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

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

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Geometric aspects of quantum computing Maris Ozols University of Waterloo Department of C&O December 10, 2007 Qubit state Qubit | = | 0 + | 1 , where , C and | | 2 + | | 2 = 1 . Qubit state Qubit |

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SLIDE 1

Geometric aspects of quantum computing

Maris Ozols University of Waterloo Department of C&O

December 10, 2007

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SLIDE 2

Qubit state

Qubit

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

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SLIDE 3

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 < 2π), such that |ψ = cos θ

2

eiϕ sin θ

2

  • .
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SLIDE 4

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 < 2π), such that |ψ = cos θ

2

eiϕ sin θ

2

  • .

Density matrix

The corresponding density matrix is: ρ = |ψ ψ| = 1 2 1 + cos θ e−iϕ sin θ eiϕ sin θ 1 − cos θ

  • .
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SLIDE 5

Bloch sphere

Bijection between S2 and CP1

|ψ = cos θ

2

eiϕ sin θ

2

    x = sin θ cos ϕ y = sin θ sin ϕ z = cos θ 0 ≤ θ ≤ π and 0 ≤ ϕ < 2π

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SLIDE 6

Pauli matrices

Density matrix of a qubit

ρ = 1 2 (I + r · σ) ,

  • r = (x, y, z),
  • σ = (σx, σy, σz).
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SLIDE 7

Pauli matrices

Density matrix of a qubit

ρ = 1 2 (I + r · σ) ,

  • r = (x, y, z),
  • σ = (σx, σy, σz).

Pauli matrices

I =

  • 1 0

0 1

  • ,

σx =

  • 0 1

1 0

  • ,

σy =

  • 0 −i

i 0

  • ,

σz =

  • 1 0

0 −1

  • .
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SLIDE 8

Pauli matrices

Density matrix of a qubit

ρ = 1 2 (I + r · σ) ,

  • r = (x, y, z),
  • σ = (σx, σy, σz).

Pauli matrices

I =

  • 1 0

0 1

  • ,

σx =

  • 0 1

1 0

  • ,

σy =

  • 0 −i

i 0

  • ,

σz =

  • 1 0

0 −1

  • .

Inner product

|ψ1|ψ2|2

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SLIDE 9

Pauli matrices

Density matrix of a qubit

ρ = 1 2 (I + r · σ) ,

  • r = (x, y, z),
  • σ = (σx, σy, σz).

Pauli matrices

I =

  • 1 0

0 1

  • ,

σx =

  • 0 1

1 0

  • ,

σy =

  • 0 −i

i 0

  • ,

σz =

  • 1 0

0 −1

  • .

Inner product

|ψ1|ψ2|2 = Tr(ρ1ρ2)

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SLIDE 10

Pauli matrices

Density matrix of a qubit

ρ = 1 2 (I + r · σ) ,

  • r = (x, y, z),
  • σ = (σx, σy, σz).

Pauli matrices

I =

  • 1 0

0 1

  • ,

σx =

  • 0 1

1 0

  • ,

σy =

  • 0 −i

i 0

  • ,

σz =

  • 1 0

0 −1

  • .

Inner product

|ψ1|ψ2|2 = Tr(ρ1ρ2) = 1 2(1 + r1 · r2).

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SLIDE 11

Pauli matrices

Density matrix of a qubit

ρ = 1 2 (I + r · σ) ,

  • r = (x, y, z),
  • σ = (σx, σy, σz).

Pauli matrices

I =

  • 1 0

0 1

  • ,

σx =

  • 0 1

1 0

  • ,

σy =

  • 0 −i

i 0

  • ,

σz =

  • 1 0

0 −1

  • .

Inner product

|ψ1|ψ2|

  • 2 = Tr(ρ1ρ2) = 1

2(1 + r1 · r2).

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SLIDE 12

Pauli matrices

Density matrix of a qubit

ρ = 1 2 (I + r · σ) ,

  • r = (x, y, z),
  • σ = (σx, σy, σz).

