UBC Faculty Research and Publications

Topological Quantum Computation Bonesteel, Nick 2010

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              F a u l t T o l e r a n t Q u a n tu m  C o m p u ta ti o n  b y  A n y o n s ,A.  Y u .  Ki t a e v ,  An n a l s  Ph y s .  3 0 3 ,  2  (2 0 0 3 ).  (q u a n t -p h / 9 7 0 7 0 2 1 )A  M o d u l a r  F u n c to r  W h i c h  i s  U n i v e r s a l  fo r  Q u a n tu m  C o m p u ta ti o n ,M . H .  F re e d m a n ,  M .  L a rs e n  a n d  Z .  W a n g ,  C o m m .  M a t h .  P h y s .  2 2 7 ,  6 0 5  (2 0 0 2 ).M a i n  o r i g i n a l  s o u r c e s :N o n -A b e l i a n  A n y o n s  a n d  T o p o l o g i c a l  Q u a n tu m  C o m p u ta t ion ,C .  N a y a k  e t  a l . ,  R e v .  M o d .  Ph y s .  8 0 ,  1 0 8 3  (2 0 0 8 ).   (a rX i v : 0 7 0 7 . 1 8 8 9 v 2 )L e c tu r e s  o n  T o p o l o g i c a l  Q u a n tu m  C o m p u ta ti o n ,J .  Pre s k i l l , A v a i l a b l e  o n l i n e  a t : w w w . t h e o ry . c a l t e c h . e d u / ~ p re s k i l l / p h 2 1 9 / t o p o l o g i c a l . p dfSo m e  e x c e l l e n t r e v i e w s :N E B ,   L .  H o r m o zi ,   G .  Z i ko s,  S . H .  S i m o n ,   P h ys.  R e v .  L e t t .  9 5  1 4 0 5 0 3  ( 2 0 0 5 ) .S . H .  S i m o n ,  N E B ,  M . F r e e d m a n ,  N ,  P e t r o vi c,  L .  H o r m o zi ,  P h ys.  R e v .  L e t t .  9 6 ,  0 7 0 5 0 3  ( 2 0 0 6 ) .L .  H o r m o zi ,  G .  Z i ko s,  N E B ,  a n d  S . H .  S i m o n ,  P h ys.  R e v .  B  7 5 ,  1 6 5 3 1 0  ( 2 0 0 7 ) .  L .  H o r m o zi ,  N E B ,  a n d  S . H .  S i m o n ,  P h ys.  R e v .  L e t t .  1 0 3 ,  1 6 0 5 0 1  ( 2 0 0 9 ) .A l s o :N i c k  Bo n e s t e e l ,    F l o ri d a  St a t e  U n i v e rs i t y            S t o n e            =  0            =  1            =  1T h e  i St o n e            T h e  i St o n e :  1  b i t            T h e  i St o n e  4 :  ~  2 0  b i t s           T h e  i Ph o n e  4 :  ~  2 . 6  x  1 011b i t s           T h e  i Po d :  ~  1 . 4  x  1 012b i t s           h t t p : / / e n . w i k i p e d i a . o rg / w i k i / H a rd _ d i s k _ d ri v e       “s p i n  u p ” “s p i n  d o w n ”A  sp i n - 1 / 2  p a r t i cl e :       M a n y  sp i n - 1 / 2  p a r t i cl e s:“s p i n  u p ” “s p i n  d o w n ”A  sp i n - 1 / 2  p a r t i cl e :       M a g n e t i c  O rd e r“s p i n  u p ” “s p i n  d o w n ”A  sp i n - 1 / 2  p a r t i cl e :       M a g n e t i c  O rd e r“s p i n  u p ” “s p i n  d o w n ”A  sp i n - 1 / 2  p a r t i cl e :       M a g n e t i c  O rd e r=  0“s p i n  u p ” “s p i n  d o w n ”A  sp i n - 1 / 2  p a r t i cl e :       M a g n e t i c  O rd e r=  1“s p i n  u p ” “s p i n  d o w n ”A  sp i n - 1 / 2  p a r t i cl e :       M a g n e t i c  O rd e r=  1“s p i n  u p ” “s p i n  d o w n ”A  sp i n - 1 / 2  p a r t i cl e :T e r r i f i c  f o r  s to r i n g  c l a s s i c a l  i n f o r m a ti o n , b u t u s e l e s s  f o r  q u a n tu m  In f o r m a ti o n .( )↓↑−↑↓=21          A  v a l e n ce  b o n d :( )↓↑−↑↓=21          A  v a l e n ce  b o n d :( )↓↑−↑↓=21          A  v a l e n ce  b o n d :M a n y  sp i n - 1 / 2  p a r t i cl e s:( )↓↑−↑↓=21          A  v a l e n ce  b o n d :U s e  p e ri o d i c  b o u n d a ryc o n d i t i o n s( )↓↑−↑↓=21          3 1 1 1A  v a l e n ce  b o n d :( )↓↑−↑↓=21          3 1 1 1O d dA  v a l e n ce  b o n d :( )↓↑−↑↓=21          31 31O d dA  v a l e n ce  b o n d :             | 0  |0| 0  | 0  | 1  | 0  | 0  | 1  | 0  | 1  | 1  | 1  | 1  | 1  | 1  | 0  C o n tr o l l e d - N o tA n y  N  q u b i t o p e r a ti o n  c a n  b e  c a r r i e d  o u t u s i n g  th e s e  tw o  g a te s .φUS i n g l e  Q u b i t R o ta ti o nψψφUMMMMaaaa1111iΨ=fΨUni v er s al  Q uantum  G ates| 0 |0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 C ont r ol l e d N otA ny  N  qubi t  ope r a t i on c a n be  c a r r i e d out  us i ng  t he s e  t w o g a t e s .