }}_{A,B}^{+}\rangle$.

Figure 12

Single selection–Double error (XZ) purification (Ss-Dp). Consumes two Bell pairs, $|{\mathrm{\Psi }}_{C,D}^{+}\rangle$ and $|{\mathrm{\Psi }}_{E,F}^{+}\rangle$ to detect X error and Z error accordingly.

Figure 13

Double selection–Single error (X) purification (Ds-Sp). Consumes a single Bell pair $|{\mathrm{\Psi }}_{C,D}^{+}\rangle$ to detect the X error on $|{\mathrm{\Psi }}_{A,B}^{+}\rangle$, and another Bell pair $|{\mathrm{\Psi }}_{E,F}^{+}\rangle$ to avoid the Z error propagation from $|{\mathrm{\Psi }}_{C,D}^{+}\rangle$ to $|{\mathrm{\Psi }}_{A,B}^{+}\rangle$.

Figure 14

Double selection–Double error (XZ) purification (Ds-Dp). Consumes two Bell pairs, $|{\mathrm{\Psi }}_{C,D}^{+}\rangle$ and $|{\mathrm{\Psi }}_{G,H}^{+}\rangle$, to detect X error and Z error on $|{\mathrm{\Psi }}_{A,B}^{+}\rangle$. Each consumed Bell pair will also be verified through purification using $|{\mathrm{\Psi }}_{E,F}^{+}\rangle$ and $|{\mathrm{\Psi }}_{I,J}^{+}\rangle$ accordingly.

Figure 15

Impact of the MeetInTheMiddle channel length on the reconstructed fidelity with 7000 measurement outcomes w/o, with purification. The steep drop in the fidelity is due to the high error rate of 1% each X, Y, and Z errors per km in the quantum channel chosen for this simulation.

Figure 16

Impact of the channel distance on the channel throughput with and without performing purification. The throughput has been calculated as $N_M/T$, where $T$ is the tomography process time to complete $N_M$ measurements. (a) Throughput over the MeetInTheMiddle link. (b) Throughput over the SenderReceiver link.

Figure 17

Recurrence purification protocol based on single selection X purification and Z purification. Each round of purification is a Rule in a RuleSet.

Figure 18

Reconstructed fidelity with actual fidelity and actual error rates, and estimated throughput of protocols over $L=10$ km MeetInTheMiddle link. ”Other errors” include memory excitation or relaxation error, and completely mixed error due to photon detector dark counts. (a) Simulation result of the RSs-Sp protocol. (b) Simulation result of the RSs-Dp protocol. (c) Simulation result of the RDs-Sp protocol. (d) Simulation result of the RDs-Dp protocol. (e) Protocol throughputs over the MeetInTheMiddle link. (f) Protocol throughputs over the SenderReceiver link.

Figure 19

Impact of the memory buffer size to the maximum fidelity purified through the RSs-Sp protocol over a 10-km MeetInTheMiddle link.

Figure 20

Reconstructed fidelity with actual fidelity and actual error rates, and estimated throughput of protocols over $L=20$ km MeetInTheMiddle link. ”Other errors” include memory excitation or relaxation error, and completely mixed error due to photon detector dark counts. (a) Simulation result of the RSs-Sp protocol. (b) Simulation result of the RSs-Dp protocol. (c) Simulation result of the RDs-Sp protocol. (d) Simulation result of the RDs-Dp protocol. (e) Protocol throughputs over the MeetInTheMiddle link. (f) Protocol throughputs over the SenderReceiver link.

Figure 21

Switching RDs-Sp to RSs-Sp over $L=20$ km. (a) Pattern 1 consists of one round RDs-Sp and four rounds of RSs-Sp. Purification methodology switches at $N_p=1$. (b) Pattern 2 consists of two rounds RDs-Sp and three rounds of RSs-Sp. Purification methodology switches at $N_p=2$.