What is the best lower bound for the fault-tolerance threshold in quantum computing?

Question

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, 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). Has this value been improved upon since then?

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Comments

Do you want a numerical or analytical value?

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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.

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Great question: also known as the 1 Million Dollar question in quantum computing. I know that there can be serious improvements when one assumes a specific "architecture" in the sense that how easy or hard it is to interact distant qubits (architecture is different from the error model) For example, see [here](http://arxiv.org/abs/0902.2658). I think the [PhD thesis of Bryan Eastin] (http://arxiv.org/abs/0710.2560) could be a good starting point to have a look at.

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@Kaveh_kh: thanks for the link. In case it isn't clear from the question, I mean the best *known* threshold.

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@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?

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@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.

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@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).

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@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.

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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.

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let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/1716/discussion-between-joe-fitzsimons-and-john-sidles)

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Answer 1

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). 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) and $1.32 \times 10^{-3}$ by myself and Ben Reichardt (arXiv:1106.2190).

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Comments

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!

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Answer 2

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

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Comments

@Pitor: Thanks for fixing the link for me. I originally posted this from mobile, but the mobile StackExchange is a bit buggy...

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The Fowler et al. result is an estimate (for depolarizing noise), not a lower bound.

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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.

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My apologies, then, for misunderstanding Fowler's results.

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Answer 3

Consider Nick Menicucci's recent result for continuous variable measurement based QC using cluster states and Gottesman-Kitaev-Preskill state ancilla, which requires a 20.5dB squeezing as the fault tolerance threshold for the intial cluster state.