Technologies quantiques & information quantique

Technologies quantiques & information quantique Edouard Brainis (Dr.) Service OPERA, Faculté des sciences appliquées, Université libre de Bruxelles Email: ebrainis@ulb.ac.be Séminaire V : Ordinateurs quantiques

Contenu de la leçon Ordinateurs à ions froid Ordinateurs optiques Rappels Porte de Cirac-Zoller Algorithme de Deutsch-Jozsa Perspectives Téléportation quantique Portes logiques avec de l'optique linéaire 7 mai 2010 2

Ordinateurs à ions froids 7 mai 2010 3

Rappels Initialisation et lecture du qubit Portes à 1 qubit Initialisation : efficacité >0.99, 1µs Lecture : efficacité >0.99, 1ms 7 mai 2010 4

Rappels Couplage au mode d'oscillation de la chaîne dans le piège 7 mai 2010 5

La porte de Cirac-Zoller 7 mai 2010 6

La porte de Cirac-Zoller 7 mai 2010 7

De la porte de Cirac-Zoller au C-NOT 7 mai 2010 8

Réalisation expérimentale du C-NOT 2003 : Schmidt-Kaller et al, Nature 422, 408-411 Temps de cohérence : 800 µs Fidélité : 73% 2006 : Riebe et al, Phys. Rev. Lett. 97, 220407 Temps de cohérence : 2 ms Fidélité : 91% 7 mai 2010 9

Algorithme de Deutsch-Jozsa Classes des fonctions : Constantes Balancées Comment les distinguer? Un algorithme classique a besoin de 2 n /2+1 appels à la fonction. Un algorithme quantique distingue les fonctions constantes des fonctions balancées en un seul appel! 7 mai 2010 gain exponentiel 10

Algorithme de Deutsch-Jozsa Classes des fonctions : Constantes Balancées Comment les distinguer? Un algorithme classique a besoin de 2 n /2+1 appels à la fonction. Un algorithme quantique distingue les fonctions constantes des fonctions balancées en un seul appel! 7 mai 2010 gain exponentiel 11

Algorithme de Deutsch-Jozsa 7 mai 2010 12

Algorithme de Deutsch-Jozsa 7 mai 2010 13

Perspectives Pièges segmentés Puce à ions 7 mai 2010 14

"Technologie quantique et information quantique" Séminaire 5 : ORDINATEURS QUANTIQUE Implémentation optique Evgueni Karpov (Nicolas Cerf) Centre for Quantum Information & Communication Université libre de Bruxelles

QUANTUM TELEPORTATION Entanglement Dense coding PORTE LOGIQUE Non-deterministic conditional sign flip

QUANTUM TELEPORTATION A remarkable use of quantum entanglement The fact that Alice and Bob share 2 entangled photons enables them to teleport the state of a particle (a qubit) 0 = 1 0 1 = 0 1 3

Entanglement of 2 qubits : ( Bell states ) ± = 1 2 ± = 1 2 00 ± 11 01 ± 10 4 maximally-entangled orthonormal states which form a basis of the 2-qubit Hilbert space Unitary operators in the 2-qubit Hilbert space z = 1 0 SIGN FLIP 0 1 x = 0 1 1 0 BIT FLIP I I = I z = I x = I x z = x z = i y = BOTH Local operation changes the whole non-local state! 0 1 1 0 x z 0 = 1 x z 1 = 0 4

Causality is respected! ± = 1 2 ± = 1 2 1 0 0 ±1 0 0 0 0 0 0 0 0 ±1 0 0 1 0 0 0 0 0 0 1 ±1 0 0 ±1 1 0 0 0 0 Tr 1 ± =Tr 1 ± = 1 2 1 0 0 1 Tr 2 ± =Tr 2 ± = 1 2 1 0 0 1 Whatever Bob does (e.g., applying any Pauli operation), the state of Alice remains the same. 5

But it enables dense coding! I I = I z = 4 possible operations (22 bits) are virtually encoded in 1 single qubit. I x = I x z = ALICE BOB 00 01 10 11 I z x x z 1 qubit Bell measurement 00 01 10 11 6

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Quantum teleportation ALICE = 0 1 = 1 0 1 00 11 2 = 1 2 00 2 0 01 = 1 2 0 1 2 x 1 2 1 0 = 1 00 11 2 applying associativity: 1 10 2 0 1 z 1 0 x z 0 11 2 BOB 0 00 = 000 = 00 0 1 9

Quantum teleportation ALICE = 0 1 = 1 2 00 11 = 1 2 z x x z BOB Bell measurement projection on one of four Bell states Result of measurement 00 01 10 11 2 bits 00 01 10 11 I z x x z... teleported state: = 0 1 10

