<p>Adiabatic quantum computing (AQC) is a computational model (as Peter said in the comments). Compare AQC with other models of computation such as: </p>

<ol>
<li>circuit-based quantum computing (CBQC) </li>
<li>Adleman-Lipton model (a model for computing using DNA)</li>
<li>Turing machine model (a model where computations are done with symbols on a tape)</li>
</ol>

<p>One can devise algorithms using the AQC model, such as this algorithm for factoring integers: <a href="http://arxiv.org/abs/1411.6758" rel="noreferrer">http://arxiv.org/abs/1411.6758</a>. </p>

<p>Quantum annealing is a physical process which attempts to implement such algorithms. Start with a set of quantum states, and a time-dependent Hamiltonian $H(t)$ such that the ground state of $H(t_F)$ at a final time $t_F$, encodes the solution to your problem. Let the states evolve according to $H(t)$, and hope that in the end one of the candidate states ends up in the ground state of $H(t_F)$.</p>

<p>If you're lucky enough to have a qubit that starts in the ground state of $H(0)$, and $H(0)$ evolves into $H(t_F)$ slowly enough, then according to the "adiabatic theorem", the qubit always remains in the ground state of $H(t)$ for all $t$, and you are guaranteed to get the right answer which is encoded in the ground state of $H(t_F)$. This is what the AQC model has in mind, but since it is experimentally difficult to start in the ground state, we sometimes try to implement AQC algorithms using quantum annealing, which unfortunately might not end up in the ground state.</p>

CC BY-SA 3.0
2015-08-14T10:41:54.950Z
New here. I wanted to contribute by editing some questions and leaving some comments, but needed reputation for it, so gave a shot at this answer.
user1271772
CC BY-SA 3.0
2015-08-14T10:40:52.193Z
2015-08-14T10:40:52.193Z
user1271772
<p>I had this impression that quantum annealing is an optimization technique which may or may not produce exact solutions. On the other hand adiabatic quantum computation always gives exact solutions when it gives a solution. But this <a href="http://dwave.files.wordpress.com/2007/08/20070810_d-wave_quantum_annealing.pdf" rel="noreferrer">D-Wave report</a> says as follows:</p>

<blockquote>
  <p>D-Wave processors compute by piggybacking on quantum annealing. A
  quantum annealing processor can be operated as a universal quantum
  computer. In this regime of operation, the computational model is
  referred to as adiabatic quantum computation (AQC), which can be
  thought of as the long-time limit of quantum annealing.</p>
</blockquote>

<p>Should I think that adiabatic quantum computation is an application of quantum annealing?</p>

CC BY-SA 3.0
2013-05-20T14:47:58.543Z
Doesn't Scott's recent blog post [D-Wave: Truth finally starts to emerge](http://www.scottaaronson.com/blog/?p=1400) answer your question?
Kaveh
CC BY-SA 3.0
2013-05-20T15:04:13.863Z
@Kaveh, let me quote Scott here, "Note that D-Wave itself now speaks about “quantum annealing” rather than “quantum adiabatic optimization.”  The difference between the two is that the adiabatic algorithm runs coherently, at zero temperature, while quantum annealing is a “messier” version in which the qubits are strongly coupled to their environment throughout, but still maintain some quantum coherence.". Probably a 2007 D-Wave report is little obsolete if I want to know the theoretical background of their work.
Omar Shehab
CC BY-SA 3.0
2013-05-20T15:09:12.390Z
Also have you checked https://en.wikipedia.org/wiki/Quantum_annealing and https://en.wikipedia.org/wiki/Adiabatic_quantum_computation?
Kaveh
CC BY-SA 3.0
2013-05-20T15:12:30.217Z
ps: I personally would prefer if you ask your questions more generally and without directly referring to D-Wave.
Kaveh
CC BY-SA 3.0
2013-05-20T15:14:33.273Z
@Kaveh, yes. Those two articles don't give a comparison. But from the definitions I understood that the former is an optimization technique while the proof of adiabatic quantum evolution tells us that it gives exact solution. I was confused by reading D-Wave's technical report. I think I should look at papers which experiments adiabatic quantum computation and not quantum annealing, what D-Wave does.
Omar Shehab
CC BY-SA 3.0
2013-05-20T15:15:04.213Z
@Kaveh, understood.
Omar Shehab
CC BY-SA 3.0
2013-05-20T18:35:24.813Z
Adiabatic quantum computation doesn't necessarily always give an exact solution.  Nor, let me note, does conventional digital computation; it depends on the algorithm you're running. The situation in adiabatic quantum computation is similar; it's not an algorithm, it's a computational model.
Peter Shor 
CC BY-SA 3.0
2013-05-22T13:59:58.690Z
@PeterShor, You said that 'adiabatic quantum computation' is not an algorithm but a computational model. From the statement of Scott quoted in my first comment, I assume that 'quantum adiabatic optimization' is not 'adiabatic quantum computation' but an optimization algorithm which probably runs on an adiabatic quantum computer. But the first line of the PhD thesis of David Gosset (dspace.mit.edu/handle/1721.1/68872) says, that the first adiabatic paper by Farhi et. al. proposed 'quantum adiabatic optimization'. Didn't the paper actually propose the computational model and not the algorithm?
Omar Shehab
CC BY-SA 3.0
2013-05-22T14:08:52.343Z
The paper proposed the computational model ***and*** the algorithm. Scott clearly means the algorithm by "quantum adiabatic optimization", while your quote from the D-Wave report (whch my comment was responding to) specifically says that "adiabatic quantum computation" is a model. Ideally, one should distinguish the two, but unfortunately "adiabatic quantum computation" is used to refer to both the algorithm and the model.
Peter Shor 
CC BY-SA 3.0
2013-05-31T15:07:09.350Z
@PeterShor, could you please point me to any paper which solves the same problem with both an adiabatic quantum algorithm and quantum annealing algorithm and compares their performances. I am yet to realize the difference between these two hence wanted to take a look at an example.
Omar Shehab
CC BY-SA 3.0
2013-05-31T18:57:25.477Z
@Omar: I don't believe there are such papers yet.
Peter Shor 
<quantum-computing><optimization><quantum-information><physics><statistical-physics>
Quantum annealing vs adiabatic quantum computation
Omar Shehab
# Quantum annealing vs adiabatic quantum computation