Pauli matrices

I =

  • 1 0

0 1

  • ,

σx =

  • 0 1

1 0

  • ,

σy =

  • 0 −i

i 0

  • ,

σz =

  • 1 0

0 −1

  • .

Inner product

|ψ1|ψ2|

  • 2 = Tr(ρ1ρ2) = 1

2(1 + r1 · r2 ).

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SLIDE 13

Pauli matrices

Density matrix of a qubit

ρ = 1 2 (I + r · σ) ,

  • r = (x, y, z),
  • σ = (σx, σy, σz).

Pauli matrices

I =

  • 1 0

0 1

  • ,

σx =

  • 0 1

1 0

  • ,

σy =

  • 0 −i

i 0

  • ,

σz =

  • 1 0

0 −1

  • .

Inner product

|ψ1|ψ2|

  • 2 = Tr(ρ1ρ2) = 1

2(1 + r1 · r2

−1

).

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SLIDE 14

General 2 × 2 unitary

Rotation around z-axis

Consider the action of U = |0 0| + eiϕ |1 1| =

  • 1

0 eiϕ

  • :
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SLIDE 15

General 2 × 2 unitary

Rotation around z-axis

Consider the action of U = |0 0| + eiϕ |1 1| =

  • 1

0 eiϕ

  • :

U |0 = |0 , U |1 = eiϕ |1 .

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SLIDE 16

General 2 × 2 unitary

Rotation around z-axis

Consider the action of U = |0 0| + eiϕ |1 1| =

  • 1

0 eiϕ

  • :

U |0 = |0 , U |1 = eiϕ |1 . Now if we act on |0+|1

√ 2 , we get:

U |0 + |1 √ 2 = |0 + eiϕ |1 √ 2 = 1 √ 2 1 eiϕ

  • .
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SLIDE 17

General 2 × 2 unitary

Rotation around z-axis

Consider the action of U = |0 0| + eiϕ |1 1| =

  • 1

0 eiϕ

  • :

U |0 = |0 , U |1 = eiϕ |1 . Now if we act on |0+|1

√ 2 , we get:

U |0 + |1 √ 2 = |0 + eiϕ |1 √ 2 = 1 √ 2 1 eiϕ

  • .

General rotation

Rotation around r by angle ϕ: U( r, ϕ) = ρ( r) + eiϕρ(− r).

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SLIDE 18

Some curiosities

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SLIDE 19

Some curiosities

Quaternions

i2 = j2 = k2 = ijk = −1.

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SLIDE 20

Some curiosities

Quaternions

i2 = j2 = k2 = ijk = −1.

Bijection

(1, i, j, k) ⇐ ⇒ (I, iσz, iσy, iσx).

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SLIDE 21

Some curiosities

Quaternions

i2 = j2 = k2 = ijk = −1.

Bijection

(1, i, j, k) ⇐ ⇒ (I, iσz, iσy, iσx).

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SLIDE 22

Some curiosities

Quaternions

i2 = j2 = k2 = ijk = −1.

Bijection

(1, i, j, k) ⇐ ⇒ (I, iσz, iσy, iσx).

Finite field of order 4

F4 = (

  • 0, 1, ω, ω2

, +, ∗), x ≡ −x, ω2 ≡ ω + 1.

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SLIDE 23

Some curiosities

Quaternions

i2 = j2 = k2 = ijk = −1.

Bijection

(1, i, j, k) ⇐ ⇒ (I, iσz, iσy, iσx).

Finite field of order 4

F4 = (

  • 0, 1, ω, ω2

, +, ∗), x ≡ −x, ω2 ≡ ω + 1.

Bijection

(0, 1, ω, ω2) ⇐ ⇒ (I, σx, σz, σy).

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SLIDE 24

Some curiosities

Quaternions

i2 = j2 = k2 = ijk = −1.

Bijection

(1, i, j, k) ⇐ ⇒ (I, iσz, iσy, iσx).

Finite field of order 4

F4 = (

  • 0, 1, ω, ω2

, +, ∗), x ≡ −x, ω2 ≡ ω + 1.