φUS i ng l e  Q ubi t  R ot a t i onψψφUMMMMaaaa1111iΨ=fΨO ne w ay  to go…L os s  a nd D i V i nc e nz o, ‘ 98|  0    =| 1    =M a ni pul a t e  e l e c t r on s pi ns  w i t h e l e c t r i c  a nd m a g ne t i c  f i e l ds  t o c a r r y  out  qua nt um  g a t e s .P r ob l e m :   Er r or s  a nd D e c ohe r e nc e !   M a y  be  s ol va bl e , but  i t  w on’ t  be  e a s y !T o p o l o g i c a l  O rd e r  (W e n & N i u ,  PR B 4 1 ,  9 3 7 7  (1 9 9 0 ))C o n v e n t i o n a l l y  O r d e r e d  S t a t e s :   M u l t i p l e  “ b r o ke n  sy m m e t r y ”  g r o u n d  st a t e s ch a r a ct e r i z e d  b y  a  l o ca l l y  o b se r v a b l e  o r d e r  p a r a m e t e r .T o p o l o g i ca l l y  O r d e r e d  S t a t e s :   M u l t i p l e  g r o u n d  st a t e s o n  t o p o l o g i ca l l y  n o n t r i v i a l  su r f a ce s w i t h  n o  l o ca l l y  o b se r v a b l e  o r d e r  p a r a m e t e r .21+==zSmoddodde v e noddodde v e ne v e ne v e nm a g n e t i z a t i o n21−==zSmm a g n e t i z a t i o nN a tu r e ’ s  c l a s s i c a l  e r r o r  c o r r e c ti n g  c o d e s  !N a tu r e ’ s  q u a n tu m  e r r o r  c o r r e c ti n g  c o d e s  ?       UUW ha t  br a i d c or r e s ponds  t o t hi s  c i r c ui t ?( )↓↑−↑↓=21          31 31O d dA  v a l e n ce  b o n d :( )↓↑−↑↓=21          31 31O d dA  v a l e n ce  b o n d :( )↓↑−↑↓=21          3 1 3 1O d dA  v a l e n ce  b o n d :( )↓↑−↑↓=21          3 1 1 1O d dA  v a l e n ce  b o n d :( )↓↑−↑↓=21          O d dA  v a l e n ce  b o n d :Q u a n t u m  s u p e rp o s i t i o no f  m a n y  v a l e n c e -b o n d  s t a t e s :   A  “ s p i n  l i q u i d .”( )↓↑−↑↓=21          2 2 0 2Ev e nA  v a l e n ce  b o n d :( )↓↑−↑↓=21          2 0 0 2Ev e nA  v a l e n ce  b o n d :( )↓↑−↑↓=21          Ev e nA  v a l e n ce  b o n d :( )↓↑−↑↓=21          3 1 3 1O d d|0A  v a l e n ce  b o n d :( )↓↑−↑↓=21          O d dA  v a l e n ce  b o n d :|0( )↓↑−↑↓=21          2 0 0 2Ev e n|1A  v a l e n ce  b o n d :( )↓↑−↑↓=21          Ev e nA  v a l e n ce  b o n d :|1                                                                                        4           4            2            2           2            4            4           2            2Ev e n                          αβ+E n v i r o n m e n t  ca n  m e a su r e  t h e  st a t e  o f  t h e  q u b i t  b y  a  l o ca l  m e a su r e m e n t  – a n y  q u a n t u m  su p e r p o si t i o n  w i l l  d e co h e r e  a l m o st  i n st a n t l y .Ba d  Q u b i t !αβ+O d dEv e nE n v i r o n m e n t  ca n  o n l y  m e a su r e  t h e  st a t e  o f  t h e  q u b i t  b y  a  g l o b a l  m e a su r e m e n t  – q u a n t u m  su p e r p o si t i o n  sh o u l d  h a v e  l o n g  co h e r e n ce  t i m e .      G o o d  Q u b i t !αβ+O d dEv e n      T o p o l o g i ca l l y  O r d e r e d  S t a t e s               ) :   M u l t i p l e  g r o u n d  st a t e s o n  t o p o l o g i ca l l y  n o n t r i v i a l  su r f a ce s w i t h  n o  l o ca l l y o b se r v a b l e  o r d e r  p a r a m e t e r .  oddodde v e noddodde v e ne v e ne v e nN a tu r e ’ s  q u a n tu m  e r r o r  c o r r e c ti n g  c o d e s  ?                                                         S p i n  f l i p :  “ q u a s ip a r t ic le ”  w i th  to ta l  Sz=+1                                                   B r e a k i n g  a  b o n d  c r e a te s  a n  e x c i ta ti o n  w i th  Sz=  1                 B r e a k i n g  a  b o n d  c r e a te s  a n  e x c i ta ti o n  w i th  Sz=  1                 B r e a k i n g  a  b o n d  c r e a te s  a n  e x c i ta ti o n  w i th  Sz=  1          Sz=  1  e x c i ta ti o n  f r a c t io n a liz e s i n to  tw o  Sz=  ½  q u a s i p a r ti c l e s .                BA  t w o dim ens ional gas  of  elec t rons  in a s t rong m agnet ic  f ield B .E l e c t r on s                BQ u an t u m  H al l  F l u i dAn i n co mp r essi b l e q u an tu m l i q u i d c an f orm  w hen t he Landau lev el f illing f rac t ion νννν =  ne l e c(h c/ eB ) is  a rat ional f rac t ion.               