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"Measurement-based quantum computation10th Canadian Summer School on QIDan BrowneDept. of Physics and AstronomyUniversity College LondonThursday, 29 July 2010What is a quantum computer?Thursday, 29 July 2010The one-way quantum computerA multi-qubit entangled \u201Cresource state\u201DE.g. cluster stateSingle-qubit measurementsChoice of bases specifycomputationAdaptive (some bases depend upon previous outcomes)+=A universal quantum computerThursday, 29 July 2010Overview of Lectures\u26051. Cluster states and graph states I\u2605What are they? Basic properties.\u26052. The one-way quantum computer\u2605What is the model? How does it work?\u26053. Cluster states and graph states II\u2605Stabilizer formalism and graphical representation\u26054.  Cluster states and graph states III\u2605How do we build them?\u26055. Beyond cluster states (If time permits)\u2605(other models of MBQC)\u26056. (Robert Raussendorf) Fault tolerant MBQCThursday, 29 July 2010Pauli group and Clifford Group\u2022Pauli group: \u2022Set of all n-fold tensor products of X, Y, Z and I, with pre-factors +1, -1. +i, -i for group closure.\u2022Clifford group:\u2022\u201CNormalizer\u201D of Set of unitaries C such that Equiv,:\u201CMaps Pauli group onto Pauli group\u201DnnC \u03C3kC\u2020= \u03C3j\u2200 \u03C3k\u2208n\u03C3j\u2208nC \u03C3k= \u03C3jC =( C \u03C3kC \u2020 ) CThursday, 29 July 20102-qubit gates and universal q.c.J J J J JJ J J J JJ J J J JThursday, 29 July 20102-qubit gates and universal q.c.J J J J JJ J J J JJ J J J JThursday, 29 July 20102-qubit gates and universal q.c.J J J J JJ J J J JJ J J J JThursday, 29 July 2010Etching with z measurementsV OLUME 86, N UMBER 22 P H Y S I C A L R E V I E W L E T T E R S 28 M AY 2001A One-W ay Quantum ComputerRober t Raussendorf and Hans J. Brie gelTheor et isc he Physik, L udwig-Maximilians-Univer sit\u00E4t M\u00FCnc hen, Ger m any( Recei v ed 25 October 2000 )W e present a scheme of quantum computation that consists entirely of one-qubit measurements on apar ticular class of entangled states, the cluster states. The measurements are used to imprint a quantumlogic circuit on the state, thereby destro ying its entanglement at the same time. Cluster states are thusone-w ay quantum computers and the measurements form the program.DOI: 10.1103 / Ph ysRe vLett.86.5188 P A CS numbers: 03.67.Lx, 03.65.UdA quantum computer promises efficient processing ofcer tain computational tasks that are intractable with clas-sical computer technology [1]. While basic principles of aquantum computer ha v e been demonstrated in the labora-tor y [2], scalability of these systems to a lar ge number ofqubits [3], essential for practical applications such as theShor algorithm, represents a formidable challenge. Mostof the current e xperiments are designed to implement se-quences of highly controlled interactions between selectedpar ticles (qubits), thereby follo wing models of a quan-tum computer as a (sequential) netw ork of quantum logicgates [4,5].Here we propose a different model of a scalable quan-tum computer . In our model, the entire resource for thequantum computation is pro vided initially in the form ofa specific entangled state (a so-called cluster state [6]) ofa lar ge number of qubits. Information is then written ontothe cluster , processed, and read out from the cluster byone-par ticle measurements only . The entangled state ofthe cluster thereby serv es as a uni v ersal \u201Csubstrate\u201D for an yquantum computation. Cluster states can be created effi-ciently in an y system with a quantum Ising-type interaction(at v er y lo w temperatures) between tw o-state par ticles ina lattice configuration.W e consider tw o- and three-dimensional arrays ofqubits that interact via an Ising-type ne xt-neighbor in-teraction [6] described by a Hamiltonian HintH33527 g H20849 t H20850 3PH20855 a ,a0H208561 1sH20849 a H20850z21 2sH20849 a0H20850z2H11061 214g H20849 t H20850PH20855 a ,a0H20856sH20849 a H20850zsH20849 a0H20850z[7] whosestrength g H20849 t H20850 can be controlled e xternally . A possiblerealization of such a system is discussed belo w . A qubit atsite a can be in tw o states j0 H20856aH11013 j0 H20856z ,aor j1 H20856aH11013 j1 H20856z ,a,the eigenstates of the P auli phase fl ip operator sH20849 a H20850zH20851sH20849 a H20850zji H20856aH33527 H208492 1 H20850iji H20856aH20852 . These tw o states form the compu-tational basis. Each qubit can equally be in an arbitrar ysuperposition state a j0 H20856 1b j1 H20856 , ja j21 jb j2H33527 1 . F orour purpose, we initially prepare all qubits in the su-perposition j1 H20856 H33527 H20849 j0 H20856 1 j1 H20856H20850H20862p2 , an eigenstate of theP auli spin fl ip operator sxH20851sxj6 H20856 H33527 6 j6 H20856H20852 . Hintisthen switched on for an appropriately chosen finite timeinterv al T , whereRT0dt g H20849 t H20850 H33527 p , by which a unitar ytransformation S is realized. Since Hintacts uniformly onthe lattice, entire clusters of neighboring par ticles becomeentangled in one single step. The quantum state jF H20856C,the state of a cluster H20849 C H20850 of neighboring qubits, which isthereby created pro vides in adv ance all entanglement thatis in v olv ed in the subsequent quantum computation. It hasbeen sho wn [6] that the cluster state jF H20856Cis characterizedby a set of eigen v alue equationssH20849 a H20850xOa0[ ngbh H20849 a H20850sH20849 a0H20850zjF H20856CH33527 6 jF H20856C, (1)where ngbh H20849 a H20850 specifies the sites of all qubits that inter-act with the qubit at site a [ C . The eigen v alues are de-termined by the distrib ution of the qubits on the lattice.The equations (1) are central for the proposed computationscheme. As an e xample, a measurement on an indi vidualqubit of a cluster has a random outcome. On the otherhand, Eqs. (1) imply that an y tw o qubits at sites a , a0[ Ccan be projected into a Bell state by measuring a subset ofthe other qubits in the cluster . This proper ty will be used tod e fi n e q u a n tu m c h a n n e ls th a t a llo w us to p r o p a g a te q u a n -tum information through a cluster .W e sho w that a cluster state jF H20856Ccan be used as a sub-strate on which an y quantum circuit can be imprinted byone-qubit measurements. In Fig. 1 this scheme is illus-trated. F or simplicity , we assume that in a cer tain re gionof the lattice each site is occupied by a qubit. This re-quirement is not essential as will be e xplained belo w [see(d)]. In the first step of the computation, a subset ofqubits is measured in the basis of szwhich effecti v elyremo v es them. In Fig. 1 these qubits are denoted by \u201C \u00D8 .\u201Dq u a n t u m  g a t einformation flo wFIG. 1. Quantum computation by measuring tw o-state par ti-cles on a lattice. Before the measurements the qubits are in thecluster state jF H20856Cof (1). Circles \u00D8 symbolize measurements ofsz, v er tical arro ws are measurements of sx, while tilted arro wsrefer to measurements in the x - y plane.5188 0031 - 9007 H20862 01 H20862 86(22) H20862 5188(4) $15.00 \u00A9 2001 The American Physical Society\u2022Figure 1in H. J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86, 5188 (2001)z-measurements\u201Cquantum wire\u201Dfor logical singlequbitThursday, 29 July 2010Measurement-patterns for gates in the one-way model5B. A u n iv e r s a l s e t o f q u a n t u m g a t e sTo p r o v id e s o m e t h in g d e fi n it e t o d is c u s s r ig h t fr o m t h ebe g in n in g , w e n o w g iv e th e p r o c e d u r e s o f h o w to r e a liz eaC N O T -gate and ag eneral one-qubit rotation viao ne-qu b it m e a s u r e m e n t s o n a c lu s t e r s t a t e . T h e e xp la n a t io nof w h y an d h o w t h e s e gat e s w ork w ill b e gi v e n in S e c -ti o n I I G .(a )123 4 5 679 10 11 12 13 14 158controltargetYYYYYYY XXXXXXCN O T- g a t e(b )123 4 5_X!_+\"+#+_(c )123 4 5X X X+#_ge n e r al r ot at ion z -r o t a t io n(d )123 4 5X YY Y(e )123 4 5X X XYHa d a m a r d -g a t e pi / 2- p h as e gat eFI G . 2 : R ea liza t io n o f el em en t a r y q u a n t u m g a t es o n t h e Q CC.Ea c h s q u a r e r e p r e s e n t s a la t t ic e q u b it . T h e s q u a r e s in t h eex t r em e lef t co lu m n m a r k ed w it h w h it e ci r cl es d en o t e t h e in -put q ubi t s , t ho s e in t he r ig h t -m o s t c o lum n de no t e t he o ut putqu b it s .AC N O T -gate can be realized on a cluster state of 15qu b it s , a s s h o w n in F ig . 2 . A ll m e a s u r e m e n t s c a n b epe r fo r m e d s im u lta n e o u s ly . T h e p r o c e d u r e to r e a liz e aCN O T- g a t e o n a c lu s t e r wi t h 1 5 q u b it s a s d is p la y e d inFi g . 2 isPr o c e d u r e 1 Re a liz a t io n o f a C N O T -g a t e a c t ing o n atw o -q u b it s t a t e |\u03C8in\u3009.1. Pre p are t h e s t at e|\u03A8in\u3009C15= |\u03C8in\u30091 ,9\u2297parenleftBigcirclemultiplytexti\u2208 C15\\{ 1 ,9 }|+ \u3009iparenrightBig.2. En t an gl e t h e 15 q u b it s of t h e c lu s t e r C15vi a t h euni t a r y o p e r a t io n S( C15).3. M e as u re al l q u b it s of C15ex cep t fo r t h e o u t p u tqu b it s 7 , 1 5 ( fo llo w in g t h e la b e lin g in F ig . 2 ) . T h eme a s u r e me n t s c a n b e p e r fo r me d s im u lt a n e o u s ly .Qu b it s 1 , 9 , 1 0 , 1 1 , 1 3 , 1 4 a r e m e a s u r e d in t h e \u03C3x-ei g en b a s is a n d q u b it s 2 -6 , 8 , 1 2 in t h e \u03C3y-e ig e n b a s is .De p e n d e n t o n t h e m e a s u r e m e n t r e s u lt s , t h e fo llo w in ggat e is t h e re b y re al iz e d :UprimeCN O T= U\u03A3 ,C N O TCN O T ( c, t ) . (2 3 )Th e r e in t h e b y p r o d u c t o p e r a t o r U\u03A3 ,C N O Tha s t he fo r mU\u03A3 ,C N O T= \u03C3( c )x\u03B3( c )x\u03C3( t )x\u03B3( t )x\u03C3( c )z\u03B3( c )z\u03C3( t )z\u03B3( t )z, wi t h\u03B3( c )x= s2+ s3+ s5+ s6\u03B3( t )x= s2+ s3+ s8+ s10+ s12+ s14\u03B3( c )z= s1+ s3+ s4+ s5+ s8+ s9+ s11+1\u03B3( t )z= s9+ s11+ s13.(2 4 )Th e r e in , t h e sire p re s e n t t h e m e as u re m e n t ou t c om e s sion t h e q u b it s i. Th e e x p r e s s io n ( 2 4 ) is m o d ifi e d if r e -dunda n t c lus t e r q ubi t s a r e pr e s e n t a nd/ o r if t he c lus t e rs t a t e o n wh ic h t h e CN O T g a t e is r e a liz e d is s p e c ifi e d b yas et { \u03BAa} di ff er en t fr o m ( 1 2 ) , s ee S ect io n I I C . T h is co n -cl u d es t h e p r es en t a t io n o f t h e C N O T g a t e, t h e p r o o f o fit s fu n c t io n in g is g iv e n in S e c t io n I I G .An a r b itr a r y r o ta ti o n URo t\u2208 SU (2 ) c a n b e r e a liz e don a c h ai n of 5 q u b it s . C on s id e r a rot at ion in it s Eu le rre p re s e n t at ionURo t[\u03BE , \u03B7 , \u03B6 ]= Ux[\u03B6 ]Uz[\u03B7 ]Ux[\u03BE ], (2 5 )wh e r e t h e r o t a t io n s a b o u t t h e x -a nd z -a x is a r eUx[\u03B1 ]= expparenleftBig\u2212 i\u03B1\u03C3x2parenrightBigUz[\u03B1 ]= expparenleftBig\u2212 i\u03B1\u03C3z2parenrightBig.(2 6 )In it ia lly , t h e fi r s t q u b it is p r e p a r e d in s o m e s t a t e |\u03C8in\u3009,wh ic h is t o b e r o t a t e d , a n d t h e o t h e r q u b it s a r e p r e p a r e din |+ \u3009.A fter the 5 qubits are entangled by the unitarytr a n s fo r m a ti o n S ,t he state |\u03C8in\u3009 ca n b e r o t a t ed b y m ea -su r in g q u b it s 1 t o 4 . A t t h e sa m e t im e , t h e st a t e is a lsos w a p p e d t o s it e 5 . Th e q u b it s 1 .. 4a re m easured in ap-pr o pr ia t e ly c ho s e n ba s e sBj( \u03D5j)=braceleftbigg|0 \u3009j+ ei\u03D5j|1 \u3009j\u221A2,|0 \u3009j\u2212 ei\u03D5j|1 \u3009j\u221A2bracerightbigg, (2 7 )wh e r e b y t h e m e a s u r e m e n t o u t c o m e s sj\u2208 { 0 , 1 } fo r j =1 .. 