A proof-of-principle experiment of eliminating photon-loss errors in cluster states

Quantum computation can be efficiently achieved by adopting the cluster-state model in principle, that is, first preparing multi-qubit cluster states and then performing single-qubit measurements. One problem encountered in this model is that qubits are easily lost, leaving the remaining qubits in a mixed state. Here, we experimentally demonstrate a simple way to eliminate such errors in one-dimensional cluster states using the polarizing beam splitter (PBS) gate, and we also perform the indirect-Z measurement to overcome qubit loss. The methods of quantum state tomography and entanglement witness are exploited to verify the performance of the schemes, and the experimental results indicate good agreement with the theoretical predictions.

Varnava et al [15] to fault-tolerantly cope with qubit loss. In section 3, we demonstrate a proof-of-principle experiment to demonstrate the key operations of these methods.

The PBS gate operation and the indirect-Z measurement
First, we review the PBS gate [14] in the following paragraph. This operation, the same as parity check [17,18], is implemented by mixing two photons at a PBS and accepting the case when each of the detectors receives one and only one photon. As shown in figure 1(a), the basic resource is two-photon Bell states which are relatively easy to obtain. This gate is conditioned on detecting one and only one photon in each output, a technique called post-selection [19], so it is naturally non-deterministic with a success probability of 50% and it has destructive characters. However, it can be useful for scalable quantum computation when exploiting 'Divide and Conquer' [14] or 'percolation' schemes [20]. Furthermore, the PBS gate can be easily converted to the well-known Type-I and Type-II fusion gates which are important for efficient one-way quantum computation [6,21].
In this paragraph, we present a description of the PBS gate similar to Benjamin's [13]. The initial state is prepared as where H, V denotes the horizontal and vertical polarizations, |X represents the state of the entire cluster state minus the two marked nodes as shown in figure 1(b). The operator σ L z ≡ σ Z 1L σ Z 2L · · · σ Z j L is the product of σ Z operators applied to each of those qubits 1 · · · j connected to one of the qubits which will go through the PBS. The σ Z R is defined the product of σ Z operators applied to each of those qubits 1 · · · j connected to the other. For a PBS, the photons from the two input modes will go to different sides if and only if both photons have the same polarization, either H H or V V . If we detect one photon in each side, the PBS will be 4 described by the projector |H H H H | + |V V V V |. After the PBS and the post-selection, the initial state will become Then, we make a Hardmard transformation of one of the qubits. The final state will be where |± = 1 √ 2 (|H ± |V ). In this state, a 'node qubit' connected to a 'leaf qubit' inherits all the bonds of the previous qubits as shown in figure 1(b). Note that, here, we consider the PBS gate as a non-destructive gate. The PBS gate above is also described in the language of stabilizer operators of cluster states by Bodiya and Duan [14].
An interesting character of the PBS gate is that it can be used to eliminate photon-loss errors naturally. As shown in the formula (3), when two nodes join together, the remaining node will inherit all the previous bonds. If the two nodes bond to a common qubit and the common qubit does not bond to other qubits, the gate will result in the state where |+ c is the common qubit, |X − 1 is the state |X minus the common qubit and so the common qubit is not connected to the cluster state any more (see figure 2(a)). This was first introduced by Benjamin [13] using the EO gate.
Next, we review the indirect-Z measurement in the following paragraph. An n-qubit cluster state can be expressed in terms of a set of n stabilizers, one for each vertex in the graph. The stabilizers have the form where i denotes a qubit, v(i) denotes the neighborhood of i, and X i and Z j denote the usual Pauli bit-flip and phase-flip operators, respectively, acting on qubits i and j. As shown in figure 2(b), utilizing the property that cluster states are eigenstates of the stabilizers, Varnava et al [15] introduce the indirect-Z measurement to overcome qubit loss. More details about indirect-Z measurements can be found in [15].

