Experimental Realization of One-Way Quantum Computing with Two-Photon Four-Qubit Cluster States
We report an experimental realization of one-way quantum computing on a two-photon four-qubit cluster state. This is accomplished by developing a two-photon cluster state source entangled both in polarization and spatial modes. With this special source, we implemented a highly efficient Grover’s search algorithm and high-fidelity two qubits quantum gates. Our experiment demonstrates that such cluster states could serve as an ideal source and a building block for rapid and precise optical quantum computation.
pacs:03.67.Lx, 03.67.Mn, 42.50.Dv
Highly entangled multipartite states, so-called cluster states, have recently raised enormous interest in quantum information processing (QIP). These sorts of states are crucial as a fundamental resource and a building block aimed at one-way universal quantum computing onewayQC (). They are also essential elements for various quantum error correction codes and quantum communication protocols ECC-qcp (). Moreover, the entanglements are shown to be robust against decoherence robust (), and persistent against loss of qubits onewayQC (), and thus are exceptionally well suited for quantum computing and many tasks onewayQC (); ECC-qcp (). Considerable efforts have been made toward generating and characterizing cluster state in linear optics browne-duan (); Weinfurter-cluster2005 (); Zeilinger-clusterbell2005 (); zeilingernature2005 (); zeilingernature2007 (); Lu-Pan2007cluster (). Recently the principal feasibility of a one-way quantum computing model has been experimentally demonstrated through 4-photon cluster states successfully zeilingernature2005 (); zeilingernature2007 (); onewayDeutsch ().
So far, preparing photonic cluster state still suffers from several serious limitations. Due to the probabilistic nature and Poissonian distribution of the parametric down-conversion process, the generation rate of 4-photon cluster states is quite low Zeilinger-clusterbell2005 (); Weinfurter-cluster2005 (); zeilingernature2005 (); zeilingernature2007 (), and largely restricts speed of computing. Besides, the quality and fidelity of prepared cluster states are relatively low Weinfurter-cluster2005 (); zeilingernature2005 (); zeilingernature2007 (), which are difficult to be improved substantially. These disadvantages consequently impose great challenges of advancement even for few-qubit quantum computing.
Fortunately, motivated by the progress that an important type of states termed hyper-entangled states have been experimentally generated AVN2005-martini (); AVN2005-pan (); Kiwat-hyper2005 (); Weinfurter-bell2006 (), we have the possibility to produce a new type of cluster state (2-photon 4-qubit cluster state) with nearly perfect fidelity and high generation rate. The hyper-entangled states have been used to test “All-Versus-Nothing” (AVN) quantum nonlocality AVN-theory (); AVN2005-martini (); AVN2005-pan (), and are shown to lead to an enhancing violation of local realism cabello2006 (); Martini-enhancingbellviolationi2006 (). The states also enable to perform complete deterministic Bell state analysis kwiat-weinfurter1998 () as demonstrated in Weinfurter-bell2006 (); Martini-bell2006 ().
In this Letter we report an experimental realization of one-way quantum computing with such a 2-photon 4-qubit cluster state. The key idea is to develop and employ a bright source which produces a 2-photon state entangled both in polarization and spatial modes. We are thus able to implement the Grover’s algorithm and quantum gates with excellent performances. The genuine four-partite entanglement and high fidelity of better than 88% are characterized by an optimal entanglement witness. Inheriting the intrinsic two-photon character, our scheme promises a brighter source by 4 orders of magnitude than the usual multi-photon source, which offers a significantly high efficiency for optical quantum computing. It thus provides a simple and fascinating alternative to complement the latter. With ease of manipulation and control, the nearly perfect quality of this source allows to perform highly faithful and precise quantum computing.
To generate the state we use the technique developed in previous experiments AVN2005-pan () with type-I spontaneous parametric down-conversion (SPDC) source SPDC-I (). The experimental setup is shown in Fig. 1a. A pulse of ultraviolet (UV) light passes twice through two contiguous beta-barium borate (BBO) with optic axes aligned in perpendicular planes to produce one polarization entangled photon pair, with one possibility in the forward direction with a state on spatial (path) modes , and another possibility in the backward direction with a state on modes . Here () stands for photons with horizontal (vertical) polarization.
Through perfect temporal overlaps of modes and and of modes and , one can obtain a state
By properly adjusting the distance between the concave mirror and the crystal such that , the generated state will be exactly the desired cluster state
if we identify the polarization and spatial modes of photon A to be qubits 2, 3, respectively and photon B’s polarization and spatial modes to be qubits 1,4 and encode logical qubits as , , , . We observe a cluster state generation rate about per second for 200mW UV pump, which is 4 order of magnitude brighter than the usual 4-photon cluster state production zeilingernature2005 (); zeilingernature2007 (); Weinfurter-cluster2005 () where only a rate of is achieved per second.