4a re obtained. H ere, sj=0 m eans that qubit j ispr o je c t e d in t o t he fir s t s t a t e o f Bj( \u03D5j). In (2 7 ) th e b a -si s st a t e s o f a ll p o ssi b le m e a su r e m e n t b a se s lie o n t h eeq u a t o r o f t h e B lo c h s p h er e, i.e. o n t h e in t er s ect io n o fth e B lo c h s p h e r e w ith th e x -y -p la n e . T h e r e fo r e , t h e m e a -su r e m e n t b a si s fo r q u b it j ca n b e s p eci fi ed b y a s in g le p a -ram e t e r, t h e m e as u re m e n t an gl e \u03D5j.T he m easurem entdi r e c t io n o f q ubi t j is t h e v e c t o r o n t h e B lo c h s p h e r ewh ic h c o r r e s p o n d s t o t h e fi r s t s t a t e in t h e m e a s u r e m e n tba s is Bj( \u03D5j). T h u s , th e m e a s u r e m e n t a n g le \u03D5jis t h e a n -gl e b e tw e e n t h e m e as u re m e n t d ire c t ion at q u b it j an dth e p o s iti v e x -a x is . In s u m m a r y , t h e p r o c e d u r e t o r e a liz ean arb it rary rot at ion URo t[\u03BE , \u03B7 , \u03B6 ], s p e c ifi e d b y it s E u le ran gl e s \u03BE , \u03B7 , \u03B6 ,i s this:R. Raussendorf, D. E. Browne and H.J. Briegel,  Phys. Rev. A 68: 022312 (2003)Thursday, 29 July 2010LC-orbit of local Clifford equivalent statesE NT ANGL E M E NT I N G RA P H S TA T E S A N D I T S A PPL I C A T I O N S 21th e g r a p h u n c h a n g e d :(4 1 ) \u03C4a: G mapsto\u2192 \u03C4a( G ): = G + Na.Wi t h t h i s n o t a t i o n t h e f o l l o w i n g r e s u l t c a n b e s t a t e d [ 3 9 , 6 0 ]:Pr o p o s i t i o n 5 ( LC - r u le ) . By l o c a l c o m p l e m e n t a t i o n o f a g r a p h G at s om e v e r te x a \u2208 V oneobt ai ns an L C - e qui v al e nt gr aph s tat e | \u03C4a( G ) \u3009 :(4 2 ) | \u03C4a( G ) \u3009 = U\u03C4a( G ) | G \u3009 ,wh e r e(4 3 ) U\u03C4a( G )= e\u2212 ipi4\u03C3axeipi4\u03C3Naz\u221DradicalbigKais a lo c a l C lif fo r d u n ita r y . F u r th e r m o r e , tw o g r a p h s ta te s | G \u3009 and | Gprime\u3009 ar e L C - e qui v al e nt iff theco r r es p o n d i n g g r a p h s a r e r el a t ed b y a s eq u en ce o f l o ca l co m p le m e n ta tio n s , i. e . Gprime= \u03C4a1\u25E6 ... \u25E6\u03C4an( G ) fo r s o m e a1,...,an\u2208 V .23412341234123412341234123412341234123412341No. 1 No. 2 No. 3 No. 4No. 5 No. 6 No. 7 No. 8No. 9 No. 10 No. 11Apply LC ! RuleFi g . 4 . \u2013 A n e x a m p l e f o r a s u c c e s s i v e a p p l i c a -tio n o f th e L C - r u le , w h ic h e x h ib its th e w h o leeq u i v al en ce cl as s as s o ci at ed w i t h g r ap h N o . 1 .Th e r u l e i s s u c c e s s i v e l y a p p l i e d t o t h e v e r t e x ,wh i c h i s c o l o r e d r e d i n t h e fi g u r e .Fi g . 4 d e p i c t s a n e x a m p l e f o r s u c h a s u c c e s s i v eap p l i cat i o n o f t h e L C - r u l e. S t ar t i n g w i t h t h e fi r s tgr a ph t he c om pl e t e or bi t c a n be obt a i ne d by a p-pl yi ng t he L C - r ul e t o t he v e r t i c e s i n t he pr e c e di nggr a ph t ha t a ppe a r a bo v e t he a r r o w of t he f ol l o w i ngdi a gr a m :No . 13\u2212\u2212 \u2212 \u2212\u2192 No . 22\u2212\u2212 \u2212 \u2212\u2192 No . 33\u2212\u2212 \u2212 \u2212\u2192No . 41\u2212\u2212 \u2212 \u2212\u2192 No . 53\u2212\u2212 \u2212 \u2212\u2192 No . 61\u2212\u2212 \u2212 \u2212\u2192No . 73\u2212\u2212 \u2212 \u2212\u2192 No . 84\u2212\u2212 \u2212 \u2212\u2192 No . 91\u2212\u2212 \u2212 \u2212\u2192No . 102\u2212\u2212 \u2212 \u2212\u2192 No . 11Pr o o f o f Pr o p o s i t i o n 5 :Le t G be a gr a ph w i t h c or r e l a t i on ope r a t or s Kban d Gprime= \u03C4a( G ) th e c o r r e s p o n d in g g r a p h u n d e rlo c a l c o m p le m e n ta tio n a t v e r te x a wi t h c o r r e l a t i o n o p e r a t o r s Kprimeb.F o r c \u2208 V \\ Nawe fi n d(4 4 ) U\u03C4aKc( U\u03C4a)\u2020= Kc= Kprimec.Fo r b \u2208 Na,w e c o m p u teU\u03C4aKb( U\u03C4a)\u2020=( \u2212 i \u03C3ax)parenleftbigi \u03C3bzparenrightbig\u03C3bx\u03C3az\u03C3Nb\\ az= \u03C3ax\u03C3Naz\u00B7 \u03C3bx\u03C3Nb+ Naz= Kprimea\u00B7 Kprimeb.E NT ANGL E M E NT I N G RA P H S TA T E S A N D I T S A PPL I C A T I O N S 21th e g r a p h u n c h a n g e d :(4 1 ) \u03C4a: G mapsto\u2192 \u03C4a(G ): = G + Na.Wi t h t h i s n o t a t i o n t h e f o l l o w i n g r e s u l t c a n b e s t a t e d [ 3 9 , 6 0 ]:Pr o p o s itio n 5 (LC -r u le ) . By lo c a l c o m p le m e n ta tio n o f a g ra p h G at s om e v e r te x a \u2208 V oneobt ai ns an L C -e qui v al e nt gr aph s tat e |\u03C4a(G )\u3009:(4 2 ) |\u03C4a(G )\u3009 = U\u03C4a(G ) |G \u3009 ,wh e re(4 3 ) U\u03C4a(G )= e\u2212 ipi4\u03C3axeipi4\u03C3Naz\u221DradicalbigKais a lo c a l C lif fo rd u n ita r y. F u r th e r m o re , tw o g ra p h s ta te s |G \u3009 and |Gprime\u3009 ar e L C -e qui v al e nt iff theco r res p o n d in g g ra p h s a re rel a ted b y a s eq u en ce o f lo ca l co m p le m e n ta tio n s , i. e . Gprime= \u03C4a1\u25E6 ... \u25E6\u03C4an(G ) fo r s o m e a1,...,an\u2208 V .23412341234123412341234123412341234123412341No. 1 No. 2 No. 3 No. 4No. 5 No. 6 No. 7 No. 8No. 9 No. 10 No. 11Apply LC ! RuleFi g . 4 . \u2013 A n e x a m p l e f o r a s u c c e s s iv e a p p l i c a -tio n o f th e L C - r u le , w h ic h e x h ib its th e w h o leeq u iv al en ce cl as s as s o ci at ed w i t h g r ap h N o . 1 .Th e r u l e i s s u c c e s s iv e l y a p p l i e d t o t h e v e r t e x ,wh i c h i s c o l o r e d r e d i n t h e fi g u r e .Fi g . 4 d e p i c t s a n e x a m p l e f o r s u c h a s u c c e s s iv eap p l i cat i o n o f t h e L C - r u l e. S t ar t i n g w i t h t h e fi r s tgr a ph t he c om pl e t e or bi t c a n be obt a i ne d by a p-pl yi ng t he L C - r ul e t o t he v e r t i c e s i n t he pr e c e di nggr a ph t ha t a ppe a r a bo v e t he a r r o w of t he f ol l o w i ngdi a gr a m :No . 13\u2212\u2212 \u2212 \u2212\u2192 No . 22\u2212\u2212 \u2212 \u2212\u2192 No . 33\u2212\u2212 \u2212 \u2212\u2192No . 41\u2212\u2212 \u2212 \u2212\u2192 No . 53\u2212\u2212 \u2212 \u2212\u2192 No . 61\u2212\u2212 \u2212 \u2212\u2192No . 73\u2212\u2212 \u2212 \u2212\u2192 No . 84\u2212\u2212 \u2212 \u2212\u2192 No . 91\u2212\u2212 \u2212 \u2212\u2192No . 102\u2212\u2212 \u2212 \u2212\u2192 No . 11Pr o o f o f Pr o p o s itio n 5 :Le t G be a gr a ph w i t h c or r e l a t i on ope r a t or s Kban d Gprime= \u03C4a(G ) th e c o r r e s p o n d in g g r a p h u n d e rlo c a l c o m p le m e n ta tio n a t v e r te x a wi t h c o r r e l a t i o n o p e r a t o r s Kprimeb.F or c \u2208 V \\ Nawe fi n d(4 4 ) U\u03C4aKc(U\u03C4a)\u2020= Kc= Kprimec.Fo r b \u2208 Na,w e co m p u teU\u03C4aKb(U\u03C4a)\u2020=( \u2212 i\u03C3ax)parenleftbigi\u03C3bzparenrightbig\u03C3bx\u03C3az\u03C3Nb\\ az= \u03C3ax\u03C3Naz\u00B7 \u03C3bx\u03C3Nb+ Naz= Kprimea\u00B7 Kprimeb.\u2022Figure 4 in M. Hein, W. D\u00FCr, J. Eisert, R. Raussendorf, M. Van den Nest, H.-J. Briegel, quant-ph/0602096 (Varena lectures)Thursday, 29 July 2010Counter-example to LU-LC conjecture11eq u iv a len t b u t n o t LC eq u iv a len t gr a p h st a tes c a n di ffer o n ly in o n e d g e . N o te th a t th e s im p lifi-ca tio n o f Q (x ) di sc u ssed a b o v e do es n o t si m p lify o r c h a n ge th e fo r m o f th e fo ur s ta b iliz e r s ta te sin F ig . 2 .FI G . 3 : C o r r e s p o n d in g G r a p h S ta te sV. C O N C L U S I O NIn s u m m a r y , w e h a v e s h ow n th a t th e L U -L C c on je c tu re is fa ls e b yg ivin g exp licit cou n terex-am p le s o f it . It is als o c le ar th at in o r d e r to d is p r o v e th e c o n je c tu r e , w e h a v e to c o n s id e r s ta bi liz e rst a tes w ith th e r a n k o f th e su p p o r t n o less th a n 6 .T his result leads us to rethink about the localeq u iv a len c e p ro b lem o f st a b iliz er s a s m o st o f th e p rev io u s w or k foc u s on pr ov in g th e c on je c tu reto b e tr u e .Th e r a n d o m p ro c e d u re g e n e r a te s c o u n te re x a m p le s o f s iz e 27 an d 35 .T hough not proved, w ebe lie v e th a t 27 is th e s m alle s t p o s s ib le s ize o f c o u n te r e x am p le s o f L U -L C. A lt h o u g h th e L U -L Cco n je ct u r e is n o w d is p r o ve d , w e s till k n o w little a b o u t th e r el a tio n o f LU a n d LC eq u iv a len c e a n ddo n o t h a v e a n ex p lic it u n der st a n di n g o f w h y o r w h en LU eq u iv a le n c e d if fe r s fr o m L C e q u iv a-le n c e . W h e n d =6 , th e r a n d o m g e n e r a tio n p r o c e d u r e c a n fin d a c o u n te r e x a m p le in sec o n ds, b u tit is in fa c t n o t a n e ffic ie n t a lg o r ith m w h e n d is lar g e o r e v e n w h e n d =7 .T hat fact is w e havene v e r o b ta ine d a v a lid c o u nt e r e xa m p le o f d =7 us in g th e r a n d o m p r o c e d ur e . F o r tun a te ly , la r g e rsc a le c o u n ter ex a m p les, in c lu di n g th o se o f d =7 ,h av e b een fo u n d [7] m o tiv ated b y th e ran d o m lygen er a ted c o u n ter ex a m p les.As L U a n d L C e q u iv a le n c e s a re n o w k n o w n to b e d iffe re n t. It is a ls o c h alle n g in g to as k w h e th e rth e r e is a n e ffic ie n t a lg o r ith m d e c id in g L U e q u iv a le n c e fo r s tab ilize r s o r w h e th e r th e r e is a g r ap hth e or e ti c a l in te r pr e ta ti on of L U e q u iv a le n t g r a ph s ta te s . T h is s e e m s to b e d iffic u lt a s lo c a l u n ita r yope r a ti on s a re m u c h le s s lin k e d u p w ith th e s ta b iliz e r for m a lis m .The two graphs (with / without the red edge) represent locally equivalent states, but are not related by the LC rule.Zhengfeng Ji et al, Quantum Inf. Comput., Vol. 10, No. 1&amp;2, 97-108, 2010Thursday, 29 July 2010Universal resources derived via graphical rulesThese regular graphs can all be mapped into square lattices via Pauli measurements - using the graphical rules. Hence all are resources for universal MBQC.M. Van den Nest, et al, New J. Phys. 9 204 (2007).CO N TE N TS 32Fi g . 2 . A c c o rd in g ly, w e fi n a lly o b ta in a sq u a re la ttic e , i.e .,a 2D cluster state. T he totalsp a tia l o v e rh e a d is e ig h t in th a t a u n it sq u a re o n (d ) is o b ta ine d fro m e ig h t he x a g o nsin (a ). S in c e th e o v e rh e a d is c o n sta n t, a ll re so u rc e sta te s a re effi ci en t u n iv er sa l. squareNo te th a t, sin c e th e se o th e r re so u rc e sta te s c a n b e tr a n sfo rme d in to th e 2 Dcl u ster sta te a lw a y s b y adapt iv e local pr oje c tiv e m eas u re m e n ts as siste d w ith c las sic alco m m u n ica tio n , a n d lo ca l (C liff ord ) op e rat ion s, th e y are al so u n iv e rsa l in th e se n se o fac onventional one-way quantum com putation.It is in te re stin g to o b se rv e th a t th e sq u a re la ttic e , to g e th er w ith th e h ex a g o n a l a n dtr ia n g u la r la tti c e s a re th e o n ly p o ssib iliti e s to o b ta in a reg u la r tilin g o f th e 2 D p la n e.Fu r t h e r m o r e , t h e K a g o m e la t t ic e is a n e x a m p le o f a u n ifo r m s e mi re g u la r 2 D tilin gwi th 2 b a sic tile s (th e tria n g le a n d th e h e x a g o n ). Th e h e x a g o na l la ttic e ha s v e rte xde g re e 3 , w hi ch le a ds to a n inc re a se d ro bus tne ss a g a ins t lo c al n oi se as c om p are d toth e 2 D c lu ste r sta te [5 3 ]. S in c e th e 1 D c lu ste r sta te (w ith u n if o rm v e rte x d e g re e 2 )ha s b e e n pr o v e d no t to b e uni v e rsa l [2 7 ], the v e rte x de g re e 3 ism inim al for universalre sou rc e on un ifo r m la ttic e stru c tu re s.We r e m a r k t h a t o t h e r u n iv e r s a l r e s o u r c e s t a t e s h a v e b e e n p r e se n te d [5 4 ] b a se don n on -u n iform lat tic e stru c tu re s (se e al so [55 ]), w h e re e a ch g a te in a u n iv e r s a l s e tof u n itary gat e s c an b e im p le m e n te d b y lo c al m e as u re m e n ts on an e le m e n tary u n itan d th e se u n its are c om b in e d (b ot tom -u p ap p roac h ). H e re w e too k , in co n tra st, a(a) (b)(c )321(d)Fi g u r e 2 . Ex a m p les o f effi ci en t u n iv er sa l res o u rce fo r M Q C . T h es e a re g rap hst a tes co rresp o n d in g to (a ) h ex a g o n a l, (b ) tria n g u la r a n d (c) K a g o m e la ttices .De term in istic L OC C tra n sfo rm a tio n fro m (a ) to (d ) (2 D clu ster sta te) v ia (b )an d (c ) is in d ic at e d , w h e re sim p le gr ap h ru le s c an b e u se d se q ue n tia lly (\u03C3yan d\u03C3zme a su reme n ts a re d isp lay ed b y square an d \u2666 ,r espectively). The spatialo verheadfo r th e tra n sfo rm a tio n is co n sta n t.Hexagonal TriangularKagome SquareThursday, 29 July 2010Universal resources derived via graphical rulesThese regular graphs can all be mapped into square lattices via Pauli measurements - using the graphical rules. Hence all are resources for universal MBQC.M. Van den Nest, et al, New J. Phys. 9 204 (2007).CO N TE N TS 32Fi g . 