Interoperability in encoded quantum repeater networks

The future of quantum repeater networking will require interoperability between various error-correcting codes. A few specific code conversions and even a generalized method are known, however, no detailed analysis of these techniques in the context of quantum networking has been performed. In this paper we analyze a generalized procedure to create Bell pairs encoded heterogeneously between two separate codes used often in error-corrected quantum repeater network designs. We begin with a physical Bell pair and then encode each qubit in a different error-correcting code, using entanglement purification to increase the fidelity. We investigate three separate protocols for preparing the purified encoded Bell pair. We calculate the error probability of those schemes between the Steane [[7,1,3]] code, a distance-3 surface code, and single physical qubits by Monte Carlo simulation under a standard Pauli error model and estimate the resource efficiency of the procedures. A local gate error rate of ${10}^{-3}$ allows us to create high-fidelity logical Bell pairs between any of our chosen codes. We find that a postselected model, where any detected parity flips in code stabilizers result in a restart of the protocol, performs the best.

Figure 1

Case for quantum repeaters building heterogeneously encoded Bell pairs. Each cloud represents a quantum autonomous system that is based on an error-correcting code or entanglement swapping and purification. Colored links are connections using those codes. Boxes are quantum repeaters building heterogeneously encoded Bell pairs. Cylinders are quantum repeaters, each of which supports only a single code. All links from a homogeneous repeater (cylinder) are the same type (color) since only quantum repeaters building heterogeneously encoded Bell pairs (boxes) can interoperate between different codes.

Figure 2

Heterogeneously encoded Bell pairs can be used to bridge quantum networks using different error-correcting mechanisms. MAN stands for the metropolitan area network. A blue circle denotes a physical qubit. A set of blue lines indicates qubits that comprise an encoded qubit. Each thin red line describes an entanglement between physical qubits. Each thick red loop outlines entanglement between encoded qubits. The half of the Bell pair encoded in the surface code can be sent to the neighboring quantum repeater by the method of Fowler et al. [29]. The other half of the Bell pair, encoded in the Steane [[7,1,3]] code, undergoes entanglement swapping with a Steane [[7,1,3]]—Steane [[7,1,3]] encoded Bell pair established via the method of Jiang et al. [28]. Therefore, this central quantum repeater can create entanglement between the two quantum repeaters in different types of networks. In the green dashed rectangle is the procedure for encoding a Bell pair heterogeneously. A qubit of a Bell pair is encoded in the Steane [[7,1,3]] code on the left side of the figure, adding six qubits. The other qubit of the Bell pair is encoded onto the surface code of distance 3, adding 24 qubits on the right side of the figure. Multiple copies are prepared, entangled, and purified. Eventually, a heterogeneously encoded logical Bell pair is achieved with high enough fidelity to enable coupling of the two networks. Ancilla qubits for syndrome measurement are depicted in the surface code, while those are not depicted in the Steane code.

Figure 3

Architecture of a scalable quantum repeater supporting routing. Each red line describes a photonic connection. The crossbar switch switches interconnections of network interface cards (NICs). Each network interface card is connected to another quantum repeater or computer at the open ends of the red lines. By using a crossbar switch, interconnections do not interrupt each other, so this architecture is scalable. The dashed green box in Fig. 2 corresponds to creating heterogeneous Bell pairs between, e.g., NIC1 and NIC2. Step 1 in Fig. 2 is the creation of physical Bell pairs via the optical crossbar switch. Step 2 takes place via local gates within the respective NICs.

Figure 4

Circuit to encode an arbitrary state to the Steane [[7,1,3]] code. Here $|\psi \rangle$ is the state to be encoded. This circuit is not fault tolerant. The KQ of this circuit is 42 because some gates can be performed simultaneously.

Figure 5

Circuit to encode an arbitrary state $|\psi \rangle$ to a distance-3 surface code [55]. This circuit is not fault tolerant. The KQ of this circuit is 250 if some gates are performed simultaneously.

Figure 6

Circuit for entanglement purification [2]. The two measured values are compared. If they disagree, the output qubits are discarded. If they agree, the output qubits are treated as a new Bell pair. At this point, the $X$ error rate of the output Bell pair is suppressed from the input Bell pairs. The Hadamard gates exchange the $X$ and $Z$ axes, so the following round of purification suppresses the $Z$ error rate. As a result, entanglement purification consumes two Bell pairs and generates a Bell pair of higher fidelity stochastically.

Figure 7

Overview of the scheme that purifies physical Bell pairs to generate an encoded Bell pair of high fidelity. First, entanglement purification is conducted between physical Bell pairs an arbitrary number of times. Second, each qubit of the purified physical Bell pair is encoded to a heterogeneous error-correcting code.

Figure 8

Overview of the scheme that purifies encoded Bell pairs to achieve an encoded Bell pair of high fidelity. In this method, first, raw physical Bell pairs are encoded into our heterogeneous error-correcting code, Second, those heterogeneously encoded Bell pairs are purified directly at the logical level.

Figure 9

Overview of the scheme that purifies encoded Bell pairs to achieve an encoded Bell pair of high fidelity with strict postselection. First, raw physical Bell pairs are encoded to a heterogeneous error-correcting code, the same as purification after encoding. Second, at measurement in purification, eigenvalues of each stabilizer are checked classically. If any eigenvalue of $-1$ is measured, the output Bell pair is discarded (in a manner similar to if the overlying purification protocol failed).

Figure 10

Results of a baseline simulation of creation of a Steane [[7,1,3]]–Steane [[7,1,3]] homogeneous Bell pair, showing residual logical error rate versus physical Bell pairs consumed. The three schemes plus the baseline case of purification of physical Bell pairs are each represented by a line. Each point along a line corresponds to the number of rounds of purification. The leftmost point represents no purification, the second point is one round of purification, and the rightmost point represents four rounds of purification (a)–(c). Improving values of local gate error rate. (d) The three cases with residual error rate of ${10}^{-3}$ or less.

Figure 11

Results of a baseline simulation of creation of a surface code distance-3–surface code distance-3 homogeneous Bell pair, showing the residual logical error rate versus physical Bell pairs consumed. Other conditions and definitions are as in Fig. 10.

Figure 12

Results of simulation of creation of a Steane [[7,1,3]]–surface code distance-3 heterogeneous Bell pair, showing residual logical error rate versus physical Bell pairs consumed. Other conditions and definitions are as in Fig. 10.