Quantum Computing - An Introduction Cris Cecka April 10, 2006 Cris - - PowerPoint PPT Presentation

Outline Quantum Bits So What? Other Topics and Open Problems

Quantum Computing - An Introduction

Cris Cecka April 10, 2006

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems

Quantum Bits Quantum Superposition Dirac Properties Orthogonality and Bases Operators So What? Interesting Operators Quantum Computing Strategies Amplitude Amplification - A Start Grover’s Algorithm Other Topics and Open Problems

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Quantum Superposition Dirac Properties Orthogonality and Bases Operators

(Qu)Bits Classical Bits (Bits)

◮ Two States:

Quantum Bits (Qubits)

◮ Two States:

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Quantum Superposition Dirac Properties Orthogonality and Bases Operators

(Qu)Bits Classical Bits (Bits)

◮ Two States:

◮ On, Up, High

Quantum Bits (Qubits)

◮ Two States:

◮ Spin up, Energy State Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Quantum Superposition Dirac Properties Orthogonality and Bases Operators

(Qu)Bits Classical Bits (Bits)

◮ Two States:

◮ On, Up, High

⇒ 1

Quantum Bits (Qubits)

◮ Two States:

◮ Spin up, Energy State

⇒ |1

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Quantum Superposition Dirac Properties Orthogonality and Bases Operators

(Qu)Bits Classical Bits (Bits)

◮ Two States:

◮ On, Up, High

⇒ 1

◮ Off, Down, Low

Quantum Bits (Qubits)

◮ Two States:

◮ Spin up, Energy State

⇒ |1

◮ Spin down, Energy State Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Quantum Superposition Dirac Properties Orthogonality and Bases Operators

(Qu)Bits Classical Bits (Bits)

◮ Two States:

◮ On, Up, High

⇒ 1

◮ Off, Down, Low

⇒

Quantum Bits (Qubits)

◮ Two States:

◮ Spin up, Energy State

⇒ |1

◮ Spin down, Energy State

⇒ |0

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Quantum Superposition Dirac Properties Orthogonality and Bases Operators

At Least It Sounds Good

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Quantum Superposition Dirac Properties Orthogonality and Bases Operators

Quantum Superposition

Definition

Qubits can be BOTH |0 and |1. In general, a single qubit is |Ψ = α|0 + β|1 = α β

• where α2 + β2 = 1.

Problem

This superposition only occurs when we aren’t “looking”. When we “look”

◮ Measure |0 with probability α2. ◮ Measure |1 with probability β2.

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Quantum Superposition Dirac Properties Orthogonality and Bases Operators

Speaking The Same Language

Definition

Kets: |x = a|0 + b|1 = a b

• Can be thought of as vectors.

Definition

Bras: x| = a∗0| + b∗1| =

• a∗

b∗ = a b ∗ = |x∗ Can be thought of as vectors. The ultimate fate of a bra is to meet a ket x|x =

• a∗

b∗ a b

• = a2 + b2

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Quantum Superposition Dirac Properties Orthogonality and Bases Operators

Orthogonality and Bases

We’ve been using {|0, |1} as a basis. Note: |0 = 1

• and |1 =

1

• are orthogonal:

0|1 = 1|0 = 0 0|0 = 1|1 = 1

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Quantum Superposition Dirac Properties Orthogonality and Bases Operators

Operators

The identity operator ˆ I = 1 1

• can be written

ˆ I = |00| + |11|

Example

ˆ I|Ψ = (|00| + |11|) (α|0 + β|1) = α|00|0 + β|00|1 + α|11|0 + β|11|1 = α|0 + β|1

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Interesting Operators Quantum Computing Strategies Amplitude Amplification - A Start Grover’s Algorithm

Consider...

Definition (Hadamard Transform)

ˆ H =

1 √ 2 (|00| + |10| − |01| + |11|) = 1 √ 2

• 1

1 −1 1

• Also called the Mixing Operator.

For example, ˆ H|0 = ˆ H ✔ 1 ✕ = 1 √ 2 ✔ 1 −1 ✕ = |0 √ 2 − |1 √ 2 ˆ H|1 = ˆ H ✔ 1 ✕ = 1 √ 2 ✔ 1 1 ✕ = |0 √ 2 + |1 √ 2 and again, ˆ H ✒ |0 √ 2 + |1 √ 2 ✓ = 1 √ 2 ✒ ˆ H ✔ 1 ✕ + ˆ H ✔ 1 ✕✓ = 1 2 ✒✔ 1 −1 ✕ + ✔ 1 1 ✕✓ = |1 Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Interesting Operators Quantum Computing Strategies Amplitude Amplification - A Start Grover’s Algorithm

Quantum Computing Strategies

◮ Use superposition to parallelize computations.

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Interesting Operators Quantum Computing Strategies Amplitude Amplification - A Start Grover’s Algorithm

Quantum Computing Strategies

◮ Use superposition to parallelize computations.