E l e c t r on( c h ar ge  = e )Q u as i p ar t i c l e s( c h ar ge  = e / 3 f or  νννν =1/ 3)W hen an elec t ron is  added t o a F QH  s t at e it  c an be fr acti o n al i z ed --- i. e. ,  it  c an break  apart  int o fr acti o n al l y  ch ar g ed  q u asi p ar ti cl es.                             4 1 ,  9 3 7 7  (1 9 9 0 ))A s i n  o u r  sp i n - l i q u i d  e x a m p l e ,  FQ H  st a t e s o n  t opol ogi c a l l y  nont r i v i a l  s ur f a c e s h a v e  d e g e n e r a t e  g r o u n d  st a t e s w h i ch  c a n onl y  be  di s t i ngui s he d by  gl oba l  m e a s ur e m e nt s .D e g e n e r a cy139Fo r  t h e  νννν = 1 / 3 st a t e :…3N1 2N                           Fr a ct i o n a l l y  ch a r g e d  q u a si p a r t i cl e sA  d e g e n e r a t e  H i l b e r t  sp a ce  w h o se  d i m e n si o n a l i t y  i s e x pone nt i a l l y  l a r ge  i n t he  num be r  of  qua s i pa r t i c l e s .S t a t e s i n  t h i s sp a ce  c a n onl y  be  di s t i ngui s he d by gl oba l  m e a s ur e m e nt s p r o v i d e d  q u a si p a r t i cl e s a r e  f a r  a p a r t .E s s e nt i a l  f e a t ur e s :A  pe r f e c t  pl a c e  t o hi de  qua nt um  i nf or m a t i on!              ( ) ( )2112,, rrrr ψλψ =( )21, rrψ( ) ( )12221,, rrrr ψλψ =T w o  e x ch a n g e s =  I d e n t i t y λ2 = 1λ =  + 1    B o so n s λ = − 1    Fe r m i o n sO n e  e x ch a n g eA  se co n d  e x ch a n g er1r2P h o t o n s,  H e4a t o m s,  G l u o n s…E l e ct r o n s,  P r o t o n s,  N e u t r o n s…                   2  sp a ce  d i m e n si o n s1  t i m ed i m e n si o nP a r t i cl e  “ w o r l d - l i n e s”  f o r m  br a i ds i n  2 + 1  ( = 3 )  d i m e n si o n sP a r t i cl e  “ w o r l d - l i n e s”  f o r m  br a i ds i n  2 + 1  ( = 3 )  d i m e n si o n sC l oc k w i s ee x c ha ngeC ount e r c l oc k w i s ee x c ha nge                                     iψiife ψψϑ=P ha s eθ =  0     B o so n s θ = π Fe r m i o n sθ = π / 3     ν = 1 / 3  q u a si p a r t i cl e sO nl y  pos s i bl e  f or  pa r t i c l e s  i n 2  s pa c e  di m e ns i ons .An y o n s                           10ψβψαψ +=i=βαψi10~~ψβψαψ +=f==βαβαψ22211211~~aaaafde ge ne r a t e  s t a t e s==βαβαψ22211211~~aaaaf=βαψiM a t r i x !M a t r i ce s f o r m  a  non- A be l i a n r e p r e se n t a t i o n  o f  t h e  br a i d gr oup .( R e l a t e d  t o  t h e  Jo n e s P o l y n o m i a l ,  T Q FT  ( W i t t e n ) ,  C o n f o r m a l  Fi e l d  T h e o r y  ( M o o r e ,  S e i b e r g ) ,  e t c. )                                           MMMMaaaa1111iΨ=fΨiΨfΨt i m e             iΨfΨMMMMaaaa1111iΨ=fΨt i m e             iΨfΨMMMMaaaa1111iΨ=fΨt i m eM a t r i x  d e p e n d s o n l y  o n  t h e  t o p o l o g y  o f  t h e  b r a i d  sw e p t  o u t  b y  a n y o n  w o r l d  l i n e s!R obus t  qua nt um  c om put a t i on?             | 0  |0| 0  | 0  | 1  | 0  | 0  | 1  | 0  | 1  | 1  | 1  | 1  | 1  | 1  | 0  C o n tr o l l e d - N o tA n y  N  q u b i t o p e r a ti o n  c a n  b e  c a r r i e d  o u t u s i n g  th e s e  tw o  g a te s .φUS i n g l e  Q u b i t R o ta ti o nψψφUMMMMaaaa1111iΨ=fΨ       UUW ha t  br a i d c or r e s ponds  t o t hi s  c i r c ui t ?              J . S .  X ia  e t a l. ,  P R L  ( 2 0 0 4 ) .              J . S .  X ia  e t a l. ,  P R L  ( 2 0 0 4 ) .ν = 5/ 2 :   P r o b a b l e  M o o r e - R e a d  P f a f f i a n  s ta te .   C h a r g e  e /4 q u a s i p a r ti c l e s  d e s c r i b e d  b y  S U ( 2 )2C h e r n - S i m o n s  T h e o r y .  N a y a k  & W i l c z e k ,  ’ 9 6              ν = 12/ 5 :  P o s s i b l e  R e a d - R e z a y i  “ P a r a f e r m i o n ”  s ta te . R e a d  & R e z a y i ,  ‘ 9 9C h a r g e  e /5 q u a s i p a r ti c l e s  d e s c r i b e d  b y  S U ( 2 )3C h e r n - S i m o n s  T h e o r y .Sl i n g e rl a n d  & Ba i s  ’ 0 1U n i v e r s a l  f o r  Q u a n tu m  C o m p u ta ti o n !F re e d m a n ,  L a rs e n  & W a n g  ’ 0 2J . S .  X ia  e t a l. ,  P R L  ( 2 0 0 4 ) .ν = 5/ 2 :   P r o b a b l e  M o o r e - R e a d  P f a f f i a n  s ta te .   C h a r g e  e /4 q u a s i p a r ti c l e s  d e s c r i b e d  b y  S U ( 2 )2C h e r n - S i m o n s  T h e o r y .  