Bijection (up to phase)

(0, 1, ω, ω2) ⇐ ⇒ (I, σx, σz, σy).

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SLIDE 25

Some curiosities

Quaternions

i2 = j2 = k2 = ijk = −1.

Bijection

(1, i, j, k) ⇐ ⇒ (I, iσz, iσy, iσx).

Finite field of order 4

F4 = (

  • 0, 1, ω, ω2

, +, ∗), x ≡ −x, ω2 ≡ ω + 1.

Bijection (up to phase)

(0, 1, ω, ω2) ⇐ ⇒ (I, σx, σz, σy).

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SLIDE 26

More curiosities

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SLIDE 27

More curiosities

Clifford group

The Clifford group of a qubit is C = {U|σ ∈ P ⇒ UσU† ∈ P}, where P = {±I, ±σx, ±σy, ±σz} – the set of Pauli matrices.

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SLIDE 28

More curiosities

Clifford group

The Clifford group of a qubit is C = {U|σ ∈ P ⇒ UσU† ∈ P}, where P = {±I, ±σx, ±σy, ±σz} – the set of Pauli matrices.

Cuboctahedron

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SLIDE 29

More curiosities

Clifford group

The Clifford group of a qubit is C = {U|σ ∈ P ⇒ UσU† ∈ P}, where P = {±I, ±σx, ±σy, ±σz} – the set of Pauli matrices.

Cuboctahedron

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SLIDE 30

Counting dimensions

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SLIDE 31

Counting dimensions

Density matrix

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SLIDE 32

Counting dimensions

Density matrix

  • 1. hermitian: ρ† = ρ,
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SLIDE 33

Counting dimensions

Density matrix

  • 1. hermitian: ρ† = ρ,
  • 2. unit trace: Tr ρ = 1,
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SLIDE 34

Counting dimensions

Density matrix

  • 1. hermitian: ρ† = ρ,
  • 2. unit trace: Tr ρ = 1,
  • 3. positive semi-definite: ρ ≥ 0.
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SLIDE 35

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 n2 − 1 degrees of freedom.

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SLIDE 36

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 n2 − 1 degrees of freedom.

Degrees of freedom for |ψ

A pure quantum state |ψ ∈ Cn has 2(n − 1) degrees of freedom.

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SLIDE 37

Qutrit

Qutrit state

|ψ = α |0 + β |1 + γ |2 .

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SLIDE 38

Qutrit

Qutrit state

|ψ = α |0 + β |1 + γ |2 .

Gell-Mann matrices

λ1 = 0 1 0

1 0 0 0 0 0

  • ,

λ2 = 0 −i 0

i 0 0 0 0 0

  • ,

λ3 = 1 0 0

0 −1 0 0 0 0

  • ,

λ4 = 0 0 1

0 0 0 1 0 0

  • ,

λ5 = 0 0 −i

0 0 0 i 0 0

  • ,

λ6 = 0 0 0

0 0 1 0 1 0

  • ,

λ7 = 0 0 0

0 0 −i 0 i 0

  • ,

λ8 =

1 √ 3

1 0 0

0 1 0 0 0 −2

  • .
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SLIDE 39

Qutrit

Qutrit state

|ψ = α |0 + β |1 + γ |2 .

Gell-Mann matrices

λ1 = 0 1 0

1 0 0 0 0 0

  • ,

λ2 = 0 −i 0

i 0 0 0 0 0

  • ,

λ3 = 1 0 0

0 −1 0 0 0 0

  • ,

λ4 = 0 0 1

0 0 0 1 0 0

  • ,

λ5 = 0 0 −i

0 0 0 i 0 0

  • ,

λ6 = 0 0 0

0 0 1 0 1 0

  • ,

λ7 = 0 0 0

0 0 −i 0 i 0

  • ,

λ8 =

1 √ 3

1 0 0

0 1 0 0 0 −2

  • .

Density matrix

ρ = |ψ ψ| = 1 3(I + √ 3 r · λ),

  • r ∈ R8.
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SLIDE 40

Qutrit

Qutrit state

|ψ = α |0 + β |1 + γ |2 .

Gell-Mann matrices

λ1 = 0 1 0

1 0 0 0 0 0

  • ,

λ2 = 0 −i 0

i 0 0 0 0 0

  • ,

λ3 = 1 0 0

0 −1 0 0 0 0

  • ,

λ4 = 0 0 1

0 0 0 1 0 0

  • ,

λ5 = 0 0 −i

0 0 0 i 0 0

  • ,

λ6 = 0 0 0

0 0 1 0 1 0

  • ,

λ7 = 0 0 0

0 0 −i 0 i 0

  • ,

λ8 =

1 √ 3

1 0 0

0 1 0 0 0 −2

  • .

Density matrix

ρ = |ψ ψ| = 1 3(I + √ 3 r · λ),

  • r ∈ R8.
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SLIDE 41

Qutrit

Qutrit state

|ψ = α |0 + β |1 + γ |2 .

Gell-Mann matrices

λ1 = 0 1 0

1 0 0 0 0 0

  • ,

λ2 = 0 −i 0

i 0 0 0 0 0

  • ,

λ3 = 1 0 0

0 −1 0 0 0 0

  • ,

λ4 = 0 0 1

0 0 0 1 0 0

  • ,

λ5 = 0 0 −i

0 0 0 i 0 0

  • ,

λ6 = 0 0 0

0 0 1 0 1 0

  • ,

λ7 = 0 0 0

0 0 −i 0 i 0

  • ,

λ8 =

1 √ 3

1 0 0

0 1 0 0 0 −2

  • .

Density matrix

ρ = |ψ ψ| = 1 3(I + √ 3 r · λ),

  • r ∈ R8.
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SLIDE 42

Qutrit

Qutrit state

|ψ = α |0 + β |1 + γ |2 .

Gell-Mann matrices

λ1 = 0 1 0

1 0 0 0 0 0

  • ,

λ2 = 0 −i 0

i 0 0 0 0 0

  • ,

λ3 = 1 0 0

0 −1 0 0 0 0

  • ,

λ4 = 0 0 1

0 0 0 1 0 0

  • ,

λ5 = 0 0 −i

0 0 0 i 0 0

  • ,

λ6 = 0 0 0

0 0 1 0 1 0

  • ,

λ7 = 0 0 0

0 0 −i 0 i 0

  • ,

λ8 =

1 √ 3

1 0 0

0 1 0 0 0 −2

  • .

Density matrix

ρ = |ψ ψ| = 1 3(I + √ 3 r · λ),

  • r ∈ R8.
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SLIDE 43

SU(n) generators

Generalized Pauli matrices

Let Pjk = |j k|.

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SLIDE 44

SU(n) generators

Generalized Pauli matrices

Let Pjk = |j k|. Define        Xjk = Pkj + Pjk, Yjk = i(Pkj − Pjk), Zj =

  • 2

j(j+1)

  • P11 + P22 + · · · + Pjj − j · Pj+1,j+1
  • ,

where 1 ≤ j < k ≤ n.

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SLIDE 45

SU(n) generators

Generalized Pauli matrices

Let Pjk = |j k|. Define        Xjk = Pkj + Pjk, Yjk = i(Pkj − Pjk), Zj =

  • 2

j(j+1)

  • P11 + P22 + · · · + Pjj − j · Pj+1,j+1
  • ,

where 1 ≤ j < k ≤ n. Then {λi} = {Xjk} ∪ {Yjk} ∪ {Zj} is the set of generalized Pauli matrices. Note that |{λi}| = n2 − 1.

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SLIDE 46

State of an n-level quantum system

General density matrix

ρ = 1 n

  • I +
  • n(n − 1)

2

  • r ·

λ

  • ,
  • r ∈ Rn2−1.
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SLIDE 47

State of an n-level quantum system

General density matrix

ρ = 1 n

  • I +
  • n(n − 1)

2

  • r ·

λ

  • ,
  • r ∈ Rn2−1.

