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Experimental quantum coding against qubit loss error

Edited by David J. Wineland, National Institute of Standards and Technology, Boulder, CO, and approved May 27, 2008

↵*C.Y.L. and W.B.G. contributed equally to this work. (received for review January 27, 2008)
Abstract
The fundamental unit for quantum computing is the qubit, an isolated, controllable twolevel system. However, for many proposed quantum computer architectures, especially photonic systems, the qubits can be lost or can leak out of the desired twolevel systems, posing a significant obstacle for practical quantum computation. Here, we experimentally demonstrate, both in the quantum circuit model and in the oneway quantum computer model, the smallest nontrivial quantum codes to tackle this problem. In the experiment, we encode singlequbit input states into highly entangled multiparticle code words, and we test their ability to protect encoded quantum information from detected 1qubit loss error. Our results prove inprinciple the feasibility of overcoming the qubit loss error by quantum codes.
 quantum information
 quantum computation
 quantum error correction
 losstolerant quantum code
 multiphoton entaglement
Quantum computers are expected to harness the strange properties of quantum mechanics such as superposition and entanglement for enhanced ways of information processing. However, it has proved extremely difficult to build such devices in practice. Arguably, the most formidable hurdle is the unavoidable decoherence caused by the coupling of the quantum computers to the environment, which destroys the fragile quantum information rapidly. It is thus of crucial importance to find ways to reduce the decoherence and carry out coherent quantum operations in the presence of noise.
Recent experiments have made progress toward this goal by demonstrating quantum error correction (1⇓⇓⇓–5), decoherencefree subspace (6⇓⇓–9) and entanglement purification (10, 11). These experiments were designed to cope with one special kind of decoherence, that is, when qubits become entangled with the environment or undergo unknown rotations in the qubit space. Such errors can be represented as linear combinations of the standard errors: no error, bitflip, phaseflip, or both.
There is, however, another significant source of error—the loss of qubits in quantum computers. The qubit, which is the basic element of standard quantum computation (QC), is supposed to be an isolated twolevel system consisting of a pair of orthonormal quantum states. However, most proposed quantum hardware are in fact multilevel systems, and the states of qubits are defined in a twolevel subspace, which may leak out of the desired qubit space and into a larger Hilbert space (12⇓–14). This problem is common in practical QC with various qubits candidates, such as Josephson junctions (15), neutral atoms in optical lattices (16), and, most notoriously, single photons that can be lost during processing or owing to inefficient photon sources and detectors (17⇓⇓⇓–21). The loss of physical qubits is detrimental to QC because the working of quantum gates, algorithms and error correction codes (see, e.g., refs. 22⇓–24) all hinge on the percept that the quantum system remains in the qubit space.
Here, we demonstrate the smallest meaningful quantum codes to protect quantum information from detected 1qubit loss. Our experiment deals with qubit loss in both the quantum circuit model and oneway quantum computer model (25). We encode singlequbit states into losstolerant codes that are multiparticle entangled states. The performances of the quantum codes are tested by determining the fidelities of the recovered states compared with the ideal original states. Our results verify that the qubit loss error could, in principle, be overcome by quantum codes.
Theoretical Schemes
We now briefly review the quantum codes designed to tackle the problem of qubit loss. A special class of quantum erasure–error correction (QEEC) code was proposed by Grassl et al. (26), where a 4qubit code is sufficient to correct a detected 1qubit loss error. The QEEC code was used by Knill et al. (18) to deal with the photonloss problem for scalable photonic QC. In recent years, extensive efforts have been devoted to devising losstolerant quantum computer architectures (27⇓⇓–30). In particular, in the quantum circuit model Ralph et al. (29) used an incremental parity encoding method to achieve efficient linear optics QC and showed that a losstolerant optical memory was possible with a loss probability <0.18. In the approach known as the oneway QC, Varnava et al. (30) exploited the inherent correlations in cluster states and introduced a scheme for faulttolerantly coping with losses in the oneway QC that can tolerate up to 50% qubit loss.