N a y a k  & W i l c z e k ,  ’ 9 6               F a u l t T o l e r a n t Q u a n tu m  C o m p u ta ti o n  b y  A n y o n s ,A.  Y u .  Ki t a e v ,  An n a l s  Ph y s .  3 0 3 ,  2  (2 0 0 3 ).  (q u a n t -p h / 9 7 0 7 0 2 1 )A  M o d u l a r  F u n c to r  W h i c h  i s  U n i v e r s a l  fo r  Q u a n tu m  C o m p u ta ti o n ,M . H .  F re e d m a n ,  M .  L a rs e n  a n d  Z .  W a n g ,  C o m m .  M a t h .  P h y s .  2 2 7 ,  6 0 5  (2 0 0 2 ).M a i n  o r i g i n a l  s o u r c e s :N o n -A b e l i a n  A n y o n s  a n d  T o p o l o g i c a l  Q u a n tu m  C o m p u ta t ion ,C .  N a y a k  e t  a l . ,  R e v .  M o d .  Ph y s .  8 0 ,  1 0 8 3  (2 0 0 8 ).   (a rX i v : 0 7 0 7 . 1 8 8 9 v 2 )L e c tu r e s  o n  T o p o l o g i c a l  Q u a n tu m  C o m p u ta ti o n ,J .  Pre s k i l l , A v a i l a b l e  o n l i n e  a t : w w w . t h e o ry . c a l t e c h . e d u / ~ p re s k i l l / p h 2 1 9 / t o p o l o g i c a l . p dfSo m e  e x c e l l e n t r e v i e w s :N E B ,   L .  H o r m o zi ,   G .  Z i ko s,  S . H .  S i m o n ,   P h ys.  R e v .  L e t t .  9 5  1 4 0 5 0 3  ( 2 0 0 5 ) .S . H .  S i m o n ,  N E B ,  M . F r e e d m a n ,  N ,  P e t r o vi c,  L .  H o r m o zi ,  P h ys.  R e v .  L e t t .  9 6 ,  0 7 0 5 0 3  ( 2 0 0 6 ) .L .  H o r m o zi ,  G .  Z i ko s,  N E B ,  a n d  S . H .  S i m o n ,  P h ys.  R e v .  B  7 5 ,  1 6 5 3 1 0  ( 2 0 0 7 ) .  L .  H o r m o zi ,  N E B ,  a n d  S . H .  S i m o n ,  P h ys.  R e v .  L e t t .  1 0 3 ,  1 6 0 5 0 1  ( 2 0 0 9 ) .A l s o :N i c k  Bo n e s t e e l ,    F l o ri d a  St a t e  U n i v e rs i t y            S t o n e            =  0            =  1            =  1T h e  i St o n e            T h e  i St o n e :  1  b i t            T h e  i St o n e  4 :  ~  2 0  b i t s           T h e  i Ph o n e  4 :  ~  2 . 6  x  1 011b i t s           T h e  i Po d :  ~  1 . 4  x  1 012b i t s           h t t p : / / e n . w i k i p e d i a . o rg / w i k i / H a rd _ d i s k _ d ri v e       “s p i n  u p ” “s p i n  d o w n ”A  sp i n - 1 / 2  p a r t i cl e :       M a n y  sp i n - 1 / 2  p a r t i cl e s:“s p i n  u p ” “s p i n  d o w n ”A  sp i n - 1 / 2  p a r t i cl e :       M a g n e t i c  O rd e r“s p i n  u p ” “s p i n  d o w n ”A  sp i n - 1 / 2  p a r t i cl e :       M a g n e t i c  O rd e r“s p i n  u p ” “s p i n  d o w n ”A  sp i n - 1 / 2  p a r t i cl e :       M a g n e t i c  O rd e r=  0“s p i n  u p ” “s p i n  d o w n ”A  sp i n - 1 / 2  p a r t i cl e :       M a g n e t i c  O rd e r=  1“s p i n  u p ” “s p i n  d o w n ”A  sp i n - 1 / 2  p a r t i cl e :       M a g n e t i c  O rd e r=  1“s p i n  u p ” “s p i n  d o w n ”A  sp i n - 1 / 2  p a r t i cl e :T e r r i f i c  f o r  s to r i n g  c l a s s i c a l  i n f o r m a ti o n , b u t u s e l e s s  f o r  q u a n tu m  In f o r m a ti o n .( )↓↑−↑↓=21          A  v a l e n ce  b o n d :( )↓↑−↑↓=21          A  v a l e n ce  b o n d :( )↓↑−↑↓=21          A  v a l e n ce  b o n d :M a n y  sp i n - 1 / 2  p a r t i cl e s:( )↓↑−↑↓=21          A  v a l e n ce  b o n d :U s e  p e ri o d i c  b o u n d a ryc o n d i t i o n s( )↓↑−↑↓=21          3 1 1 1A  v a l e n ce  b o n d :( )↓↑−↑↓=21          3 1 1 1O d dA  v a l e n ce  b o n d :( )↓↑−↑↓=21          31 31O d dA  v a l e n ce  b o n d :             | 0  |0| 0  | 0  | 1  | 0  | 0  | 1  | 0  | 1  | 1  | 1  | 1  | 1  | 1  | 0  C o n tr o l l e d - N o tA n y  N  q u b i t o p e r a ti o n  c a n  b e  c a r r i e d  o u t u s i n g  th e s e  tw o  g a te s .φUS i n g l e  Q u b i t R o ta ti o nψψφUMMMMaaaa1111iΨ=fΨUni v er s al  Q uantum  G ates| 0 |0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 C ont r ol l e d N otA ny  N  qubi t  ope r a t i on c a n be  c a r r i e d out  us i ng  t he s e  t w o g a t e s .φUS i ng l e  Q ubi t  R ot a t i onψψφUMMMMaaaa1111iΨ=fΨO ne w ay  to go…L os s  a nd D i V i nc e nz o, ‘ 98|  0    =| 1    =M a ni pul a t e  e l e c t r on s pi ns  w i t h e l e c t r i c  a nd m a g ne t i c  f i e l ds  t o c a r r y  out  qua nt um  g a t e s .P r ob l e m :   Er r or s  a nd D e c ohe r e nc e !   