(Asked by: username_0 on 2013-05-20)

I had this impression that quantum annealing is an optimization technique which may or may not produce exact solutions. On the other hand adiabatic quantum computation always gives exact solutions when it gives a solution. But this [D-Wave report](http://dwave.files.wordpress.com/2007/08/20070810_d-wave_quantum_annealing.pdf) says as follows:

> D-Wave processors compute by piggybacking on quantum annealing. A
> quantum annealing processor can be operated as a universal quantum
> computer. In this regime of operation, the computational model is
> referred to as adiabatic quantum computation (AQC), which can be
> thought of as the long-time limit of quantum annealing.

Should I think that adiabatic quantum computation is an application of quantum annealing?

**Tags:** quantum-computing, optimization, quantum-information, physics, statistical-physics
> **Comments:**
> **username_1**: Doesn't Scott's recent blog post [D-Wave: Truth finally starts to emerge](http://www.scottaaronson.com/blog/?p=1400) answer your question?  
> **username_0**: @Kaveh, let me quote Scott here, "Note that D-Wave itself now speaks about “quantum annealing” rather than “quantum adiabatic optimization.”  The difference between the two is that the adiabatic algorithm runs coherently, at zero temperature, while quantum annealing is a “messier” version in which the qubits are strongly coupled to their environment throughout, but still maintain some quantum coherence.". Probably a 2007 D-Wave report is little obsolete if I want to know the theoretical background of their work.  
> **username_1**: Also have you checked https://en.wikipedia.org/wiki/Quantum_annealing and https://en.wikipedia.org/wiki/Adiabatic_quantum_computation?  
> **username_1**: ps: I personally would prefer if you ask your questions more generally and without directly referring to D-Wave.  
> **username_0**: @Kaveh, yes. Those two articles don't give a comparison. But from the definitions I understood that the former is an optimization technique while the proof of adiabatic quantum evolution tells us that it gives exact solution. I was confused by reading D-Wave's technical report. I think I should look at papers which experiments adiabatic quantum computation and not quantum annealing, what D-Wave does.  
> **username_0**: @Kaveh, understood.  
> **username_2**: Adiabatic quantum computation doesn't necessarily always give an exact solution.  Nor, let me note, does conventional digital computation; it depends on the algorithm you're running. The situation in adiabatic quantum computation is similar; it's not an algorithm, it's a computational model.  
> **username_0**: @PeterShor, You said that 'adiabatic quantum computation' is not an algorithm but a computational model. From the statement of Scott quoted in my first comment, I assume that 'quantum adiabatic optimization' is not 'adiabatic quantum computation' but an optimization algorithm which probably runs on an adiabatic quantum computer. But the first line of the PhD thesis of David Gosset (dspace.mit.edu/handle/1721.1/68872) says, that the first adiabatic paper by Farhi et. al. proposed 'quantum adiabatic optimization'. Didn't the paper actually propose the computational model and not the algorithm?  
> **username_2**: The paper proposed the computational model ***and*** the algorithm. Scott clearly means the algorithm by "quantum adiabatic optimization", while your quote from the D-Wave report (whch my comment was responding to) specifically says that "adiabatic quantum computation" is a model. Ideally, one should distinguish the two, but unfortunately "adiabatic quantum computation" is used to refer to both the algorithm and the model.  
> **username_0**: @PeterShor, could you please point me to any paper which solves the same problem with both an adiabatic quantum algorithm and quantum annealing algorithm and compares their performances. I am yet to realize the difference between these two hence wanted to take a look at an example.  
> **username_2**: @Omar: I don't believe there are such papers yet.  

---

## 1 answers

### Answer by username_3 on 2015-08-14. Score: 8 | Accepted answer

Adiabatic quantum computing (AQC) is a computational model (as Peter said in the comments). Compare AQC with other models of computation such as:

1. circuit-based quantum computing (CBQC)
2. Adleman-Lipton model (a model for computing using DNA)
3. Turing machine model (a model where computations are done with symbols on a tape)

One can devise algorithms using the AQC model, such as this algorithm for factoring integers: <http://arxiv.org/abs/1411.6758>.

Quantum annealing is a physical process which attempts to implement such algorithms. Start with a set of quantum states, and a time-dependent Hamiltonian $H(t)$ such that the ground state of $H(t\_F)$ at a final time $t\_F$, encodes the solution to your problem. Let the states evolve according to $H(t)$, and hope that in the end one of the candidate states ends up in the ground state of $H(t\_F)$.

If you're lucky enough to have a qubit that starts in the ground state of $H(0)$, and $H(0)$ evolves into $H(t\_F)$ slowly enough, then according to the "adiabatic theorem", the qubit always remains in the ground state of $H(t)$ for all $t$, and you are guaranteed to get the right answer which is encoded in the ground state of $H(t\_F)$. This is what the AQC model has in mind, but since it is experimentally difficult to start in the ground state, we sometimes try to implement AQC algorithms using quantum annealing, which unfortunately might not end up in the ground state.

> **Comments:**
> **username_3**: New here. I wanted to contribute by editing some questions and leaving some comments, but needed reputation for it, so gave a shot at this answer.  

---