QEEC Codes
To show the principle of the QEEC codes (26), let us start with a specific example: a 4qubit code that is able to protect a logical qubit from loss of a physical qubit. Here, a logical qubit ψ〉_{l} = a_{0}0〉_{l}+a_{1}1〉_{l} is encoded in the subspace with four physical qubits as This code can also be viewed as a combination of parity and redundant encoding, which is the basic module in Ralph's scheme of losstolerant optical QC (29).
We can consider the effect of a qubit loss as an unintended measurement from which we learn no information. The main feature of the code shown as Eq. 1 is that the detected loss of any one of the physical qubits will not destroy the information of the logical qubit but merely yields a recoverable Pauli error. Suppose, for example, qubit 1 is lost. We first measure qubit 2 in the computational (0/1〉) basis. With its measurement result (q_{2} = 0 or 1), we can obtain a pure quantum state ψ′〉_{l} = a_{0}(0〉_{3}0〉_{4} + 1〉_{3}1〉_{4}) + (−1)^{q2}a_{1}(0〉_{3}0〉_{4} − 1〉_{3}1〉_{4}). With similar reasoning, morequbit loss can also be corrected by increasing the size of losstolerant codes in the form of ψ〉_{l} = a_{0}(^{⊗n} + 1^{⊗n})^{⊗m} + b_{0}(0〉^{⊗n} − 1〉^{⊗n})^{⊗m,} which can be created, e.g., by the incremental encoding scheme proposed in ref. 29.
Demonstration of the QEEC Code
A quantum circuit to implement the encoding of the 4qubit QEEC code is shown in Fig. 1. To implement this, we design a linear optics network (see Fig. 2A). The physical qubits are encoded by the polarizations of photons, with 0 corresponding to the horizontal (H) polarization and 1 to the vertical (V). As shown in refs. 32 and 33, such an encoding method naturally incorporates a lossdetection mechanism and may enable highfidelity linear optical QC. Our experimental setup is illustrated in Fig. 2B. We use spontaneous parametric downconversion (34) to create the primary photonic qubits, which are then coherently manipulated by linear optical elements to implement the coding circuit and read out by using singlephoton detectors (see the legend of Fig. 2B and Methods).
To demonstrate that the quantum codes work for general unknown states, we test three different input states: V〉, +〉, and R〉 = (H〉+ iV〉)/ , which are encoded into the 4qubit QEEC codes, respectively, as (normalizations omitted) where the subscript denotes the spatial mode. Interestingly they show three distinct types of entanglement: V〉_{l} is a product state of two Einstein–Podolsky–Rosen (EPR) pairs (38), +〉_{l} is a 4qubit Greenberger–Horne–Zeilinger (GHZ) state (39), whereas R〉_{l} is locally equivalent to a cluster state (40).
We test the performance of the encoding process by determining fidelities of the encoded 4qubit states. The fidelities are judged by the overlap of the experimentally produced state with the ideal one: F = 〈ψρ_{exp}ψ〉. To do so, we first decompose ρ = ψ〉〈ψ into locally measurable observables that are products of Pauli operators [the detailed constructions are shown in supporting information (SI) SI Text]. For the states V〉_{l}, +〉_{l} and R〉_{l}, we need to take 9, 5, and 9 settings of fourphoton polarization correlation measurements, respectively, each composed of 2^{4} coincidence detections, to determine the probabilities of different outcome combinations. From the data shown in Figs. S1–S3, the fidelities of the QEEC codewords are: F_{V} = 0.620 ± 0.017, F_{+} = 0.566 ± 0.020, F_{R} = 0.554 ± 0.017. The fidelities of the 4qubit GHZ state and cluster state are above the threshold of 0.5, thus they are confirmed to contain genuine fourpartite entanglement (41, 42). The imperfections of the fidelities are caused mainly by highorder emissions of entangled photons and remaining distinguishability of independent photons overlapping on the PBSs. Finally, it should be noted that for the purpose of “benchmarking” quantum computers, more settings of measurements will be needed to infer the average fidelity of the quantum coding (2).
“LossandRecovery” Test
Now we test the codes' ability to protect the logical qubit information from one detected physical qubit loss through a “lossandrecovery” process. Here, we simulate the loss of a photon by detecting the photon without knowing its polarization information, which only tells us that the photon is lost. Experimentally, this is done by placing no polarizer or PBS in front of the detector.