M a y  be  s ol va bl e , but  i t  w on’ t  be  e a s y !T o p o l o g i c a l  O rd e r  (W e n & N i u ,  PR B 4 1 ,  9 3 7 7  (1 9 9 0 ))C o n v e n t i o n a l l y  O r d e r e d  S t a t e s :   M u l t i p l e  “ b r o ke n  sy m m e t r y ”  g r o u n d  st a t e s ch a r a ct e r i z e d  b y  a  l o ca l l y  o b se r v a b l e  o r d e r  p a r a m e t e r .T o p o l o g i ca l l y  O r d e r e d  S t a t e s :   M u l t i p l e  g r o u n d  st a t e s o n  t o p o l o g i ca l l y  n o n t r i v i a l  su r f a ce s w i t h  n o  l o ca l l y  o b se r v a b l e  o r d e r  p a r a m e t e r .21+==zSmoddodde v e noddodde v e ne v e ne v e nm a g n e t i z a t i o n21−==zSmm a g n e t i z a t i o nN a tu r e ’ s  c l a s s i c a l  e r r o r  c o r r e c ti n g  c o d e s  !N a tu r e ’ s  q u a n tu m  e r r o r  c o r r e c ti n g  c o d e s  ?       UUW ha t  br a i d c or r e s ponds  t o t hi s  c i r c ui t ?( )↓↑−↑↓=21          31 31O d dA  v a l e n ce  b o n d :( )↓↑−↑↓=21          31 31O d dA  v a l e n ce  b o n d :( )↓↑−↑↓=21          3 1 3 1O d dA  v a l e n ce  b o n d :( )↓↑−↑↓=21          3 1 1 1O d dA  v a l e n ce  b o n d :( )↓↑−↑↓=21          O d dA  v a l e n ce  b o n d :Q u a n t u m  s u p e rp o s i t i o no f  m a n y  v a l e n c e -b o n d  s t a t e s :   A  “ s p i n  l i q u i d .”( )↓↑−↑↓=21          2 2 0 2Ev e nA  v a l e n ce  b o n d :( )↓↑−↑↓=21          2 0 0 2Ev e nA  v a l e n ce  b o n d :( )↓↑−↑↓=21          Ev e nA  v a l e n ce  b o n d :( )↓↑−↑↓=21          3 1 3 1O d d|0A  v a l e n ce  b o n d :( )↓↑−↑↓=21          O d dA  v a l e n ce  b o n d :|0( )↓↑−↑↓=21          2 0 0 2Ev e n|1A  v a l e n ce  b o n d :( )↓↑−↑↓=21          Ev e nA  v a l e n ce  b o n d :|1                                                                                        4           4            2            2           2            4            4           2            2Ev e n                          αβ+E n v i r o n m e n t  ca n  m e a su r e  t h e  st a t e  o f  t h e  q u b i t  b y  a  l o ca l  m e a su r e m e n t  – a n y  q u a n t u m  su p e r p o si t i o n  w i l l  d e co h e r e  a l m o st  i n st a n t l y .Ba d  Q u b i t !αβ+O d dEv e nE n v i r o n m e n t  ca n  o n l y  m e a su r e  t h e  st a t e  o f  t h e  q u b i t  b y  a  g l o b a l  m e a su r e m e n t  – q u a n t u m  su p e r p o si t i o n  sh o u l d  h a v e  l o n g  co h e r e n ce  t i m e .      G o o d  Q u b i t !αβ+O d dEv e n      T o p o l o g i ca l l y  O r d e r e d  S t a t e s               ) :   M u l t i p l e  g r o u n d  st a t e s o n  t o p o l o g i ca l l y  n o n t r i v i a l  su r f a ce s w i t h  n o  l o ca l l y o b se r v a b l e  o r d e r  p a r a m e t e r .  oddodde v e noddodde v e ne v e ne v e nN a tu r e ’ s  q u a n tu m  e r r o r  c o r r e c ti n g  c o d e s  ?                                                         S p i n  f l i p :  “ q u a s ip a r t ic le ”  w i th  to ta l  Sz=+1                                                   B r e a k i n g  a  b o n d  c r e a te s  a n  e x c i ta ti o n  w i th  Sz=  1                 B r e a k i n g  a  b o n d  c r e a te s  a n  e x c i ta ti o n  w i th  Sz=  1                 B r e a k i n g  a  b o n d  c r e a te s  a n  e x c i ta ti o n  w i th  Sz=  1          Sz=  1  e x c i ta ti o n  f r a c t io n a liz e s i n to  tw o  Sz=  ½  q u a s i p a r ti c l e s .                BA  t w o dim ens ional gas  of  elec t rons  in a s t rong m agnet ic  f ield B .E l e c t r on s                BQ u an t u m  H al l  F l u i dAn i n co mp r essi b l e q u an tu m l i q u i d c an f orm  w hen t he Landau lev el f illing f rac t ion νννν =  ne l e c(h c/ eB ) is  a rat ional f rac t ion.               