Preparation of the state
In this section, we demonstrate a proof-of-principle experiment of how the methods in section 2 can be used. First, we prepare a four-photon cluster state with which we demonstrate the indirect-Z method. Applying a PBS gate to construct a four-qubit cluster state with two pairs of Bell states is the most simple but fundamental case as depicted in figure 1(b). A schematic drawing of our experimental setup is shown in figure 3. Two pairs of entangled photons in mode 1-2 and mode 3-4 are produced as the primary source by passing an ultraviolet laser pulse through two β-barium borate (BBO) crystals. The UV laser with a central wavelength of 394 nm has a pulse duration of 120 fs, a repetition rate of 76 MHz, and an average pump   figure 1(b). The sources are two pairs of maximally entangled photons produced by type-II spontaneous parametric down-conversion. The HWP before the PBS1 is used to transform the photon from H/V basis to +/− basis. Quarterwave plates (QWPs), HWPs and polarizers before detectors are used for necessary polarization analysis. In the experiment, we managed to obtain an average two-fold coincidence of 25000 s −1 .
6 power of 540 mW. After proper birefringence compensation and local unitary transformation with HWPs and nonlinear crystals, the initial state is presented as We then superpose the photon-2 and photon-3 at the PBS1. Their path lengths are adjusted so that they arrive simultaneously. To achieve good spatial and temporal overlaps, the outputs are spectrally filtered ( U = 3.2 nm) and monitored by fiber-coupled single-photon detectors. The filtering process stretches the coherence time to about 648 fs, substantially larger than the pump pulse duration [22]. This effectively erases any possibility to distinguish the two photons and subsequently leads to interference. After the post-selection, the final state will be which is the state in figure 1(b). It is equivalent to a four-photon Greenberger-Horne-Zeilinger (GHZ) state up to single-qubit unitary transformations. We exploit the method of entanglement witness to prove its multipartite entanglement [23]. Entanglement witness is an observable which has a positive expectation value on all biseparable states, thus a negative expectation value proves the presence of genuine multipartite entanglement. For |C 12 3 4 , we use witness [24,25] |C 12 3 4 C 12 3 4 | is decomposed into locally measurable observables as [26,27] |C 12 3 4 C 12 3 4 | = 1 where X, Y and Z are short notations for the Pauli matrices. To implement this witness, nine measurement settings are required. Figure 4(a) depicts the measurement results, yielding Tr(W C4 ρ exp ) = −0.24 ± 0.01, which is negative by 24 standard derivations and thus proves the presence of genuine four-partite entanglement. From the expectation values of the witness, we can directly calculate the obtained fidelity as C 12 3 4 | ρ exp |C 12 3 4 = 0.74 ± 0.01.
After photon-1 is projected to |H , the state remains a three-qubit linear cluster state with which we demonstrate the PBS gate method against qubit-loss errors. We decompose its witness as From figure 5(b), the expectation value of W C3 results to be Tr(W C3 ρ exp ) = −0.31 ± 0.01 and the fidelity of the state is 0.81 ± 0.01.
The expectation values of five measurement settings of the witness in equation (10), Each of them is expected to be +1 or −1, and the error bars represent one standard deviation, deduced from propagated Poissonian statistics of the raw detection events.

Eliminating photon-loss errors in cluster states
After the preparation of the basic state, we next demonstrate experimentally the key operations of eliminating photon-loss errors. Because the post-selection technique is exploited in the preparation of multi-photon states in our experiment, an equivalence is assumed here, that is, we simulate the loss of a photon by detecting the photon without knowing its polarization information, which tells us that the photon is lost. Experimentally, this is done by placing no polarizer or PBS in front of the detector, and it does not prevent an in-principle verification of the schemes to eliminate photon-loss errors. Suppose photon 3 of the cluster state |C 2 3 4 is lost, which is experimentally performed by placing no polarizer in front of the detector 3, the damaged state remains to be a mixed state As shown in figure 5(a), the state tomography is performed to depict the density matrix of |D 2 4 . In the output modes, HWPs, QWPs and freely-rotatable polarizers are used to make projections onto the polarizations {H, V, +, R}. We perform each of these 16 correlation measurements for 105 s using all combinations of {H, V, +, R}. A maximum of 664 two-fold coincidence counts in 105 s are measured in the case of the setting ++. Instead of a direct linear combination of measurement results, which can lead to unphysical density matrices, we use a maximum-likelihood reconstruction technique [28]- [30]. From figure 5(a), we can see that the state is very consistent with the highly mixed state.
If the lost-qubit is connected to a leaf qubit in the preparation step, we will be able to delete it with the method of indirect-Z measurement. Concretely, in the experiment, if qubit 3 has been connected to qubit 1 before it is lost, just as depicted in figure 1(b), we can eliminate qubit 3 under an X-measurement of qubit 1. The experimental result is shown in figure 5  agree that photons 2 and 4 have no entanglement correlation. These results conclusively give the proof that qubit 3 has been eliminated and demonstrate the underlying principle of indirect-Z measurements against qubit-loss errors.
In the next step, we exploit the method of the PBS gate to remove the photon-loss errors. Now, we start from the cluster state |C 2 3 4 . Experimentally, as depicted in figure 3, photons 2 and 4 are passed through PBS2 to remove qubit 3 from the cluster state, leaving a smaller but pure state. After the post-selection, the final state is expected to be an entangled state To characterize the quality of the experimental process, we depict here the output density matrix of the final state in figure 5(c), where the maximum-likelihood reconstruction technique is also exploited. The agreement with the theory can be quantified from the fidelity F = R 2 4 |ρ exp |R 2 4 = 0.72 ± 0.03. Again we have performed the measurements of the polarization correlation between photon 2 and photon 4. The experimental results can be found in figure 6(b), from which we can see that the final state is an entanglement state. This demonstrates the underlying principle of the method of the PBS gate against qubit-loss errors.
In conclusion, we experimentally demonstrate the methods of eliminating photon-loss errors in cluster states by using the PBS gate and performing indirect-Z measurements. The multi-photon states in this experiment are created probabilistically and a post-selection technique is exploited here. The photon-loss error is simulated by detecting it without knowing its polarization information. Although it does not defeat the proof-of-principle experiment here, on-demand entangled photon sources will be required for future application. The proposed schemes can be extended to other fields if we replace the PBS gate with general parity measurements. In addition, it remains an interesting question as to how the qubit-loss errors can be eliminated most efficiently, and our results can be seen as contributing to this effort.