# Quantum walk

Quantum walks are considered to be quantum analogs of random walks. The study of quantum walks started to get attention around 2000 in relation to a quantum computer.

Contents

### Discrete-time quantum walk on the line ｏ

#### Probability amplitude ｏ

The whole system of a discrete-time 2-state quantum walk on the line is described by probability amplitudes. The probability amplitude at position at time is expressed as a 2-component complex vector , as shown in the following figure.

#### Time evolution ｏ

Assuming is a unitary matrix, the time evolution is determined by a reccurence

where

and are complex numbers.

#### Probability disributionｏ

The probability that the walker can be observed at position at time is defined by

where is a random variable and denotes the position of the quantum walker.

• Example
• Quantum walk v.s. Random walk

#### Limit theorem ｏ

For complex numbers , we take initial conditions to be

under the condition . The symbol means the transposed operator.

If we assume that the unitary matrix satisfies the condition , then we have

where

and

• Example

In the case of a simple random walk, we have

where

### Continuous-time quantum walk on the line ｏ

#### Probability amplitude ｏ

The whole system of a continuous-time quantum walk on the line is also described by probability amplitudes. The probability amplitude at position at time is expressed as a complex number , as shown in the following figure.

#### Time evolution ｏ

The time evolution is defined by a discrete-space Schrödinger equation,

where , and is a positive constant.

#### Probability distribution ｏ

The quantum walker can be observed at position at time with probability

• Example

#### Limit theorem ｏ

We take the initial conditions

Then we have

where

and

• Example