In principle, the QEEC code works when only one and any one of the four physical qubits is lost. In our experiment, we test individually all possible cases where any single one of the four photons is lost. For instance, if we assume photon 2 is lost, the experimental procedure goes as follows. We erase the photon 2, perform a measurement in H/V basis on photon 3 and in +/− basis on photon 4. Depending on different measurement results: H〉_{3}+〉_{4}, V〉_{3}+〉_{4}, H〉_{3}−〉_{4}, V〉_{3}−〉_{4}, correction operations: H_{d}, H_{d}X, H_{d}Z, H_{d}XZ should be applied on photon 5. As a proofofprinciple, here, we apply corrections for every individual outcome of photons 3 and 4, and determine the state fidelity of photon 5 compared with the original input state. For an explicit example, if we fix the polarizers in front of D_{3} and D_{4} in V〉 and +〉 polarization, we should apply H_{d}X to photon 5 and then measure its fidelity. Each measurement is flagged by a fivefold coincidence event where all five detectors fire simultaneously.
Fig. 3 shows the measured recovery fidelities compared with original input state for all possible combinations. For input states V〉, +〉, and R〉, the recovery fidelities averaged over all possible measurement outcomes are found to be 0.832 ± 0.012, 0.764 ± 0.014, and 0.745 ± 0.015, respectively, which establish the effectiveness of the 4qubit QEEC codes. It can be noticed that the encoding and recovery fidelities for the state V〉 is higher than those for +〉 and R〉. We believe this is because in our setup, the coding process for V〉 requires interference of photons only on PBS_{2} and PBS_{3}, whereas for the latter cases, the interferences involve all three PBSs. Also it can be seen from Fig. 3A that for the input state V〉, the recovery fidelities when we simulate photons 4 or 5 are lost are considerably better than those when photons 2 or 3 are lost. We note that this is because in the former case, the coding involves interference between dependent photons c and d (from the same EPR pair) on the PBS_{2}, whereas the latter case requires interference of independent photons b and e on the PBS_{3}.
LossTolerant OneWay QC
Now we consider how to overcome the qubit loss in the oneway QC model (25). In this model, QC is achieved by singlequbit measurements on prepared highly entangled cluster states (40), where the orders and choices of measurements determine the algorithm computed. It is important to note that the loss detection is naturally incorporated in the measurement step in this QC model.
To tackle fault tolerance in this architecture, protocols have been developed by exploiting the builtin properties of the entangled cluster states that provide natural resilience to decoherence (30, 43). In particular, Varnava et al. (30) used the treeshaped graph states and analyzed that a high error threshold of 0.5 exists for qubit loss error. The salient feature of this scheme is illustrated in Fig. 4. Briefly, the tree graph state takes advantage of the perfect correlations in the cluster states and embodies two useful features that enable reduction of the effective qubit loss rate. First, it allows multiple attempts to do the desired measurement on an encoded qubit. Second, it is designed such that any given qubit within the cluster could be removed (see the legend of Fig. 4).
Creation of a 5Qubit Cluster State
We use a 5qubit cluster state for demonstration of Varnava et al.'s scheme (30). As shown in Fig. 4D, it can be thought of as being reduced from a 7qubit cluster after two adjacent X measurements. Alternatively, the cluster can be grown directly, as we do in our experiment.
Starting from an EPR pair and three single photons, we use the linear optical network shown in Fig. 2C and the experimental setup shown in Fig. 2B to create the fivephoton cluster state (see Methods). When each of the five output modes registers a photon, the five photons are in the highly entangled cluster state where the subscript labels the photon's spatial mode (see Fig. 2C). The state φ_{5}〉 is local unitarily equivalent to the 5qubit linear cluster state shown in Fig. 4D under the H_{d} transformations on photon 1, 3, and 5. This is, to our best knowledge, the longest onedimensional cluster state realized so far.
To determine the fidelity of the fivephoton cluster state, we decompose the projector φ_{5}〉〈φ_{5} into 15 local measurable observables (see SI Text), each takes 2^{5} fivefold coincidence measurements. The experimental results are shown in Fig. 5, yielding F_{c} = 〈φ_{5}ρ_{exp}φ_{5}〉 = 0.564 ± 0.015. Because the fidelity of the cluster state exceeds 0.5, the presence of true fivepartite entanglement of our cluster state is also confirmed (42).
