Geometric aspects of quantum computing Maris Ozols University of - PowerPoint PPT Presentation

Geometric aspects of quantum computing Maris Ozols University of Waterloo Department of C&O December 10, 2007

Qubit state Qubit | ψ � = α | 0 � + β | 1 � , where α, β ∈ C and | α | 2 + | β | 2 = 1 .

Qubit state Qubit | ψ � = α | 0 � + β | 1 � , where α, β ∈ C and | α | 2 + | β | 2 = 1 . Parametrization We can find θ ( 0 ≤ θ ≤ π ) and ϕ ( 0 ≤ ϕ < 2 π ), such that � cos θ � 2 | ψ � = . e iϕ sin θ 2

Qubit state Qubit | ψ � = α | 0 � + β | 1 � , where α, β ∈ C and | α | 2 + | β | 2 = 1 . Parametrization We can find θ ( 0 ≤ θ ≤ π ) and ϕ ( 0 ≤ ϕ < 2 π ), such that � cos θ � 2 | ψ � = . e iϕ sin θ 2 Density matrix The corresponding density matrix is: � 1 + cos θ e − iϕ sin θ � ρ = | ψ � � ψ | = 1 . e iϕ sin θ 1 − cos θ 2

Bloch sphere Bijection between S 2 and C P 1 � cos θ � | ψ � = 2 e iϕ sin θ 2 �  x = sin θ cos ϕ   y = sin θ sin ϕ   z = cos θ 0 ≤ θ ≤ π and 0 ≤ ϕ < 2 π

Pauli matrices Density matrix of a qubit ρ = 1 r · � 2 ( I + � σ ) , � r = ( x, y, z ) , � σ = ( σ x , σ y , σ z ) .

Pauli matrices Density matrix of a qubit ρ = 1 r · � 2 ( I + � σ ) , � r = ( x, y, z ) , � σ = ( σ x , σ y , σ z ) . Pauli matrices � � � � � � � � 0 − i 1 0 1 0 0 1 I = , σ x = , σ y = , σ z = . 0 1 1 0 0 − 1 i 0

Pauli matrices Density matrix of a qubit ρ = 1 r · � 2 ( I + � σ ) , � r = ( x, y, z ) , � σ = ( σ x , σ y , σ z ) . Pauli matrices � � � � � � � � 0 − i 1 0 1 0 0 1 I = , σ x = , σ y = , σ z = . 0 1 1 0 0 − 1 i 0 Inner product |� ψ 1 | ψ 2 �| 2

Pauli matrices Density matrix of a qubit ρ = 1 r · � 2 ( I + � σ ) , � r = ( x, y, z ) , � σ = ( σ x , σ y , σ z ) . Pauli matrices � � � � � � � � 0 − i 1 0 1 0 0 1 I = , σ x = , σ y = , σ z = . 0 1 1 0 0 − 1 i 0 Inner product |� ψ 1 | ψ 2 �| 2 = Tr( ρ 1 ρ 2 )

Pauli matrices Density matrix of a qubit ρ = 1 r · � 2 ( I + � σ ) , � r = ( x, y, z ) , � σ = ( σ x , σ y , σ z ) . Pauli matrices � � � � � � � � 0 − i 1 0 1 0 0 1 I = , σ x = , σ y = , σ z = . 0 1 1 0 0 − 1 i 0 Inner product |� ψ 1 | ψ 2 �| 2 = Tr( ρ 1 ρ 2 ) = 1 2(1 + � r 1 · � r 2 ) .

Pauli matrices Density matrix of a qubit ρ = 1 r · � 2 ( I + � σ ) , � r = ( x, y, z ) , σ = ( σ x , σ y , σ z ) . � Pauli matrices � � � � � � � � 0 − i 1 0 1 0 0 1 I = , σ x = , σ y = , σ z = . 0 1 1 0 0 − 1 i 0 Inner product 2 = Tr( ρ 1 ρ 2 ) = 1 |� ψ 1 | ψ 2 �| 2(1 + � r 1 · � r 2 ) . � �� � 0

Pauli matrices Density matrix of a qubit ρ = 1 r · � 2 ( I + � σ ) , � r = ( x, y, z ) , � σ = ( σ x , σ y , σ z ) . Pauli matrices � � � � � � � � 0 − i 1 0 1 0 0 1 I = , σ x = , σ y = , σ z = . 0 1 1 0 0 − 1 i 0 Inner product 2 = Tr( ρ 1 ρ 2 ) = 1 |� ψ 1 | ψ 2 �| 2(1 + � r 1 · � � �� � ) . r 2 � �� � 0

