Linear-optical Quantum Computing

1. Linear-optical Quantum Computing Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China

2. Outline • Why Linear-optical quantum computing? • What is quantum computation? • Photonic qubits • Why linear-optical quantum computing? (advantages and disadvantages) • Manipulating photonic qubits with linear optics • Linear optical logic gates • A theoretical idea • Experimental realization • Cluster states vs one-way quantum computing • Linear-optical one-way quantum computing • single photons + linear optics + measurements • An interface of photons and atomic ensembles: Quantum memory for polarization qubits • Summary and outlook

3. Why linear-optical quantum computing? What is quantum computation, first of all? A quantum computation can be considered as a physical process that transforms an input quantum state into an output state and as such, it respects quantum laws. The "information flow" in quantum computing process is carried by qubits which are subject to a designed unitary evolution. … … To perform general transformations relies on the ability to engineer arbitrary interactions between the qubits. Fortunately, this task can be greatly simplified by the following powerful theorem for universal quantum computation (Barenco et al., 1995; Lloyd, 1995): Any unitary transformation of an N-qubit system can be implemented with single-qubit operations and quantum controlled-NOT (CNOT) gates or equivalent two-qubit gates.

4. Why linear-optical quantum computing? Qubits: the building block of a quantum computer Coherent superposition “0” and “1” Entanglement Each individual qubit carries no information! The fact that qubits can be in coherent superposition and entangled states gives the extraordinary power to a quantum computer to outperform its classical counterpart. Any two-level quantum system with an SU(2) symmetry represents a qubit. There are many different ways of implementing qubits. Examples include electron’s spin, atoms with two relevant energy levels, superconducting Josephson junctions, and photon’s polarization or spatial modes, etc. Photonic realization of qubits is one of the most promising candidates not only for quantum communication but also for quantum computing.

5. Why linear-optical quantum computing? An individual photon possesses a few physical degrees of freedom (DoF), each of which can in principle be used to carry information under appropriate experimental arrangements. These DoF of photons include the internal polarization, orbital angular momentum, spatial mode, emission time and frequency, etc. Two most important realizations of photonic qubits: “H” and “V” Polarization encoding: polarization qubits “here” and “there” Path encoding: spatial qubits

6. Why linear-optical quantum computing? • Why photonic encoding, or why linear-optical quantum computing? • First of all, the question is interesting in its own right. • Quantum states of photons can be easily manipulated by simple linear-optical elements with a high precision at about 99.9% level. • Easy realization of any single-qubit rotations. • Robust to the environmental noises (photons have no charge!) • Photons are the fastest information carriers, which is important for quantum communication and distributed quantum information processing • Challenges for linear-optical quantum computing • Difficulty of realizing two-qubit gates for photons due to the lack of photon-photon interaction (except for photons in certain nonlinear medium which can induce tiny optical nonlinearity) • Difficulty of storing photons for a reasonable long time Is linear-optical quantum computing possible, anyway?

7. Why linear-optical quantum computing? Yes, in principle! • Knill, Laflamme and Milburn (KLM), using the dual-rail encoding (spatial qubits), has shown that nondeterministic quantum logic operations can be performed using linear optical elements, additional photons (ancilla) and postselection based on the output of single-photon detectors. • They further demonstrated that the success rate of the quantum logic can be arbitrarily close to one by using more additional ancilla and detectors. • This implies that the nondeterministic quantum logic gate based on linear optics can be used as a basic block for quantum information protocols, even for efficient quantum computation. For details, see E. Knill, R. Laflamme, and G. J. Milburn, Nature 409, 46 (2001); Pieter Kok et al., quant-ph/0512071 (a review).

8. Why linear-optical quantum computing? • The KLM scheme provides the surprising possibility. • The scheme itself is complicated in its use of complex interferometers, resource-consuming, … • Yet, it is a real breakthrough and motives many subsequent study on linear-optical quantum information protocols. • For many of these later development in the context of quantum computing, please refer to a nice review (Pieter Kok et al., quant-ph/0512071) • For pedagogical purposes, this lecture will show you how to implement • a linear-optical CNOT gate • linear-optical one-way quantum computing

9. Some useful notions Universal set of quantum logic gates Graphical representations of Hadamard and CNOT gates. Here, a+b denotes addition modulo 2.