E l e c t r on( c h ar ge  = e )Q u as i p ar t i c l e s( c h ar ge  = e / 3 f or  νννν =1/ 3)W hen an elec t ron is  added t o a F QH  s t at e it  c an be fr acti o n al i z ed --- i. e. ,  it  c an break  apart  int o fr acti o n al l y  ch ar g ed  q u asi p ar ti cl es.                             4 1 ,  9 3 7 7  (1 9 9 0 ))A s i n  o u r  sp i n - l i q u i d  e x a m p l e ,  FQ H  st a t e s o n  t opol ogi c a l l y  nont r i v i a l  s ur f a c e s h a v e  d e g e n e r a t e  g r o u n d  st a t e s w h i ch  c a n onl y  be  di s t i ngui s he d by  gl oba l  m e a s ur e m e nt s .D e g e n e r a cy139Fo r  t h e  νννν = 1 / 3 st a t e :…3N1 2N                           Fr a ct i o n a l l y  ch a r g e d  q u a si p a r t i cl e sA  d e g e n e r a t e  H i l b e r t  sp a ce  w h o se  d i m e n si o n a l i t y  i s e x pone nt i a l l y  l a r ge  i n t he  num be r  of  qua s i pa r t i c l e s .S t a t e s i n  t h i s sp a ce  c a n onl y  be  di s t i ngui s he d by gl oba l  m e a s ur e m e nt s p r o v i d e d  q u a si p a r t i cl e s a r e  f a r  a p a r t .E s s e nt i a l  f e a t ur e s :A  pe r f e c t  pl a c e  t o hi de  qua nt um  i nf or m a t i on!              ( ) ( )2112,, rrrr ψλψ =( )21, rrψ( ) ( )12221,, rrrr ψλψ =T w o  e x ch a n g e s =  I d e n t i t y λ2 = 1λ =  + 1    B o so n s λ = − 1    Fe r m i o n sO n e  e x ch a n g eA  se co n d  e x ch a n g er1r2P h o t o n s,  H e4a t o m s,  G l u o n s…E l e ct r o n s,  P r o t o n s,  N e u t r o n s…                   2  sp a ce  d i m e n si o n s1  t i m ed i m e n si o nP a r t i cl e  “ w o r l d - l i n e s”  f o r m  br a i ds i n  2 + 1  ( = 3 )  d i m e n si o n sP a r t i cl e  “ w o r l d - l i n e s”  f o r m  br a i ds i n  2 + 1  ( = 3 )  d i m e n si o n sC l oc k w i s ee x c ha ngeC ount e r c l oc k w i s ee x c ha nge                                     iψiife ψψϑ=P ha s eθ =  0     B o so n s θ = π Fe r m i o n sθ = π / 3     ν = 1 / 3  q u a si p a r t i cl e sO nl y  pos s i bl e  f or  pa r t i c l e s  i n 2  s pa c e  di m e ns i ons .An y o n s                           10ψβψαψ +=i=βαψi10~~ψβψαψ +=f==βαβαψ22211211~~aaaafde ge ne r a t e  s t a t e s==βαβαψ22211211~~aaaaf=βαψiM a t r i x !M a t r i ce s f o r m  a  non- A be l i a n r e p r e se n t a t i o n  o f  t h e  br a i d gr oup .( R e l a t e d  t o  t h e  Jo n e s P o l y n o m i a l ,  T Q FT  ( W i t t e n ) ,  C o n f o r m a l  Fi e l d  T h e o r y  ( M o o r e ,  S e i b e r g ) ,  e t c. )                                           MMMMaaaa1111iΨ=fΨiΨfΨt i m e             iΨfΨMMMMaaaa1111iΨ=fΨt i m e             iΨfΨMMMMaaaa1111iΨ=fΨt i m eM a t r i x  d e p e n d s o n l y  o n  t h e  t o p o l o g y  o f  t h e  b r a i d  sw e p t  o u t  b y  a n y o n  w o r l d  l i n e s!R obus t  qua nt um  c om put a t i on?             | 0  |0| 0  | 0  | 1  | 0  | 0  | 1  | 0  | 1  | 1  | 1  | 1  | 1  | 1  | 0  C o n tr o l l e d - N o tA n y  N  q u b i t o p e r a ti o n  c a n  b e  c a r r i e d  o u t u s i n g  th e s e  tw o  g a te s .φUS i n g l e  Q u b i t R o ta ti o nψψφUMMMMaaaa1111iΨ=fΨ       UUW ha t  br a i d c or r e s ponds  t o t hi s  c i r c ui t ?              J . S .  X ia  e t a l. ,  P R L  ( 2 0 0 4 ) .              J . S .  X ia  e t a l. ,  P R L  ( 2 0 0 4 ) .ν = 5/ 2 :   P r o b a b l e  M o o r e - R e a d  P f a f f i a n  s ta te .   C h a r g e  e /4 q u a s i p a r ti c l e s  d e s c r i b e d  b y  S U ( 2 )2C h e r n - S i m o n s  T h e o r y .  N a y a k  & W i l c z e k ,  ’ 9 6              ν = 12/ 5 :  P o s s i b l e  R e a d - R e z a y i  “ P a r a f e r m i o n ”  s ta te . R e a d  & R e z a y i ,  ‘ 9 9C h a r g e  e /5 q u a s i p a r ti c l e s  d e s c r i b e d  b y  S U ( 2 )3C h e r n - S i m o n s  T h e o r y .Sl i n g e rl a n d  & Ba i s  ’ 0 1U n i v e r s a l  f o r  Q u a n tu m  C o m p u ta ti o n !F re e d m a n ,  L a rs e n  & W a n g  ’ 0 2J . S .  X ia  e t a l. ,  P R L  ( 2 0 0 4 ) .ν = 5/ 2 :   P r o b a b l e  M o o r e - R e a d  P f a f f i a n  s ta te .   C h a r g e  e /4 q u a s i p a r ti c l e s  d e s c r i b e d  b y  S U ( 2 )2C h e r n - S i m o n s  T h e o r y .  