Pauli matrices Density matrix of a qubit ρ = 1 r · � 2 ( I + � σ ) , � r = ( x, y, z ) , � σ = ( σ x , σ y , σ z ) . Pauli matrices � � � � � � � � 0 − i 1 0 1 0 0 1 I = , σ x = , σ y = , σ z = . 0 1 1 0 0 − 1 i 0 Inner product 2 = Tr( ρ 1 ρ 2 ) = 1 |� ψ 1 | ψ 2 �| 2(1 + � r 1 · � ) . r 2 � �� � � �� � − 1 0

General 2 × 2 unitary Rotation around z -axis � � Consider the action of U = | 0 � � 0 | + e iϕ | 1 � � 1 | = 1 0 : 0 e iϕ

General 2 × 2 unitary Rotation around z -axis � � Consider the action of U = | 0 � � 0 | + e iϕ | 1 � � 1 | = 1 0 : 0 e iϕ U | 1 � = e iϕ | 1 � . U | 0 � = | 0 � ,

General 2 × 2 unitary Rotation around z -axis � � Consider the action of U = | 0 � � 0 | + e iϕ | 1 � � 1 | = 1 0 : 0 e iϕ U | 1 � = e iϕ | 1 � . U | 0 � = | 0 � , Now if we act on | 0 � + | 1 � 2 , we get: √ � 1 = | 0 � + e iϕ | 1 � � U | 0 � + | 1 � 1 √ √ = √ . e iϕ 2 2 2

General 2 × 2 unitary Rotation around z -axis � � Consider the action of U = | 0 � � 0 | + e iϕ | 1 � � 1 | = 1 0 : 0 e iϕ U | 1 � = e iϕ | 1 � . U | 0 � = | 0 � , Now if we act on | 0 � + | 1 � 2 , we get: √ � 1 = | 0 � + e iϕ | 1 � � U | 0 � + | 1 � 1 √ √ = √ . e iϕ 2 2 2 General rotation Rotation around � r by angle ϕ : r ) + e iϕ ρ ( − � U ( � r, ϕ ) = ρ ( � r ) .

Some curiosities

Some curiosities Quaternions i 2 = j 2 = k 2 = ijk = − 1 .

Some curiosities Quaternions i 2 = j 2 = k 2 = ijk = − 1 . Bijection (1 , i, j, k ) ⇐ ⇒ ( I, iσ z , iσ y , iσ x ) .

Some curiosities Quaternions i 2 = j 2 = k 2 = ijk = − 1 . Bijection (1 , i, j, k ) ⇐ ⇒ ( I, iσ z , iσ y , iσ x ) .

Some curiosities Quaternions i 2 = j 2 = k 2 = ijk = − 1 . Bijection (1 , i, j, k ) ⇐ ⇒ ( I, iσ z , iσ y , iσ x ) . Finite field of order 4 � 0 , 1 , ω, ω 2 � ω 2 ≡ ω + 1 . F 4 = ( , + , ∗ ) , x ≡ − x,

Some curiosities Quaternions i 2 = j 2 = k 2 = ijk = − 1 . Bijection (1 , i, j, k ) ⇐ ⇒ ( I, iσ z , iσ y , iσ x ) . Finite field of order 4 � 0 , 1 , ω, ω 2 � ω 2 ≡ ω + 1 . F 4 = ( , + , ∗ ) , x ≡ − x, Bijection (0 , 1 , ω, ω 2 ) ⇐ ⇒ ( I, σ x , σ z , σ y ) .

Some curiosities Quaternions i 2 = j 2 = k 2 = ijk = − 1 . Bijection (1 , i, j, k ) ⇐ ⇒ ( I, iσ z , iσ y , iσ x ) . Finite field of order 4 � 0 , 1 , ω, ω 2 � ω 2 ≡ ω + 1 . F 4 = ( , + , ∗ ) , x ≡ − x, Bijection (up to phase) (0 , 1 , ω, ω 2 ) ⇐ ⇒ ( I, σ x , σ z , σ y ) .

Some curiosities Quaternions i 2 = j 2 = k 2 = ijk = − 1 . Bijection (1 , i, j, k ) ⇐ ⇒ ( I, iσ z , iσ y , iσ x ) . Finite field of order 4 � 0 , 1 , ω, ω 2 � ω 2 ≡ ω + 1 . F 4 = ( , + , ∗ ) , x ≡ − x, Bijection (up to phase) (0 , 1 , ω, ω 2 ) ⇐ ⇒ ( I, σ x , σ z , σ y ) .

