• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


New features!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

123 submissions , 104 unreviewed
3,600 questions , 1,219 unanswered
4,602 answers , 19,547 comments
1,470 users with positive rep
423 active unimported users
More ...

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

+ 22 like - 0 dislike

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?

This post has been migrated from (A51.SE)
asked Nov 2, 2011 in Theoretical Physics by Joe Fitzsimons (3,555 points) [ no revision ]
retagged Mar 18, 2014 by dimension10
Most voted comments show all comments
Do you want a numerical or analytical value?

This post has been migrated from (A51.SE)
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.

This post has been migrated from (A51.SE)
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.

This post has been migrated from (A51.SE)
@Kaveh_kh: thanks for the link. In case it isn't clear from the question, I mean the best *known* threshold.

This post has been migrated from (A51.SE)
@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?

This post has been migrated from (A51.SE)
Most recent comments show all comments
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.

This post has been migrated from (A51.SE)
let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/1716/discussion-between-joe-fitzsimons-and-john-sidles)

This post has been migrated from (A51.SE)

3 Answers

+ 12 like - 0 dislike

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).

This post has been migrated from (A51.SE)
answered Nov 2, 2011 by Adam Paetznick (120 points) [ no revision ]
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!

This post has been migrated from (A51.SE)
+ 5 like - 0 dislike

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%.

This post has been migrated from (A51.SE)
answered Nov 2, 2011 by Chris Granade (260 points) [ no revision ]
@Pitor: Thanks for fixing the link for me. I originally posted this from mobile, but the mobile StackExchange is a bit buggy...

This post has been migrated from (A51.SE)
The Fowler et al. result is an estimate (for depolarizing noise), not a lower bound.

This post has been migrated from (A51.SE)
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.

This post has been migrated from (A51.SE)
My apologies, then, for misunderstanding Fowler's results.

This post has been migrated from (A51.SE)
+ 2 like - 0 dislike

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.

answered Dec 6, 2014 by nir [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights