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Defining standard qubit gates for qutrits

Why is the decomposition of a qubit-qutrit Hamiltonian in terms of Pauli and Gell-Mann matrices not unique?How to apply single and two qubit gates to 2 qubits multiple times?The general form of unitary operations on a single qubitQutrit TeleportationWhat is the matrix for the operator that implements a function to tell the parity of its argument?Matrix representation and CX gateHow does quantum function parallelism work?What's the point of quantum gates being 'continuous'?How do you implement the gates for Grover's algorithm for more than 4 elements?Making a maximally mixed 2-qubit state in the IBM Q

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3

$begingroup$

I am actually working on quantum computing with qutrits. I am trying to define standard qubit gates for qutrits. The CNOT gate for qubits is defined as follows: $$|x,yrangle to |x,y+x bmod 2rangle.$$

So the matrix is $$mathrmUCN =
beginpmatrix
1 & 0 & 0 & 0\
0 & 1 & 0 & 0\
0 & 0 & 0 & 1\
0 & 0 & 1 & 0
endpmatrix.$$

For the definition of the same type of gate for qutrit I am left with two choices:

$$|x,yrangle to |x,y+x bmod 3rangle$$
$$|x,yrangle to |x,y+2x bmod 3rangle$$

Does anyone know if one of the two is more common or more usually defined?

PS: I know that applying one two times is equivalent to apply the other one time.

quantum-gate qutrit

share|improve this question

edited May 7 at 16:14

Sanchayan Dutta♦

7,12041658

asked May 7 at 14:08

Marco FalsoMarco Falso

184

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Hi, Marco. Welcome to Quantum Computing SE! It is preferable that you use MathJax to typeset your posts. Review How to write a good question?. I've edited the question on your behalf.
  $endgroup$
  – Sanchayan Dutta♦
  May 7 at 16:18
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Hi Sanchayan, thanks a lot for the edit and the advice. I will make sure that I write a good question next time.
  $endgroup$
  – Marco Falso
  May 7 at 16:21
  

  
  
  
  

add a comment | 

3

$begingroup$

I am actually working on quantum computing with qutrits. I am trying to define standard qubit gates for qutrits. The CNOT gate for qubits is defined as follows: $$|x,yrangle to |x,y+x bmod 2rangle.$$

So the matrix is $$mathrmUCN =
beginpmatrix
1 & 0 & 0 & 0\
0 & 1 & 0 & 0\
0 & 0 & 0 & 1\
0 & 0 & 1 & 0
endpmatrix.$$

For the definition of the same type of gate for qutrit I am left with two choices:

$$|x,yrangle to |x,y+x bmod 3rangle$$
$$|x,yrangle to |x,y+2x bmod 3rangle$$

Does anyone know if one of the two is more common or more usually defined?

PS: I know that applying one two times is equivalent to apply the other one time.

quantum-gate qutrit

share|improve this question

edited May 7 at 16:14

Sanchayan Dutta♦

7,12041658

asked May 7 at 14:08

Marco FalsoMarco Falso

184

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Hi, Marco. Welcome to Quantum Computing SE! It is preferable that you use MathJax to typeset your posts. Review How to write a good question?. I've edited the question on your behalf.
  $endgroup$
  – Sanchayan Dutta♦
  May 7 at 16:18
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Hi Sanchayan, thanks a lot for the edit and the advice. I will make sure that I write a good question next time.
  $endgroup$
  – Marco Falso
  May 7 at 16:21
  

  
  
  
  

add a comment | 

$begingroup$

I am actually working on quantum computing with qutrits. I am trying to define standard qubit gates for qutrits. The CNOT gate for qubits is defined as follows: $$|x,yrangle to |x,y+x bmod 2rangle.$$

So the matrix is $$mathrmUCN =
beginpmatrix
1 & 0 & 0 & 0\
0 & 1 & 0 & 0\
0 & 0 & 0 & 1\
0 & 0 & 1 & 0
endpmatrix.$$

For the definition of the same type of gate for qutrit I am left with two choices:

$$|x,yrangle to |x,y+x bmod 3rangle$$
$$|x,yrangle to |x,y+2x bmod 3rangle$$

Does anyone know if one of the two is more common or more usually defined?

PS: I know that applying one two times is equivalent to apply the other one time.

quantum-gate qutrit

share|improve this question

edited May 7 at 16:14

Sanchayan Dutta♦

7,12041658

asked May 7 at 14:08

Marco FalsoMarco Falso

184

$endgroup$

share|improve this question

edited May 7 at 16:14

Sanchayan Dutta♦

7,12041658

asked May 7 at 14:08

Marco FalsoMarco Falso

184

share|improve this question

edited May 7 at 16:14

Sanchayan Dutta♦

7,12041658

asked May 7 at 14:08

Marco FalsoMarco Falso

184

edited May 7 at 16:14

Sanchayan Dutta♦

7,12041658

edited May 7 at 16:14

Sanchayan Dutta♦

7,12041658

asked May 7 at 14:08

Marco FalsoMarco Falso

184

asked May 7 at 14:08

Marco FalsoMarco Falso

184

1 Answer
1

active

oldest

votes

3

$begingroup$

Your first definition,

$$|x,yrangle to |x,y+x bmod 3rangle$$

is both more common and, IMO, easier to use. Especially as you point out that the second definition can be simulated by repeated application of the first. You can see the authors have used this version of the qutrit CNOT in
Construction of two qutrit entanglement by using magnetic resonance selective pulse sequences (Çorbaci et al., 2016) and in
Controlled Not (Cnot) Gate For Two Qutrit Systems, (KOCAKOÇ & TAPRAMAZ, 2018).

