Qubit and Quantum GatesSchool on Quantum Computing @Yagami

Day 1, Lesson 19:00-10:00, March 22, 2005

Eisuke AbeDepartment of Applied Physics and Physico-Informatics,

and CREST-JST, Keio University

From classical to quantum

Information is physical- Rolf Landauer

QUANTUM information or quantum INFORMATION?It depends on your background (physics or information science)Ultimately, you need bothAt the beginning, it would be better to keep one perspective (physics here)

References

Quantum Computation and Quantum Information (and references therein), M. A. Nielsen and I. L. Chuang, Cambridge University Press (2000)

Day 1, Lesson 1- Day 2, Lesson 2Physical Review A 65, 012320 (2001), N. D. Mermin

Day 1, Lesson 2L. M. K. Vandersypen, Ph. D Thesis (available at arXiv: quant-ph/0205193)

Day 2, Lesson 2

Outline

Rules of the gameQuantum bit (State space)Quantum gate (Unitary evolution)

NOT (X), Y, Z, Hadamard (H)MeasurementMultiple-qubit (Tensor product)

CNOT, SWAP, Controlled-Z, Toffoli

Quantum bitFor physicists, “quantum bit (qubit)” is a synonym for “quantum mechanical two-level system”

Superposition0≡g 1≡e

Quantum bit

Vector notation for computational basis states

⎥⎦

⎤⎢⎣

⎡=

01

0 ⎥⎦

⎤⎢⎣

⎡=

10

1

⎥⎦

⎤⎢⎣

⎡=+=

βα

βαψ 10

1||||,

22 =+∈

βαβα C : Probability amplitude

: Probabilities sum to 1

POSTULATE State space (Hilbert space)

Unitary evolutionPOSTULATE

The evolution of a qubit system is described by a unitary transformation such as

)()( 1122 tUt ψψ =

Hermitian conjugate: A† = (AT)*

Hermitian (self-adjoint): A = A†

Unitary: UU† = I⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡**

**†

dbca

dcba

Unitary evolutionConnection with the Schrödinger equation

)()()(exp)( 112112

2 tUtttiHtHdt

di ψψψψ

ψ≡⎥⎦

⎤⎢⎣⎡ −−

=⇒=h

h

H: Hamiltonian of the qubit system (Hermitian)

Exponential operator = unitary

Any unitary operator U can be realized in the form U = exp(iH) where H is some Hermitian operator

For now, actual physical systems that realize necessary Hamiltonians are NOT our interest

Quantum gate)()( 1122 tUt ψψ =

Input

U12

Output

)( 1tψ )( 2tψ

Time

ψ1U )( 12 ψUU

LU1 U2 U3ψ ψ ′Un

Successive implementation

ψ12UUUnL

Time

Quantum gate

Input Output

U )10( βαψ +=UU10 βαψ +=

0

1

0U

1U

ψβα UUU =+ )1()0(U

U

U

Superposition principle

We have infinite inputs, but it suffices to consider only the computational basis states

NOT gate

Input output0 11 0

Classical NOTa 1⊕≡ aa

01111100

=⊕==⊕=The only non-trivial

one-bit gate in the classical case

Xor

a aaX =

Quantum NOT

⎥⎦

⎤⎢⎣

⎡=⇔

==

==

0110

011100

XXX

Matrix representation

Matrix representation

⎥⎦

⎤⎢⎣

⎡=⇔

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

⇔⎪⎩

⎪⎨⎧

==

==

0110

01

10

10

01

011100

XX

X

XX

⎥⎦

⎤⎢⎣

⎡=

0110

X

The first column represents the final state of |0⟩

The second column represents the final state of |1⟩

Pauli-X, Y, Z gates

⎥⎦

⎤⎢⎣

⎡=

0110

XXa a

⎥⎦

⎤⎢⎣

⎡ −=⇔

−==

00

0110

ii

YiY

iYY aia)1(−a

⎥⎦

⎤⎢⎣

⎡−

=⇔−=

=10

0111

00Z

ZZ

Z aa)1(−a

iYXZiXZY

iZYXXYYX

2],[2],[

2],[

==

=−=IZYX === 222

0},{0},{0},{

=+===

XZZXXZZYYX

Hermitian Commutation relations

Hadamard gate

21)1(0

)1(2

11,0

a

b

ba b−+

=−∑=

⋅Ha

⎥⎦

⎤⎢⎣

⎡−

=⇔

⎥⎦

⎤⎢⎣

⎡−

=⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⇔

−=−=

+==

∑

∑

=

=

1111

21

11

21

10

,11

21

01

210

)1(2

11

210

210

1,0

1,0

H

HHbH

bH

b

b

b

XHZHYHYHZHXH =−==

IH =2

Hermitian

Circuit identities

Measurement gate

POSTULATE

10 βαψ +=Classical bitQuantum bit

a

0 with probability |α |2, or1 with probability |β |2

Mathematical descriptionGeneral measurementProjective measurementPOVM

Multiple-qubitHow do we describe multiple-qubit states?