◮ Problem is when we measure the qubit, the result is one state,

not the entire superposition.

◮ Want a lot more information than is available. Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Interesting Operators Quantum Computing Strategies Amplitude Amplification - A Start Grover’s Algorithm

Quantum Computing Strategies

◮ Use superposition to parallelize computations.

◮ Problem is when we measure the qubit, the result is one state,

not the entire superposition.

◮ Want a lot more information than is available.

◮ A good strategy: encode solutions and make wrong answers

“cancel out”

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Interesting Operators Quantum Computing Strategies Amplitude Amplification - A Start Grover’s Algorithm

Quantum Computing Strategies

◮ Use superposition to parallelize computations.

◮ Problem is when we measure the qubit, the result is one state,

not the entire superposition.

◮ Want a lot more information than is available.

◮ A good strategy: encode solutions and make wrong answers

“cancel out”

◮ Problem is this is really hard. ◮ Since we can’t see the entire quantum state, most algorithms

have to be oblivious.

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Interesting Operators Quantum Computing Strategies Amplitude Amplification - A Start Grover’s Algorithm

Quantum Computing Strategies

◮ Use superposition to parallelize computations.

◮ Problem is when we measure the qubit, the result is one state,

not the entire superposition.

◮ Want a lot more information than is available.

◮ A good strategy: encode solutions and make wrong answers

“cancel out”

◮ Problem is this is really hard. ◮ Since we can’t see the entire quantum state, most algorithms

have to be oblivious.

◮ Amplitude amplification: amplify correct answers.

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Interesting Operators Quantum Computing Strategies Amplitude Amplification - A Start Grover’s Algorithm

Consider the operator

ˆ M = ˆ H

• 2|00| − ˆ

I

• ˆ

H = 2 N

• i,j

|ij| − ˆ I

• n a general state |Ψ =

k αk|k.

ˆ M|Ψ =   2 N

• i,j

|ij| − ˆ I  

• k

αk|k

• = 2

N  

i,j

αj|i   −

• k

αk|k = 2

j αj

N

• i

|i −

• k

αk|k =

• k

(2α − αk) |k

This operator flips all the amplitudes about their average!

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Interesting Operators Quantum Computing Strategies Amplitude Amplification - A Start Grover’s Algorithm

Quantum Searching

◮ Given an oracle operator

ˆ Of |x = (−1)f (x)|x

◮ f is like a search

function.

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Interesting Operators Quantum Computing Strategies Amplitude Amplification - A Start Grover’s Algorithm

Quantum Searching

◮ Given an oracle operator

ˆ Of |x = (−1)f (x)|x

◮ f is like a search

function.

◮ Start with

|1|1|1 · · · = |1⊗n

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Interesting Operators Quantum Computing Strategies Amplitude Amplification - A Start Grover’s Algorithm

Quantum Searching

◮ Given an oracle operator

ˆ Of |x = (−1)f (x)|x

◮ f is like a search

function.

◮ Start with

|1|1|1 · · · = |1⊗n

◮ Apply ˆ

H to make a big superposition.

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Interesting Operators Quantum Computing Strategies Amplitude Amplification - A Start Grover’s Algorithm

Quantum Searching

◮ Given an oracle operator

ˆ Of |x = (−1)f (x)|x

◮ f is like a search

function.

◮ Start with

|1|1|1 · · · = |1⊗n

◮ Apply ˆ

H to make a big superposition.

◮ Apply the oracle.

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Interesting Operators Quantum Computing Strategies Amplitude Amplification - A Start Grover’s Algorithm

Quantum Searching

◮ Given an oracle operator

ˆ Of |x = (−1)f (x)|x

◮ f is like a search

function.

◮ Start with

|1|1|1 · · · = |1⊗n

◮ Apply ˆ

H to make a big superposition.

◮ Apply the oracle. ◮ Apply mean inversion, ˆ

M.

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems Interesting Operators Quantum Computing Strategies Amplitude Amplification - A Start Grover’s Algorithm

Quantum Searching

◮ Given an oracle operator

ˆ Of |x = (−1)f (x)|x

◮ f is like a search

function.

◮ Start with

|1|1|1 · · · = |1⊗n

◮ Apply ˆ

H to make a big superposition.

◮ Apply the oracle. ◮ Apply mean inversion, ˆ

M.

◮ Repeat O(

√ N) times.

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems

Other Topics and Open Problems

Quantum Computation Introduction Topics

◮ Deutsch-Jozsa Decision Problem ◮ Zero-Failure Grover Algorithm ◮ Quantum Fourier Transform ◮ Shor’s Factoring Algorithm ◮ Super Dense Coding ◮ Perfect Encryption

Cris Cecka Quantum Computing - An Introduction

Outline Quantum Bits So What? Other Topics and Open Problems

Questions?

Questions?

Cris Cecka Quantum Computing - An Introduction