→ 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

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"

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"

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

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. . .

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±

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

→

RoleotQuANTUMMECHANM ← 1197 - - 1473

,O -- Ey . - y

' }

in : 14)

Pfk 1147 ,M ) =/ 4%14712

Q : NORMAULATION ?

Q : MEASURING ONI 50Mt QUBITS ?

MANIPULATING QUANTUM REGISTERSIN ANALOGY TO CLASSICAL CIRCUITS-

"

LEGAL"

OPERATIONS

CLASSICAL : Any f :{ 0,13"

→ { 0,13M

PROBABILISTIC CLASSICAL : ANY LINEAR STOCHASTIC MAP

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

- --REMINDER

. . .

" UNITARY"

014) = 14'

) [ Linear ⇒ Matrix ! ]

a mi:D

14 's -- [ di;] ( pic ) : Exe; g) ← eth

0,02143=040414) - -- Toft D=

Reminder

o'

;00"-

- o' 'o=H=[j !!)

" t ::÷÷iv±o÷i÷÷÷UNITARITY : U is unitary if Utu" [ Uut=utu=H]

Norm - preserving . -

"

complex rotations"

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-

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 :]

EVOLUTION ( MANIPULATION ) & MEASUREMENTS

"c"-

- { 107,1173 "o"

107 -Dz- H - I :3 -' L''

oh, tape

117 -Dz- LT] -s → Isn "

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 )

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

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

Its

" t÷÷÷"

Hi)

↳FOR THE INTERESTED !

"

BLOCH SPHERE"

GATES CONTINUED

TTTIG - GATE % :[ to e!pfiiqy) FIRST LESSON

STOPPED HERE" Z"- rotations

pilot -- RI - [ to a:(iq)s -- F [ self : ) )Tigers

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

• 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

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} )

MEASUREMENTS TURN STATES INTO"

PROBABILITY Distributions "IN AN EXPERIMENT ( ONE RUN ! ) ONLY ONE SAMPLE IS REALISED

C.f . RANDOMIZED ALGORITHM . -

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":*?. . .

CORRELATIONS : TWO -QUBIT GATES

CNOT,CONTROLLED -NOT

iii.Iii::*. cnn.io.

a :*

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÷÷:

EXAMPLE CONTINUED

xon.io#i::tt::i::i :

i:÷ :

CNOT is NOT FACTORIZABLE :t"I÷"¥jABHisea•

ft-④--B-

-

, - L