OneWay QC in the Presence of Loss
With the cluster state prepared, now we demonstrate its losstolerant feature by simulation of a quantum circuit in the presence of loss. First, let us briefly review how QC is done by measurements in the oneway model. The measurement is chosen in basis B_{j}(α) = {+α〉_{j},−α〉_{j}}, where ±α〉_{j} = (0〉_{j} ± e^{iα}1〉_{j})/ , which realizes the singlequbit rotation R_{z}(α) = exp(−iασ_{z}/2), followed by a Hadamard operation on the encoded qubit in the cluster. We define the outcome s_{j} = 0 if the measurement on the physical qubit j yields +α〉_{j}, and s_{j} = 1 if it is −α〉_{j}. When s_{j} = 0, the computation proceeds without error, whereas when s_{j} = 1, a known Pauli error is introduced that has to be compensated for (see refs. 25, 45, and 46 for more details).
The 2qubit cluster shown in Fig. 6A can implement a simple circuit, rotating an encoded input qubit +〉 to an output state: ψ_{out}〉 = X^{sa}H_{d}R_{z}(α)+〉. With this 2qubit cluster, however, one can only have a oneshot A measurement on the qubit a, that is, if this measurement fails, then the whole computation fails. As a comparison, the 5qubit cluster state we prepared can be used to realize the circuit in a more robust fashion. It provides two alternative and equivalent attempts to do the A measurement as depicted in Fig. 6 B and C. And if any one of the qubits (2, 3, 4, or 5) for the A measurement is lost, we can always find a suitable indirect Z measurement to remove the damaged qubit. For example, if the A measurement on qubit 2 fails, we can try to remove it from the cluster by an X measurement on qubit 3, and then proceed to make the A measurement on qubit 4. It can be checked that as long as no more than one physical qubit is lost, the computation will be successful.
Now we demonstrate this experimentally. To verify the scheme depicted in Fig. 6B, we erase photon 2, which makes the remaining cluster in a mixed state. Then we make an X measurement on photon 3—this should effectively remove the loss error from the cluster, leaving it as a smaller but pure quantum cluster state. Next the redundant photon 5 is measured in the Z basis. Note that because the actual state φ_{5}〉 is unitarily equivalent to the 5qubit linear cluster state shown in Fig. 4D under the H_{d} transformations on photons 1, 3, and 5, the X (Z) measurement on photon 3 (or 5) is performed on a rotated laboratory basis of H/V (+/−). Depending on the measurement outcomes of photon 3 and 5 (H〉_{3}+〉_{5}, V〉_{3}+〉_{5}, H〉_{3}−〉_{5}, V〉_{3}−〉_{5}), Pauli corrections (I, X, Z, XZ) are applied to photon 4. After that, measurements in the basis B_{j}(α) are applied on photon 4 to implement the rotation. We choose α to be three different values, 0, −π/2, −π/3, so that theoretically the output states will be +〉, R〉, and S〉 = (0〉 + e^{iπ/3}1〉)/ , respectively. Then we readout the polarizations of photon 1 and determine its fidelities compared with the ideal states. The scheme of Fig. 6C is also tested in a similar manner.
In Fig. 6 D and E, we show the experimental results of oneway QC in the presence of 1qubit loss. In the case of photon 2 lost, we find a fidelity of 0.738 ± 0.029, 0.750 ± 0.030, and 0.765 ± 0.028 for target output state +〉, R〉, and S〉, respectively. In the case of photon 4 lost, the fidelity is 0.865 ± 0.021, 0.792 ± 0.029, and 0.767 ± 0.030 for target output state +〉, R〉, and S〉. Here, the difference of the fidelity performance is caused by similar reasons as in the QEEC codes. For instance, the case for target output state +〉 corresponds to Fig. 3A, where the input state is in the state of V〉. These results conclusively demonstrate the underlying principle of losstolerant oneway QC.
Discussion
As in all current linear optical QC experiments, the multiphoton code states here are created probabilistically and conditioned on that there is one and only one photon out of each output, a technique called postselection (35). Although this does not prevent an inprinciple verification of the losstolerant quantum codes, we note that eventual scalable implementations will need significant improvements such as ondemand entangled photon sources and highefficiency singlephoton detectors.
