Graph States EPIT Mehdi Mhalla (Calgary, Canada) Simon Perdrix (Grenoble, France)

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1 Grph Sttes EPIT 2005 Mehdi Mhll (Clgry, Cnd) Simon Perdrix (Grenole, Frne)

2 Grph Stte: Introdution A grph-sed representtion of the entnglement of some (lrge) quntum stte. Verties: quits Edges: entnglement etween the quits Grph Sttes p.1

3 Grph Sttes Introdution Complexity of grph stte preprtion Grph Sttes p.2

4 A Construtive Definition For given grph G, preprtion of the orresponding grph stte G onsists: in ssoiting with eh vertex quit in the stte = 0 1 2, then Grph Sttes p.3

5 A Construtive Definition For given grph G, preprtion of the orresponding grph stte G onsists: in ssoiting with eh vertex quit in the stte = 0 1 2, then in pplying, for eh edge etween two quits nd, the unitry trnsformtion C Z on the quits nd C Z = = Grph Sttes p.4

6 A Construtive Definition For given grph G, preprtion of the orresponding grph stte G onsists: in ssoiting with eh vertex quit in the stte = 0 1 2, then in pplying, for eh edge etween two quits nd, the unitry trnsformtion C Z on the quits nd. Grph Sttes p.5

7 A Construtive Definition For given grph G, preprtion of the orresponding grph stte G onsists: in ssoiting with eh vertex quit in the stte = 0 1 2, then in pplying, for eh edge etween two quits nd, the unitry trnsformtion C Z on the quits nd. Grph Sttes p.6

8 A Construtive Definition For given grph G, preprtion of the orresponding grph stte G onsists: in ssoiting with eh vertex quit in the stte = 0 1 2, then in pplying, for eh edge etween two quits nd, the unitry trnsformtion C Z on the quits nd. Grph Sttes p.7

9 A Construtive Definition For given grph G, preprtion of the orresponding grph stte G onsists: in ssoiting with eh vertex quit in the stte = 0 1 2, then in pplying, for eh edge etween two quits nd, the unitry trnsformtion C Z on the quits nd. Grph Sttes p.8

10 A Construtive Definition For given grph G, preprtion of the orresponding grph stte G onsists: in ssoiting with eh vertex quit in the stte = 0 1 2, then in pplying, for eh edge etween two quits nd, the unitry trnsformtion C Z on the quits nd. G Grph Sttes p.9

11 Entnglement Property: Two quntum sttes "hve the sme entnglement" iff they re LU-equivlent. Definition [LU-equivlene] : φ LU ψ iff there exists lol unitry trnsformtion U (i.e U = U 1... U n, where eh U i is 1-quit unitry) suh tht φ = U ψ. Definition [LC-equivlene] : φ LC ψ iff there exists lol Clifford trnsformtion C (i.e. C = C 1... C n, where C i H,S ) suh tht φ = C ψ. Grph Sttes p.10

12 Properties Grph-sed representtion of entnglement is not unique: G,G / G LU G nd G G LU Grph Sttes p.11

13 Properties Conjeture: G LU G G LC G Theorem [Vn den Nest, 2004]: G LC G iff there exists sequene of lol omplementtions whih trnsforms G into G. Lol Complementtion ording to : G = G K N() Grph Sttes p.12

14 Properties Conjeture: G LU G G LC G Theorem [Vn den Nest, 2004]: G LC G iff there exists sequene of lol omplementtions whih trnsforms G into G. Lol Complementtion ording to : G = G K N() Grph Sttes p.13

15 Properties Conjeture: G LU G G LC G Theorem [Vn den Nest, 2004]: G LC G iff there exists sequene of lol omplementtions whih trnsforms G into G. Lol Complementtion ording to : G = G K N() Grph Sttes p.14

16 Properties Conjeture: G LU G G LC G Theorem [Vn den Nest, 2004]: G LC G iff there exists sequene of lol omplementtions whih trnsforms G into G. Lol Complementtion ording to : G = G K N() Grph Sttes p.15

17 Clss of equivlene d d d d Grph Sttes p.16

18 Miniml degree under lo. ompl. Definition [Miniml Degree]: For given grph G = (V,E), δ(g) = min v V δ(v) Definition [Miniml Degree under Lol Complementtion]:. δ lo (G) = min G LC G δ(g ) Grph Sttes p.17

19 Grph Sttes Introdution Complexity of grph stte preprtion Grph Sttes p.18

20 Grph Stte Preprtion An lgorithm of preprtion inputs grph G, nd outputs quntum iruit C G suh tht C G LC G. Complexity of Grph Stte Preprtion: T : depth of C G S: width of C G Remrk: The lssil prt of the preprtion must e relized in polynomil time. Grph Sttes p.19

