# 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 †

#### Probability amplitude †

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 †

Assuming is a unitary matrix, the time evolution is determined by a reccurence where and are complex numbers.

#### Probability disribution†

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   Suppliment PDF file

• Quantum walk v.s. Random walk #### Limit theorem †

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 †

#### Probability amplitude †

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 †

The time evolution is defined by a discrete-space Schrödinger equation, where , and is a positive constant.

#### Probability distribution †

The quantum walker can be observed at position at time with probability • Example #### Limit theorem †

We take the initial conditions Then we have where and • Example ### Interesting probability distributions of quantum walks † QIC, 21(1&2), 19-36 (2021) QIP, 19, 296 (2020) QIP, 17(9), 241 (2018) IJQI, 16(3), 1850023 (2018) QIP, 15(8), 3101-3119 (2016)  QIC, 16(5&6), 515-529 (2016) PRA, 92, 062307 (2015)  IJQI, 13(7), 1550054 (2015)  QIP, 14(5), 1539-1558 (2015) QIC, 15(5&6), 406-418 (2015)  QIC, 15(1&2), 50-60 (2015)  QIC, 15(13&14), 1248-1258 (2015)    QIC, 13(5&6), 430-438 (2013)    IJQI, 11(5), 1350053 (2013) PRA, 84, 042337 (2011)      JCTN, 10(7), 1571-1578 (2013) IJQI, 9(3), 863-874 (2011) QIC, 10(11&12), 1004-1017 (2010) IWNC 2009 Proc. ICT, 2, 226-235 (2010)

### See also †

#### Further reading †

• Julia Kempe (2003). Quantum random walks – an introductory overview. Contemporary Physics 44 (4): 307–327. DOI:10.1080/00107151031000110776.
• Viv Kendon (2007). Decoherence in quantum walks – a review. Mathematical Structures in Computer Science 17 (6): 1169–1220. DOI:10.1017/S0960129507006354.
• Norio Konno (2008). Quantum Walks. Volume 1954 of Lecture Notes in Mathematics. Springer-Verlag, (Heidelberg) pp. 309–452. DOI:10.1007/978-3-540-69365-9.
• Salvador Elías Venegas-Andraca (2008). Quantum Walks for Computer Scientists. Morgan & Claypool Publishers. DOI:10.2200/S00144ED1V01Y200808QMC001.
• Salvador Elías Venegas-Andraca (2012). Quantum walks: a comprehensive review. Quantum Information Processing 11 (5): 1015–1106. DOI:10.1007/s11128-012-0432-5.

#### External link † 