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Computazione quantistica con i fotoni -P. Mataloni

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Computazione quantistica con i fotoni
P. Mataloni
Quantum Optics Group, Dipartimento di Fisica dell’Università “La Sapienza”, Roma, 00185, Italy

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Computazione quantistica con i fotoni -P. Mataloni

  1. 1. Computazione quantistica con i fotoni P. Mataloni Quantum Optics Group, Dipartimento di Fisica dell’Università “La Sapienza”, Roma, 00185, Italy http://quantumoptics.phys.uniroma1.it
  2. 2. 10000 4M # Transistors per chip 16 M 64 M 1000 256 M Electrons per device 1G 4G - Because of the properties of quantum states 100 Why quantum computation? - Because of the high power guaranteed G the entanglement 16 by 10 ? 1 1988 1992 1996 2000 2004 2008 2012 2016 2020 By 2015 a single electron can be confined in a transistor Example: factorizing a 1024-digit number: - Classical computer takes a period > universe lifetime - Quantum computer could find the answer in 1sec.... (P.W. Shor 1994)
  3. 3. - They are easy to generate, manipulate, transmit and detect - Have low interaction with the enviroment low decoherence - Possible to encode the information in different degrees of freedom of the photons (polarization, momentum, frequency….) Why quantum computation with photons? It has been demonstrated that a universal quantum computer can be realized by photons and standard linear optical devices (beam splitters, polarizers, waveplates…..) KLM, Nature 2001
  4. 4. Outline - Basic elements quantum bit, quantum register, logic gates, entanglement... - Cluster States of Photons properties, One-Way Quantum Computation - Spontaneous Parametric Down Conversion the Roma source, tools for measurements with photons - One-Way Quantum Computation with photons single qubit rotations, C-NOT gate, Grover’s search algorithm - Optical Quantum Computing in the near future doing now, perspectives
  5. 5. Quantum bit (Qubit) Coherent superposition of the orthogonal states |0> and |1> > > |Q> = α|0> + β|1> (|α|2 + |β|2 = 1) > > >| Example: - photon passing through a Mach-Zehnder interferometer: |Q> = α|Path 1> + β|Path 2> > > > - superposition of H and V polarization: |Q> = α|H> + β|V> > > >
  6. 6. Quantum register (3-bit register) Classical: can store exactly one of the eight different numbers, 000, 001, 010, ….., 111 Quantum: can store up to 8 numbers in a quantum superposition → N qubits: up to 2N numbers at once Classical Bit Quantum Bit Classical Register Quantum Register 000 001 010 011 0 or 1 0, 1, 101 01 100 101 110 111
  7. 7. Logic gates (1) Single qubit gate: linear operator in a 2-dimension space Complex 2x2 unitary matrix U NOT: X = σx Z = σZ Y = σY  0 1  α   β   0 − i  α   − β   − β   1 0  α   α   1 0  β  =  α   i 0  β  = i α  =  α   0 − 1 β  =  − β                                  Any kind of qubit rotation in the Bloch sphere can be realized by combining in different ways the three Pauli matrices Hadamard gate: 0 +1 0→ ≡+ 1 1 1  2 H=  1 − 1  0 −1 2  1→ ≡− 2
  8. 8. Logic gates (2) Two qubit gates: unitary 4x4 matrices U Quantum vs. classic - Classical case: any kind of logic gate can be realized by suitable combinations of the NAND gate. - Quantum case: any N-qubit logic gate can be realized by 1-qubit gates and one 2-qubit gate, (C-PHASE, C-NOT) 1 0 C-PHASE: 00   0 0 10 0 ⊗ 1t + 1 c 1 ⊗ (σ Z ) t CP = 0 0 0 1 0 c   0 0 0 −1  