21 Algorithm of Preprtion d d Grph Sttes p.20

22 Algorithm of Preprtion d d d d Grph Sttes p.21

23 Algorithm of Preprtion d d d d Grph Sttes p.22

24 Algorithm of Preprtion d d T = χ Limittion: Two opertions n e relized in prllel only if they t on different quntum systems. Edge-hromti numer χ : miniml numer of olors needed to edge-olor grph suh tht two djent edges hve different olors. Complexity: T = O(χ ) = O( ), ( χ 1) S = n. Grph Sttes p.23

25 Expnsion - Contrtion v Expnsion L R L v R Contrtion Property: Any grph G n e expnded into G, suh tht (G ) 3 u w Grph Sttes p.24

26 Grph Stte Contrtion L v u w Contrtion v R L R If u nd w re X mesured Grph Sttes p.25

27 Grph Stte Contrtion L v u w Contrtion v R L R If u nd w re X mesured Algorithm to prepre grph stte G : (Clssil prt) G is expnded into G suh tht (G ) 3, G is prepred with the previous lgorithm: T 1 = O( (G )) = O(1), All the nillry quits re X-mesured in prllel: T 2 = 1. Grph Sttes p.26

28 Grph Stte Contrtion L v u w Contrtion v R L R If u nd w re X mesured Algorithm to prepre grph stte G : (Clssil prt) G is expnded into G suh tht (G ) 3, G is prepred with the previous lgorithm: T 1 = O( (G )) = O(1), All the nillry quits re X-mesured in prllel: T 2 = 1. Complexity: T = T 1 T 2 = O(1) S = O(m), where m is the numer of edges of the grph to prepre. Grph Sttes p.27

29 Trdeoff For given 2 k m/n, Algorithm to prepre grph stte G : (Clssil prt) G is expnded into G suh tht (G ) < k, G is prepred with the previous lgorithm: T 1 = k O(1), All the nillry quits re X-mesured in prllel: T 2 = 1. Complexity: T = k O(1) S = O(m/k n) = O(m/k) TS = O(m) Grph Sttes p.28

30 Mesurement-sed preprtion Grph stte preprtion without unitry trnsformtion without nillry quit (S = n) Size of the projetive mesurements required to relize this preprtion. Lemm [Mhll, Perdrix]: Any grph stte G n e prepred without nillry quit, y mens of projetive mesurements on t most δ lo (G) 1 quits. Grph Sttes p.29

31 Lower Bound Theorem [Mhll, Perdrix]: For ny G, mesurement-sed preprtion of G, without nillry quit, requires mesurements on t lest δ lo (G) 1 quits. Proof: Chrteriztion of the miniml degree under lol omplementtion, Contrdition: ssuming tht the lst mesurement of the preprtion is on δ lo (G) quits (insted of δ lo (G) 1) Grph Sttes p.30

32 σ-gme Given G = (V,E), nd D V, the verties in D re leled with 0 or 1. A onfigurtion is desription of the lels: (000), (101), (011). Plying on vertex outside D flips the inside djent verties. Grph Sttes p.31

33 σ-gme Given G = (V,E), nd D V, the verties in D re leled with 0 or 1. A onfigurtion is desription of the lels: (000), (101), (011). Plying on vertex outside D flips the inside djent verties. Grph Sttes p.32

34 σ-gme Given G = (V,E), nd D V, the verties in D re leled with 0 or 1. A onfigurtion is desription of the lels: (000), (101), (011). Plying on vertex outside D flips the inside djent verties. Grph Sttes p.33

35 Chrteriztion of δ lo (G) Lemm [Mhll, Perdrix]: Given G = (V,E), for ny D V, if D δ lo (G), ll the 2 D onfigurtions n e rehed. Grph Sttes p.34

36 Lst Mesurement Given G = (V,E), Assume the lst mesurement O of the preprtion is on δ lo (G) quits. Grph Sttes p.35

37 Let s ply with Grph stte Simultion of σ-gme: Propriétés: D V the set of the lst mesured quits. D = δ lo (G). The onfigurtion is 0 = (0... 0) (onfigure = signture of the stilizer of G ). some Puli opertions X re pplied outside D, these pplitions modify the onfigurtion s in σ-gme. onfigurtion. the quits outside D re mesured ording to Z. quntum stte φ on D quits.,, if, φ φ = 0., O φ = φ. Grph Sttes p.36

38 Let s ply with Grph stte Simultion of σ-gme: Propriétés: D V the set of the lst mesured quits. D = δ lo (G). The onfigurtion is 0 = (0... 0) (onfigure = signture of the stilizer of G ). some Puli opertions X re pplied outside D, these pplitions modify the onfigurtion s in σ-gme. onfigurtion. the quits outside D re mesured ording to Z. quntum stte φ on D quits.,, if, φ φ = 0., O φ = φ. Sine ll the 2 D onfigurtions n e rehed, { φ } is sis. O = Id Grph Sttes p.37

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