2 . A c c o rd in g ly, w e fi n a lly o b ta in a sq u a re la ttic e , i.e .,a 2D cluster state. T he totalsp a tia l o v e rh e a d is e ig h t in th a t a u n it sq u a re o n (d ) is o b ta ine d fro m e ig h t he x a g o nsin (a ). S in c e th e o v e rh e a d is c o n sta n t, a ll re so u rc e sta te s a re effi ci en t u n iv er sa l. squareNo te th a t, sin c e th e se o th e r re so u rc e sta te s c a n b e tr a n sfo rme d in to th e 2 Dcl u ster sta te a lw a y s b y adapt iv e local pr oje c tiv e m eas u re m e n ts as siste d w ith c las sic alco m m u n ica tio n , a n d lo ca l (C liff ord ) op e rat ion s, th e y are al so u n iv e rsa l in th e se n se o fac onventional one-way quantum com putation.It is in te re stin g to o b se rv e th a t th e sq u a re la ttic e , to g e th er w ith th e h ex a g o n a l a n dtr ia n g u la r la tti c e s a re th e o n ly p o ssib iliti e s to o b ta in a reg u la r tilin g o f th e 2 D p la n e.Fu r t h e r m o r e , t h e K a g o m e la t t ic e is a n e x a m p le o f a u n ifo r m s e mi re g u la r 2 D tilin gwi th 2 b a sic tile s (th e tria n g le a n d th e h e x a g o n ). Th e h e x a g o na l la ttic e ha s v e rte xde g re e 3 , w hi ch le a ds to a n inc re a se d ro bus tne ss a g a ins t lo c al n oi se as c om p are d toth e 2 D c lu ste r sta te [5 3 ]. S in c e th e 1 D c lu ste r sta te (w ith u n if o rm v e rte x d e g re e 2 )ha s b e e n pr o v e d no t to b e uni v e rsa l [2 7 ], the v e rte x de g re e 3 ism inim al for universalre sou rc e on un ifo r m la ttic e stru c tu re s.We r e m a r k t h a t o t h e r u n iv e r s a l r e s o u r c e s t a t e s h a v e b e e n p r e se n te d [5 4 ] b a se don n on -u n iform lat tic e stru c tu re s (se e al so [55 ]), w h e re e a ch g a te in a u n iv e r s a l s e tof u n itary gat e s c an b e im p le m e n te d b y lo c al m e as u re m e n ts on an e le m e n tary u n itan d th e se u n its are c om b in e d (b ot tom -u p ap p roac h ). H e re w e too k , in co n tra st, a(a) (b)(c )321(d)Fi g u r e 2 . Ex a m p les o f effi ci en t u n iv er sa l res o u rce fo r M Q C . T h es e a re g rap hst a tes co rresp o n d in g to (a ) h ex a g o n a l, (b ) tria n g u la r a n d (c) K a g o m e la ttices .De term in istic L OC C tra n sfo rm a tio n fro m (a ) to (d ) (2 D clu ster sta te) v ia (b )an d (c ) is in d ic at e d , w h e re sim p le gr ap h ru le s c an b e u se d se q ue n tia lly (\u03C3yan d\u03C3zme a su reme n ts a re d isp lay ed b y square an d \u2666 ,r espectively). The spatialo verheadfo r th e tra n sfo rm a tio n is co n sta n t.Hexagonal TriangularKagome SquareThursday, 29 July 2010Measurement-patterns for gates in the one-way model5B. A u n iv e r s a l s e t o f q u a n t u m g a t e sTo p r o v id e s o m e t h in g d e fi n it e t o d is c u s s r ig h t fr o m t h ebe g in n in g , w e n o w g iv e th e p r o c e d u r e s o f h o w to r e a liz eaC N O T -gate and ag eneral one-qubit rotation viao ne-qu b it m e a s u r e m e n t s o n a c lu s t e r s t a t e . T h e e xp la n a t io nof w h y an d h o w t h e s e gat e s w ork w ill b e gi v e n in S e c -ti o n I I G .(a )123 4 5 679 10 11 12 13 14 158controltargetYYYYYYY XXXXXXCN O T- g a t e(b )123 4 5_X!_+\"+#+_(c )123 4 5X X X+#_ge n e r al r ot at ion z -r o t a t io n(d )123 4 5X YY Y(e )123 4 5X X XYHa d a m a r d -g a t e pi / 2- p h as e gat eFI G . 2 : R ea liza t io n o f el em en t a r y q u a n t u m g a t es o n t h e Q CC.Ea c h s q u a r e r e p r e s e n t s a la t t ic e q u b it . T h e s q u a r e s in t h eex t r em e lef t co lu m n m a r k ed w it h w h it e ci r cl es d en o t e t h e in -put q ubi t s , t ho s e in t he r ig h t -m o s t c o lum n de no t e t he o ut putqu b it s .AC N O T -gate can be realized on a cluster state of 15qu b it s , a s s h o w n in F ig . 2 . A ll m e a s u r e m e n t s c a n b epe r fo r m e d s im u lta n e o u s ly . T h e p r o c e d u r e to r e a liz e aCN O T- g a t e o n a c lu s t e r wi t h 1 5 q u b it s a s d is p la y e d inFi g . 2 isPr o c e d u r e 1 Re a liz a t io n o f a C N O T -g a t e a c t ing o n atw o -q u b it s t a t e |\u03C8in\u3009.1. Pre p are t h e s t at e|\u03A8in\u3009C15= |\u03C8in\u30091 ,9\u2297parenleftBigcirclemultiplytexti\u2208 C15\\{ 1 ,9 }|+ \u3009iparenrightBig.2. En t an gl e t h e 15 q u b it s of t h e c lu s t e r C15vi a t h euni t a r y o p e r a t io n S( C15).3. M e as u re al l q u b it s of C15ex cep t fo r t h e o u t p u tqu b it s 7 , 1 5 ( fo llo w in g t h e la b e lin g in F ig . 2 ) . T h eme a s u r e me n t s c a n b e p e r fo r me d s im u lt a n e o u s ly .Qu b it s 1 , 9 , 1 0 , 1 1 , 1 3 , 1 4 a r e m e a s u r e d in t h e \u03C3x-ei g en b a s is a n d q u b it s 2 -6 , 8 , 1 2 in t h e \u03C3y-e ig e n b a s is .De p e n d e n t o n t h e m e a s u r e m e n t r e s u lt s , t h e fo llo w in ggat e is t h e re b y re al iz e d :UprimeCN O T= U\u03A3 ,C N O TCN O T ( c, t ) . (2 3 )Th e r e in t h e b y p r o d u c t o p e r a t o r U\u03A3 ,C N O Tha s t he fo r mU\u03A3 ,C N O T= \u03C3( c )x\u03B3( c )x\u03C3( t )x\u03B3( t )x\u03C3( c )z\u03B3( c )z\u03C3( t )z\u03B3( t )z, wi t h\u03B3( c )x= s2+ s3+ s5+ s6\u03B3( t )x= s2+ s3+ s8+ s10+ s12+ s14\u03B3( c )z= s1+ s3+ s4+ s5+ s8+ s9+ s11+1\u03B3( t )z= s9+ s11+ s13.(2 4 )Th e r e in , t h e sire p re s e n t t h e m e as u re m e n t ou t c om e s sion t h e q u b it s i. Th e e x p r e s s io n ( 2 4 ) is m o d ifi e d if r e -dunda n t c lus t e r q ubi t s a r e pr e s e n t a nd/ o r if t he c lus t e rs t a t e o n wh ic h t h e CN O T g a t e is r e a liz e d is s p e c ifi e d b yas et { \u03BAa} di ff er en t fr o m ( 1 2 ) , s ee S ect io n I I C . T h is co n -cl u d es t h e p r es en t a t io n o f t h e C N O T g a t e, t h e p r o o f o fit s fu n c t io n in g is g iv e n in S e c t io n I I G .An a r b itr a r y r o ta ti o n URo t\u2208 SU (2 ) c a n b e r e a liz e don a c h ai n of 5 q u b it s . C on s id e r a rot at ion in it s Eu le rre p re s e n t at ionURo t[\u03BE , \u03B7 , \u03B6 ]= Ux[\u03B6 ]Uz[\u03B7 ]Ux[\u03BE ], (2 5 )wh e r e t h e r o t a t io n s a b o u t t h e x -a nd z -a x is a r eUx[\u03B1 ]= expparenleftBig\u2212 i\u03B1\u03C3x2parenrightBigUz[\u03B1 ]= expparenleftBig\u2212 i\u03B1\u03C3z2parenrightBig.(2 6 )In it ia lly , t h e fi r s t q u b it is p r e p a r e d in s o m e s t a t e |\u03C8in\u3009,wh ic h is t o b e r o t a t e d , a n d t h e o t h e r q u b it s a r e p r e p a r e din |+ \u3009.A fter the 5 qubits are entangled by the unitarytr a n s fo r m a ti o n S ,t he state |\u03C8in\u3009 ca n b e r o t a t ed b y m ea -su r in g q u b it s 1 t o 4 . A t t h e sa m e t im e , t h e st a t e is a lsos w a p p e d t o s it e 5 . Th e q u b it s 1 .. 4a re m easured in ap-pr o pr ia t e ly c ho s e n ba s e sBj( \u03D5j)=braceleftbigg|0 \u3009j+ ei\u03D5j|1 \u3009j\u221A2,|0 \u3009j\u2212 ei\u03D5j|1 \u3009j\u221A2bracerightbigg, (2 7 )wh e r e b y t h e m e a s u r e m e n t o u t c o m e s sj\u2208 { 0 , 1 } fo r j =1 .. 4a re obtained. H ere, sj=0 m eans that qubit j ispr o je c t e d in t o t he fir s t s t a t e o f Bj( \u03D5j). In (2 7 ) th e b a -si s st a t e s o f a ll p o ssi b le m e a su r e m e n t b a se s lie o n t h eeq u a t o r o f t h e B lo c h s p h er e, i.e. o n t h e in t er s ect io n o fth e B lo c h s p h e r e w ith th e x -y -p la n e . T h e r e fo r e , t h e m e a -su r e m e n t b a si s fo r q u b it j ca n b e s p eci fi ed b y a s in g le p a -ram e t e r, t h e m e as u re m e n t an gl e \u03D5j.T he m easurem entdi r e c t io n o f q ubi t j is t h e v e c t o r o n t h e B lo c h s p h e r ewh ic h c o r r e s p o n d s t o t h e fi r s t s t a t e in t h e m e a s u r e m e n tba s is Bj( \u03D5j). T h u s , th e m e a s u r e m e n t a n g le \u03D5jis t h e a n -gl e b e tw e e n t h e m e as u re m e n t d ire c t ion at q u b it j an dth e p o s iti v e x -a x is . In s u m m a r y , t h e p r o c e d u r e t o r e a liz ean arb it rary rot at ion URo t[\u03BE , \u03B7 , \u03B6 ], s p e c ifi e d b y it s E u le ran gl e s \u03BE , \u03B7 , \u03B6 ,i s this:R. Raussendorf, D. E. Browne and H.J. Briegel,  Phys. Rev. A 68: 022312 (2003)Generate the full Clifford groupThursday, 29 July 2010Graphical \u201CGottesman-Knill Theorem\u201D2021 22 23 24 25 26 27 28 29 30 31 32 3317 18 19 2010 11 12 13 14 156754321No. 18921 25 3317 1912 146531816161029No. 2FI G . 1 6 : T h e g r a p h a s s o c ia te d w ith th e Q FT o n 3 q u b its in th e o n e- w ay q u an t u m co m p u t er i s r ep r es en t ed i n g r ap h N o . 1 , w h er e the box e sde not e t he i nput ( l e f t ) a nd out put ( r i ght ) v e r t i c e s . G r a ph N o. 2 i s obt a i ne d f r om t he fi r s t a f t e r pe r f or m i ng a l l P a ul i m e a s ur e m e nt s a c c or di ngto th e p r o to c o l in R e f . [ 3 ] , e x c e p t f r o m th e \u03C3x-m e a s u re m e n ts a t th e in p u t v e rt ic e s . M o re p re c is e ly , it is o bt a i ne d f r om gr a ph N o. 1 a f t e r\u03C3y-m e a s u re m e n ts o n th e v e rt ic e s 22 , 23 , 24 , 26 , 27 , 28 , 30 , 31 , 32 an d \u03C3x-m e a s u re m e n ts o n th e v e rt ic e s 2 , 4 , 7 , 9 , 11 , 13 , 15 , 18 , 20 ha v e be e npe r f or m e d.ro r c o rre c tio n , q u a n tu m c o m m u n ic a tio n , a n d q u a n tu m c o m -put a tion in the c ont e xt of the one - w a y qua nt um c om put e r. T heSc h m id t m e a s u r e is ta ilo r e d f o r a c o m p a r a b ly d e ta ile d a c c o u nton the qua nt um c or r e la tions gr a s pi ng ge nui ne m ul ti- pa r tic leen tan g lem en t, y et it tu r n s o u t to b e co m p u tab le f o r m an ygr a ph s ta te s . W e ha v e pr e s e nt e d a num be r of ge ne r a l r ul e sth a t c a n b e a p p lie d w h e n a p p r o a c h in g th e p r o b le m o f e v a lu -at in g th e S ch m id t m eas u r e f o r g en er al g r ap h s tat es , w h ich ar est a te d m o st ly in g r a p h th e o r e tic a l te r m s. T h e se r u le s h a v e the nbe e n a ppl ie d to a num be r of gr a ph s ta te s tha t a ppe a r in qua n-tu m c o m p u ta tio n a n d e r r o r c o r r e c tio n . A ls o , a ll c o n n e c te dgr a phs w ith up to s e v e n v e r tic e s ha v e be e n di s c us s e d in de ta il.Th e f o r m a lis m th a t w e p r e s e n t h e r e a b s tr a c ts f r o m th e a c tu a lphys ic a l r e a lis a tion, b ut a s ha s be e n poi nt e d out in s e v e r a l in -st a n c e s, a n u m b e r o f w e ll- c o n tr o lla b le p h y si c a l sy st e m s su chas n eu tr al at o m s in o p tical lat tices s er v e as p o ten tial can d i-da te s to r e a liz e s uc h gr a ph s ta te s [ 45 , 46 ] .In th is p a p e r, th e S c h m id t m e a s u re h a s b e e n e m p lo y e d toqua nt if y the de gr e e of e nt a ngl e m e nt , a s a ge ne r a liz a tion ofth e S c h m id t r a n k in th e b ip a r tite s e ttin g . T h is m e a s u r e issu f fi c ie n tly c o a r se to b e a c c e ssi b le f o r sy st e m s c o n si st in g ofma n y c o n s titu e n ts a n d to a llo w f o r a n a p p r o p r ia te d is c u s s io nof m ul ti- pa r tic le e nt a ngl e m e nt in gr a ph s ta te s . T he a ppr oa chof qua nt if yi ng e nt a ngl e m e nt in te r m s of r a te s of a s ym pt ot icre v e rs ib le s ta te tra n s fo rm a tio n s , a s a n a lte rn a tiv e , a p p e ar s u n -fe a s ib le in th e m a n y -p a rt ite s e ttin g . T h e q u e s tio n o f th e m in-im a l r e v e r s ib le e n ta n g lin g g e n e r a tin g s e t ( M R E G S ) in m u lti -pa r tite s ys te m s r e m a ins unr e s ol v e d to da te , e v e n f or qua nt u msy st e m s c o n si st in g o f th r e e q u b its, a n d d e sp ite c o n si d e r a b lere s e a rc h e ffo rt [4 7 , 4 8 ]. T h e s e M R E G S a re th e (n o t n e c e s s a r-Standard-form measurement patternQuantum Fourier TransformGraph state after all Pauli measurements performed2021 22 23 24 25 26 27 28 29 30 31 32 3317 18 19 2010 11 12 13 14 156754321No. 18921 25 3317 1912 146531816161029No. 2FI G . 