Error-correcting entanglement swapping using a practical logical photon encoding

Several emerging quantum technologies, including quantum networks, and modular and fusion-based quantum computing, rely crucially on the ability to perform photonic Bell state measurements. Therefore, photon losses and the 50% success probablity upper bound of Bell state measurements pose a critical limitation to photonic quantum technologies. Here, we develop protocols that overcome these two key challenges through logical encoding of photonic qubits. Our approach uses a tree graph state logical encoding, which can be produced deterministically with a few quantum emitters, and achieves near-deterministic logical photonic Bell state measurements while also protecting against errors including photon losses, with a record loss-tolerance threshold.

Figure 1

Two-way (top) and one-way (bottom) QR protocol using deterministic tree graph state generation [42, 50, 52] with a few matter qubits and the logical BSM protocols studied in this paper (see Appendix pp1 for the deterministic generation procedure).

Figure 2

(a) Tree graph state and notations. ${\mathcal{N}}_v$ and ${\mathcal{C}}_v$ denote the set of neighbor qubits and child qubits of vertex $v$. Indirect $Z$ measurements are also illustrated. (b), (c) Logical BSM at the logical level (left panel) and at the physical level (right panel) for the static (b) and the dynamic (c) protocols. (SPM: single-photon measurements.)

Figure 3

(a) Success probability of a BSM as a function of the single-photon detection efficiency $\eta$ for the static and the dynamic protocols for a tree of branching vector $\vec{b}=(15,15,2)$. Dashed curves correspond to ${\mathrm{P}}_{\mathrm{BSM}}={\eta }^2$ (red), $\eta =1/2$ (blue), and $\eta =\sqrt{2/3}$ (green). (b) Logical BSM error as a function of the single-qubit depolarization error rate $\mathcal{E}$ for the static and dynamic protocols. (c) Performance of the static (top) and dynamic (bottom) protocols as a function of the number of photons in the tree, for $\eta =95\%$ and $\mathcal{E}={10}^{-5}$. Red point: smallest tree which is both error correcting and loss tolerant. Green point: smallest tree which is loss tolerant. Points circled in black (uncircled): depth-3 (depth-2) trees. In all these figures, pink regions: no advantage over a two-photon BSM.

Figure 4

Protocol for the deterministic generation of a logical Bell pair using matter qubits.

Figure 5

(a) Tree encoding of a qubit. Left panel: The target qubit is attached to a tree via a $CZ$ gate and then $X$ measurements are performed on the target qubit and the root qubit. Middle panel: The resulting graph and (right panel) graphical notation used for a tree-encoded logical qubit. (b) Optical setup that actively switches between a two-photon BSM, which measures $Z{Z}^{\prime }$ unambiguously and $X{X}^{\prime }$ partially, and two single-photon $X$ or $Z$ measurements. The two photons arrive in inputs 1 and 2. BS: beam splitter; PBS: polarization BS; HWP: half wave plate (rotated by ${45}^{\circ }$); Det: single-photon detectors.