N a y a k  & W i l c z e k ,  ’ 9 6 Braid Group on n-Strands:  BnBraid Group on n-Strands:  BnSome Elements of B4Braid Group on n-Strands:  BnSome Elements of B4Not a Braid:Braid Group on n-Strands:  BnSome Elements of B4Group Multiplication×Braid Group on n-Strands:  BnSome Elements of B4Group Multiplication× =Braid Group on n-Strands:  BnSome Elements of B4Group Multiplication× =Elementary Braid Operationsσi :  Braid ith strand over i+1st strand1σ= 2σ= 3σ=1342σ1−1 σ2 σ1−1 σ2 σ3   σi `s and their inverses generate the braid groupBraid Relations111 +++ = iiiiii σσσσσσ=2||, ≥−= jiijji σσσσMatrix Rep of B3from Fib Anyonsa1time?1 =σ132Matrix Rep of B3from Fib Anyonsa1time== −5/35/41 00pipiσ iieeR132Matrix Rep of B3from Fib Anyonsa1time== −5/35/41 00pipiσ iieeR132a1132 ?2 =σMatrix Rep of B3from Fib Anyonsa1time== −5/35/41 00pipiσ iieeR132a1132 −−==−−−ττττσpipipi5/35/35/2 iiieeeFRFElementary Braid  Matricesa1 −−− ττ pipi 5/35/ ii ee= −5/35/41 00pipiσ iiee −== − ττσσ pi 5/312 ieFFσ1−1 σ2 σ1−1 σ2     =   MiΨ fΨ M T= iΨa11 1Qubit Encoding0 1Non-Computational StateQubit States1 0=0State of qubit is determined byq-spin of two leftmost particlesTransitions to this state areleakage errors1 1=1Initializing a QubitPull two quasiparticle-quasihole pairs out of the “vacuum”.0 00These three particles have total q-spin 1Initializing a QubitPull two quasiparticle-quasihole pairs out of the “vacuum”.0 10Initializing a QubitPull two quasiparticle-quasihole pairs out of the “vacuum”.0 10Measuring a QubitTry to fuse the leftmost quasiparticle-quasihole pair.10 βα +? 1Measuring a QubitIf they fuse back into the “vacuum” the result of the measurement is 0.100Measuring a QubitIf they cannot fuse back into the “vacuum” the  result of the measurement is 11111Single Qubit:  The Bloch Sphere0θ10 i−+1φ2210 i12sin02cos φθθψ ie−+=Single Qubit Operations: Rotations0ψαα = rotation vectorDirection of α is the rotation axis12sin02cos φθθψ ie−+=1Magnitude of α is the rotation angleSingle Qubit Operations: Rotations0ψα = rotation vectorα exp σααarrowrightnosparrowrightnosparrowrightnosp ⋅= iU1ψαarrowrightnospUψ ψαarrowrightnospU αarrowrightnospU2Single Qubit Operations: Rotations2 pi2 pi−2 piThe set of all single qubit rotations lives in a solid sphere of α−2 piradius 2pi.ψ ψαarrowrightnospU αarrowrightnospUSingle Qubit Operations: RotationsImportant consequence:  As long as we braid within a qubit, there is no leakage error.General rule:  Braiding inside an oval does not change the total topological charge of the enclosed particles.Can we do arbitrary single qubit rotations this way?1 12 pi2 pi−2 piσ12σ22−2 piσ1-2σ2-2N = 1N = 2N = 3N = 4N = 5N = 6N = 7N = 8N = 9N = 10N = 11Brute Force Search)10(10000003−+Oii=Brute Force Search)10(10000003−+Oii=ε“error”Braid LengthBraid Length |ln| εFor brute force search: |lnε |L. Hormozi, G. Zikos, NEB, S.H. Simon, PRB ‘07Brute Force Search)10(10000003−+Oii=ε“error”Brute force searching rapidly becomes infeasible as braids get longer.Fortunately, a clever algorithm due to Solovayand Kitaevallows for systematic improvement of the braid given a sufficiently dense covering of SU(2).Solovay-Kitaev Construction)10(10000004−+Oiiε“error”Braid Length c|ln| ε 4≈c,What About Two Qubit Gates?a1??b1 ??Problems:1. We are pulling quasiparticles out of qubits: Leakage error!2. 87 dimensional search space (as opposed to 3 for three-particle braids).  Straightforward “brute force” search is problematic.Two Qubit Controlled Gatesb1Control qubita1Goal: Find a braid in which some rotation is performed on the target qubit only if the control qubit is in the state 1.  (b=1)Target qubit“Weaving” a Two Qubit GateWeave a pair of anyons from the control qubit between anyons in the target qubit.  controlpairb1Important Rule: Braiding a q-spin 0 object does not induce transitions.Target qubit is only affected if control qubit is in state   1(b = 1)a1“Weaving” a Two Qubit GateOnly nontrivial case is when the control pair has q-spin 1.  controlpair1aWe’ve reduced the problem to weaving one anyon around three others.   