Inner product

|ψ1|ψ2|2 = 1 n

  • 1 + (n − 1)

r1 · r2

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SLIDE 48

State of an n-level quantum system

General density matrix

ρ = 1 n

  • I +
  • n(n − 1)

2

  • r ·

λ

  • ,
  • r ∈ Rn2−1.

Inner product

0 ≤ |ψ1|ψ2|2 = 1 n

  • 1 + (n − 1)

r1 · r2

  • ≤ 1,
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SLIDE 49

State of an n-level quantum system

General density matrix

ρ = 1 n

  • I +
  • n(n − 1)

2

  • r ·

λ

  • ,
  • r ∈ Rn2−1.

Inner product

0 ≤ |ψ1|ψ2|2 = 1 n

  • 1 + (n − 1)

r1 · r2

  • ≤ 1,

− 1 n − 1 ≤ r1 · r2 ≤ 1.

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SLIDE 50

State of an n-level quantum system

General density matrix

ρ = 1 n

  • I +
  • n(n − 1)

2

  • r ·

λ

  • ,
  • r ∈ Rn2−1.

Inner product

0 ≤ |ψ1|ψ2|2 = 1 n

  • 1 + (n − 1)

r1 · r2

  • ≤ 1,

− 1 n − 1 ≤ r1 · r2 ≤ 1.

Orthogonal states

  • r1 ·

r2 = cos θ = − 1 n − 1.

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SLIDE 51

State of an n-level quantum system

General density matrix

ρ = 1 n

  • I +
  • n(n − 1)

2

  • r ·

λ

  • ,
  • r ∈ Rn2−1.

Inner product

0 ≤ |ψ1|ψ2|2 = 1 n

  • 1 + (n − 1)

r1 · r2

  • ≤ 1,

− 1 n − 1 ≤ r1 · r2 ≤ 1.

Orthogonal states

  • r1 ·

r2 = cos θ = − 1 n − 1.

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SLIDE 52

Unit simplex

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SLIDE 53

Unit simplex

Geometry

Height Hn, circumradius Rn, inradius rn: Hn =

  • n+1

2n ,

Rn =

  • n

2(n+1),

rn =

1

√

2n(n+1).

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SLIDE 54

Unit simplex

Geometry

Height Hn, circumradius Rn, inradius rn: Hn =

  • n+1

2n ,

Rn =

  • n

2(n+1),

rn =

1

√

2n(n+1).

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SLIDE 55

Unit simplex

Geometry

Height Hn, circumradius Rn, inradius rn: Hn =

  • n+1

2n ,

Rn =

  • n

2(n+1),

rn =

1

√

2n(n+1).

Hn = Rn + rn

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SLIDE 56

Unit simplex

Geometry

Height Hn, circumradius Rn, inradius rn: Hn =

  • n+1

2n ,

Rn =

  • n

2(n+1),

rn =

1

√

2n(n+1).

Hn = Rn + rn Rn rn = n

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SLIDE 57

Unit vectors

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SLIDE 58

Unit vectors

Simplex of unit vectors

Unit vectors { v0, v1, . . . , vn} in Rn form a regular simplex if

  • vi ·

vj = − 1 n (i = j).

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SLIDE 59

Unit vectors

Simplex of unit vectors

Unit vectors { v0, v1, . . . , vn} in Rn form a regular simplex if

  • vi ·

vj = − 1 n (i = j).

Orthonormal basis in Cn

An orthonormal basis B = {|ψ1 , |ψ2 , . . . , |ψn} of Cn satisfy: |ψi|ψj|2 = 0,

  • vi ·

vj = − 1 n − 1 (i = j).

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SLIDE 60

Unit vectors

Simplex of unit vectors

Unit vectors { v0, v1, . . . , vn} in Rn form a regular simplex if

  • vi ·

vj = − 1 n (i = j).

Orthonormal basis in Cn

An orthonormal basis B = {|ψ1 , |ψ2 , . . . , |ψn} of Cn satisfy: |ψi|ψj|2 = 0,

  • vi ·

vj = − 1 n − 1 (i = j). It is a regular simplex in n − 1 dimensional subspace of Rn2−1!