10. Some useful notions A simple network for the Bell-state preparation and measurement The four Bell state The reversed quantum network (right-hand side of the figure) can be used to implement the so-called Bell-state measurement on the two qubits by disentangling the Bell states. • Currently, photonic Bell states can be prepared via SPDC. – previous lecture • Partial Bell-state measurement can be done with linear optics. – previous lecture • The Bell-state preparation and measurement are doable if we have photonic CNOT (though nondeterministic) – see below

11. Manipulating photonic qubits with linear optics Linear-optical elements include beam splitter (BS), polarizing BS (PBS), half- and quarter-wave plates and phase shifters and respect the conservation of total photon number unitarity the reflection and transmission coefficients 50:50 beam splitter beam splitter

12. Manipulating photonic qubits with linear optics PBS is used to separate horizontal and vertical polarizations: it transmits only photon's horizontal polarization component and reflects the vertical component. The function of a PBS. (a) The PBS transmits horizontal, and reflects vertical, polarization. (b) If the two photons incident onto the PBS have identical polarization, then they will always go out along different directions, so there will be one photon in each of the two output modes. (c) On the other hand, if the two incident photons have opposite polarization, then they will always go out along the same direction, so there will be two photons in one of the two outputs and none in the other. In essence, a PBS can thus be used as a polarization comparer (Pan and Zeilinger, 1998; Pan et al., 2001). Polarizing beam splitter

13. Linear optical logic gates • For optical quantum computing, single-qubit rotations are trivial. The problem is how to realize the nontrivial two-qubit gates. • Solution: ancilla photons + measurements → measurement-induced nonlinearity that is able to entangle two input photonic qubits • For scalable optical quantum computing, a crucial requirement is the classical feed-forwardability: • It must be in principle possible to detect when the gate has succeeded by measurement of ancilla photons in some appropriate states. This information can then be feed-forward in such a way as to condition future operations on the photon modes. • A non-destructive CNOT gate for two independent photons is sufficient for this purpose. T.B. Pittman, B.C. Jacobs, and J.D. Franson, Phys. Rev. A 64, 062311 (2001); Z. Zhao et al., Phys. Rev. Lett. 94, 030501 (2005).

14. Linear optical logic gates T.B. Pittman, B.C. Jacobs, and J.D. Franson, Phys. Rev. A 64, 062311 (2001); Z. Zhao et al., Phys. Rev. Lett. 94, 030501 (2005).

15. Linear optical logic gates The calculation: The cases in which there is one and only one photon in each of the output modes: T.B. Pittman, B.C. Jacobs, and J.D. Franson, Phys. Rev. A 64, 062311 (2001); Z. Zhao et al., Phys. Rev. Lett. 94, 030501 (2005).

16. Linear optical logic gates

17. Linear optical logic gates • The experimental fidelity of achieving the CNOT logic table is estimated to be 0.78±0.05. • A measured interference fringe with a visibility of 0.58±0.09, which is consistent with the prediction of the interference fringe for .

18. Cluster states vs one-way quantum computing • A significant step in quantum computing is the discovery of "one-way quantum computing" [R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86, 5188 (2001); R. Raussendorf, D.E. Browne, and H.J. Briegel, Phys. Rev. A 68, 022312 (2003)], which is based on the preparation of highly entangled multi-qubit states, the so-called "cluster states" [H.J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86, 910 (2001)] and on simple adaptive one-qubit measurements. Here we will focus on the following issues: • What is the cluster states, and how to prepare them? • Why one-way quantum computing is a universal quantum computing? • The computational model of one-way quantum computing. • A linear optical implementation of one-way quantum computing.

19. Cluster states vs one-way quantum computing What is the cluster states?

20. Cluster states vs one-way quantum computing What is the cluster states?

21. Cluster states vs one-way quantum computing How to prepare cluster states? tunable Ising interaction

22. Cluster states vs one-way quantum computing Examples of cluster states: The cluster states on a chain of 2, 3, and 4 qubits (all with eigenvalue 0) A Bell state A three-qubit Greenberger-Horne-Zeilinger (GHZ) state Not equivalent to four-qubit GHZ states; entangled even after destroying a single qubit

23. Cluster states vs one-way quantum computing An intuitive graphical representation of cluster states and the properties of linear cluster under single-qubit measurements

24. Cluster states vs one-way quantum computing A cluster state on a two-dimensional cluster of rectangular shape, say, is a resource that allows for any computation that fits on the cluster. Actually, the CNOT gate and general one-qubit rotations, which form a universal set of quantum logic gates, can all be implemented via one-qubit measurements on a cluster state. That is, one-way quantum computing is a universal one. For a proof, see R. Raussendorf, D.E. Browne, and H.J. Briegel, Phys. Rev. A 68, 022312 (2003)

25. Cluster states vs one-way quantum computing How an one-way quantum computer works? The computational model of one-way quantum computing:

26. Cluster states vs one-way quantum computing

27. Cluster states vs one-way quantum computing • The entire quantum computation consists only of a sequence one-qubit projective measurements in a particular pattern of measurement eigenbases and in a particular order on this entangled state; Different algorithms require only a different pattern of adapted single-qubit operations on a sufficiently large cluster state. • The entanglement in the cluster state is destroyed by the one-qubit measurements and therefore it can only be used once, hence the name “one-way quantum computer”. • The computational model of the one-way quantum computer makes no reference to the concept of unitary evolution. • The information that is processed is extracted from the measurement outcomes and is thus a purely classical quantity. • The one-way quantum computer is dramatically different from the standard one based on sequences of unitary quantum logic gates that process qubits. • The very possibility of one-way quantum computing might well change our understanding of the requirements for quantum computation and more generally how we think about quantum physics.

