Quantum bits – Single qubit

Introduction to quantum computer

October 7, 2003

Outline

• Quantum bits – single qubit – multiple qubit

• Quantum computation – Single qubit gates – Multiple qubit gates – Quantum circuits

Quantum bits – Single qubit

What is a Qubit ?

• a qubit is a vector in 2D complex vector space

• a classicl bit has a state - either 0 or 1

• a qubit can in a state other |0i or |1i

it can in a linear combination of state : superposition

|ψi = α|0i + β|1i

• |0i and |1i are two orthenormal basis of the 2D vector space (|0i, |1i) = h0|1i = 0

• matrix representation:

|0i =

"

1 0

#

|1i =

"

0 1

#

|ψi = α|0i + β|1i =

"

α 0

#

+

"

0 β

#

=

"

α β

#

Measurement

• A measurement on the qubit

|ψi = α|0i + β|1i would give EITHER

|0i with the probability |α|² ,or

|1i with the probability |β|² .

• normalization : |α|² + |β|² = 1

• the state becomes what you measured after measurement

multiple qubit

How about 2 Qubits ?

• classically, 4 possible states 00, 01, 10, and 11

• QM: a superposition of 4 states |00i, |01i, |10i, and |11i

• assuming the state vector describing 2 qubits is

|ψi = α00|00i + α01|01i + α10|10i + α11|11i

• normalization: ^P_x∈{0,1}² |α^x|² = 1

• measuring the 1st qubit give 0 with the probability

|α00|² + |α⁰¹|²

• the post-measurement state

|ψ⁰i = α₀₀|00i + α01|01i

q

|α00|² + |α01|²

Bell state or EPR pair

An important two qubit state

|00i + |11i

√2

• measuring 1st qubit gives 2 possible results

– 0 with the probability 1/2, and the post-measurement state |00i

– 1 with the probability 1/2, and the post-measurement state |11i

• measuring 2nd qubit ALWAYS gives the same result with the 1st measuement

N qubits

A superposition of the 2ⁿ states

|ψi = ^X

x_n=0,or1

α_···|x1x₂ · · · xⁿi

Quantum computation

Single qubit gates

• classical NOT gate:

0 → 1 , and 1 → 0

• quantum NOT gate:

|0i → |1i , |1i → |0i

• how about superposition state ?

α|0i + β|1i → β|0i + α|1i

Matrix representation

• the matrix representation of quantum NOT gate is:

X _≡

"

0 1 1 0

#

•

X_|0_{i =}

"

0 1 1 0

# "

1 0

#

=

"

0 1

#

= |1_i

•

X_(α|0_{i + β|}1_{i) =}

"

0 1 1 0

# "

α β

#

=

"

β α

#

= β|0_{i + α|}1_i

What kinds of matrix can be a quantum gate ?

• We requires the normalization condition

|α|² + |β|² = 1 , for |ψi = α|0i + β|1i

• This will be hold after acting of the quantum.

|ψ⁰i = α⁰|0i + β⁰|1i

• It turns out the matrix repersenting the gate is the unitary matrix U

U^†U = I

Another single qubit gates

• Z _gate

Z _≡

"

1 0 0 −1

#

• Hadamard gate

H _{≡ √}¹ 2

"

1 1 1 −1

#

•

H_(|0_{i) =} ^{|0i + |1i}_√ 2 H_(|1_{i) =} ^{|0i − |1i}_√

2

•

H₍H_{|ψi) = (}HH_{)|ψi =} I_{|ψi = |ψi}

multiple gubit gates

the controlled-NOT(CNOT) gate

• if the control qubit is set to 0, then the target qubit left alone.

• if the control qubit is set to 1, then the target qubit is flipped.

• |00i → |00i ; |01i → |01i ; |10i → |11i ; |11i → |10i .

• |A, Bi → |A, B ⊕ Ai, where ⊕ is addition modulo two

Controlled-NOT gate

matrix representation

U_CN =











1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0











U_CN^† U_CN = I

Measurement gate

• a measurement gate performs the measurement

• measurement of a sigle qubit in the state α|0i + β|1i yields the result 0 or 1

• the state after measurement becomes |0i or |1i

• the respective probabilities is |α|² and |β|²

Quantum circuits

• swap circuit

|a, bi → |a, a ⊕ bi

→ |a ⊕ (a ⊕ b), a ⊕ bi = |b, a ⊕ bi

→ |b, (a ⊕ b) ⊕ bi = |b, ai

Quantum circuits

• no clone theory

– if gate U could clone any quantum state |αi U (|αi|0i) = |αi|αi

– U did not depend on |αi alone

U (|βi|0i) = |βi|βi

– what will happen if we want to clone |γi = |αi + |βi ? U (|γi|0i) = U(|αi + |βi)|0i = |αi|αi + |βi|βi 6= |γi|γi

Quantum circuits

• example: circuit to creat Bell state

• target state

|β00i = |00i + |11i

√2

|β01i = |01i + |10i

√2

|β10i = |00i − |11i

√2

|β11i = |01i − |10i

√2