<p>There isn't a one word answer to your question, but you can have a look at</p>

<p>Xiaotong Ni's <a href="http://www.mpq.mpg.de/Theorygroup/CIRAC/wiki/images/d/d4/Ni_Master_thesis_2012.pdf" rel="nofollow">master thesis</a>,</p>

<p>where commuting circuits with several restrictions are considered and compared to classical classes. 
There you can also find the definition of the class <strong>IQP</strong>, which is a subclass of polynomial size commuting Pauli circuits. </p>

CC BY-SA 3.0
2014-08-30T20:12:13.547Z
Commuting *Pauli* circuits are pretty limited, doesn't **IQP** have to do with circuits of gates commuting with $X$?
Niel de Beaudrap
CC BY-SA 3.0
2014-08-30T20:40:26.410Z
@NieldeBeaudrap Maybe there is ambiguity in the definition of Pauli circuits. I meant the same definition used in Xiaotong's thesis.
Alessandro Cosentino
CC BY-SA 3.0
2014-08-30T20:46:49.157Z
Anyway, I just mentioned that class because the OP asked for a specific class whose definition concerns commutativity of the gates.
Alessandro Cosentino
CC BY-SA 3.0
2018-04-04T02:11:30.033Z
@AlessandroCosentino - Please accept my invitation to: https://quantumcomputing.stackexchange.com/ .
Rob
CC BY-SA 3.0
2014-08-30T16:05:47.117Z
2014-08-30T16:05:47.117Z
Alessandro Cosentino
<p>BQP is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances. In quantum computer allowed operations can be represented by unitary matrices. In general, unitary matrices do not commute with each other.</p>

<p>My question is what would be BQP analog for a quantum computer where only unitary matrices that commute with each other are allowed? </p>

<quantum-computing>
Complexity class for quantum computer with commutative gates
Oleg Stroganov
# Complexity class for quantum computer with commutative gates

(Asked by: username_0 on 2014-08-30)

BQP is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances. In quantum computer allowed operations can be represented by unitary matrices. In general, unitary matrices do not commute with each other.

My question is what would be BQP analog for a quantum computer where only unitary matrices that commute with each other are allowed?

**Tags:** quantum-computing


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## 1 answers

### Answer by username_1 on 2014-08-30. Score: 4 | Accepted answer

There isn't a one word answer to your question, but you can have a look at

Xiaotong Ni's [master thesis](http://www.mpq.mpg.de/Theorygroup/CIRAC/wiki/images/d/d4/Ni_Master_thesis_2012.pdf),

where commuting circuits with several restrictions are considered and compared to classical classes.
There you can also find the definition of the class **IQP**, which is a subclass of polynomial size commuting Pauli circuits.

> **Comments:**
> **username_2**: Commuting *Pauli* circuits are pretty limited, doesn't **IQP** have to do with circuits of gates commuting with $X$?  
> **username_1**: @NieldeBeaudrap Maybe there is ambiguity in the definition of Pauli circuits. I meant the same definition used in Xiaotong's thesis.  
> **username_1**: Anyway, I just mentioned that class because the OP asked for a specific class whose definition concerns commutativity of the gates.  
> **username_3**: @AlessandroCosentino - Please accept my invitation to: https://quantumcomputing.stackexchange.com/ .  

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