Still too hard for brute force approach!1Try Weaving Around Just Two AnyonscontrolpairWe’re back to B3, so this is numerically feasible.1Question:  Can we find a weave which does not lead to leakage errors?a1A Trick:  Effective BraidingActual Weaving Effective BraidingThe effect of weaving the blue anyon through the two green anyons has approximately the same effect as braiding the twogreen anyons twice.Controlled-“Knot” GateEffective braiding is all within the target qubit         No leakage!Not a CNOT, but sufficient for universal quantum computation.SK Improved Controlled-“Knot” GateAnother Trick: Injection Weaving100010001≈controlpairStep 1:  Inject the control pair into the target qubit.Another Trick: Injection Weavingcontrolpair1000000ii≈Step 2: Weave the control pair inside the injected target qubit.controlpairAnother Trick: Injection WeavingStep 3: Extract the control pair from the target using the inverse of the injection weave.Putting it all together we have a CNOT gate:Injection Rotation ExtractionSK Improved Controlled-NOT GateUniversal Set of GatesφarrowrightnospUψ ψφarrowrightnospUSingle qubit rotations:Controlled NOT:NEB,  L. Hormozi,  G. Zikos, S.H. Simon,  Phys. Rev. Lett. 95 140503 (2005)Quantum CircuitUUWhat braid corresponds to this circuit?Quantum CircuitUUBraidTurning any Braid into a WeaveTurning any Braid into a WeaveITurning any Braid into a WeaveIITurning any Braid into a WeaveIII-1Turning any Braid into a WeaveI I-1II-1We know it is possible to carry out universal quantum computation by moving only a single particle.Can we find an efficient CNOT construction in which only a single particle is woven through the other particles?Another Useful Braid: The F-Braid10111010=10000ττττ−F-Matrix:F-Braid:110110000ττττ−iaacc≈Single Particle Weave Gate: Part 11a1bSingle Particle Weave Gate: Part 11aF-Braid1bSingle Particle Weave Gate: Part 11aaF-Braid1bccSingle Particle Weave Gate: Part 11aa b0 00 1ba111101011b bIntermediate StateSingle Particle Weave Gate: Part 11aaa b0 00 1b111101011b bSingle Particle Weave Gate: Part 2aa b0 00 1b11110101Phase-1-1+1-1100010001−−b≈Phase Braidb’ = 1 b’ = 0Single Particle Weave Gate: Part 2a aa b0 00 1b11110101-1-1-1+1b bSingle Particle Weave Gate: Part 2a a0 00 1b11110101Phase-1-1+1-1a bb bSingle Particle Weave Gate: Part 3a1a0 00 1b11110101Phase-1-1+1-1a bb1bControlled-Phase GateF-Braid Inverse of F-BraidPhase-Braida a111000010000100001−−=U + O(10-3)b’ab b11IntermediatestateFinal resultSK Improved Controlled-Phase GateUniversal “One-Particle Weave” GatesφarrowrightnospUψ ψφarrowrightnospUSingle qubit rotations:Controlled-Phase gate:L. Hormozi, G. Zikos, NEB, and S.H. Simon, Phys. Rev. B 75, 165310 (2007).-ZHow Big is Shor’s Braid?How many elementary braids are required to factor a K-bit number N using Shor’s algorithm?Bottleneck:  Modular Exponentiation requires ~ K3 gates.oaioi NxaaU )(mod0expmod =Specific requirements:~ 40 K3 NOT gates~ 28 K3 CNOT gates~ 92 K3 CCNOT (Toffoli) gatesBeckman, Chari, Devabhaktuni, Preskill, PRA 54, 1034 (1996).~   3 K    QubitsQuantum Gates for Modular ExpNOT Gate:Length (measured in elementary braids) grows logarithmically with 0110=decreasing error:ε10log18≈NOTLRoughly same scaling seen for all“three-weaves”G. Zikos, et al., Int. J. Mod. Phys. B 23, 2727 (2009).Quantum Gates for Modular ExpCNOT Gate:ε10log905 ≈≈ NOTCNOT LL -Z=CNOT is constructed using 3 three-weaves plus 2 single qubit rotations for a total of 5 three-weaves.-Z=R(pi/2 y ) R(-pi/2 y )ε10log905 ≈≈ NOTCNOT LLQuantum Gates for Modular ExpCCNOT (Toffoli) Gate: (from http://www.cl.cam.ac.uk/teaching/0607/QuantComp/lecture4.pdf )=CCNOT can be constructed using 6 CNOTs (up to single qubit rotations on the target) and 9 single qubit rotations.  So 6x3 = 18 “CNOT” three-weaves + 9 “single qubit rotation” three-weaves = 27 three-weaves.ε10log48627 ≈≈ NOTCCNOT LL310 |log|000,50 KLShor ε≈Number of Elementary BraidsTotal number of elementary braids:2 1~|| εFor a finite probability that no error occurs, we require:3000,50 KTo factor a 128-bit number:6103~ −×ε 11106 ×≈Number of Fibonacci anyons 1000≈Number of elementary braidsM. Baraban, NEB, and S. H. Simon, PRA 81, 062317 (2010)

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