<p>The best that I am aware of is in the surface code proposal due to Fowler et al (<a href="http://arxiv.org/abs/0803.0272">arXiv:0803.0272</a>), where it is shown that they achieve a bound of 0.75%.</p>

CC BY-SA 3.0
2011-11-02T21:53:53.713Z
@Pitor: Thanks for fixing the link for me. I originally posted this from mobile, but the mobile StackExchange is a bit buggy...
Chris Granade
CC BY-SA 3.0
2011-11-03T00:22:24.320Z
The Fowler et al. result is an estimate (for depolarizing noise), not a lower bound.
Adam Paetznick
CC BY-SA 3.0
2011-11-03T04:28:14.733Z
Yes, I'm aware of lots of estimates in this range (Raussendorf, Harrington and Goyal's papers, Knill's 3% paper etc.) but what I'm looking for is proven lower bounds.
Joe Fitzsimons
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2011-11-03T12:50:46.417Z
My apologies, then, for misunderstanding Fowler's results.
Chris Granade
CC BY-SA 3.0
2011-11-02T20:35:57.290Z
2011-11-02T20:54:31.730Z
Chris Granade
<p>The highest threshold lower bound for for independent stochastic noise of which I am aware is $1.04 \times 10^{-3}$ by Aliferis, Gottesman and Preskill (<a href="http://arxiv.org/abs/quant-ph/0703264" rel="nofollow">quant-ph/0703264</a>).  They analyze Knill's teleportation-based scheme with postselection.</p>

<p>If you are willing to consider independent depolarizing noise, then I know of two slightly higher lower bounds: $1.25\times 10^{-3}$ by Aliferis and Preskill (<a href="http://arxiv.org/abs/0809.5063" rel="nofollow">arXiv:0809.5063</a>) and $1.32 \times 10^{-3}$ by myself and Ben Reichardt (<a href="http://arxiv.org/abs/1106.2190" rel="nofollow">arXiv:1106.2190</a>).</p>

CC BY-SA 3.0
2011-11-02T18:41:22.753Z
Depolarizing noise is a little less general than what I was looking for. The paper by Aliferis, Gottesman and Preskill you mention seems to be the answer to my question. Weirdly, now that you mention it and summarize the paper, it seems that I did see that paper when it came out, but it had drifted from my memory. Thanks, your answer is extremely helpful!
Joe Fitzsimons
CC BY-SA 3.0
2011-11-02T18:33:32.357Z
2011-11-02T20:55:50.650Z
Adam Paetznick
<p>It is well established that there exists a noise threshold for quantum computation, such that below this threshold, the computation can be encoded in such a way that it yields the correct result with bounded probability (with at most polynomial computational overhead). This threshold depends on the encoding used and the exact nature of the noise, and it is the case that results from simulation often give thresholds much higher than what can be proved for adversarial noise models.</p>

<blockquote>
  <p>So my question is simply what is the highest lower bound that has been
  proved for independent stochastic noise?</p>
</blockquote>

<p>The noise model I am referring to is the one dealt with in <a href="http://arxiv.org/abs/quant-ph/0504218">quant-ph/0504218</a>, where Aliferis, Gottesman and Preskill prove a lower bound $2.73 \times 10^{-5}$. Note, however, I do not care which type of encoding is used, and it need not be restricted to the code considered in that paper. The highest I'm aware of is $1.94 \times 10^{-4}$ due to Aliferis and Cross (<a href="http://arxiv.org/abs/quant-ph/0610063">quant-ph/0610063</a>). Has this value been improved upon since then?</p>

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2011-11-02T10:31:16.783Z
Do you want a numerical or analytical value?
Matty Hoban
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2011-11-02T10:38:13.770Z
I'm happy with either as long as it is actually a proven lower bound, without making further assumptions on the noise other than the maximum probability of error.
Joe Fitzsimons
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2011-11-02T13:47:45.260Z
@Kaveh_kh: thanks for the link. In case it isn't clear from the question, I mean the best *known* threshold.
Joe Fitzsimons
CC BY-SA 3.0
2011-11-02T17:31:33.777Z
@Joe, a comparably well-posed question, having both practical and fundamental implications in simulation science, is "What quantum computer architecture has the lowest proved lower bound for independent stochastic noise, such that PTIME simulation of the (noisy) computation process is possible for all error rates above the bound?"   Perhaps Joe Fitzsimons might consider adjoining some version of this question to the original question?
John Sidles
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2011-11-02T17:41:26.930Z
@JohnSidles: That is indeed an interesting question (though I assume you want the lowest upper bound on the noise rather than a lower bound). May I suggest you ask it as a separate question? The reason I suggest that rather than amending this one is that I specifically want an answer to this question as it relates to a problem I'm working on.
Joe Fitzsimons
CC BY-SA 3.0
2011-11-02T19:32:48.907Z
@JoeFitzsimons, good suggestion ... I'll go ahead and post it as a question *after* seeing what answers you get ... an meanwhile I'll be thinking of the most natural permutation/definition of "lowest/highest lower/upper simulable error rate" (right now it seems "lowest lower" is the interesting case).
John Sidles
CC BY-SA 3.0
2011-11-02T19:41:00.697Z
@JohnSidles: Is it? Surely 0 is a lower bound on the lowest level of noise above which the system is classically simulable, but it is not interesting because it is too low.
Joe Fitzsimons
CC BY-SA 3.0
2011-11-02T20:29:19.653Z
I had in mind to study the class of (error-corrected) quantum computers; each member of that class having some lowest-but-nonzero error rate for which it could be classically simulated; thus a particularly interesting case is the lowest-of-the-low-but-nonzero simulable error values.  To say it in a concisely confusing way, I reckon we'd be looking for the lowest upper-bound on the lowest lower-bound of simulable error rates.
John Sidles
CC BY-SA 3.0
2011-11-03T06:36:30.277Z
let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/1716/discussion-between-joe-fitzsimons-and-john-sidles)
Joe Fitzsimons
<quantum-computing><reference-request><fault-tolerance>
What is the best lower bound for the fault-tolerance threshold in quantum computing?
Joe Fitzsimons
# What is the best lower bound for the fault-tolerance threshold in quantum computing?