To evaluate the quality of the state, we apply an optimal entanglement witness toth-guehne2005 (). The witness is of form
where is a 2-dimensional identity matrix while are Pauli matrices. A negative value for the witness implies 4-partite entanglement for a state close to and will be optimally as -1 for a perfect cluster state. Two experimental settings of and are needed. can be attained by measuring in the basis for the polarization in each output arm after apparatus (i) in Fig. 1b. while can be realized by measuring in the basis after apparatus (ii). This is because the beam splitter (BS) acts exactly as a Hadamard transformation for the path modes to change basis to basis for measurement, namely, , . All of the observables for evaluating the witness are listed in Table 1.
Substituting their experimental values into Eq. (3) yields , which clearly proves the genuine four-partite entanglement by about standard deviations. As shown in toth-guehne2005 (), one can obtain a lower bound for fidelity of experimental prepared state to
This proves to be a better source than the ones in zeilingernature2005 (); zeilingernature2007 (); Weinfurter-cluster2005 () where fidelities are about 0.63 zeilingernature2005 (); zeilingernature2007 () and 0.74 Weinfurter-cluster2005 () respectively. We attribute impurity of our state to imperfect overlapping on BS, deviations of BS from 50%, as well as imperfections in the polarization and path modes analysis devices. To get a qualitative depiction for these imperfections, we scan the concave mirror with nanometer displacements and observe interference after BS. By measuring along basis in each output arm, we have obtained visibility of for coincidences among detectors D1-D2,D1-D4,D3-D2 and D3-D4 respectively.
A cluster state can be represented by an array of nodes, where each node is initially in the state of . Every connected line between nodes experiences a controlled-phase (CPhase) gates acting as onewayQC (). For a given cluster state, consecutive single-qubit measurements in basis will define a quantum computing in addition to feed-forward of measurement outputs, where (). A measurement output of means ‘0’ while signifies ‘1’. This measurement basis determines a rotation , followed by a Hadamard operation of encoded qubits. The state can be represented by a box type graph shown in Fig. 2a, up to a local unitary transformation.
Grover’s algorithm. For an unsorted database with entries, Grover’s search algorithm gives a quadratic speed-up for with consultations on average groveralgorithm (). Striking linear optics implementations have been achieved in kwiat-groveralgorithm (); Groverclassical (), although it is questionable whether the algorithm is truly ‘quantum’ due to a demonstration Groverclassical () based on interference of classical waves. One-way realizations have been carried out zeilingernature2005 (); zeilingernature2007 () recently. In the case of four entries , a single quantum search will find the marked element An execution goes as follows: an oracle encodes a desired entry by changing its sign through a black box with initial state . After an inversion-about-the-mean operation, the labeled element will be found with certainty by readout. It is shown in zeilingernature2005 () that this can be exactly finished with the box cluster state in Fig. 2a. For demonstration, we experimentally tag the element on qubits and make the readout on qubits all along basis . Because of the fact that the state Eq. (2) distinguishes the box cluster from an transformation on every qubit and a swap between qubits 2 and 3, this amounts to measuring along the basis after apparatus (iii) in Fig. 1b. Two polarizing beam splitters (PBS) here are for interfering, to ensure the desired cluster state. In the meantime they are acting as polarization measurement devices, which is equivalent to use apparatus (i) in this case. The outputs of the algorithm are two bits in lab basis by feed-forwarding outcomes of qubits 2,3, where are measurement outcomes on qubits . The experimental results are sketched in Fig. 2b.
Quantum gates. Non-trivial two-qubit quantum gates such as the CPhase gate are at the heart of universal quantum computation, that can be realized by cluster states conveniently. Depending on the initial cluster state and measurement basis, states with different degrees of entanglement can be generated. The horseshoe or box cluster shown in Fig. 3a and 3c can realize such important gates. For the case of horseshoe cluster in Fig. 3a, depending on the outcomes when measuring along basis and , the output state on qubits 1,4 would be where . The state is always a maximal entangled state. Taking and consider outcomes ‘00’ in qubits 2,3. This implies a final Bell state of . Note that the horseshoe cluster state is equivalent to the state Eq. (2) up to a transformation, in lab basis this amounts to the fact that the output state is exactly , that is symmetric under transformation. To characterize quality of quantum gates outputs, we put a birefringent crystal in path to make a transformation for polarization. After BS, all the Bell states on qubits 1,4 will change as
which can be completely and deterministically discriminated by measuring along basis. The fidelities of the output states in the lab basis to the ideal Bell state are shown in Fig. 3b. Similarly, for the box cluster state shown in Fig. 3c, measurements on qubits 2,3 along basis will give an output state on qubits 1,4 with CPhaseCPhase which is a product state when and . Since we can completely distinguish 4 different products states, output fidelities can be obtained directly, as shown in Fig. 3d. By employing the techniques developed in zeilingernature2007 () with active feed-forward, one can expect to achieve deterministically quantum computing with excellent quality outputs.
