Experimental realization of one-way quantum search

The state |C4i is very useful for our experimental demonstration because |C4i is equivalent to the four-qubit box cluster state up to four-qubit local unitary operations. To give a concrete demonstration, we experimentally mark the element |00i in qubits 2, 3 and make the final readout measurements on qubits 1, 4 all along basis B(π). By noting the fact that the state Eq. (10.8) differs from the box cluster state up to a H transformation on every qubit and a swap between qubits 2 and 3, this amounts to measure along the {|V i , |Hi}

basis for the polarization in each output arm after PBS1 and PBS2. The output of the algorithm is two bits (s3⊕ s⁴, s1⊕ s²) in lab basis by feed-forwarding outcomes of qubits 2,3. The experimental result of this example is shown in Fig. 10.9.

PHOTONS AND EXPERIMENTAL REALIZATION OF ONE-WAY QUANTUM COMPUTING

Figure 10.9: A successful identification probability of (96.1±0.2)% is achieved determinis-tically with feed-forward, while it is (24.9 ±0.4)% without feed-forward. This depicts that our source of cluster state is ideally suited for such a sort of algorithm’s implementation.

10.5 Conclusion

We have developed a scheme for preparation of a two-photon four-qubit cluster state.

With such a source, we have designed and demonstrated the first proof-of-principle ex-perimental realization of one-way quantum computing. The excellent quality of the state with fidelity better than 88% is characterized by an optimal witness without using of a full state tomography. Moreover, high count rates of the state creation enable more efficient quantum computing by 4 orders of magnitude than previous methods. We have thus achieved implementation of Grover’s algorithm with a successful probability of about 96%. In addition, non-trivial two-qubit quantum gates such as the CPhase gate are im-plemented with high fidelities through the approach developed. Refer to [54] for detailed discussions.

Summary and Outlook

11.1 Summary

In this thesis we have presented novel approaches to correlation structure of multipartite entanglement, entanglement detection, entanglement generation, entanglement purifica-tion, quantum error corrections, quantum search algorithm, and, furthermore, experimen-tal advance towards one-way quantum computation. Our research has covered several im-portant subjects involved in the field of quantum information and quantum computation and mainly associates with the key processes of quantum information processing.

Through the correlation criteria of multipartite entanglement, one can construct robust entanglement witness operators to detect many-qubit stabilizer, four-qubit singlet, three-qubit W, generalized many-three-qubit GHZ, two-qudit Bell, two-qudit singlet, four-ququat supersinglet, many-qudit GHZ states with fewer local measurement settings. The entan-gled states under study are all important for entanglement-based quantum information processing. In addition to detections of entanglement, the criteria proposed help to an-alyze the correlation structures of Bell inequalities and to find their connections with entanglement witness operators.

An idea of hybrid maps is proposed to establish standard entanglement purification protocols which guarantee to purify any distillable state to a desired maximally entangled

pure state all by the standard purification local operations and classical communications.

The protocols proposed in this work, in which two state transformations are used, perform better than the IBM and Oxford protocols in the sense that they require fewer operation times in yielding a same amount of the desired pure state. One of the proposed protocols in this work can even lead to a higher improved output yield when it is combined with the hashing protocol, as compared with the combined algorithm consisting of the Oxford and the hashing protocol.

Simpler encoding and decoding networks are necessary for more reliable quantum error-correcting codes. The simplification of the encoder-decoder circuit for a perfect five-qubit quantum correcting code can be derived analytically if the quantum error-correcting code is converted from its equivalent one-way entanglement purification proto-col. In our study, the analytical method to simplify the encoder-decoder circuit is intro-duced and a circuit that is as simple as the existent simplest circuits is presented as an example. The encoder-decoder circuit presented here involves nine single- and two-qubit unitary operations, only six of which are controlled-NOT gates.

A study on the cause of multi-particle entanglement is also presented in this thesis.

We show how dot-like single quantum well excitons, which are coupled to single-mode cavity photon, evolve into maximally entangled state as a series of conditional measure-ments are taken on the cavity field state. Generation of multi-particle entangled states is derived analytically. Application to quantum teleportation is also pointed out, and may be achieved with current technologies.

We have analyzed the quantum search algorithm in detail. First, a general quantum search algorithm with arbitrary unitary transformations and an arbitrary initial state is considered in this work. To search a marked state with certainty, we have derived, using an SU(2) representation: (1) the matching condition relating the phase rotations in the algorithm, (2) a concise formula for evaluating the required number of iterations for the search, and (3) the final state after the search, with a complex phase in its amplitude.

Moreover, the optimal choices and modifications of the phase angles in the Grover kernel

are also studied. As the matching condition in Grover search algorithm is transgressed due to inevitable errors in phase inversions, it gives a reduction in maximum probability of success. With a given degree of maximum success, we have derived the generalized and improved criterion for tolerated error and corresponding size of quantum database under the inevitable gate imperfections. The vanished inaccuracy to this condition has also been shown. A concise formula for evaluating minimum number of iterations is also presented. Furthermore, a family of algorithms is recently addressed for sure-success quantum search problems. When the phase inversion operations of these algorithms are identical to those of the standard Grover algorithm, we found that this family of algorithms is of robustness against inevitable phase imperfections. Finally, an analog analogue of Grover’s quantum search algorithm was studied. A generalized Hamiltonian driving the evolution of quantum state in the analog search system was derived. Equations relating all parameters considered in the present problem were given according to the required maximal probability for finding the marked state. By these equations, both the measuring time and the system energy gap suitable for a quantum search with or without certainty can thus be evaluated. It was shown that in an efficient quantum search computation, the measuring time should be proportional to the square root of the size of database.

We perform the first experimental realization of one-way quantum computation on a 2-photon four-qubit cluster state that is entangled both in polarization and spatial modes. Through solving a quantum search problem, the experiment illustrates a high-speed quantum computation in one-way realization. The experimental demonstration shows the hyper-entangled cluster states can provide an ideal source for rapidly and precisely optical quantum information processing.