In summary, we have demonstrated, in both the quantum circuit model and in the oneway model, the smallest meaningful quantum codes to protect quantum information from qubitloss error. These quantum codes are the key modules for the losstolerant quantum computer architectures (29, 30) and can, in principle, be extended to larger numbers of qubits. Our results verify that it is possible to overcome the qubitloss error, a major decoherence mechanism common in many physical systems, and thus constitute a necessary step toward scalable quantum information processing, especially for photonbased QC. The losstolerant quantum codes can be further concatenated with standard quantum errorcorrection codes (22⇓–24) or decoherencefree space (6⇓⇓–9) to correct multiple errors and may become a useful part of future implementations of quantum algorithms (45⇓⇓⇓⇓–50).
Methods
Experimental Implementation.
We first prepare two entangled photon pairs in spatial modes a–b and c–d with an average coincidence count of ≈6.2 × 10^{4}s^{−1}, and a pseudosingle photon source in mode e, which has a very small probability (p) of containing a single photon for each pulse. The p value is carefully chosen to be ≈0.07 experimentally to get an optimal visibility (36). Photon a serves as a trigger, indicating that the pair photon b is under way; Photon b is prepared in different input states; Photons c, d, nd e are initialized in the +45° linear polarization state +〉 = (H〉 + V〉)/ . We superpose photons b, c, d, and e on the three PBSs step by step. For alignment on the PBS_{1}, for instance, we first initialize the two photons b and c at state +〉 before they enter into the PBS_{1}, and, by making fine adjustment of delay Δd_{1}, we are able to observe the twophoton Hong–Ou–Mandel type dip (37) after the PBS_{1} by performing polarization measurements in both output modes in the ±〉 = (H〉 ± V〉)/ basis. Similarly, optimal superpositions of the photons on the other two PBSs are also achieved. The optimal interference occurs at the regime of zero delay, where our experimental measurements are performed. Note that some PBSs may have undesired phase shifts, that is, when two input photons that were prepared in the states of +〉 are superposed on the PBSs, the output photons are not in the expected state of (H〉H〉 + V〉V〉)/ but in the state of (H〉H〉 + e^{iθ}V〉V〉)/ with a θ phase shift. This can be overcome by a compensator (see Fig. 2B) that can introduce a phase delay between H〉 and V〉.
FivePhoton Cluster State Preparation.
An EPR photon pair as prepared in the spatial mode a–b in the state (H〉_{a}H〉_{b} + V〉_{a}V〉_{b})/ with a visibility of ≈92% in the +/− basis. We note that this nonperfect visibility may add additional noise into the fivephoton cluster state compared with the fourphoton states by using the scheme as shown in Fig. 2A. To get a better fidelity for the cluster state, we lower the pump power and obtain an average coincidence of ≈5.0 × 10^{4}s^{−1} in modes a–b and c–d, such that the rate of double pair emission of entangled photons is diminished. Meanwhile, the p value for the pseudosingle photon source is also reduced to ≈0.06 accordingly.
Acknowledgments
We thank T. Rudolph, D. Browne, O. Gühne, T. Ralph, and M. Zukowski for helpful discussions. This work was supported by the National Natural Science Foundation of China, the Chinese Academy of Sciences, and National Fundamental Research Program Grant 2006CB921900. This work was also supported by the Alexander von Humboldt Foundation and a Marie Curie Excellence Grant of the European Union.
Footnotes
 ↵^{‡}To whom correspondence may be addressed. Email: cylu{at}mail.ustc.edu.cn or jianwei.pan{at}physi.uniheidelberg.de

Author contributions: C.Y.L., W.B.G., and J.W.P. designed research; C.Y.L., W.B.G., J.Z., X.Q.Z., T.Y., and J.W.P. performed research; C.Y.L. and W.B.G. contributed new reagents/analytic tools; C.Y.L. and W.B.G. analyzed data; and C.Y.L. and J.W.P. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0800740105/DCSupplemental.
 Received January 27, 2008.
 © 2008 by The National Academy of Sciences of the USA
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