1 6 : T h e g r a p h a s s o c ia te d w ith th e Q FT o n 3 q u b its in th e o n e- w ay q u an tu m co m p u ter is r ep r es en ted in g r ap h N o . 1 , w h er e the box e sde not e t he i nput ( l e f t ) a nd out put ( r i ght ) v e r t i c e s . G r a ph N o. 2 i s obt a i ne d f r om t he fi r s t a f t e r pe r f or m i ng a l l P a ul i m e a s ur e m e nt s a c c or di ngto th e p r o to c o l in R e f . [ 3 ] , e x c e p t f r o m th e \u03C3x-m e a s u re m e n ts a t th e in p u t v e rt ic e s . M o re p re c is e ly , it is o bt a i ne d f r om gr a ph N o. 1 a f t e r\u03C3y-m e a s u re m e n ts o n th e v e rt ic e s 22 , 23 , 24 , 26 , 27 , 28 , 30 , 31 , 32 an d \u03C3x-m e a s u re m e n ts o n th e v e rt ic e s 2 , 4 , 7 , 9 , 11 , 13 , 15 , 18 , 20 ha v e be e npe r f or m e d.ro r c o rre c tio n , q u a n tu m c o m m u n ic a tio n , a n d q u a n tu m c o m -put a tion in the c ont e xt of the one - w a y qua nt um c om put e r. T heSc h m id t m e a s u r e is ta ilo r e d f o r a c o m p a r a b ly d e ta ile d a c c o u nton the qua nt um c or r e la tions gr a s pi ng ge nui ne m ul ti- pa r tic leen tan g lem en t, y et it tu r n s o u t to b e co m p u tab le f o r m an ygr a ph s ta te s . W e ha v e pr e s e nt e d a num be r of ge ne r a l r ul e sth a t c a n b e a p p lie d w h e n a p p r o a c h in g th e p r o b le m o f e v a lu -at in g th e S ch m id t m eas u r e f o r g en er al g r ap h s tat es , w h ich ar est a te d m o st ly in g r a p h th e o r e tic a l te r m s. T h e se r u le s h a v e the nbe e n a ppl ie d to a num be r of gr a ph s ta te s tha t a ppe a r in qua n-tu m c o m p u ta tio n a n d e r r o r c o r r e c tio n . A ls o , a ll c o n n e c te dgr a phs w ith up to s e v e n v e r tic e s ha v e be e n di s c us s e d in de ta il.Th e f o r m a lis m th a t w e p r e s e n t h e r e a b s tr a c ts f r o m th e a c tu a lphys ic a l r e a lis a tion, b ut a s ha s be e n poi nt e d out in s e v e r a l in -st a n c e s, a n u m b e r o f w e ll- c o n tr o lla b le p h y si c a l sy st e m s su chas n eu tr al at o m s in o p tical lat tices s er v e as p o ten tial can d i-da te s to r e a liz e s uc h gr a ph s ta te s [ 45 , 46 ] .In th is p a p e r, th e S c h m id t m e a s u re h a s b e e n e m p lo y e d toqua nt if y the de gr e e of e nt a ngl e m e nt , a s a ge ne r a liz a tion ofth e S c h m id t r a n k in th e b ip a r tite s e ttin g . T h is m e a s u r e issu f fi c ie n tly c o a r se to b e a c c e ssi b le f o r sy st e m s c o n si st in g ofma n y c o n s titu e n ts a n d to a llo w f o r a n a p p r o p r ia te d is c u s s io nof m ul ti- pa r tic le e nt a ngl e m e nt in gr a ph s ta te s . T he a ppr oa chof qua nt if yi ng e nt a ngl e m e nt in te r m s of r a te s of a s ym pt ot icre v e rs ib le s ta te tra n s fo rm a tio n s , a s a n a lte rn a tiv e , a p p e ar s u n -fe a s ib le in th e m a n y -p a rt ite s e ttin g . T h e q u e s tio n o f th e m in-im a l r e v e r s ib le e n ta n g lin g g e n e r a tin g s e t ( M R E G S ) in m u lti -pa r tite s ys te m s r e m a ins unr e s ol v e d to da te , e v e n f or qua nt u msy st e m s c o n si st in g o f th r e e q u b its, a n d d e sp ite c o n si d e r a b lere s e a rc h e ffo rt [4 7 , 4 8 ]. T h e s e M R E G S a re th e (n o t n e c e s s a r-M. Hein,  J. Eisert and H.J. Briegel, Phys. Rev. A 69, 062311 (2004)Thursday, 29 July 2010Valence bond PEPS to MPS\u201CVirtual Quditpairs\u201Dd\uFFFFn =1| n \uFFFF | n \uFFFFPEPS \u201Cprojector\u201DConstructs a Matrix Product State MPSM ( j )m, n= Ajp,q| \u03C8 \uFFFF =\uFFFFs1,...snTr [ M ( s1) M ( s2) .. .M ( sn)] | s1\uFFFF | s2\uFFFF \u00B7 \u00B7\u00B7| sn\uFFFFD\uFFFFj =1d\uFFFFp,q =1Ajp,q| j \uFFFF\uFFFF p | \uFFFF q |Thursday, 29 July 2010References\u2022Progress Review\u2022H. J. Briegel, D. E. Browne, W. D\u00FCr, R. Raussendorf, M. Van den Nest,  Nature Physics 5 1, 19-26 (2009)\u2022Tutorials\u2022M. Hein, W. D\u00FCr, J. Eisert, R. Raussendorf, M. Van den Nest, H.-J. Briegel, quant-ph/0602096 (Varena lectures on graph states)\u2022D. E. Browne and H. J. Briegel, quant-ph/0603226\u2022M.A. Nielsen, quant-ph/0504097\u2022And many more...\u2022Search arxiv for One-way, MBQC, Cluster States, Graph States, ....Thursday, 29 July 2010 Measurement-based quantum computation10th Canadian Summer School on QIDan BrowneDept. of Physics and AstronomyUniversity College LondonThursday, 29 July 2010What is a quantum computer?Thursday, 29 July 2010The one-way quantum computerA multi-qubit entangled \u201Cresource state\u201DE.g. cluster stateSingle-qubit measurementsChoice of bases specifycomputationAdaptive (some bases depend upon previous outcomes)+=A universal quantum computerThursday, 29 July 2010Overview of Lectures\u26051. Cluster states and graph states I\u2605What are they? Basic properties.\u26052. The one-way quantum computer\u2605What is the model? How does it work?\u26053. Cluster states and graph states II\u2605Stabilizer formalism and graphical representation\u26054.  Cluster states and graph states III\u2605How do we build them?\u26055. Beyond cluster states (If time permits)\u2605(other models of MBQC)\u26056. (Robert Raussendorf) Fault tolerant MBQCThursday, 29 July 2010Pauli group and Clifford Group\u2022Pauli group: \u2022Set of all n-fold tensor products of X, Y, Z and I, with pre-factors +1, -1. +i, -i for group closure.\u2022Clifford group:\u2022\u201CNormalizer\u201D of Set of unitaries C such that Equiv,:\u201CMaps Pauli group onto Pauli group\u201DnnC \u03C3kC\u2020= \u03C3j\u2200 \u03C3k\u2208n\u03C3j\u2208nC \u03C3k= \u03C3jC =( C \u03C3kC \u2020 ) CThursday, 29 July 20102-qubit gates and universal q.c.J J J J JJ J J J JJ J J J JThursday, 29 July 20102-qubit gates and universal q.c.J J J J JJ J J J JJ J J J JThursday, 29 July 20102-qubit gates and universal q.c.J J J J JJ J J J JJ J J J JThursday, 29 July 2010Etching with z measurementsV OLUME 86, N UMBER 22 P H Y S I C A L R E V I E W L E T T E R S 28 M AY 2001A One-W ay Quantum ComputerRober t Raussendorf and Hans J. Brie gelTheor et isc he Physik, L udwig-Maximilians-Univer sit\u00E4t M\u00FCnc hen, Ger m any( Recei v ed 25 October 2000 )W e present a scheme of quantum computation that consists entirely of one-qubit measurements on apar ticular class of entangled states, the cluster states. The measurements are used to imprint a quantumlogic circuit on the state, thereby destro ying its entanglement at the same time. Cluster states are thusone-w ay quantum computers and the measurements form the program.DOI: 10.1103 / Ph ysRe vLett.86.5188 P A CS numbers: 03.67.Lx, 03.65.UdA quantum computer promises efficient processing ofcer tain computational tasks that are intractable with clas-sical computer technology [1]. While basic principles of aquantum computer ha v e been demonstrated in the labora-tor y [2], scalability of these systems to a lar ge number ofqubits [3], essential for practical applications such as theShor algorithm, represents a formidable challenge. Mostof the current e xperiments are designed to implement se-quences of highly controlled interactions between selectedpar ticles (qubits), thereby follo wing models of a quan-tum computer as a (sequential) netw ork of quantum logicgates [4,5].Here we propose a different model of a scalable quan-tum computer . In our model, the entire resource for thequantum computation is pro vided initially in the form ofa specific entangled state (a so-called cluster state [6]) ofa lar ge number of qubits. Information is then written ontothe cluster , processed, and read out from the cluster byone-par ticle measurements only . The entangled state ofthe cluster thereby serv es as a uni v ersal \u201Csubstrate\u201D for an yquantum computation. Cluster states can be created effi-ciently in an y system with a quantum Ising-type interaction(at v er y lo w temperatures) between tw o-state par ticles ina lattice configuration.W e consider tw o- and three-dimensional arrays ofqubits that interact via an Ising-type ne xt-neighbor in-teraction [6] described by a Hamiltonian HintH33527 g H20849 t H20850 3PH20855 a ,a0H208561 1sH20849 a H20850z21 2sH20849 a0H20850z2H11061 214g H20849 t H20850PH20855 a ,a0H20856sH20849 a H20850zsH20849 a0H20850z[7] whosestrength g H20849 t H20850 can be controlled e xternally . A possiblerealization of such a system is discussed belo w . A qubit atsite a can be in tw o states j0 H20856aH11013 j0 H20856z ,aor j1 H20856aH11013 j1 H20856z ,a,the eigenstates of the P auli phase fl ip operator sH20849 a H20850zH20851sH20849 a H20850zji H20856aH33527 H208492 1 H20850iji H20856aH20852 . These tw o states form the compu-tational basis. Each qubit can equally be in an arbitrar ysuperposition state a j0 H20856 1b j1 H20856 , ja j21 jb j2H33527 1 . F orour purpose, we initially prepare all qubits in the su-perposition j1 H20856 H33527 H20849 j0 H20856 1 j1 H20856H20850H20862p2 , an eigenstate of theP auli spin fl ip operator sxH20851sxj6 H20856 H33527 6 j6 H20856H20852 . Hintisthen switched on for an appropriately chosen finite timeinterv al T , whereRT0dt g H20849 t H20850 H33527 p , by which a unitar ytransformation S is realized. Since Hintacts uniformly onthe lattice, entire clusters of neighboring par ticles becomeentangled in one single step. The quantum state jF H20856C,the state of a cluster H20849 C H20850 of neighboring qubits, which isthereby created pro vides in adv ance all entanglement thatis in v olv ed in the subsequent quantum computation. It hasbeen sho wn [6] that the cluster state jF H20856Cis characterizedby a set of eigen v alue equationssH20849 a H20850xOa0[ ngbh H20849 a H20850sH20849 a0H20850zjF H20856CH33527 6 jF H20856C, (1)where ngbh H20849 a H20850 specifies the sites of all qubits that inter-act with the qubit at site a [ C . The eigen v alues are de-termined by the distrib ution of the qubits on the lattice.The equations (1) are central for the proposed computationscheme. As an e xample, a measurement on an indi vidualqubit of a cluster has a random outcome. On the otherhand, Eqs. (1) imply that an y tw o qubits at sites a , a0[ Ccan be projected into a Bell state by measuring a subset ofthe other qubits in the cluster . This proper ty will be used tod e fi n e q u a n tu m c h a n n e ls th a t a llo w us to p r o p a g a te q u a n -tum information through a cluster .W e sho w that a cluster state jF H20856Ccan be used as a sub-strate on which an y quantum circuit can be imprinted byone-qubit measurements. In Fig. 1 this scheme is illus-trated. F or simplicity , we assume that in a cer tain re gionof the lattice each site is occupied by a qubit. This re-quirement is not essential as will be e xplained belo w [see(d)]. In the first step of the computation, a subset ofqubits is measured in the basis of szwhich effecti v elyremo v es them. In Fig. 1 these qubits are denoted by \u201C \u00D8 .