description - Leiden Universityliacs.leidenuniv.nl/~dunjkov/QAlg/Lecture2/... · PREVIOUS LESSON...
Transcript of description - Leiden Universityliacs.leidenuniv.nl/~dunjkov/QAlg/Lecture2/... · PREVIOUS LESSON...
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→ STATE OF n- QUBIT REGISTER DESCRIBED BY 2"
complex numbers
ACCESSING INFORMATION ?
Q : HOW DO WE ACCESS THE INFO OF h- COIN stochastic SYSTEM ?
→ description Also exponential . - 2"
probable ties ?- -
description . . . us . -state
- s
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QUANTUM MEASUREMENT
1.)"
computational basis measurement"
I 4) = { Nba . - bn Ibn - - bn ) ;by . . bn
⑧
14 ) →DM→ outcome bi - - bn fpmfbn.ba/l47)=/dbn--b#/"observation of Q
. system"
.
- ..
"
observation of hidden coins"
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QUANTUM MEASUREMENT
1.)"
computational basis measurement"
C- 006. . O
← 00. . A
14 ) -- finna . -bulb. - - bn ) ; MD - f!!! ) :l,
e 11117
14 ) -so/④= outcome by. . . byNORM- SQUARED . . .
"observation of Q. system
"
Recall normalization- .
.
"
observation of hidden coins"
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MANY TYPES of MEASUREMENTS . .
→ COMPLETE PROJECTIVE MEASUREMENTS
→ SPECIFIED BY A RASE HILLYER't SPACE 117
VECTOR
s,It>
.
{107,473 ; Hifi> + us , #¥ ' "
t>t) :=fz( lo) - H ) )
span ,c{ 107,11 ) ) = span ,c{ It) ,I-7 }
"c"-basis
"
H"
- basis
Z -basis X- basis
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147=4107 TBH ) what is
14) -I I1451147
,
""" ) = ?
Method a) express 14 ) as linear combination
of It ),I -7
,and as usual
felon )
O →"It)
" EQUIVALENTLY
1 -s"
I - s"
b) inner products
( 147,147 ) -- fair.t
i: ate. . .
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147=4107 TBH ) what is
P(5,1147,
"H" ) = ?
14) -Dmt? I
Method a) express 14 ) as linear combination
of It ),I - D
,and as usual
before ) EQUIVALENTLYµ ,
nothin
O →"Its
" b) inner productsµ,inner
prom"
I →"
I -g"
( 147,14 ) ) = Eri
i÷i÷÷÷÷i I "iiii±
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P ( 5-0/14) ) = 14+147/2-
ordering matters.
Choose 0 It)
Plan 114 HI -- K- 14712 / rest -xbecause Hadamard-
EXAMPLE INNER PRODUCT
bilinearKy . .14h ) -- alt ) t pl - 7
( actually sesgnuilinearity 142) -- 8$07 t 811 )as ( ahh, → a- 1141,147)
( 4h14) -- ( Ict HTC- 1) (81071-81171=28410) t -- Fs th)- w
YE -
tn
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→
RoleotQuANTUMMECHANM ← 1197 - - 1473
,O -- Ey . - y
' }
in : 14)
Pfk 1147 ,M ) =/ 4%14712
Q : NORMAULATION ?
Q : MEASURING ONI 50Mt QUBITS ?
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MANIPULATING QUANTUM REGISTERSIN ANALOGY TO CLASSICAL CIRCUITS-
"
LEGAL"
OPERATIONS
CLASSICAL : Any f :{ 0,13"
→ { 0,13M
PROBABILISTIC CLASSICAL : ANY LINEAR STOCHASTIC MAP
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MANIPULATING QUANTUM REGISTERS
QUANTUM : ALL • LlNEARMA• MAPPING Q . STATES TO Q. STATES-
LINEAR : ⑨ ( NY)tpl9) ) = 20/147 ) tp ⑧ ( MS )
VALID : NORM - PRESERVING
THEOREM : LINEAR t NP = ? ⑧ IS UNITARY
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- --REMINDER
. . .
" UNITARY"
014) = 14'
) [ Linear ⇒ Matrix ! ]
a mi:D
14 's -- [ di;] ( pic ) : Exe; g) ← eth
0,02143=040414) - -- Toft D=
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Reminder
o'
;00"-
- o' 'o=H=[j !!)