  9. 9. Logic gates (3) C − NOT ⇒ U t ≡ (σ X ) t Control Target Control Target 1 0 0 0   0 0 0 0 0 1 0 0 = C NOT 0 1 0 1 0 0 1 0   1 0 1 1 0 0 1 0   1 1 1 0 C NOT ( + 1 ) = C NOT  1  1 0+ 1 1 = 2 2   1  1     C-NOT can generate entanglement:  1  0  = 1  0  = 0.0 + 1,1 C NOT  2  1  2 0 2      0 1      
  10. 10. Circuital model of a quantum computer 1 1 1 ∑ ...∑∑ c Ψ= in −1 ⊗ ... ⊗ i0 Superposition: n −1...i1 ,io in−1 = 0 i1 = 0 i0 = 0 Parallelism Ψ Unitary evolution of |Ψ> based on single and two qubit logic gates ... Linear Optics Quantum Computation: based on single photon qubits, linear optics devices for single qubit rotations and two qubit gates (KLM, Nature ‘01)
  11. 11. Entanglement |0>a > |0>b > Left: particle “a” carries the information “0”, or vice versa. S Right: particle “b” carries the information “1”, or vice versa. |1>a > |1>b > 0 a1b±1a 0 Ψ = b ab 2 can not be expressed by the product of single qubit states |Ψ>a and |Ψ>b Neither of the two qubits carries a definite value: as soon as one qubit is measured randomly, the other one will immediately be found to carry the opposite value, independently of the relative distance (quantum nonlocality)
  12. 12. Quantum nonlocality Singlet state: ( ) ALICE 1 − Ψ −V H V H a b a b ab 2 a BOB Alice measures photon a with 50% probability to detect: b - H or V (|0> or |1>): , - 45° or -45° (|+> or |->): ,  - L or R: , Perfect correlations in any basis!
  13. 13. Cluster states in Quantum Information Particular graph states associated to a n-dimensional lattice Each dots correspond to the qubit: Each link corresponds to a Control σZ gate Create a genuine multiqubit entanglement Robust entanglement against single qubit measurements Fundamental resource for one-way quantum computation
  14. 14. 4-qubit linear cluster states ( + 00 + + + 01 − + − 10 + − − 11 − ) 1 C4 = 2 1 2 3 4 ( + 0 + − 1 )⊗ 1 ( 0 + − 1 − ) 1 ≠ C4 2 2 1 2 3 (3-qubit) Linear cluster 1 2 3 4 (4-qubit) Linear cluster Not factorizable! 2 1 Horseshoe cluster 3 4 2 1 Horseshoe cluster (rotated 180°) 4 3 1 2 Box cluster 3 4
  15. 15. One-way quantum computation (Briegel et al. PRL 01) Initialization - Preparation of the cluster state Manipulation - Algorithm: pattern of single qubit measurements Qubit j measured in the bases: - Feed forward measurements - Irreversibility (one-way) Read out - Feed forward corrections - Not measured qubit: output
  16. 16. Building blocks of the logical operations
  17. 17. Logical operation: example
  18. 18. Entangled states with photons Allows to generate photon pairs by the spontaneous parametric down conversion (SPDC) process Twin photons created over conical regions, at different wavelengths, with polarization orthogonal to that of the pump
  19. 19. SPDC features Low probability (≅ 10-9) Non-deterministic process Energy matching: Phase matching: Degenerate emission:
  20. 20. The Roma source: polarization – momentum hyperentanglement of 2 photons [ ] [ ] 1 1 H a H b + e iθ VaVb la rb + e iφ ra lb = Φ ⊗ ψ Π= ⊗ 2 2 Barbieri et al. PRA 05 Cinelli et al. PRL 05 2 photons → 4 qubits Barbieri et al. PRL 06
  21. 21. Polarization – momentum entanglement Quantum Bell-CHSH inequality test: σ 213-σ violation Classical Bell-CHSH inequality test: σ 170-σ violation
  22. 22. Photon cluster states 4-photon cluster states (based on the simultaneous generation of 2 photon pairs [Zeilinger et al., Nature (05, 07)] [ ] 1 H a H b H c H d + H a H bVcVd + VaVb H c H d − VaVbVcVd 4 Problems: - Generation/detection rate ∼ 1 Hz - Limited purity of the state - Need of post-selection Alternative: Generate cluster states starting from 2-photon hyperentangled states