\u201Dq u a n t u m  g a t einformation flo wFIG. 1. Quantum computation by measuring tw o-state par ti-cles on a lattice. Before the measurements the qubits are in thecluster state jF H20856Cof (1). Circles \u00D8 symbolize measurements ofsz, v er tical arro ws are measurements of sx, while tilted arro wsrefer to measurements in the x - y plane.5188 0031 - 9007 H20862 01 H20862 86(22) H20862 5188(4) $15.00 \u00A9 2001 The American Physical Society\u2022Figure 1in H. J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86, 5188 (2001)z-measurements\u201Cquantum wire\u201Dfor logical singlequbitThursday, 29 July 2010Measurement-patterns for gates in the one-way model5B. A u n iv e r s a l s e t o f q u a n t u m g a t e sTo p r o v id e s o m e t h in g d e fi n it e t o d is c u s s r ig h t fr o m t h ebe g in n in g , w e n o w g iv e th e p r o c e d u r e s o f h o w to r e a liz eaC N O T -gate and ag eneral one-qubit rotation viao ne-qu b it m e a s u r e m e n t s o n a c lu s t e r s t a t e . T h e e xp la n a t io nof w h y an d h o w t h e s e gat e s w ork w ill b e gi v e n in S e c -ti o n I I G .(a )123 4 5 679 10 11 12 13 14 158controltargetYYYYYYY XXXXXXCN O T- g a t e(b )123 4 5_X!_+\"+#+_(c )123 4 5X X X+#_ge n e r al r ot at ion z -r o t a t io n(d )123 4 5X YY Y(e )123 4 5X X XYHa d a m a r d -g a t e pi / 2- p h as e gat eFI G . 2 : R ea liza t io n o f el em en t a r y q u a n t u m g a t es o n t h e Q CC.Ea c h s q u a r e r e p r e s e n t s a la t t ic e q u b it . T h e s q u a r e s in t h eex t r em e lef t co lu m n m a r k ed w it h w h it e ci r cl es d en o t e t h e in -put q ubi t s , t ho s e in t he r ig h t -m o s t c o lum n de no t e t he o ut putqu b it s .AC N O T -gate can be realized on a cluster state of 15qu b it s , a s s h o w n in F ig . 2 . A ll m e a s u r e m e n t s c a n b epe r fo r m e d s im u lta n e o u s ly . T h e p r o c e d u r e to r e a liz e aCN O T- g a t e o n a c lu s t e r wi t h 1 5 q u b it s a s d is p la y e d inFi g . 2 isPr o c e d u r e 1 Re a liz a t io n o f a C N O T -g a t e a c t ing o n atw o -q u b it s t a t e |\u03C8in\u3009.1. Pre p are t h e s t at e|\u03A8in\u3009C15= |\u03C8in\u30091 ,9\u2297parenleftBigcirclemultiplytexti\u2208 C15\\{ 1 ,9 }|+ \u3009iparenrightBig.2. En t an gl e t h e 15 q u b it s of t h e c lu s t e r C15vi a t h euni t a r y o p e r a t io n S( C15).3. M e as u re al l q u b it s of C15ex cep t fo r t h e o u t p u tqu b it s 7 , 1 5 ( fo llo w in g t h e la b e lin g in F ig . 2 ) . T h eme a s u r e me n t s c a n b e p e r fo r me d s im u lt a n e o u s ly .Qu b it s 1 , 9 , 1 0 , 1 1 , 1 3 , 1 4 a r e m e a s u r e d in t h e \u03C3x-ei g en b a s is a n d q u b it s 2 -6 , 8 , 1 2 in t h e \u03C3y-e ig e n b a s is .De p e n d e n t o n t h e m e a s u r e m e n t r e s u lt s , t h e fo llo w in ggat e is t h e re b y re al iz e d :UprimeCN O T= U\u03A3 ,C N O TCN O T ( c, t ) . (2 3 )Th e r e in t h e b y p r o d u c t o p e r a t o r U\u03A3 ,C N O Tha s t he fo r mU\u03A3 ,C N O T= \u03C3( c )x\u03B3( c )x\u03C3( t )x\u03B3( t )x\u03C3( c )z\u03B3( c )z\u03C3( t )z\u03B3( t )z, wi t h\u03B3( c )x= s2+ s3+ s5+ s6\u03B3( t )x= s2+ s3+ s8+ s10+ s12+ s14\u03B3( c )z= s1+ s3+ s4+ s5+ s8+ s9+ s11+1\u03B3( t )z= s9+ s11+ s13.(2 4 )Th e r e in , t h e sire p re s e n t t h e m e as u re m e n t ou t c om e s sion t h e q u b it s i. Th e e x p r e s s io n ( 2 4 ) is m o d ifi e d if r e -dunda n t c lus t e r q ubi t s a r e pr e s e n t a nd/ o r if t he c lus t e rs t a t e o n wh ic h t h e CN O T g a t e is r e a liz e d is s p e c ifi e d b yas et { \u03BAa} di ff er en t fr o m ( 1 2 ) , s ee S ect io n I I C . T h is co n -cl u d es t h e p r es en t a t io n o f t h e C N O T g a t e, t h e p r o o f o fit s fu n c t io n in g is g iv e n in S e c t io n I I G .An a r b itr a r y r o ta ti o n URo t\u2208 SU (2 ) c a n b e r e a liz e don a c h ai n of 5 q u b it s . C on s id e r a rot at ion in it s Eu le rre p re s e n t at ionURo t[\u03BE , \u03B7 , \u03B6 ]= Ux[\u03B6 ]Uz[\u03B7 ]Ux[\u03BE ], (2 5 )wh e r e t h e r o t a t io n s a b o u t t h e x -a nd z -a x is a r eUx[\u03B1 ]= expparenleftBig\u2212 i\u03B1\u03C3x2parenrightBigUz[\u03B1 ]= expparenleftBig\u2212 i\u03B1\u03C3z2parenrightBig.(2 6 )In it ia lly , t h e fi r s t q u b it is p r e p a r e d in s o m e s t a t e |\u03C8in\u3009,wh ic h is t o b e r o t a t e d , a n d t h e o t h e r q u b it s a r e p r e p a r e din |+ \u3009.A fter the 5 qubits are entangled by the unitarytr a n s fo r m a ti o n S ,t he state |\u03C8in\u3009 ca n b e r o t a t ed b y m ea -su r in g q u b it s 1 t o 4 . A t t h e sa m e t im e , t h e st a t e is a lsos w a p p e d t o s it e 5 . Th e q u b it s 1 .. 4a re m easured in ap-pr o pr ia t e ly c ho s e n ba s e sBj( \u03D5j)=braceleftbigg|0 \u3009j+ ei\u03D5j|1 \u3009j\u221A2,|0 \u3009j\u2212 ei\u03D5j|1 \u3009j\u221A2bracerightbigg, (2 7 )wh e r e b y t h e m e a s u r e m e n t o u t c o m e s sj\u2208 { 0 , 1 } fo r j =1 .. 4a re obtained. H ere, sj=0 m eans that qubit j ispr o je c t e d in t o t he fir s t s t a t e o f Bj( \u03D5j). In (2 7 ) th e b a -si s st a t e s o f a ll p o ssi b le m e a su r e m e n t b a se s lie o n t h eeq u a t o r o f t h e B lo c h s p h er e, i.e. o n t h e in t er s ect io n o fth e B lo c h s p h e r e w ith th e x -y -p la n e . T h e r e fo r e , t h e m e a -su r e m e n t b a si s fo r q u b it j ca n b e s p eci fi ed b y a s in g le p a -ram e t e r, t h e m e as u re m e n t an gl e \u03D5j.T he m easurem entdi r e c t io n o f q ubi t j is t h e v e c t o r o n t h e B lo c h s p h e r ewh ic h c o r r e s p o n d s t o t h e fi r s t s t a t e in t h e m e a s u r e m e n tba s is Bj( \u03D5j). T h u s , th e m e a s u r e m e n t a n g le \u03D5jis t h e a n -gl e b e tw e e n t h e m e as u re m e n t d ire c t ion at q u b it j an dth e p o s iti v e x -a x is . In s u m m a r y , t h e p r o c e d u r e t o r e a liz ean arb it rary rot at ion URo t[\u03BE , \u03B7 , \u03B6 ], s p e c ifi e d b y it s E u le ran gl e s \u03BE , \u03B7 , \u03B6 ,i s this:R. Raussendorf, D. E. Browne and H.J. Briegel,  Phys. Rev. A 68: 022312 (2003)Thursday, 29 July 2010LC-orbit of local Clifford equivalent statesE NT ANGL E M E NT I N G RA P H S TA T E S A N D I T S A PPL I C A T I O N S 21th e g r a p h u n c h a n g e d :(4 1 ) \u03C4a: G mapsto\u2192 \u03C4a( G ): = G + Na.Wi t h t h i s n o t a t i o n t h e f o l l o w i n g r e s u l t c a n b e s t a t e d [ 3 9 , 6 0 ]:Pr o p o s i t i o n 5 ( LC - r u le ) . By l o c a l c o m p l e m e n t a t i o n o f a g r a p h G at s om e v e r te x a \u2208 V oneobt ai ns an L C - e qui v al e nt gr aph s tat e | \u03C4a( G ) \u3009 :(4 2 ) | \u03C4a( G ) \u3009 = U\u03C4a( G ) | G \u3009 ,wh e r e(4 3 ) U\u03C4a( G )= e\u2212 ipi4\u03C3axeipi4\u03C3Naz\u221DradicalbigKais a lo c a l C lif fo r d u n ita r y . F u r th e r m o r e , tw o g r a p h s ta te s | G \u3009 and | Gprime\u3009 ar e L C - e qui v al e nt iff theco r r es p o n d i n g g r a p h s a r e r el a t ed b y a s eq u en ce o f l o ca l co m p le m e n ta tio n s , i. e . Gprime= \u03C4a1\u25E6 ... \u25E6\u03C4an( G ) fo r s o m e a1,...,an\u2208 V .23412341234123412341234123412341234123412341No. 1 No. 2 No. 3 No. 4No. 5 No. 6 No. 7 No. 8No. 9 No. 10 No. 11Apply LC ! RuleFi g . 4 . \u2013 A n e x a m p l e f o r a s u c c e s s i v e a p p l i c a -tio n o f th e L C - r u le , w h ic h e x h ib its th e w h o leeq u i v al en ce cl as s as s o ci at ed w i t h g r ap h N o . 1 .Th e r u l e i s s u c c e s s i v e l y a p p l i e d t o t h e v e r t e x ,wh i c h i s c o l o r e d r e d i n t h e fi g u r e .Fi g . 4 d e p i c t s a n e x a m p l e f o r s u c h a s u c c e s s i v eap p l i cat i o n o f t h e L C - r u l e. S t ar t i n g w i t h t h e fi r s tgr a ph t he c om pl e t e or bi t c a n be obt a i ne d by a p-pl yi ng t he L C - r ul e t o t he v e r t i c e s i n t he pr e c e di nggr a ph t ha t a ppe a r a bo v e t he a r r o w of t he f ol l o w i ngdi a gr a m :No . 13\u2212\u2212 \u2212 \u2212\u2192 No . 22\u2212\u2212 \u2212 \u2212\u2192 No . 33\u2212\u2212 \u2212 \u2212\u2192No . 41\u2212\u2212 \u2212 \u2212\u2192 No . 53\u2212\u2212 \u2212 \u2212\u2192 No . 61\u2212\u2212 \u2212 \u2212\u2192No . 73\u2212\u2212 \u2212 \u2212\u2192 No . 84\u2212\u2212 \u2212 \u2212\u2192 No . 91\u2212\u2212 \u2212 \u2212\u2192No . 102\u2212\u2212 \u2212 \u2212\u2192 No . 11Pr o o f o f Pr o p o s i t i o n 5 :Le t G be a gr a ph w i t h c or r e l a t i on ope r a t or s Kban d Gprime= \u03C4a( G ) th e c o r r e s p o n d in g g r a p h u n d e rlo c a l c o m p le m e n ta tio n a t v e r te x a wi t h c o r r e l a t i o n o p e r a t o r s Kprimeb.F o r c \u2208 V \\ Nawe fi n d(4 4 ) U\u03C4aKc( U\u03C4a)\u2020= Kc= Kprimec.Fo r b \u2208 Na,w e c o m p u teU\u03C4aKb( U\u03C4a)\u2020=( \u2212 i \u03C3ax)parenleftbigi \u03C3bzparenrightbig\u03C3bx\u03C3az\u03C3Nb\\ az= \u03C3ax\u03C3Naz\u00B7 \u03C3bx\u03C3Nb+ Naz= Kprimea\u00B7 Kprimeb.E NT ANGL E M E NT I N G RA P H S TA T E S A N D I T S A PPL I C A T I O N S 21th e g r a p h u n c h a n g e d :(4 1 ) \u03C4a: G mapsto\u2192 \u03C4a(G ): = G + Na.Wi t h t h i s n o t a t i o n t h e f o l l o w i n g r e s u l t c a n b e s t a t e d [ 3 9 , 6 0 ]:Pr o p o s itio n 5 (LC -r u le ) . By lo c a l c o m p le m e n ta tio n o f a g ra p h G at s om e v e r te x a \u2208 V oneobt ai ns an L C -e qui v al e nt gr aph s tat e |\u03C4a(G )\u3009:(4 2 ) |\u03C4a(G )\u3009 = U\u03C4a(G ) |G \u3009 ,wh e re(4 3 ) U\u03C4a(G )= e\u2212 ipi4\u03C3axeipi4\u03C3Naz\u221DradicalbigKais a lo c a l C lif fo rd u n ita r y. F u r th e r m o re , tw o g ra p h s ta te s |G \u3009 and |Gprime\u3009 ar e L C -e qui v al e nt iff theco r res p o n d in g g ra p h s a re rel a ted b y a s eq u en ce o f lo ca l co m p le m e n ta tio n s , i. e . Gprime= \u03C4a1\u25E6 ... \u25E6\u03C4an(G ) fo r s o m e a1,...,an\u2208 V .23412341234123412341234123412341234123412341No. 1 No. 2 No. 3 No. 4No. 5 No. 6 No. 7 No. 8No. 9 No. 10 No. 11Apply LC ! RuleFi g . 4 . \u2013 A n e x a m p l e f o r a s u c c e s s iv e a p p l i c a -tio n o f th e L C - r u le , w h ic h e x h ib its th e w h o leeq u iv al en ce cl as s as s o ci at ed w i t h g r ap h N o . 1 .Th e r u l e i s s u c c e s s iv e l y a p p l i e d t o t h e v e r t e x ,wh i c h i s c o l o r e d r e d i n t h e fi g u r e .Fi g . 4 d e p i c t s a n e x a m p l e f o r s u c h a s u c c e s s iv eap p l i cat i o n o f t h e L C - r u l e. S t ar t i n g w i t h t h e fi r s tgr a ph t he c om pl e t e or bi t c a n be obt a i ne d by a p-pl yi ng t he L C - r ul e t o t he v e r t i c e s i n t he pr e c e di nggr a ph t ha t a ppe a r a bo v e t he a r r o w of t he f ol l o w i ngdi a gr a m :No . 13\u2212\u2212 \u2212 \u2212\u2192 No . 22\u2212\u2212 \u2212 \u2212\u2192 No . 33\u2212\u2212 \u2212 \u2212\u2192No . 41\u2212\u2212 \u2212 \u2212\u2192 No . 53\u2212\u2212 \u2212 \u2212\u2192 No . 61\u2212\u2212 \u2212 \u2212\u2192No . 73\u2212\u2212 \u2212 \u2212\u2192 No . 84\u2212\u2212 \u2212 \u2212\u2192 No . 91\u2212\u2212 \u2212 \u2212\u2192No . 102\u2212\u2212 \u2212 \u2212\u2192 No . 11Pr o o f o f Pr o p o s itio n 5 :Le t G be a gr a ph w i t h c or r e l a t i on ope r a t or s Kban d Gprime= \u03C4a(G ) th e c o r r e s p o n d in g g r a p h u n d e rlo c a l c o m p le m e n ta tio n a t v e r te x a wi t h c o r r e l a t i o n o p e r a t o r s Kprimeb.F or c \u2208 V \\ Nawe fi n d(4 4 ) U\u03C4aKc(U\u03C4a)\u2020= Kc= Kprimec.Fo r b \u2208 Na,w e co m p u teU\u03C4aKb(U\u03C4a)\u2020=( \u2212 i\u03C3ax)parenleftbigi\u03C3bzparenrightbig\u03C3bx\u03C3az\u03C3Nb\\ az= \u03C3ax\u03C3Naz\u00B7 \u03C3bx\u03C3Nb+ Naz= Kprimea\u00B7 Kprimeb.