" t ::÷÷iv±o÷i÷÷÷UNITARITY : U is unitary if Utu" [ Uut=utu=H]
Norm - preserving . -
"
complex rotations"
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SINGLE QUBIT UNITARIES"
single qubit quantum gates"
1) IDENTITY ; U 14) I 14 )
u -- to :] -- H-
2) BIT FLIP ,NOT
.PAULI - X
,6th
,*
Not -- H ) ) x -- Iff )X11) -- lo )
-E-
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3) PHASE FUD,PAULI -2,82 ,
2h22
2107=104
211) e - my
Z - Iff)
-1¥
4) HADAMARD GATE H
tho) if (lost us )= . It)
HH-c-klioseuy.it, Haiti :]
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EVOLUTION ( MANIPULATION ) & MEASUREMENTS
"c"-
- { 107,1173 "o"
107 -Dz- H - I :3 -' L''
oh, tape
117 -Dz- LT] -s → Isn "
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EVOLUTION ( MANIPULATION ) & MEASUREMENTS
"C"-
- { 107,1173 "o"
107 -HEH - I 'd -' E'ok, tape
117 -Dz- LT] -s → th "
€ "c"i{ lol ,H}
WORK IT OUT
µ¥④t= lose [ f ]→Last 'm )
m -Du-Det'T"
µµ=f, th )-ZHH-efft.o.lk/--- tf )
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z.tk#---Llo:llH---ftIH---f( f ! ) H 107 - IH : fr (lol thy
HM=H=:fn)H-kHl=#
HELMI -- HEEM - if :D . ⇒±
=L HH - I. Heel - fifth - HIKE cod -- El
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EVOLUTION ( MANIPULATION ) & MEASUREMENTS
"c"-
- { 107,1173 "o"
107 -HEH - I :3 -still, laps
117 -Dz- - LT] -s → th "
"c':{ 1041173
to> -De-TE④t=M HZµ -Du-Dz-Ff" "7
Note : Htt't⇒ HH -- H
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Its
" t÷÷÷"
Hi)
↳FOR THE INTERESTED !
"
BLOCH SPHERE"
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GATES CONTINUED
TTTIG - GATE % :[ to e!pfiiqy) FIRST LESSON
STOPPED HERE" Z"- rotations
pilot -- RI - [ to a:(iq)s -- F [ self : ) )Tigers
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PREVIOUS LESSON SUMMARY
• DESCRIPTION OF tATEOFQUANTUMRE.CI- R ( QUANTUM STATE OF n- qubits )(1, . . -t )
14 " ) c- ¢"
14 "'
) -- E x.. . . ..
Ibn . . .ba> 1114"' >Hitbi -- bn -- ) - vector .(01 . - - O ) " o . . -on
"
is a label
-¢"I span,c{ 100--07,100--017,100--107t_ . - IM- - my}
lui's" Hi
,
i
inner product
frmh qubits → 2
"
complex numbers ,one for eachconfiguration ( bitsking )
Cf . n coins → 2"
probabilities ,one for each configuration
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• Putting systems together : XO
a÷:i÷÷i:* t.÷÷i" Kronecker product
"
" Tensor product'
, ¢"
①2M€ qzhtmFACTOR12Ability ?-
BASIS "C"
-
• Measurement : 14 > E span£1000 - -07,. - Ibn - bn )
,- - 11111))
Wrt "C"
14) -l±fb . . .by,P (bi - - 4114) ) -- Mlb . -ubn 14112
I 19bar - bn 12
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Basis B
-
• Measurement : 14 > e span.LI 407,197 - -- 1% . .)}Wrt B T t' I'
index IEOO [ "outcomes " ]can be bit-strings
147 .*D=i Pfi 1147,131=14%14712--
BORN RULE.
Eg .
11-7=12/1071-11 ) ) H - { Htt - - t) , Itt . - t - y1-7=-1 ( lol -117 ) It - - t - t) - - - K - - - y}
-
- ten"
.tn". - - in} )
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MEASUREMENTS TURN STATES INTO"
PROBABILITY Distributions "IN AN EXPERIMENT ( ONE RUN ! ) ONLY ONE SAMPLE IS REALISED
C.f . RANDOMIZED ALGORITHM . -
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I
•• EVOLUTION- ALL UNITARY MAPS ( Ut = @ * IT ; wt - Utu -- tf )" SINGLE QUBIT GATES
" A1 = [too ]
X - Eff ] NO) - 113,11117=10)Z=LII¥stk¥ffh]HHiHjHHinto .i÷÷÷÷:÷÷÷÷÷:÷÷÷÷:÷'s":*?. . .
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CORRELATIONS : TWO -QUBIT GATES
CNOT,CONTROLLED -NOT
iii.Iii::*. cnn.io.
a :*
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Two -QUBIT GATES FROM SINGLE - QUBIT GATES
r-
T
1%47511 ⇒bimetal.---
i
f
4-dim I 4-dim EXAMPLE
valor'
I've" YYIEIY.FI?m)uotbnoxttnb4brD--tnhHotbn7lbr7"""""" t.si: i:*:*:i;*ii:i÷÷:
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EXAMPLE CONTINUED
xon.io#i::tt::i::i :
i:÷ :
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CNOT is NOT FACTORIZABLE :t"I÷"¥jABHisea•
ft-④--B-
-
, - L
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