More curiosities

More curiosities Clifford group The Clifford group of a qubit is C = { U | σ ∈ P ⇒ UσU † ∈ P } , where P = {± I, ± σ x , ± σ y , ± σ z } – the set of Pauli matrices.

More curiosities Clifford group The Clifford group of a qubit is C = { U | σ ∈ P ⇒ UσU † ∈ P } , where P = {± I, ± σ x , ± σ y , ± σ z } – the set of Pauli matrices. Cuboctahedron

More curiosities Clifford group The Clifford group of a qubit is C = { U | σ ∈ P ⇒ UσU † ∈ P } , where P = {± I, ± σ x , ± σ y , ± σ z } – the set of Pauli matrices. Cuboctahedron

Counting dimensions

Counting dimensions Density matrix

Counting dimensions Density matrix 1. hermitian: ρ † = ρ ,

Counting dimensions Density matrix 1. hermitian: ρ † = ρ , 2. unit trace: Tr ρ = 1 ,

Counting dimensions Density matrix 1. hermitian: ρ † = ρ , 2. unit trace: Tr ρ = 1 , 3. positive semi-definite: ρ ≥ 0 .

Counting dimensions Density matrix 1. hermitian: ρ † = ρ , 2. unit trace: Tr ρ = 1 , 3. positive semi-definite: ρ ≥ 0 . Degrees of freedom for ρ For an n × n density matrix there are n 2 − 1 degrees of freedom.

Counting dimensions Density matrix 1. hermitian: ρ † = ρ , 2. unit trace: Tr ρ = 1 , 3. positive semi-definite: ρ ≥ 0 . Degrees of freedom for ρ For an n × n density matrix there are n 2 − 1 degrees of freedom. Degrees of freedom for | ψ � A pure quantum state | ψ � ∈ C n has 2( n − 1) degrees of freedom.

Qutrit Qutrit state | ψ � = α | 0 � + β | 1 � + γ | 2 � .

Qutrit Qutrit state | ψ � = α | 0 � + β | 1 � + γ | 2 � . Gell-Mann matrices � 0 1 0 � 0 − i 0 � 1 0 0 � � � λ 1 = λ 2 = λ 3 = , , , 1 0 0 0 − 1 0 i 0 0 0 0 0 0 0 0 0 0 0 � 0 0 1 � 0 0 − i � 0 0 0 � � � λ 4 = , λ 5 = , λ 6 = , 0 0 0 0 0 0 0 0 1 1 0 0 i 0 0 0 1 0 � 0 0 0 � 1 0 0 � � 1 λ 7 = λ 8 = , √ . 0 0 − i 0 1 0 3 0 0 − 2 0 i 0

Qutrit Qutrit state | ψ � = α | 0 � + β | 1 � + γ | 2 � . Gell-Mann matrices � 0 1 0 � 0 − i 0 � 1 0 0 � � � λ 1 = λ 2 = λ 3 = , , , 1 0 0 0 − 1 0 i 0 0 0 0 0 0 0 0 0 0 0 � 0 0 1 � 0 0 − i � 0 0 0 � � � λ 4 = , λ 5 = , λ 6 = , 0 0 0 0 0 0 0 0 1 1 0 0 i 0 0 0 1 0 � 0 0 0 � 1 0 0 � � 1 λ 7 = λ 8 = , √ . 0 0 − i 0 1 0 3 0 0 − 2 0 i 0 Density matrix √ ρ = | ψ � � ψ | = 1 r · � r ∈ R 8 . 3( I + 3 � λ ) , �

Qutrit Qutrit state | ψ � = α | 0 � + β | 1 � + γ | 2 � . Gell-Mann matrices � 0 1 0 � 0 − i 0 � 1 0 0 � � � λ 1 = λ 2 = λ 3 = , , , 1 0 0 0 − 1 0 i 0 0 0 0 0 0 0 0 0 0 0 � 0 0 1 � 0 0 − i � 0 0 0 � � � λ 4 = , λ 5 = , λ 6 = , 0 0 0 0 0 0 0 0 1 1 0 0 i 0 0 0 1 0 � 0 0 0 � 1 0 0 � � 1 λ 7 = λ 8 = , √ . 0 0 − i 0 1 0 3 0 0 − 2 0 i 0 Density matrix √ ρ = | ψ � � ψ | = 1 r · � r ∈ R 8 . 3( I + 3 � λ ) , �