\u2022Figure 4 in M. Hein, W. D\u00FCr, J. Eisert, R. Raussendorf, M. Van den Nest, H.-J. Briegel, quant-ph/0602096 (Varena lectures)Thursday, 29 July 2010Counter-example to LU-LC conjecture11eq u iv a len t b u t n o t LC eq u iv a len t gr a p h st a tes c a n di ffer o n ly in o n e d g e . N o te th a t th e s im p lifi-ca tio n o f Q (x ) di sc u ssed a b o v e do es n o t si m p lify o r c h a n ge th e fo r m o f th e fo ur s ta b iliz e r s ta te sin F ig . 2 .FI G . 3 : C o r r e s p o n d in g G r a p h S ta te sV. C O N C L U S I O NIn s u m m a r y , w e h a v e s h ow n th a t th e L U -L C c on je c tu re is fa ls e b yg ivin g exp licit cou n terex-am p le s o f it . It is als o c le ar th at in o r d e r to d is p r o v e th e c o n je c tu r e , w e h a v e to c o n s id e r s ta bi liz e rst a tes w ith th e r a n k o f th e su p p o r t n o less th a n 6 .T his result leads us to rethink about the localeq u iv a len c e p ro b lem o f st a b iliz er s a s m o st o f th e p rev io u s w or k foc u s on pr ov in g th e c on je c tu reto b e tr u e .Th e r a n d o m p ro c e d u re g e n e r a te s c o u n te re x a m p le s o f s iz e 27 an d 35 .T hough not proved, w ebe lie v e th a t 27 is th e s m alle s t p o s s ib le s ize o f c o u n te r e x am p le s o f L U -L C. A lt h o u g h th e L U -L Cco n je ct u r e is n o w d is p r o ve d , w e s till k n o w little a b o u t th e r el a tio n o f LU a n d LC eq u iv a len c e a n ddo n o t h a v e a n ex p lic it u n der st a n di n g o f w h y o r w h en LU eq u iv a le n c e d if fe r s fr o m L C e q u iv a-le n c e . W h e n d =6 , th e r a n d o m g e n e r a tio n p r o c e d u r e c a n fin d a c o u n te r e x a m p le in sec o n ds, b u tit is in fa c t n o t a n e ffic ie n t a lg o r ith m w h e n d is lar g e o r e v e n w h e n d =7 .T hat fact is w e havene v e r o b ta ine d a v a lid c o u nt e r e xa m p le o f d =7 us in g th e r a n d o m p r o c e d ur e . F o r tun a te ly , la r g e rsc a le c o u n ter ex a m p les, in c lu di n g th o se o f d =7 ,h av e b een fo u n d [7] m o tiv ated b y th e ran d o m lygen er a ted c o u n ter ex a m p les.As L U a n d L C e q u iv a le n c e s a re n o w k n o w n to b e d iffe re n t. It is a ls o c h alle n g in g to as k w h e th e rth e r e is a n e ffic ie n t a lg o r ith m d e c id in g L U e q u iv a le n c e fo r s tab ilize r s o r w h e th e r th e r e is a g r ap hth e or e ti c a l in te r pr e ta ti on of L U e q u iv a le n t g r a ph s ta te s . T h is s e e m s to b e d iffic u lt a s lo c a l u n ita r yope r a ti on s a re m u c h le s s lin k e d u p w ith th e s ta b iliz e r for m a lis m .The two graphs (with / without the red edge) represent locally equivalent states, but are not related by the LC rule.Zhengfeng Ji et al, Quantum Inf. Comput., Vol. 10, No. 1&amp;2, 97-108, 2010Thursday, 29 July 2010Universal resources derived via graphical rulesThese regular graphs can all be mapped into square lattices via Pauli measurements - using the graphical rules. Hence all are resources for universal MBQC.M. Van den Nest, et al, New J. Phys. 9 204 (2007).CO N TE N TS 32Fi g . 2 . A c c o rd in g ly, w e fi n a lly o b ta in a sq u a re la ttic e , i.e .,a 2D cluster state. T he totalsp a tia l o v e rh e a d is e ig h t in th a t a u n it sq u a re o n (d ) is o b ta ine d fro m e ig h t he x a g o nsin (a ). S in c e th e o v e rh e a d is c o n sta n t, a ll re so u rc e sta te s a re effi ci en t u n iv er sa l. squareNo te th a t, sin c e th e se o th e r re so u rc e sta te s c a n b e tr a n sfo rme d in to th e 2 Dcl u ster sta te a lw a y s b y adapt iv e local pr oje c tiv e m eas u re m e n ts as siste d w ith c las sic alco m m u n ica tio n , a n d lo ca l (C liff ord ) op e rat ion s, th e y are al so u n iv e rsa l in th e se n se o fac onventional one-way quantum com putation.It is in te re stin g to o b se rv e th a t th e sq u a re la ttic e , to g e th er w ith th e h ex a g o n a l a n dtr ia n g u la r la tti c e s a re th e o n ly p o ssib iliti e s to o b ta in a reg u la r tilin g o f th e 2 D p la n e.Fu r t h e r m o r e , t h e K a g o m e la t t ic e is a n e x a m p le o f a u n ifo r m s e mi re g u la r 2 D tilin gwi th 2 b a sic tile s (th e tria n g le a n d th e h e x a g o n ). Th e h e x a g o na l la ttic e ha s v e rte xde g re e 3 , w hi ch le a ds to a n inc re a se d ro bus tne ss a g a ins t lo c al n oi se as c om p are d toth e 2 D c lu ste r sta te [5 3 ]. S in c e th e 1 D c lu ste r sta te (w ith u n if o rm v e rte x d e g re e 2 )ha s b e e n pr o v e d no t to b e uni v e rsa l [2 7 ], the v e rte x de g re e 3 ism inim al for universalre sou rc e on un ifo r m la ttic e stru c tu re s.We r e m a r k t h a t o t h e r u n iv e r s a l r e s o u r c e s t a t e s h a v e b e e n p r e se n te d [5 4 ] b a se don n on -u n iform lat tic e stru c tu re s (se e al so [55 ]), w h e re e a ch g a te in a u n iv e r s a l s e tof u n itary gat e s c an b e im p le m e n te d b y lo c al m e as u re m e n ts on an e le m e n tary u n itan d th e se u n its are c om b in e d (b ot tom -u p ap p roac h ). H e re w e too k , in co n tra st, a(a) (b)(c )321(d)Fi g u r e 2 . Ex a m p les o f effi ci en t u n iv er sa l res o u rce fo r M Q C . T h es e a re g rap hst a tes co rresp o n d in g to (a ) h ex a g o n a l, (b ) tria n g u la r a n d (c) K a g o m e la ttices .De term in istic L OC C tra n sfo rm a tio n fro m (a ) to (d ) (2 D clu ster sta te) v ia (b )an d (c ) is in d ic at e d , w h e re sim p le gr ap h ru le s c an b e u se d se q ue n tia lly (\u03C3yan d\u03C3zme a su reme n ts a re d isp lay ed b y square an d \u2666 ,r espectively). The spatialo verheadfo r th e tra n sfo rm a tio n is co n sta n t.Hexagonal TriangularKagome SquareThursday, 29 July 2010Universal resources derived via graphical rulesThese regular graphs can all be mapped into square lattices via Pauli measurements - using the graphical rules. Hence all are resources for universal MBQC.M. Van den Nest, et al, New J. Phys. 9 204 (2007).CO N TE N TS 32Fi g . 2 . A c c o rd in g ly, w e fi n a lly o b ta in a sq u a re la ttic e , i.e .,a 2D cluster state. T he totalsp a tia l o v e rh e a d is e ig h t in th a t a u n it sq u a re o n (d ) is o b ta ine d fro m e ig h t he x a g o nsin (a ). S in c e th e o v e rh e a d is c o n sta n t, a ll re so u rc e sta te s a re effi ci en t u n iv er sa l. squareNo te th a t, sin c e th e se o th e r re so u rc e sta te s c a n b e tr a n sfo rme d in to th e 2 Dcl u ster sta te a lw a y s b y adapt iv e local pr oje c tiv e m eas u re m e n ts as siste d w ith c las sic alco m m u n ica tio n , a n d lo ca l (C liff ord ) op e rat ion s, th e y are al so u n iv e rsa l in th e se n se o fac onventional one-way quantum com putation.It is in te re stin g to o b se rv e th a t th e sq u a re la ttic e , to g e th er w ith th e h ex a g o n a l a n dtr ia n g u la r la tti c e s a re th e o n ly p o ssib iliti e s to o b ta in a reg u la r tilin g o f th e 2 D p la n e.Fu r t h e r m o r e , t h e K a g o m e la t t ic e is a n e x a m p le o f a u n ifo r m s e mi re g u la r 2 D tilin gwi th 2 b a sic tile s (th e tria n g le a n d th e h e x a g o n ). Th e h e x a g o na l la ttic e ha s v e rte xde g re e 3 , w hi ch le a ds to a n inc re a se d ro bus tne ss a g a ins t lo c al n oi se as c om p are d toth e 2 D c lu ste r sta te [5 3 ]. S in c e th e 1 D c lu ste r sta te (w ith u n if o rm v e rte x d e g re e 2 )ha s b e e n pr o v e d no t to b e uni v e rsa l [2 7 ], the v e rte x de g re e 3 ism inim al for universalre sou rc e on un ifo r m la ttic e stru c tu re s.We r e m a r k t h a t o t h e r u n iv e r s a l r e s o u r c e s t a t e s h a v e b e e n p r e se n te d [5 4 ] b a se don n on -u n iform lat tic e stru c tu re s (se e al so [55 ]), w h e re e a ch g a te in a u n iv e r s a l s e tof u n itary gat e s c an b e im p le m e n te d b y lo c al m e as u re m e n ts on an e le m e n tary u n itan d th e se u n its are c om b in e d (b ot tom -u p ap p roac h ). H e re w e too k , in co n tra st, a(a) (b)(c )321(d)Fi g u r e 2 . Ex a m p les o f effi ci en t u n iv er sa l res o u rce fo r M Q C . T h es e a re g rap hst a tes co rresp o n d in g to (a ) h ex a g o n a l, (b ) tria n g u la r a n d (c) K a g o m e la ttices .De term in istic L OC C tra n sfo rm a tio n fro m (a ) to (d ) (2 D clu ster sta te) v ia (b )an d (c ) is in d ic at e d , w h e re sim p le gr ap h ru le s c an b e u se d se q ue n tia lly (\u03C3yan d\u03C3zme a su reme n ts a re d isp lay ed b y square an d \u2666 ,r espectively). The spatialo verheadfo r th e tra n sfo rm a tio n is co n sta n t.Hexagonal TriangularKagome SquareThursday, 29 July 2010Measurement-patterns for gates in the one-way model5B. A u n iv e r s a l s e t o f q u a n t u m g a t e sTo p r o v id e s o m e t h in g d e fi n it e t o d is c u s s r ig h t fr o m t h ebe g in n in g , w e n o w g iv e th e p r o c e d u r e s o f h o w to r e a liz eaC N O T -gate and ag eneral one-qubit rotation viao ne-qu b it m e a s u r e m e n t s o n a c lu s t e r s t a t e . T h e e xp la n a t io nof w h y an d h o w t h e s e gat e s w ork w ill b e gi v e n in S e c -ti o n I I G .(a )123 4 5 679 10 11 12 13 14 158controltargetYYYYYYY XXXXXXCN O T- g a t e(b )123 4 5_X!_+\"+#+_(c )123 4 5X X X+#_ge n e r al r ot at ion z -r o t a t io n(d )123 4 5X YY Y(e )123 4 5X X XYHa d a m a r d -g a t e pi / 2- p h as e gat eFI G . 2 : R ea liza t io n o f el em en t a r y q u a n t u m g a t es o n t h e Q CC.Ea c h s q u a r e r e p r e s e n t s a la t t ic e q u b it . T h e s q u a r e s in t h eex t r em e lef t co lu m n m a r k ed w it h w h it e ci r cl es d en o t e t h e in -put q ubi t s , t ho s e in t he r ig h t -m o s t c o lum n de no t e t he o ut putqu b it s .AC N O T -gate can be realized on a cluster state of 15qu b it s , a s s h o w n in F ig . 2 . A ll m e a s u r e m e n t s c a n b epe r fo r m e d s im u lta n e o u s ly . T h e p r o c e d u r e to r e a liz e aCN O T- g a t e o n a c lu s t e r wi t h 1 5 q u b it s a s d is p la y e d inFi g . 2 isPr o c e d u r e 1 Re a liz a t io n o f a C N O T -g a t e a c t ing o n atw o -q u b it s t a t e |\u03C8in\u3009.1. Pre p are t h e s t at e|\u03A8in\u3009C15= |\u03C8in\u30091 ,9\u2297parenleftBigcirclemultiplytexti\u2208 C15\\{ 1 ,9 }|+ \u3009iparenrightBig.2. En t an gl e t h e 15 q u b it s of t h e c lu s t e r C15vi a t h euni t a r y o p e r a t io n S( C15).3. M e as u re al l q u b it s of C15ex cep t fo r t h e o u t p u tqu b it s 7 , 1 5 ( fo llo w in g t h e la b e lin g in F ig . 2 ) . T h eme a s u r e me n t s c a n b e p e r fo r me d s im u lt a n e o u s ly .Qu b it s 1 , 9 , 1 0 , 1 1 , 1 3 , 1 4 a r e m e a s u r e d in t h e \u03C3x-ei g en b a s is a n d q u b it s 2 -6 , 8 , 1 2 in t h e \u03C3y-e ig e n b a s is .De p e n d e n t o n t h e m e a s u r e m e n t r e s u lt s , t h e fo llo w in ggat e is t h e re b y re al iz e d :UprimeCN O T= U\u03A3 ,C N O TCN O T ( c, t ) . (2 3 )Th e r e in t h e b y p r o d u c t o p e r a t o r U\u03A3 ,C N O Tha s t he fo r mU\u03A3 ,C N O T= \u03C3( c )x\u03B3( c )x\u03C3( t )x\u03B3( t )x\u03C3( c )z\u03B3( c )z\u03C3( t )z\u03B3( t )z, wi t h\u03B3( c )x= s2+ s3+ s5+ s6\u03B3( t )x= s2+ s3+ s8+ s10+ s12+ s14\u03B3( c )z= s1+ s3+ s4+ s5+ s8+ s9+ s11+1\u03B3( t )z= s9+ s11+ s13.