Teleportation does not violate no-cloning theorem (Alice's qubit is destroyed) Bob's qubit contains no information (not causally linked with Alice's qubit) Transmitted classical bits contain no information (otherwise qubit would be disturbed) Teleportation dual to dense coding: EPR +1 1 1 EPR +1 B U 1 qubit 1 qubit 1 qubit 2 bits 2 bits 2 bits U B 11

Performing a Bell measurement using a balanced beam splitter a V a H b V ' b V b H ' b H a V ' a H ' Horizontally polarized mode: a H '= 1 2 a H b H b H '= 1 2 a H b H Vertically polarized mode: a V ' = 1 2 a V b V b V '= 1 2 a V b V Impinging Bell state? = 1 2 a H b H a V b V 0 a V = 1 Inverse relations: a H = 1 2 a H ' b H ' b H = 1 2 a H ' b H ' 2 a V ' b V ' b V = 1 2 a V ' b V ' 12

= 1 2 a H b H a V b V 0 = 1 2 2 a H ' b H ' a H ' b H ' a V ' b V ' a V ' b V ' 0 = 1 2 2 a H ' 2 b H ' 2 a V ' 2 b V ' 2 0 Outcome: 2 photons of same polarization in same port = 1 2 a H b H a V b V 0 = Outcome: 2 photons of same polarization in same port thus, indistinguishable (without further interference) or 13

= 1 2 a H b V a V b H 0 = 1 2 2 a H ' b H ' a V ' b V ' a V ' b V ' a H ' b H ' 0 = 1 2 a H ' a V ' b H ' b V ' 0 Outcome: 2 photons of opposite polarizations in same port thus, indistinguishable (without polarization meas.) 14

= 1 2 a H b V a V b H 0 = 1 2 2 a H ' b H ' a V ' b V ' a V ' b V ' a H ' b H ' 0 = 1 2 a V ' b H ' a H ' b V ' 0 Outcome: 2 photons of opposite polarizations in different ports The singlet state can be discriminated via a coincidence 15

Experimental quantum teleportation teleporting the polarization state of a single photon ( Vienna experiment, 1997 ) 2 passages via nonlinear crystal discriminating using a beam splitter success probability = 25 % (keep only events) 16

Linear Optics Quantum Computation Basic elements bosonic qubits states of optical modes Optical mode superposition of the number states Basic states of a bosonic qubit encoded in two modes: State preparation single photon source n 0 1 Computation manipulation of quantum states via optical elements Simplest optical elements are phase shifters and beam splitters all one-qubit rotations Full power of computation is achieved by adding a two-qubit gate 0 0 1 1 1 0 Example: Conditional sign flip a b 1 ab a b a,b=0, 1 How to implement with optical elements? - come back later Readout measurement with photodetectors 17

Readout with photodetectors Measuring photon number: Avalanche photodetector discriminates vacuum state 0 n 0 two possible outcomes: or, n unknown In LOQC: n 4............... N Probability of undercounting: n P= n n 1 / 2N 18

Basic operations Implementing Conditional sign flip Evolution implementable by passive linear optics preserves the number of photons U 0 l = 0 l Unitary operation is given by a unitary matrix U 1 l = k u kl 1 l Phase shifter u(p θ ) = e iθ Beam splitter Action on bosonic qubits: exp (- iθσ z /2) u(b θ,φ ) = cos(θ) - e iφ sin(θ) e -iφ sin(θ) cos(θ) exp (- iθσ y /2) for φ=0 19

Implementing Conditional sign flip Nondeterministic nonlinear sign change NS: 0 0 1 1 2 2 0 0 1 1 2 2 is a particular case of nonlinear phase shift of one mode NS x implemented with phase shifters, beam splitters and post-selection x= 1 θ 1 = 22.5, φ 1 = 0 θ 2 = 65.5302, φ 2 = 0 θ 3 = - 22.5, φ 3 = 0 Φ 4 = 180 Post-selection: keeping only the outcomes with 1 2 0 3 Probability of success 0.18082 20

Implementing Conditional sign flip Nondeterministic Conditional sign flip Two independent applications of NS achieves a conditional sign flip with success probability 1/16 Bosonic modes First beamsplitter: 1 1 1 3 2 1 0 3 0 1 2 3 Second beamsplitter: 2 1 0 3 0 1 2 3 1 1 1 3 21

Improving nondeterministic conditional sign flip Quantum gates by teleportation : using 4 ancillas, success probability 1/4 Using near-deterministic quantum teleportation : increasing the number of ancillas, success probability n 2 /(n+1) 2 Using less detection in order to return the state into the qubit space 22

Conclusion Possibility of implementing a universal quantum computation via linear passive optics no particle interaction Hidden nonlinearity all nonlinear elements are effectively put into state preparation (single photon source) measurement (photo detectors) post selection (non-deterministic gates) Possibility of implementing a universal quantum computation via linear passive optics no particle interaction Scalability? 23