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SLIDE 61

Unit vectors

Simplex of unit vectors

Unit vectors { v0, v1, . . . , vn} in Rn form a regular simplex if

  • vi ·

vj = − 1 n (i = j).

Orthonormal basis in Cn

An orthonormal basis B = {|ψ1 , |ψ2 , . . . , |ψn} of Cn satisfy: |ψi|ψj|2 = 0,

  • vi ·

vj = − 1 n − 1 (i = j). It is a regular simplex in n − 1 dimensional subspace of Rn2−1!

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SLIDE 62

MUBs

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SLIDE 63

MUBs

Mutually Unbiased Bases

A set of n + 1 orthonormal bases Bi =

  • ψi

1

  • ,
  • ψi

2

  • , . . . ,
  • ψi

n

  • f

Cn, such that for any |ψi ∈ Bi and |ψj ∈ Bj |ψi|ψj|2 = 1 n,

  • vi ·

vj = 0 (i = j).

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SLIDE 64

MUBs

Mutually Unbiased Bases

A set of n + 1 orthonormal bases Bi =

  • ψi

1

  • ,
  • ψi

2

  • , . . . ,
  • ψi

n

  • f

Cn, such that for any |ψi ∈ Bi and |ψj ∈ Bj |ψi|ψj|2 = 1 n,

  • vi ·

vj = 0 (i = j).

slide-65
SLIDE 65

MUBs

Mutually Unbiased Bases

A set of n + 1 orthonormal bases Bi =

  • ψi

1

  • ,
  • ψi

2

  • , . . . ,
  • ψi

n

  • f

Cn, such that for any |ψi ∈ Bi and |ψj ∈ Bj |ψi|ψj|2 = 1 n,

  • vi ·

vj = 0 (i = j). Bz =

  • 1
  • ,
  • 1
  • ,

Bx =

  • 1

√ 2

  • 1

1

  • ,

1 √ 2

  • 1

−1

  • ,

By =

  • 1

√ 2

  • 1

i

  • ,

1 √ 2

  • 1

−i

  • .
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SLIDE 66

Duality

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SLIDE 67

Duality

Unbiased = ⇒ Orthogonal

If |ψi and |ψj are from different bases, then |ψi|ψj|2 = 1 n,

  • vi ·

vj = 0 (i = j).

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SLIDE 68

Duality

Unbiased = ⇒ Orthogonal

If |ψi and |ψj are from different bases, then |ψi|ψj|2 = 1 n,

  • vi ·

vj = 0 (i = j).

Orthogonal = ⇒ Unbiased

If |ψi and |ψj are from the same basis, then |ψi|ψj|2 = 0,

  • vi ·

vj = − 1 n − 1 (i = j).

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SLIDE 69

SIC POVM

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SLIDE 70

SIC POVM

Symmetric Informationally Complete POVM

A set of n2 unit vectors |ψi ∈ Cn, such that |ψi|ψj|2 = 1 n + 1,

  • vi ·

vj = − 1 n2 − 1 (i = j).

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SLIDE 71

SIC POVM

Symmetric Informationally Complete POVM

A set of n2 unit vectors |ψi ∈ Cn, such that |ψi|ψj|2 = 1 n + 1,

  • vi ·

vj = − 1 n2 − 1 (i = j).

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SLIDE 72

SIC POVM

Symmetric Informationally Complete POVM

A set of n2 unit vectors |ψi ∈ Cn, such that |ψi|ψj|2 = 1 n + 1,

  • vi ·

vj = − 1 n2 − 1 (i = j). |ψ1 =

  • 1
  • ,

|ψ2 =

1 √ 3

  • 1

√ 2

  • ,

|ψ3 =

1 √ 3

  • 1

e+iϕ√ 2

  • ,

|ψ4 =

1 √ 3

  • 1

e−iϕ√ 2

  • ,

where ϕ = 2π

3 .

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SLIDE 73

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