(2 4 )Th e r e in , t h e sire p re s e n t t h e m e as u re m e n t ou t c om e s sion t h e q u b it s i. Th e e x p r e s s io n ( 2 4 ) is m o d ifi e d if r e -dunda n t c lus t e r q ubi t s a r e pr e s e n t a nd/ o r if t he c lus t e rs t a t e o n wh ic h t h e CN O T g a t e is r e a liz e d is s p e c ifi e d b yas et { \u03BAa} di ff er en t fr o m ( 1 2 ) , s ee S ect io n I I C . T h is co n -cl u d es t h e p r es en t a t io n o f t h e C N O T g a t e, t h e p r o o f o fit s fu n c t io n in g is g iv e n in S e c t io n I I G .An a r b itr a r y r o ta ti o n URo t\u2208 SU (2 ) c a n b e r e a liz e don a c h ai n of 5 q u b it s . C on s id e r a rot at ion in it s Eu le rre p re s e n t at ionURo t[\u03BE , \u03B7 , \u03B6 ]= Ux[\u03B6 ]Uz[\u03B7 ]Ux[\u03BE ], (2 5 )wh e r e t h e r o t a t io n s a b o u t t h e x -a nd z -a x is a r eUx[\u03B1 ]= expparenleftBig\u2212 i\u03B1\u03C3x2parenrightBigUz[\u03B1 ]= expparenleftBig\u2212 i\u03B1\u03C3z2parenrightBig.(2 6 )In it ia lly , t h e fi r s t q u b it is p r e p a r e d in s o m e s t a t e |\u03C8in\u3009,wh ic h is t o b e r o t a t e d , a n d t h e o t h e r q u b it s a r e p r e p a r e din |+ \u3009.A fter the 5 qubits are entangled by the unitarytr a n s fo r m a ti o n S ,t he state |\u03C8in\u3009 ca n b e r o t a t ed b y m ea -su r in g q u b it s 1 t o 4 . A t t h e sa m e t im e , t h e st a t e is a lsos w a p p e d t o s it e 5 . Th e q u b it s 1 .. 4a re m easured in ap-pr o pr ia t e ly c ho s e n ba s e sBj( \u03D5j)=braceleftbigg|0 \u3009j+ ei\u03D5j|1 \u3009j\u221A2,|0 \u3009j\u2212 ei\u03D5j|1 \u3009j\u221A2bracerightbigg, (2 7 )wh e r e b y t h e m e a s u r e m e n t o u t c o m e s sj\u2208 { 0 , 1 } fo r j =1 .. 4a re obtained. H ere, sj=0 m eans that qubit j ispr o je c t e d in t o t he fir s t s t a t e o f Bj( \u03D5j). In (2 7 ) th e b a -si s st a t e s o f a ll p o ssi b le m e a su r e m e n t b a se s lie o n t h eeq u a t o r o f t h e B lo c h s p h er e, i.e. o n t h e in t er s ect io n o fth e B lo c h s p h e r e w ith th e x -y -p la n e . T h e r e fo r e , t h e m e a -su r e m e n t b a si s fo r q u b it j ca n b e s p eci fi ed b y a s in g le p a -ram e t e r, t h e m e as u re m e n t an gl e \u03D5j.T he m easurem entdi r e c t io n o f q ubi t j is t h e v e c t o r o n t h e B lo c h s p h e r ewh ic h c o r r e s p o n d s t o t h e fi r s t s t a t e in t h e m e a s u r e m e n tba s is Bj( \u03D5j). T h u s , th e m e a s u r e m e n t a n g le \u03D5jis t h e a n -gl e b e tw e e n t h e m e as u re m e n t d ire c t ion at q u b it j an dth e p o s iti v e x -a x is . In s u m m a r y , t h e p r o c e d u r e t o r e a liz ean arb it rary rot at ion URo t[\u03BE , \u03B7 , \u03B6 ], s p e c ifi e d b y it s E u le ran gl e s \u03BE , \u03B7 , \u03B6 ,i s this:R. Raussendorf, D. E. Browne and H.J. Briegel,  Phys. Rev. A 68: 022312 (2003)Generate the full Clifford groupThursday, 29 July 2010Graphical \u201CGottesman-Knill Theorem\u201D2021 22 23 24 25 26 27 28 29 30 31 32 3317 18 19 2010 11 12 13 14 156754321No. 18921 25 3317 1912 146531816161029No. 2FI G . 1 6 : T h e g r a p h a s s o c ia te d w ith th e Q FT o n 3 q u b its in th e o n e- w ay q u an t u m co m p u t er i s r ep r es en t ed i n g r ap h N o . 1 , w h er e the box e sde not e t he i nput ( l e f t ) a nd out put ( r i ght ) v e r t i c e s . G r a ph N o. 2 i s obt a i ne d f r om t he fi r s t a f t e r pe r f or m i ng a l l P a ul i m e a s ur e m e nt s a c c or di ngto th e p r o to c o l in R e f . [ 3 ] , e x c e p t f r o m th e \u03C3x-m e a s u re m e n ts a t th e in p u t v e rt ic e s . M o re p re c is e ly , it is o bt a i ne d f r om gr a ph N o. 1 a f t e r\u03C3y-m e a s u re m e n ts o n th e v e rt ic e s 22 , 23 , 24 , 26 , 27 , 28 , 30 , 31 , 32 an d \u03C3x-m e a s u re m e n ts o n th e v e rt ic e s 2 , 4 , 7 , 9 , 11 , 13 , 15 , 18 , 20 ha v e be e npe r f or m e d.ro r c o rre c tio n , q u a n tu m c o m m u n ic a tio n , a n d q u a n tu m c o m -put a tion in the c ont e xt of the one - w a y qua nt um c om put e r. T heSc h m id t m e a s u r e is ta ilo r e d f o r a c o m p a r a b ly d e ta ile d a c c o u nton the qua nt um c or r e la tions gr a s pi ng ge nui ne m ul ti- pa r tic leen tan g lem en t, y et it tu r n s o u t to b e co m p u tab le f o r m an ygr a ph s ta te s . W e ha v e pr e s e nt e d a num be r of ge ne r a l r ul e sth a t c a n b e a p p lie d w h e n a p p r o a c h in g th e p r o b le m o f e v a lu -at in g th e S ch m id t m eas u r e f o r g en er al g r ap h s tat es , w h ich ar est a te d m o st ly in g r a p h th e o r e tic a l te r m s. T h e se r u le s h a v e the nbe e n a ppl ie d to a num be r of gr a ph s ta te s tha t a ppe a r in qua n-tu m c o m p u ta tio n a n d e r r o r c o r r e c tio n . A ls o , a ll c o n n e c te dgr a phs w ith up to s e v e n v e r tic e s ha v e be e n di s c us s e d in de ta il.Th e f o r m a lis m th a t w e p r e s e n t h e r e a b s tr a c ts f r o m th e a c tu a lphys ic a l r e a lis a tion, b ut a s ha s be e n poi nt e d out in s e v e r a l in -st a n c e s, a n u m b e r o f w e ll- c o n tr o lla b le p h y si c a l sy st e m s su chas n eu tr al at o m s in o p tical lat tices s er v e as p o ten tial can d i-da te s to r e a liz e s uc h gr a ph s ta te s [ 45 , 46 ] .In th is p a p e r, th e S c h m id t m e a s u re h a s b e e n e m p lo y e d toqua nt if y the de gr e e of e nt a ngl e m e nt , a s a ge ne r a liz a tion ofth e S c h m id t r a n k in th e b ip a r tite s e ttin g . T h is m e a s u r e issu f fi c ie n tly c o a r se to b e a c c e ssi b le f o r sy st e m s c o n si st in g ofma n y c o n s titu e n ts a n d to a llo w f o r a n a p p r o p r ia te d is c u s s io nof m ul ti- pa r tic le e nt a ngl e m e nt in gr a ph s ta te s . T he a ppr oa chof qua nt if yi ng e nt a ngl e m e nt in te r m s of r a te s of a s ym pt ot icre v e rs ib le s ta te tra n s fo rm a tio n s , a s a n a lte rn a tiv e , a p p e ar s u n -fe a s ib le in th e m a n y -p a rt ite s e ttin g . T h e q u e s tio n o f th e m in-im a l r e v e r s ib le e n ta n g lin g g e n e r a tin g s e t ( M R E G S ) in m u lti -pa r tite s ys te m s r e m a ins unr e s ol v e d to da te , e v e n f or qua nt u msy st e m s c o n si st in g o f th r e e q u b its, a n d d e sp ite c o n si d e r a b lere s e a rc h e ffo rt [4 7 , 4 8 ]. T h e s e M R E G S a re th e (n o t n e c e s s a r-Standard-form measurement patternQuantum Fourier TransformGraph state after all Pauli measurements performed2021 22 23 24 25 26 27 28 29 30 31 32 3317 18 19 2010 11 12 13 14 156754321No. 18921 25 3317 1912 146531816161029No. 2FI G . 1 6 : T h e g r a p h a s s o c ia te d w ith th e Q FT o n 3 q u b its in th e o n e- w ay q u an tu m co m p u ter is r ep r es en ted in g r ap h N o . 1 , w h er e the box e sde not e t he i nput ( l e f t ) a nd out put ( r i ght ) v e r t i c e s . G r a ph N o. 2 i s obt a i ne d f r om t he fi r s t a f t e r pe r f or m i ng a l l P a ul i m e a s ur e m e nt s a c c or di ngto th e p r o to c o l in R e f . [ 3 ] , e x c e p t f r o m th e \u03C3x-m e a s u re m e n ts a t th e in p u t v e rt ic e s . M o re p re c is e ly , it is o bt a i ne d f r om gr a ph N o. 1 a f t e r\u03C3y-m e a s u re m e n ts o n th e v e rt ic e s 22 , 23 , 24 , 26 , 27 , 28 , 30 , 31 , 32 an d \u03C3x-m e a s u re m e n ts o n th e v e rt ic e s 2 , 4 , 7 , 9 , 11 , 13 , 15 , 18 , 20 ha v e be e npe r f or m e d.ro r c o rre c tio n , q u a n tu m c o m m u n ic a tio n , a n d q u a n tu m c o m -put a tion in the c ont e xt of the one - w a y qua nt um c om put e r. T heSc h m id t m e a s u r e is ta ilo r e d f o r a c o m p a r a b ly d e ta ile d a c c o u nton the qua nt um c or r e la tions gr a s pi ng ge nui ne m ul ti- pa r tic leen tan g lem en t, y et it tu r n s o u t to b e co m p u tab le f o r m an ygr a ph s ta te s . W e ha v e pr e s e nt e d a num be r of ge ne r a l r ul e sth a t c a n b e a p p lie d w h e n a p p r o a c h in g th e p r o b le m o f e v a lu -at in g th e S ch m id t m eas u r e f o r g en er al g r ap h s tat es , w h ich ar est a te d m o st ly in g r a p h th e o r e tic a l te r m s. T h e se r u le s h a v e the nbe e n a ppl ie d to a num be r of gr a ph s ta te s tha t a ppe a r in qua n-tu m c o m p u ta tio n a n d e r r o r c o r r e c tio n . A ls o , a ll c o n n e c te dgr a phs w ith up to s e v e n v e r tic e s ha v e be e n di s c us s e d in de ta il.Th e f o r m a lis m th a t w e p r e s e n t h e r e a b s tr a c ts f r o m th e a c tu a lphys ic a l r e a lis a tion, b ut a s ha s be e n poi nt e d out in s e v e r a l in -st a n c e s, a n u m b e r o f w e ll- c o n tr o lla b le p h y si c a l sy st e m s su chas n eu tr al at o m s in o p tical lat tices s er v e as p o ten tial can d i-da te s to r e a liz e s uc h gr a ph s ta te s [ 45 , 46 ] .In th is p a p e r, th e S c h m id t m e a s u re h a s b e e n e m p lo y e d toqua nt if y the de gr e e of e nt a ngl e m e nt , a s a ge ne r a liz a tion ofth e S c h m id t r a n k in th e b ip a r tite s e ttin g . T h is m e a s u r e issu f fi c ie n tly c o a r se to b e a c c e ssi b le f o r sy st e m s c o n si st in g ofma n y c o n s titu e n ts a n d to a llo w f o r a n a p p r o p r ia te d is c u s s io nof m ul ti- pa r tic le e nt a ngl e m e nt in gr a ph s ta te s . T he a ppr oa chof qua nt if yi ng e nt a ngl e m e nt in te r m s of r a te s of a s ym pt ot icre v e rs ib le s ta te tra n s fo rm a tio n s , a s a n a lte rn a tiv e , a p p e ar s u n -fe a s ib le in th e m a n y -p a rt ite s e ttin g . T h e q u e s tio n o f th e m in-im a l r e v e r s ib le e n ta n g lin g g e n e r a tin g s e t ( M R E G S ) in m u lti -pa r tite s ys te m s r e m a ins unr e s ol v e d to da te , e v e n f or qua nt u msy st e m s c o n si st in g o f th r e e q u b its, a n d d e sp ite c o n si d e r a b lere s e a rc h e ffo rt [4 7 , 4 8 ]. T h e s e M R E G S a re th e (n o t n e c e s s a r-M. Hein,  J. Eisert and H.J. Briegel, Phys. Rev. A 69, 062311 (2004)Thursday, 29 July 2010Valence bond PEPS to MPS\u201CVirtual Quditpairs\u201Dd\uFFFFn =1| n \uFFFF | n \uFFFFPEPS \u201Cprojector\u201DConstructs a Matrix Product State MPSM ( j )m, n= Ajp,q| \u03C8 \uFFFF =\uFFFFs1,...snTr [ M ( s1) M ( s2) .. .M ( sn)] | s1\uFFFF | s2\uFFFF \u00B7 \u00B7\u00B7| sn\uFFFFD\uFFFFj =1d\uFFFFp,q =1Ajp,q| j \uFFFF\uFFFF p | \uFFFF q |Thursday, 29 July 2010References\u2022Progress Review\u2022H. J. Briegel, D. E. Browne, W. D\u00FCr, R. Raussendorf, M. Van den Nest,  Nature Physics 5 1, 19-26 (2009)\u2022Tutorials\u2022M. Hein, W. D\u00FCr, J. Eisert, R. Raussendorf, M. Van den Nest, H.-J. Briegel, quant-ph/0602096 (Varena lectures on graph states)\u2022D. E. Browne and H. J. Briegel, quant-ph/0603226\u2022M.A. Nielsen, quant-ph/0504097\u2022And many more...\u2022Search arxiv for One-way, MBQC, Cluster States, Graph States, ....Thursday, 29 July 2010 Fault-tolerant quantum computationwith cluster statesRobert Raussendorf, UBCQI10 @ UBC, July 29, 2010Our settingqubitelementary cell 2D/3D: Nearest-neighbor translation-invariant interaction. High fault-tolerance thresholdOur settingFault-tolerant quantum computationTask: Maintain the quantum speedup in the presence of deco-herence.Solution:Fault-tolerance theorem : If for a universal quantum computerthe noise per elementary operation is below a constant non-zero error threshold  then arbitrarily long quantum computa-tions can be performed e ciently with arbitrary accuracy.