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How to cooka quantum computer

A. Cabello, L. Danielsen, A. Lpez Tarrida, P. Moreno,J. R. Portillo

University of Seville, SpainUniversity of Bergen, Norway

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Playa del Carmen, Mexico. November 2010 09/12/10 04:52 AM

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How to cooka quantum graph state



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Playa del Carmen, Mexico. November 2010 09/12/10 04:52 AM

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Optimal preparationof quantum graph states



ACCOTAPlaya del Carmen, Mexico. November 2010

04:52 AM

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Some previous ideas

Bit vs. qubit

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Some ideas

Bit vs. qubit

Quantum states: superposition and entaglementStabilizer statesgraph states

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Some ideas

Bit vs. qubit

Quantum states: superposition and entaglementStabilizer statesgraph states Oh! Graph TheoryOh! Graph Theory

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Some ideas

Bit vs. qubit

Entaglement measuresRepresentative graph state

Quantum computers are made with graph states, but are unstable

Quantum states: superposition and entaglementStabilizer states

graph states Oh! Graph TheoryOh! Graph Theory

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Bit

0 and 1 (on/off, true/false, yes/no).

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Qubit

2-dimensional quantum physic system,

Hilbert space isomorphic to C 2 .

Schumacher, 1995

spin particle.E.g.,

Photon polarization.

Two relevant states physic system.

0

1BASIC STATE VECTORS

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Qubit

PHYSICS MATHEMATICS

10

= :0 C 2

0

1= :1 C 2

isomorphic toC 2

10

= :0 C 2

0

1= :1 C 2

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Quantum Mechanics: superpositions

If it is posible and

then

IRL

Photons:

Atoms:

laser

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Qubit

=0 1 C 2

qubit INFINITE PURE STATES:

Lineal superposition (coherent) of basic states:

BLOCH's sphere =cos

20 ei sen

21

Or:

2

2=1

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Complex SYSTEMS

ENTAGLEMENT STATES:

PRODUCT STATES:

PHYSCIS MATHEMATICS

10

10

= :00 C 2 C 2

01

01 = :11 C

2 C

2

1

0

1

0

0

1

0

1

= :00 11 C 2 C 2

H 1 H 2

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Classification of states by entaglement

Entagled states CANNOTbe preparated with local dispositives .

much stronger correlated than all possible classic systems.

Quantum Mechanics => ENTAGLEMENTS

Theory / Applications

Pure state of a multipartite quantum system is ENTAGLED if it is NOT a product of states .

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CRITERIA(pure states, multipartites)

L O C C L U L O C CInfinite classes, (bipartites too).

Equivalent entaglement:

S L O C C L U S L O C CInfinite classes, (three parts or more).

Equivalent entaglement:

W. Dr, G. Vidal and J. I. Cirac, Phys. Rev. A 62, 062314 (2000).F. Verstraete et al ., Phys. Rev. A 65, 052112 (2002).

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n>3 qubits: INFINITE amountof different, INEQUIVALENT

classes of ENTAGLED STATES

Subsets of states:

Graph states


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Stabilizer states

n- qubits stabilizer state:Simultaneous by n independent operators of Pauli group of order n

S

Stabilizer state by an operator A if :

A =

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Pauli group. Stabilizer state

N -QUBITS STABILIZER STATE

M jS =S M j= j M 1

j M n

j , j= 1, j= 1, , n.

M = M M 1 M n M i{ 0 , x , y , z} M = 1, i

PAULI GROUP

0= =1 00 1

X = X

=0 11 0

Y = Y =0 i

i 0

Z = Z

=1 00 1

PAULI MATRICES

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graph state

An n-qubits graph state is a special kind of stabilizer state .

S G

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graph state

An n-qubits graph state isa pure quantum state asociated to a simple connected graph G(V,E).

Each vertex represents a qubit and each edge a qubits entaglement

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graph state? Definition

GOnly state satisfying:G V ,E

giG =G , i=1,... ,n

gi:= X i

i , j E Z j Generator operator

S= g1 ,.. . , gn ={s j}j=

1

2nstabilizer

V = {1,... ,n} E V V

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graph state



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graph state



Applications:Quantum computation based on measures (cluster states)Quantum correction of errorsSecret sharing protocolsProof of Bell's Theorem (e.g.; all-versus-nothing)Reduction of communication complexityTeletransportation...

Theory of entaglement.

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graph states in REAL LIFE (lab)?

6-qubits 4-photons graph states

Now, we can:

8-qubits 4-photons graph states

10- qubits 5 -photons graph states

n-qubits n-photons graph states up to n = 6 .

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graph states in REAL LIFE (lab)?

Futur:

30 qbits 10 teraflops

1000 clasical computers

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graph state? Constructive definition

G V ,E G

STEP 1

= 1 20 1

Asociate each vertex with a qubit in the state:

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What is a graph state? CONSTRUCTIVE .

G V ,E G

C Z = 00 00 01 01 10 10 11 11 =

1 0 0 0

0 1 0 0

0 1 1 0

0 1 0 1

STEP 2Apply, for each edge, controlled-Z to the qbits:

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What is a graph state? CONSTRUCTIVE .