28. Linear-optical one-way quantum computing • M.A. Nielsen, Phys. Rev. Lett. 93, 040503 (2004): A deterministic entangling quantum gate may be performed using, on average, a few hundred coherently interacting optical elements (BS, phase shifters, single photon sources, and photodetectors with feedforward). This scheme combines ideas from the KLM scheme and the abstract cluster-state model of quantum computation. • D.E. Browne and T. Rudolph, Phys. Rev. Lett. 95, 010501 (2005): Cluster states may be efficiently generated from pairs of maximally polarization entangled photons using linear optical elements. • Requirement of stable interferometry over only the coherence length of the photons. • A much greater degree of efficiency and a simpler implementation than previous proposals. • Redundant encoding of qubits + type-I/II two-qubit fusion. • Without proving the overall efficiency of the scheme. • L.-M. Duan and R. Raussendorf, Phys. Rev. Lett. 95, 080503 (2005): Efficient quantum computation can be constructed even if all the entangling quantum gates only succeed with an arbitrarily small probability p. • A linear optical one-way quantum computing scheme combining the Browne-Rudolph and the Duan-Raussendorf schemes, to be explained below. (Z.-B. Chen, B. Zhao, and J.-W. Pan, unpublished)

29. Linear-optical one-way quantum computing D.E. Browne and T. Rudolph, Phys. Rev. Lett. 95, 010501 (2005)

30. Linear-optical one-way quantum computing Redundant encoding of qubits + type-I/II two-qubit fusion.

31. Linear-optical one-way quantum computing A strategy of preparing a two-dimensional cluster: entangled photon pairs are resource and can be created, e.g., using single photons and linear optics. Z.-B. Chen, B. Zhao, and J.-W. Pan, unpublished)

32. Linear-optical one-way quantum computing

33. Linear-optical one-way quantum computing • The above strategy can finally prepare a square-lattice cluster of N qubits with a temporal overhead scaling logarithmically with N, and with an operational overhead (i.e., number of fusion operations) scaling as ~NlnN (Chen et al., unpublished). • The described protocol is a linear-optical realization of the Duan-Raussendorf proposal (2005), but combines the advantages of the Browne-Rudolph scheme, whose overall efficiency is thereby demonstrated. • Following the above protocol, complex multi-party quantum network can in principle be created very efficiently with the help of quantum memory (to be briefly explained below). This provides an exciting perspective for manipulating a huge number of photons applicable to quantum computing.

34. Linear-optical one-way quantum computing A linear-optical polarization entangler with four single photons: selecting the case where there is one and only one photon is each of the four output modes (a,A;b,B). This occurs with the probability of ¼. The total success probability is 25%*50% (two out of four Bell states can be distinguished)=1/8

35. Linear-optical one-way quantum computing So far, there is only one experiment, by P. Walther et al. [Nature 434, 169 (2005)], demonstrating the basic idea of the one-way quantum computing.

36. Quantum memory for polarization qubits P. Walther et al., Nature 434, 169 (2005).

37. Quantum memory for polarization qubits • Let us recall • the challenges for linear-optical quantum computing • Difficulty of realizing two-qubit gates for photons due to the lack of photon-photon interaction • Solution: linear optics + quantum measurements → nondeterministic two-qubit gates which are sufficient for efficient quantum computing √ • Difficulty of storing photons for a reasonable long time • Solution: an interface of photons and atomic ensembles, namely, quantum memory for polarization qubits with atomic ensembles, which is relatively easy to implement under current technology • Why quantum memory? • Photons travel at the speed of light!!! • Photonic two-qubit gates are probabilistic. With memory, we can keep applying the probabilistic gates until we succeed and do certain operations in parallel. Otherwise, optical quantum computing cannot be efficient.

38. Quantum memory for polarization qubits Atomic-ensemble-based quantum memoryis used to transfer the photonic states to the excitation in atomic internal states so that it can be stored, and after the storage of a programmable time, it should be possible to read out the excitation to photons without change of its quantum state. M.D. Lukin et al., Phys. Rev. Lett. 84, 4232 (2000); M. Fleischhauer and M.D. Lukin, Phys. Rev. Lett. 84, 5094 (2000).

39. Quantum memory for polarization qubits purely photon-like state (release of the single photon purely atom-like state (storageof the single photon)

40. Quantum memory for polarization qubits

41. Summary and Outlook • Photons can be a strong candidate not only for long-distance quantum communication, but also for large-scale quantum computing. • Given several atomic-ensemble-based technologies, linear-optical quantum information processing has a brilliant future. The interface of photons and atomic ensembles offers the fascinating playground for integrating the linear-optical and atomic-ensemble-based techniques for long-distance quantum communication and scalable optical quantum computing. • This would open up an exciting perspective for manipulating quantum states of a huge number of photons, with unexpected applications in future. • … Thanks!