*: Aharonov &amp; Ben-Or (1996), Kitaev (1997), Knill &amp; La amme &amp; Zurek(1998), Aliferis &amp; Gottesman &amp; Preskill (2005)Talk outline1. Universal cluster state computation. The scheme: computation by local measurements Cluster states: creation, de nition, experiment2. General introduction to quantum error-correction3. Making cluster state computation fault-tolerantPart I:Cluster state quantum computation1.1 Cluster state quantum computationmeasurement of Z ( ), X (\"), cos X + sin Y (%) Universal computational resource: cluster state. Information written onto the cluster, processed andread out by one-qubit measurements only.R. Raussendorf and H.J Briegel, PRL 86, 5188 (2001).1.2 Cluster states - creation1. Prepare product state Oa2Cj0ia +j1iap2 on d-dimensional qubitlattice C.2. Apply the Ising interaction for a  xed time T:UIsing = e igT hPhi;ji (i)z  (j)z, withgT h = 4: Interaction time T independent of cluster size.1.2 Cluster states - simple examplesj i2 =j0i1j+i2 +j1i1j i2Bell statej i3 =j+i1j0i2j+i3 +j i1j1i2j i3GHZ-statej i4 =j0i1j+i2j0i3j+i4 +j0i1j i2j1i3j i4 ++j1i1j i2j0i3j+i4 +j1i1j+i2j1i3j i4Number of terms exponential in number of qubits!1.2 Cluster states - de nitionZZZZ XA cluster statej iC on a clusterC is the single common eigenstateof the stabilizer operators fKag,Kaj iC =j iC; 8a2C;withKa = Xa Ob2N(a)Zb; 8a2C: (1)Therein, b2N(a) if a,b are spatial next neighbors in C.1.2 Cluster states - experimentCold atoms in opticallattices [1,2]The QCC with photons [3].1: Greiner, Mandel, Esslinger, H ansch, and Bloch, Nature 415, 39-44 (2002),2: Greiner, Mandel, H ansch and Bloch, Nature, 419, 51-54 (2002).3: P. Walther et al., Nature 434, 169 (2005).Part II:Introduction to quantum error correction... take a break from cluster states2.1 Quantum vs. classical bitszxy\u03A8\u03A8  = \u03B1     +\u03B20 1E01s = 1,0V01quantum bit classical bit Measurement a ects state  Mmnt does not a ect state Set of states continuous  Set of states discreteDespite the di erences: Quantum error-correction (QEC) is possible. QEC is based on classical error correction.2.2 Starting point: Classical ECAn example: the repitition code.encoded state 0erroneous statesencoded state10 0 00 1 00 1 11 1 1bit   1     2     3   Individually read out all bits   Perform majority vote   &amp; Procedure on n-bit code corrects bn 12 c errors. Error-correction procedure learns encoded state.If state measurement does not help in classical EC why botherthat it can\u2019t be done in QEC?2.2 Starting point: Classical ECSame e ect without state measurement: Read out parities only.encoded state 0erroneous statesencoded state10 0 00 1 00 1 11 1 1bit   1     2     301()11()1 1 00 1 1( )=0SyndromeParity check     matrixcode word w. errors Syndrome only reveals error, not encoded state:Sy(c) = 0; 8 codewords c.Sy(E c) Sy(E): (2)Learning the state is not crucial for classical error-correction.2.3 How Quantum Error Correction worksClassical-to-quantum dictionary:c2f000;111g  ! j i=  j000i+ j111iErrors: bit  ip  ! spin &amp; phase  ips  x,  y,  zParity check matrixz }| { 1 1 00 1 1!  ! stabilizer operatorsZ1 Z2;Z2 Z3Syndrome  ! Measured eigenvalues ofstabilizer operators.2.3 How Quantum Error Correction works Repeated measurement of the stabilizer operators, and con-ditional correction. Correctable errors anti-commute with at least one stabilizeroperator ! error-syndrome. Syndrome informs about an error, not the encoded state.Emergence of the error thresholdResidual error after correctionerror probability psmall codelarge codeinfinite-sizecodeerror thresholdFault-tolerance theorem: For a universal quantum computer,an error per gate &lt; is e ectively as good as zero error.So far... Have explained the basics of quantum error-correction. Have ignored:{ Errors introduced by error-correction itself.{ Computation.... but that can be  xedPart III:Fault-tolerant quantum computation with 3D clusterstatesPart III outline3.1 Topological quantum error-correction with 3D cluster states3.2 Topologocal quantum gates3.3 Fault-tolerance threshold, overhead scaling, mapping to 2DKnown threshold valuesgeometric constraintno constraint[1][2][3][4]0.03, est.101010-3-4-5, est., bd., est.1D2D[7]10-8, bd.[6]2 10-5, bd..[5] 7 10-3, est.. Error sources:j+i-Preparation,  (Z)-gates, Hadamard gates, measurement.[1] Knill, (2005); [2] Zalka (1999); [3] Dawson &amp; Nielsen (2005); [4] Aliferis &amp; Gottesman &amp;Preskill (2005), [5] Raussendorf &amp; Harrington, quant-ph/0610062; [6] Svore &amp; DiVincenzo&amp; Terhal, quant-ph/0604090, [7] Aharonov &amp; Ben-Or (1999)Main idea3D cluster state = fault-tolerant substrateGates from non-trivialcluster topologyMacroscopic viewThree cluster regions:V (Vacuum), D (Defect) and S (Singular qubits).Qubits q2V: local X-measurements,Qubits q2D: local Z-measurements,Qubits q2S: local measurements of X Y.R. Raussendorf, J. Harrington and K. Goyal, Ann. Phys. 321, 2242 (2006).Microscopic view012012zxcluster edgeselementary cell of Lqubit location: (even, odd, odd) - face of L,qubit location: (odd, odd, even) - edge of L,syndrome location: (odd, odd, odd) - cube of L,syndrome location: (even, even, even) - site of L.Lattice duality L !LTranslation by vector (1;1;1)T: Cluster C invariant, L (primal)  !L (dual).face of L  edge of L;edge of L  face of L;site of L  cube of L;cube of L  site of L;(3) Many objects in this scheme exist as \u2018primal\u2019 and \u2018dual\u2019.Part III.1:Quantum Error-correction in 3D cluster states3.1 Measuring the cluster state stabilizerXZZZZX-measurementZ-measurement12345 K1 = X1Z2Z3Z4Z5But ...3.1 Measuring the cluster state stabilizerXZZZZX-measurementZ-measurement12345 K1 = X1Z2Z3Z4Z5 X;1  Z;2  Z;3  Z;4  Z;5 = +1: 1  1  1  1  1Measure eigenvalue of K1 by local measurements on qubits 1 - 5.But ...3.1 Measuring the cluster state stabilizerXZZZZX-measurementZ-measurement12345But ... all measurements in cluster region V are in the X-basis.Are there stabilizer elements that we can measure by local X-measurements only?Criterion:KJ = Oa2JXa:3.1 Measuring the cluster state stabilizerCriterion: KJ = Na2JXa:Such stabilizer elements exist!Example: XXX-measurementXXXX123456X1X2X3X4X5X6 = K1K2K3K4K5K6Correlation of measured eigenvalues: X;1  X;2  X;3  X;4  X;5  X;6 = +1, if no error: 1  1  1  1  1  13.1 Measuring the cluster state stabilizerXXX-measurementXXXX123456 X;1 X;2 X;3 X;4 X;5 X;6| {z }Error syndrome= -1 indicates an error: One bit of error syndrome per lattice cell.3.1 Measuring the cluster state stabilizerError    non-trivialerror syndrome            -1    non-trivialerror syndrome            -1Z-error on face qubit yieldsnon-trivial syndrome on ad-jacent cells. Each error leaves characteristic signature in the syndrome. Identify error by that syndrome.3.1 Geometry and topologyErrorError syndromesupported on closed surfacelocated on dual edgemultiple edges  =  chain3.1 Geometry and topologyf f 2 e 1 error chain Error syndrome 1 Error syndrome 2 1  An error chain Z(e) is detected by a syndrome Sy(f) if eand f interesect an even number of times. Intersection number is a topological invariant.3.1 Geometry and topologyf f 2 e 1 e 2 f error chain equivalent error chain Error syndrome 1 Error syndrome 2 1  Homologically equivalent error chains have same e ect onthe computation:e2 = e1 +@f  ! Z(e2) Z(e1): Only need to identify the homology class of the error.3.1 Topological error-correction Topological error-correction in 3D cluster states describedby Random plaquette Z2-gauge model (RPGM) [1]. FT quantum memory with toric code described by RPGM as well [1].[1] Dennis et al., quant-ph/0110143 (2001).3.1 Phase diagram of the RPGMMap error correction to statistical mechanics:ECno ECNishimori linepToptimalError correction [1]Minimum weightchain matching [2]3% Have an error budget of 3%.[1] T. Ohno et al., quant-ph/0401101 (2004). [2] E. Dennis et al., quant-ph/0110143(2001); J. Edmonds, Canadian J. Math. 17, 449 (1965).Part III.2:Topological quantum gates3.2 Encoded quantum gatesZ-measurementremoves qubitfrom the cluster Local Z-measurements remove the qubits in region D fromthe cluster. Remaining cluster has non-trivial topology.3.2 Encoded quantum gatesSurface perpendicular to \\time\" supports a quantum code3.2 Surface codes Storage capacity of the code depends upon the topology ofthe code surface.3.2 The surface codeharmlesserrorsyndrome at endpointharmfulerrorZZZXXXXZ ZZZ ZZ ZZZXXXXXXXXXZ==plaquettestabilizersite stabilizerOne qubit located on every edgeZXAsBpj i= Asj i= Bpj i; 8j i2HC;8s;p: (4) Surface codes are stabilizer codes associated with 2D lattices. Only the homology class of an error chain matters.A. Kitaev,quant-ph/9707021 (1997).3.2 The surface codesyndrome at endpointharmfulerrorNon-correctable error: small weight-distance awayfrom non-trivial cycle.3.2 Surface code on plane with holessite stabilizer not enforcedXXXXplaquette stabilizer not enforcedZZZZdual holeprimal hole There are two types of holes: primal and dual. A pair of same-type holes constitutes a qubit.3.2 Surface code on plane with holesSurface code with boundary:dualholedualholerough boundaryZdXdprimal holeprimal holesmooth boundaryXpZpprimal qubitdual qubit X-chain cannot end in primal hole, can end in dual hole. Z-chain can end in primal hole, cannot end in dual hole.3.2 Encoded quantum gatesDefect D = worldline of hole.3.2 Encoded quantum gatesTopological quantum gates are encoded in the wayworldlines of primal and dual holes are braided.3.2 A CNOT-gateXtXXccdualholedualholerough boundaryZdXdprimal holeprimal holesmooth boundaryXpZpprimal qubitdual qubit Propagation relation: Xc !Xc Xt. Remaining prop rel Zc!Zc, Xt!Xt, Zt!Zc Zt forCNOT derived analogously.3.2 Topological quantum gates3.2 Universal gate set Need one non-Cli ord element:fault-tolerant preparation of jAi:= X+Yp2 jAi.Encoding of jAi. FT prep. of jAi provided through realization of magic statedistillation .*: S. Bravyi and A. Kitaev, Phys. Rev. A 71, 022316 (2005).Part III.3:Threshold and Overhead Scaling3.3 Sequential cluster state creationmeasurementconveyor beltfor sequentialcluster statecreationdepth dHow large or small can the depth d be?3.3 Sequential cluster state creationHow large?measurementconveyor beltfor sequentialcluster statecreationdepth dLimitation: Temporal order of measurements  ! accumulating memoryerror.3.3 Sequential cluster state creationHow small?measurementconveyor beltfor sequentialcluster statecreationdepth d d 1.- d = 1: mapping to circuit model, d 2: cluster state.3.3 Error model Locality: Errors are assocoated with the elementary gatesof a quantum computer. Errors act where the gates act. Independence: Errors associated with di erent gates arestochastically independent. Probabilistic error-model: The elementary errors are prob-abilistic Pauli- ips  x,  y,  z on all qubits.All these assumptions can be relaxed.Example: \\Fault-tolerance with long-range correlated noise\" (Dorit Aharonov,Alexei Kitaev, John Preskill, Phys. Rev. Lett. 96 (2006) 050504)3.3 Error Model Error sources:2D (d = 1) 3D (d 2)1. j+i-preparation2. cPhase-gates3. Hadamard-gates4. Local measurement1. j+i-preparation2. cPhase-gates3. Memory error4. Local measurement Every quantum operation has same error p. Instant classical processing.3.3 Fault-tolerance thresholdTopological threshold in cluster region V:pc = 7:5 10 3 (2D);pc = 6:8 10 3 (3D): (5)Puri cation threshold for fault-tolerant jAi-preparation:pc = 3:7 10 2: (6)Topological EC sets the overall threshold.3.3 Fault-tolerance threshold 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.007  0.0072  0.0074  0.0076  0.0078  0.008fidelity of error correctionerror probability p  l  =  3  l  =  5  l  =  7  l  =  9  l  = 11  l  = 13Numerical estimate of the fault-tolerance threshold in 2D.3.3 Overhead and Robustness Denote by S (S0) the bare (encoded) size of a quantum cir-cuit. Then, for the described method:S0 Slog3S: (7) The threshold is robust against variations in the error modelsuch as higher weight elemetary errors, long-distance errors.3.3 Overhead in absolute termsOperational cost of a fault-tolerant gate, at 1/3 threshold.Summary The 3D cluster state has error-correction built-in Encoded gates by topology High error threshold of 0.7%Reading:Raussendorf and Harrington, Phys. Rev. Lett. 98, 190504 (2007).Raussendorf, Harrington and Goyal, New J. Phys. 9, 199 (2007)."@en
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