  23. 23. From hyperentangled to cluster states HW [ H ala a aH b rbb b+ H aaraa Hbblbb + Vaala bVbb rb +a Va rab bVblb ] 1 1[H l H r + H r H l + V l V r − V r V l ] a a 4 4 - High generation rate (~1000 coincidences per sec detected) - High purity of the states Vallone et al. PRL 07 - No post-selection required
  24. 24. Measurement tools Polarization (p) Momentum (k) observables observables l+r sx = x l-r 50/50 BS sy = y BS Glass plate l sz= z r
  25. 25. Single qubit rotation By choosing α and β any arbitrary single qubit rotation can be performed up to Pauli errors (corrected by feed-forward) α β Rz (α ) Rx (β ) Z X
  26. 26. Measurement setup: probabilistic QC - Measurements done by spatial mode matching on a common 50:50 BS - Qubit rotations performed by using either π or k as output qubit Vallone et al. LPL 08
  27. 27. πB πA kA kB Polarization output qubit s2 = s3 = 0 : output state: πA πB kA kB Linear momentum output qubit s2 = s3 = 0 : output state:
  28. 28. Experimental results with probabilistic QC
  29. 29. Measurement setup: deterministic QC Vallone et al. PRL 08
  30. 30. Experimental results with deterministic QC
  31. 31. 2-qubit gates
  32. 32. C-NOT gate Control: linear momentum of photon B; target: polarization of photon B. Realized by the 4-qubit horseshoe (180° rotated) cluster state. Qubit 1 measured in the basis |0> , |1> (or |+>, |->) 1/√ Qubit 4 measured in the basis |α±> = 1/√2 [|0> ± e-iα|1>)] α ( ) Ψout = H t C − NOT O + c ⊗ Rz (α ) + t Qubit 1 measured in the basis ( |0> , |1> ) Qubit 1 measured in the basis ( |+>, |- > )
  33. 33. Grover’s search algorithm Allows to identify the tagged item in a database within 2M possible solutions (encoded in M qubits). Right solution found within √2M steps (classical: 2M /2 ) Vallone et al. PRA, in press
  34. 34. Conclusion and Perspectives One-Way Quantum Computation with 2-photon 4-qubit cluster states Low decoherence High repetition rates High fidelity of the algorithms Need to increase the computational power by using more qubits Different strategies: Use more degrees of freedom Use more photons Hybrid approach (more photons + more degreees of freedom)
  35. 35. 6-qubit cluster state (based on triple entanglement of two photons) 2-crystal geometry Kint π Kext LC6
  36. 36. Integrated system of GRIN lenses with single mode optical fibers Allows efficient coupling of SPDC radiation belonging to many optical modes Multipath Entanglement
  37. 37. Measurement setup Rossi et al. ar-Xiv: quant-ph 08
  38. 38. 70 cm
  39. 39. An important result: use of integrated optics O’ Brien, Science ‘08 Optical waveguide: allows the propagation of light in single modes of e.m. field Miniaturized circuits realized by directional couplers 5 mm Completely integrated C-NOT gate Future quantum circuit architectures on chips are now possible
  40. 40. What we need more?? - More and more qubits to put in a cluster state (more photons, more degrees of freedom…) - More efficient and compact sources of entangled photons (to be integrated on waveguide chips) - New optical tools to manipulate photons (i.e. quantum converters between different degrees of freedom) - Efficient error corrections but, in particular, a REAL, deterministic, high repetition rate source of n- photn Fock states (in particular single photons Photon gun)
  41. 41. The team Pino Vallone Alessandro Rossi post-doc undergr. student Raino Ceccarelli undergr. student Previous members Marco Barbieri, post-doc Enrico Pomarico Quantum Technology Lab. GAP-Optique University University of Queensland of Geneva Brisbane Institute d’Optique, Paris Chiara Cinelli, laurea ENEL

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