(Asked by: username_0 on 2011-11-02)

It is well established that there exists a noise threshold for quantum computation, such that below this threshold, the computation can be encoded in such a way that it yields the correct result with bounded probability (with at most polynomial computational overhead). This threshold depends on the encoding used and the exact nature of the noise, and it is the case that results from simulation often give thresholds much higher than what can be proved for adversarial noise models.

> So my question is simply what is the highest lower bound that has been
> proved for independent stochastic noise?

The noise model I am referring to is the one dealt with in [quant-ph/0504218](http://arxiv.org/abs/quant-ph/0504218), where Aliferis, Gottesman and Preskill prove a lower bound $2.73 \times 10^{-5}$. Note, however, I do not care which type of encoding is used, and it need not be restricted to the code considered in that paper. The highest I'm aware of is $1.94 \times 10^{-4}$ due to Aliferis and Cross ([quant-ph/0610063](http://arxiv.org/abs/quant-ph/0610063)). Has this value been improved upon since then?

**Tags:** quantum-computing, reference-request, fault-tolerance

> **Comments:**
> **username_1**: Do you want a numerical or analytical value?  
> **username_0**: I'm happy with either as long as it is actually a proven lower bound, without making further assumptions on the noise other than the maximum probability of error.  
> **username_0**: @Kaveh_kh: thanks for the link. In case it isn't clear from the question, I mean the best *known* threshold.  
> **username_2**: @Joe, a comparably well-posed question, having both practical and fundamental implications in simulation science, is "What quantum computer architecture has the lowest proved lower bound for independent stochastic noise, such that PTIME simulation of the (noisy) computation process is possible for all error rates above the bound?"   Perhaps username_0 might consider adjoining some version of this question to the original question?  
> **username_0**: @JohnSidles: That is indeed an interesting question (though I assume you want the lowest upper bound on the noise rather than a lower bound). May I suggest you ask it as a separate question? The reason I suggest that rather than amending this one is that I specifically want an answer to this question as it relates to a problem I'm working on.  
> **username_2**: @JoeFitzsimons, good suggestion ... I'll go ahead and post it as a question *after* seeing what answers you get ... an meanwhile I'll be thinking of the most natural permutation/definition of "lowest/highest lower/upper simulable error rate" (right now it seems "lowest lower" is the interesting case).  
> **username_0**: @JohnSidles: Is it? Surely 0 is a lower bound on the lowest level of noise above which the system is classically simulable, but it is not interesting because it is too low.  
> **username_2**: I had in mind to study the class of (error-corrected) quantum computers; each member of that class having some lowest-but-nonzero error rate for which it could be classically simulated; thus a particularly interesting case is the lowest-of-the-low-but-nonzero simulable error values.  To say it in a concisely confusing way, I reckon we'd be looking for the lowest upper-bound on the lowest lower-bound of simulable error rates.  
> **username_0**: let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/1716/discussion-between-joe-fitzsimons-and-john-sidles)  

---

## 2 answers

### Answer by username_4 on 2011-11-02. Score: 15 | Accepted answer

The highest threshold lower bound for for independent stochastic noise of which I am aware is $1.04 \times 10^{-3}$ by Aliferis, Gottesman and Preskill ([quant-ph/0703264](http://arxiv.org/abs/quant-ph/0703264)). They analyze Knill's teleportation-based scheme with postselection.

If you are willing to consider independent depolarizing noise, then I know of two slightly higher lower bounds: $1.25\times 10^{-3}$ by Aliferis and Preskill ([arXiv:0809.5063](http://arxiv.org/abs/0809.5063)) and $1.32 \times 10^{-3}$ by myself and Ben Reichardt ([arXiv:1106.2190](http://arxiv.org/abs/1106.2190)).

> **Comments:**
> **username_0**: Depolarizing noise is a little less general than what I was looking for. The paper by Aliferis, Gottesman and Preskill you mention seems to be the answer to my question. Weirdly, now that you mention it and summarize the paper, it seems that I did see that paper when it came out, but it had drifted from my memory. Thanks, your answer is extremely helpful!  

---

### Answer by username_3 on 2011-11-02. Score: 6

The best that I am aware of is in the surface code proposal due to Fowler et al ([arXiv:0803.0272](http://arxiv.org/abs/0803.0272)), where it is shown that they achieve a bound of 0.75%.

> **Comments:**
> **username_3**: @Pitor: Thanks for fixing the link for me. I originally posted this from mobile, but the mobile StackExchange is a bit buggy...  
> **username_4**: The Fowler et al. result is an estimate (for depolarizing noise), not a lower bound.  
> **username_0**: Yes, I'm aware of lots of estimates in this range (Raussendorf, Harrington and Goyal's papers, Knill's 3% paper etc.) but what I'm looking for is proven lower bounds.  
> **username_3**: My apologies, then, for misunderstanding Fowler's results.  

---