G V ,E G

1 2

3 4

1

2

3

4

G

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Graph states equivalence

LU (local unitary) equivalence: LU U =U 1 U n =U

Graph states are entaglement-equivalent iff are LU-equivalent.

LC (local Clifford) equivalence:

LC C =C 1

C n ,CiH ,S =C

L U

L C

conjecture LU LC: H = 1 21 11 1

, S= 1 00 i

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Graph states equivalence

L U L C Conjecture LU LC: FALSEZ. Ji, J. Chen, Z. Wei y M. Ying; arXiv: 0709.1266

But

True for small n. Small known counterexamples: 27 qubits. Probably inferior limit .Z. Ji, J. Chen, Z. Wei y M. Ying; arXiv: 0709.1266

True for some classes of graph states.

M. Van den Nest et al ., Phys. Rev. A 71, 062323 (2005)

B. Zeng et al ., Phys. Rev. A 75, 032325 (2007)

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LC equivalence and local complementation

Theorem (M. Van den Nest et al ., Phys. Rev. A 69 022316 (2004) ):

G LCG' There exists a sequence of local

complementation operator that maps

graph G into graph G .

G'

G G '

LC

LC LC LC LC

G

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LC equivalence and local complementation

j j

Theorem (M. Van den Nest et al ., Phys. Rev. A 69 022316 (2004) ):

G LCG' There exists a sequence of local

complementation operator that maps graph G into graph G .

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RBIT (LC class)

LC equivalence and local complementation. Orbit.

LC equivalence class. ORBIT:

REPRESENTANTIVE?

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ORBIT

LC equivalence and local complementation. Orbit.

LC equivalence class = orbit:

#Orbit: 802 non isomorphgraphs

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Entaglements in Graph states. Classification

n

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Entaglements in Graph states. Classification

n

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Entaglements in Graph states. Classification.

n

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Sort criteria n

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Sort criteria n

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Sort criteria

SCHMIDT RANKS

Rank index (Schmidt rank of all bipartite splits).

H A H B

=i= 1 R

i

i

A

i

B iC , i= 1, ,R

i j

H j

, j= A,B

r = Rm n= SR A G

RI p= p p , , 1

p =[ j p]j= p1

j p # SR A G = j , with A= p.

RANK INDEX

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Schmidt measure bounds

MAXIMUM SCHMIDT RANK

SRm x G E S G PP G V C G

PAULI PERSISTENCE

MINIMAL VERTEX COVER

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Graph states entaglement. Classifition

n

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Graph states entaglements. Clasification

n=8:

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Entrelazamiento en Graph states. Clasificacin

NO DISTINCTION

NO EQUIVALENT CLASS!

ATTENTION:PROBLEM!!!!

Solved (n

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Sort criteria n9)

EXPERIMENTAL corresponds to: Minimum #controlled-Z gates . Minimum preparation deepth (time units).

n Download Size8 entanglement8 101 graphs9 entanglement9 440 graphs10 entanglement10 3132 graphs (509 KB)11 entanglement11.bz2 40,457 graphs (1.2 MB compressed)12 entanglement12.bz2 1,274,068 graphs (45 MB compressed)

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Graph states entaglements. Clasification

n

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Cooking graph states

A few invariants for 9

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Cooking graph states

If we need prepare a GRAPH STATE

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CONCLUSIONS

Extended up to 12 qubits g raph states entaglement classification.

Best (in the sense of minimum time preparation and/or minimum work)

representative of each new 1300000+ LC equivalence class.

An (almost) complete sort criteria and new invariants for labeling class.

Help to new proofs (AVN type) of Bell's theorem .

Research of non-locality.

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CONCLUSIONS

Procedure for the optimal preparationof 1.65 101.65 10 1111 graph states with up to 12 qubits

Procedure for the optimal preparationof 1.65 101.65 10 1111 graph states with up to 12 qubits

OPTIMAL:minimum number of entangling gates

minimum number of time steps

OPTIMAL:minimum number of entangling gates

minimum number of time steps

Main goal:Main goal: to provide in a single package all thetools needed to rapidly identify the entanglement classthe target state belongs to, and then easily find thecorresponding optimal circuit(s) of entangling gates, andfinally the explicit additional one-qubit gates needed toprepare the target

Main goal:Main goal: to provide in a single package all thetools needed to rapidly identify the entanglement classthe target state belongs to, and then easily find thecorresponding optimal circuit(s) of entangling gates, andfinally the explicit additional one-qubit gates needed toprepare the target

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PUBLISHED

Arxiv: http://arxiv.org/abs/1011.5464Nov 24, 2010

Arxiv: http://arxiv.org/abs/1011.5464Nov 24, 2010

This workhas been submitted to Physical Review A

Nov 25, 2010

This workhas been submitted to Physical Review A

Nov 25, 2010

Slides: http://slidesha.re/eEauJgSlides: http://slidesha.re/eEauJg
http://arxiv.org/abs/1011.5464http://arxiv.org/abs/1011.5464http://arxiv.org/abs/1011.5464

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Thanks for your attention!

Gracias por escucharme!

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