Additionally, you may want to consider the context of your qutrit gates. For example, section 2 of arXiv:1105.5485 first defines three versions of the qutrit X gate and then discusses what a controlled X would mean in these cases.

share|improve this answer

edited May 7 at 23:12

Sanchayan Dutta♦

7,12041658

answered May 7 at 16:44

Noah DNoah D

462

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks a lot for the articles. None of them uses the Hadamard gate I am using but at least I have some articles to relate to about computation with qutrits. I am convinced that X gate and Hadamard gate must change all the states and not be only extensions of the two dimensional ones.
  $endgroup$
  – Marco Falso
  May 7 at 17:40
  

  
  
  
  

add a comment | 

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3

$begingroup$

Your first definition,

$$|x,yrangle to |x,y+x bmod 3rangle$$

is both more common and, IMO, easier to use. Especially as you point out that the second definition can be simulated by repeated application of the first. You can see the authors have used this version of the qutrit CNOT in
Construction of two qutrit entanglement by using magnetic resonance selective pulse sequences (Çorbaci et al., 2016) and in
Controlled Not (Cnot) Gate For Two Qutrit Systems, (KOCAKOÇ & TAPRAMAZ, 2018).

Additionally, you may want to consider the context of your qutrit gates. For example, section 2 of arXiv:1105.5485 first defines three versions of the qutrit X gate and then discusses what a controlled X would mean in these cases.

share|improve this answer

edited May 7 at 23:12

Sanchayan Dutta♦

7,12041658

answered May 7 at 16:44

Noah DNoah D

462

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks a lot for the articles. None of them uses the Hadamard gate I am using but at least I have some articles to relate to about computation with qutrits. I am convinced that X gate and Hadamard gate must change all the states and not be only extensions of the two dimensional ones.
  $endgroup$
  – Marco Falso
  May 7 at 17:40
  

  
  
  
  

add a comment | 

3

$begingroup$

Your first definition,

$$|x,yrangle to |x,y+x bmod 3rangle$$

is both more common and, IMO, easier to use. Especially as you point out that the second definition can be simulated by repeated application of the first. You can see the authors have used this version of the qutrit CNOT in
Construction of two qutrit entanglement by using magnetic resonance selective pulse sequences (Çorbaci et al., 2016) and in
Controlled Not (Cnot) Gate For Two Qutrit Systems, (KOCAKOÇ & TAPRAMAZ, 2018).

Additionally, you may want to consider the context of your qutrit gates. For example, section 2 of arXiv:1105.5485 first defines three versions of the qutrit X gate and then discusses what a controlled X would mean in these cases.

share|improve this answer

edited May 7 at 23:12

Sanchayan Dutta♦

7,12041658

answered May 7 at 16:44

Noah DNoah D

462

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks a lot for the articles. None of them uses the Hadamard gate I am using but at least I have some articles to relate to about computation with qutrits. I am convinced that X gate and Hadamard gate must change all the states and not be only extensions of the two dimensional ones.
  $endgroup$
  – Marco Falso
  May 7 at 17:40
  

  
  
  
  

add a comment | 

$begingroup$

Your first definition,

$$|x,yrangle to |x,y+x bmod 3rangle$$

is both more common and, IMO, easier to use. Especially as you point out that the second definition can be simulated by repeated application of the first. You can see the authors have used this version of the qutrit CNOT in
Construction of two qutrit entanglement by using magnetic resonance selective pulse sequences (Çorbaci et al., 2016) and in
Controlled Not (Cnot) Gate For Two Qutrit Systems, (KOCAKOÇ & TAPRAMAZ, 2018).

Additionally, you may want to consider the context of your qutrit gates. For example, section 2 of arXiv:1105.5485 first defines three versions of the qutrit X gate and then discusses what a controlled X would mean in these cases.

share|improve this answer

edited May 7 at 23:12

Sanchayan Dutta♦

7,12041658

answered May 7 at 16:44

Noah DNoah D

462

$endgroup$

share|improve this answer

edited May 7 at 23:12

Sanchayan Dutta♦

7,12041658

answered May 7 at 16:44

Noah DNoah D

462

edited May 7 at 23:12

Sanchayan Dutta♦

7,12041658

edited May 7 at 23:12

Sanchayan Dutta♦

7,12041658

answered May 7 at 16:44

Noah DNoah D

462

answered May 7 at 16:44

Noah DNoah D

462