Qubit and Quantum Gates - 物理情報工学科 | Dept. … Quantum Computation and Quantum...
Transcript of Qubit and Quantum Gates - 物理情報工学科 | Dept. … Quantum Computation and Quantum...
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