We remark that other 2-qubit states can be generated, by suitable measurements on qubits 2,3. However, an arbitrary single-qubit rotation needs generally 3 single-qubit measurements on a cluster for one-way implementation zeilingernature2005 (); zeilingernature2007 (), which is a big consuming of resource. Fortunately, this rotation can be easily attained by linear optical components both for polarization and spatial modes. Therefore a hybrid framework would be more practical with one-way realization of two-qubit gates and the usual single-qubit gates. Due to low efficiency for producing multi-photon and concurrent occupations for polarization-spatial degrees of freedom of the photons, our source is not yet scalable, the same as the multi-photon source zeilingernature2007 (). However, the scheme developed here leads to quantum computing with a quality and efficiency at present largely unmatched by previous methods.
In summary, we have developed a scheme for preparation of a 2-photon 4-qubit cluster state, designed and demonstrated the first proof-of-principle realization of one-way quantum computing employing such a source. The excellent quality of the state with fidelity better than is achieved. The high count rates enable quantum computing by 4 orders of magnitude more efficient than previous methods. We have implemented the Grover’s algorithm with a successful probability of about and quantum gates with high fidelities of about on average. Our scheme helps to make a significant advancement of QIP, and the source constitutes a promising candidate for efficient and high quality one-way optical quantum computing. By using more photons and more degrees of freedom, one can expand our ability to generate many-qubit cluster states for performing quantum computing and other complex tasks. Our results can also find rapid applications in quantum error correction codes, multi-partite quantum communication protocols ECC-qcp (), as well as novel types of AVN tests for nonlocality AVN-theory ().
This work was supported by the Marie Curie Excellence Grant of the EU, the Alexander von Humboldt Foundation, the CAS and the National Fundamental Research Program (Grant No. 2006CB921900).
Note added.– During preparation of our manuscript, we are aware of one related experiment for realization of a linear cluster state Martiniprl2007 ().
- (1) H.J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86, 910 (2001); R. Raussendorf and H. J. Briegel, ibid. 86, 5188 (2001); R. Raussendorf, D.E. Browne, and H.J. Briegel, Phys. Rev. A 68, 022312 (2003).
- (2) D. Schlingemann and R.F. Werner, Phys. Rev. A 65, 012308 (2001); R. Cleve, D. Gottesman, and H.-K. Lo, Phys. Rev. Lett. 83, 648 (1999).
- (3) M. Hein, W. Dür, and H.-J. Briegel, Phys. Rev. A 71, 032350 (2005).
- (4) D.E. Browne, and T. Rudolph, Phys. Rev. Lett. 95, 010501 (2005); T.P. Bodiya, and L.-M. Duan, ibid. 97, 143601 (2006).
- (5) P. Walther, M. Aspelmeyer, K.J. Resch, and A. Zeilinger, Phys. Rev. Lett. 95, 020403 (2005).
- (6) N. Kiesel et al., Phys. Rev. Lett. 95, 210502 (2005).
- (7) P. Walther et al., Nature (London) 434, 169 (2005).
- (8) R. Prevedel et al., Nature (London) 445, 65 (2007).
- (9) C.-Y. Lu et al., Nature Physics 3, 91-95 (2007).
- (10) M.S. Tame et al., Phys. Rev. Lett. 98, 140501 (2007).
- (11) C. Cinelli et al., Phys. Rev. Lett. 95, 240405 (2005);
- (12) T. Yang et al., Phys. Rev. Lett. 95, 240406 (2005).
- (13) J.T. Barreiro, N.K. Langford, N.A. Peters, and P.G. Kwiat, Phys. Rev. Lett. 95, 260501 (2005).
- (14) C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, Phys. Rev. Lett. 96, 190501 (2006).
- (15) A. Cabello, Phys. Rev. Lett. 87, 010403 (2001); Z.-B. Chen et al., ibid. 90, 160408 (2003).
- (16) A. Cabello, Phys. Rev. Lett. 97, 140406 (2006).
- (17) M. Barbieri et al., Phys. Rev. Lett. 97, 140407 (2006).
- (18) P.G. Kwiat, and H. Weinfurter, Phys. Rev. A 58, R2623 (1998).
- (19) M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, Phys. Rev. A 75, 042317 (2007).
- (20) P.G. Kwiat et al., Phys. Rev. A 60, R773 (1999).
- (21) G. Tóth, O. Gühne, Phys. Rev. A 72, 022340 (2005).
- (22) L.K. Grover, Phys. Rev. Lett. 79, 325 (1997).
- (23) P.G. Kwiat, J.R. Mitchell, P.D.D. Schwindt, and A. G. White, J. Mod. Opt. 47, 257 (2000).
- (24) N. Bhattacharya, H.B. van Linden van den Heuvell, and R.J.C. Spreeuw, Phys. Rev. Lett. 88, 137901 (2002).
- (25) G. Vallone et al., Phys. Rev. Lett. 98, 180502 (2007).