Computational basis states for two-qubit states may be written as |00⟩, |01⟩, |10⟩, |11⟩We require them to be orthogonal, so they may be written as

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

1000

11

0100

10

0010

01

0001

00

Speculation…

A multiple-qubit state is the tensor product of the component qubit systems

POSTULATE

Tensor product

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡×

⎥⎦

⎤⎢⎣

⎡×

=⎥⎦

⎤⎢⎣

⎡⊗⎥

⎦

⎤⎢⎣

⎡=⊗

22

12

21

11

2

12

2

11

2

1

2

1

babababa

bb

a

bb

a

bb

aa

ba

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡×

⎥⎦

⎤⎢⎣

⎡×

=⊗==

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡×

⎥⎦

⎤⎢⎣

⎡×

=⊗==

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡×

⎥⎦

⎤⎢⎣

⎡×

=⊗==

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡×

⎥⎦

⎤⎢⎣

⎡×

=⊗==

1000

10

1

10

0111111

0010

10

0

10

1101001

0100

01

1

01

0010110

0001

01

0

01

1000000

Computational basis set for 2-qubit states

Matrix representation

Multiple-qubit gates

U

1a

2a

3a

naM M

naaaaU L321

|a1a2…an⟩ : 2n-dimensional vector

U : 2n by 2n unitary matrix

Independent gatesH

X H

Z

YH

X

Z

H

Ya aYH

bHX

cZ

b

c

cZbHXaYHcbaZHXYHcba ⊗⊗=⊗⊗→ )()()(

U (8 by 8 unitary matrix)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=⎥⎦

⎤⎢⎣

⎡××××

=⎥⎦

⎤⎢⎣

⎡⊗⎥

⎦

⎤⎢⎣

⎡=⊗

44244222

34143212

43234121

33133111

42

31

42

31

42

31

babababababababababababababababa

BaBaBaBa

bbbb

aaaa

BA

Controlled-U gates

U

a

b

aControl bit

10=

=abU

abif

ifbU aTarget bit

U can be an arbitrary single-qubit gateU works only when a = 1Ua|b⟩ is just a formal expression

CNOT gate

X

a

b

a

1100

=⊕==⊕=

abbabb

ifif

abbX a ⊕=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=⇔

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⇔

=⊕==⊕==⊕==⊕=

0100100000100001

0100

1000

,1000

0100

,0010

0010

,0001

0001

1011111111011001010010000000

1212121212

12

12

12

12

CCCCC

CCCC Frequently used

CNOT gate

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

0010010010000001

21C

a

b

ba⊕

b

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⇔

=⊕==⊕==⊕==⊕=

0010

1000

,0100

0100

,1000

0010

,0001

0001

011111110001101111001

0000000

21212121

21

21

21

21

CCCC

CCCC

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

≠

1000010000010010

21CNever mistake…

SWAP gatea b

a ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

1000001001000001

SWAP

b

abbabbCabbababaC

abaCba

=⊕⊕⎯⎯→⎯

⊕=⊕⊕⊕⎯⎯ →⎯

⊕⎯⎯→⎯

)(

)(12

21

12 0=⊕ aa 1

2 3

4To implement SWAP, we need to…1. Encode information on |a⟩ into 2nd qubit2. Erase information on |a⟩ from 1st qubit3. Encode information on |b⟩ into 1st qubit4. Erase information on |b⟩ from 2nd qubit

Controlled-Z gate

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

=

1000010000100001

CZ

a a

Zb bba⋅− )1(

baCZba ba⋅−⎯⎯ →⎯ )1(

Z

Controlled-Z is nonlocal

Toffoli

a

b

c

a

b

abc⊕⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

0100000010000000001000000001000000001000000001000000001000000001

Toffoli

111 =∧=⇔= baab

Toffoli is often referred to as “controlled-controlled-NOT (C2-NOT)”

QuizProve the following circuit identity

aaZbaH a

b

ba )1(,)1(2

1−=−= ∑ ⋅

XHZHIH

==2

ababaC ⊕=12

Also prove the followings

Use the following expressions for quantum gates

Answer

acbababcbaCabcbaabcaabaC

abcabaCcabaCcba

⊕=⊕⊕⊕⎯⎯ →⎯

⊕⊕=⊕⊕⊕⊕⎯⎯→⎯

⊕⊕⊕⎯⎯ →⎯

⊕⎯⎯→⎯

)(

)(23

12

23

12 Cascade implementation

Cascade erasure

Answer

( ) aaaaa

aa

cc

bHaHH

b

b

b c

bca

b c

cbba

b

ba

=−++=

−+=

−=⎟⎠

⎞⎜⎝

⎛−−=

⎟⎠

⎞⎜⎝

⎛−=

∑

∑∑∑ ∑

∑⋅+⋅⋅

⋅

21

))1((21

)1(21)1(

21)1(

21

)1(2

1

)(

⎩⎨⎧

==

=⋅+)()(0

)(acbac

bca

Constructive and destructive interferences

Answer

( ) aaaaa

aa

c

bH

bHZaHZH

b

b

b c

bca

b

bba

b

ba

=−++=

−+=

−=

⎟⎠

⎞⎜⎝

⎛−=

⎟⎠

⎞⎜⎝

⎛−=

∑

∑∑

∑

∑

⋅+

+⋅

⋅

21

))1((21

)1(21

)1(2

1

)1(2

1

)(

⎩⎨⎧

==

=⋅+)()(0

)(acbac

bca

bababba ⋅=⋅+=+⋅ )1(

Constructive and destructive interferences