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This position is for the 2020-2021 school year
Speech and Language Pathoogist
Licensed/Certified Speech Language Pathologist in the State of Wyoming
Successful experience as a Speech and Language Pathologist. Willingness and ability to travel. Good public relations.
Identified supervisor(s) and under the general supervision of the Executive Director.
Maintains confidentiality with all aspects of our program, including personnel issues, clients and client files.
Work with other staff to implement county child find activities.
Work as part of a team to evaluate speech and language deficiencies, using appropriate diagnostic techniques, includes submitting a written report summarizing the results of the evaluation within the guidelines of IDEA.
Interpret speech and language evaluations results to staff and parents.
Assist team in developing and implementing Individualized Educational Programs (IEP)/
Individualized Family Service Plans (IFSP), including specific materials and teaching methods within the guidelines of IDEA.
Prepare for meetings by entering proper documentation on special education forms
Conduct MDT/IEP/IFSP meetings as assigned.
Provide speech correction and language development activities to individual students or groups of students as prescribed in each student's IEP or IFSP.
Adopt/adapt a data collection system for the highest priority objectives for each child
Develop and maintain therapy schedules with individual students.
Maintain and ensure each child's monthly therapy notes are completed.
Provide other staff members with ongoing information regarding each child's current status on IEP/IFSP modifications.
Coordinate services with other agencies involved in the child's program, such as PHN, DFS, local school districts, etc. as assigned.
Provide in-service to other staff on teaching/therapy approaches.
Participate in child study meetings.
Will participate in staff meetings regarding program operation.
Will supervise Speech/Language Pathologist Assistant (SLPA)
- Will meet with SLPA who conduct individual/small group activities to teach specific skills no less than once a month to discuss specific techniques and program modifications
- Will provide aides with written instructions when they are teaching specific skills
Evaluate all staff you are assigned supervision of on an annual basis, in line with the policy and procedures set forth by the Board of Directors.
Will complete a written report on a child following an accident.
Comply with state and federal regulations as prescribed in PL 94-142, Part B and Part C Rules and Regulation and Department of Family Services (DFS) Rules and Regulations.
Will attend required meetings, trainings and conferences.
Maintain and increase professional growth through attendance at various workshops and in-services relating to profession, as well as keeping abreast of new literature in the field.
Must maintain State of Wyoming Board of Examiners of Speech Pathology and Audiology license.
Will maintain a valid driver's license.
Able to travel/drive to required scheduled assignments.
Provide all necessary information to file Medicaid/Kid Care (Title XIX).
Maintain regular and continuous liaison with the Executive Director to keep them informed of significant events in the program, staffing's and personnel, community and family issues.
Be responsible for maintaining the general cleanliness and upkeep of the physical facility and equipment and notify the Executive Director of the need for additional attention to repairs or maintenance.
Adhere to the established policies, procedures and regulations of the facility.
Be an advocate for children and families.
Other duties as assigned by supervisor and Executive Director.
Negotiable by Child Development Services Board of Directors.
As per Child Development Services policies and procedures.
Click on the link below to complete our online application. You will have the option to upload your resume and cover letter.
Speech/Language Pathologist Application
Alternatively, click the link at the bottom of the page to download an application. Please fill this out, attach your resume and cover letter and mail to:
CDS Human Resources
P.O. Box 593
Lander, WY 82520.
Thank you for your interest in CDS!


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Our latest JSA TV Virtual CEO Roundtable, 'M&A: Reviewing the Big Deals of 2016 & Predictions for 2017' is now available On Demand. The virtual roundtable, moderated by Robert Powell, Founder and Chief Editor at Telecom Ramblings, and esteemed panelists Rick Calder, President and CEO of GTT; Ed Mullane, TMT Editor of Mergermarket; and Greg Weiner, Co-Founder of Vertix Consulting reviewed the major M&A activity impacting the tech and telecom space, as well as provided analysis and predictions of deals to come. The panelists discuss the challenges they saw in 2016, and you won't believe what they said to look out for in 2017. Click here to watch the Virtual CEO Roundtable.
The JSA TV CEO Virtual Roundtable series provides an outlet for multiple viewpoints on critical issues facing the telecom industry today, in an informative video format distributed to the JSA Community and all Telecom Exchange (TEX) attendees. The next Telecom Exchange event takes place June 20-21, 2017 at Cipriani Wall Street in New York City.

subset: commoncrawl


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Plants > Climbers
A small-flowered climber with abundant bell-shaped deep purple-blue flowers with creamy yellow anthers. H&S: 3m x 1m. Flowers June to September.
0800 612 2643
Copyright © 2012 Coolings Nurseries Ltd.Registered Office Rushmore Hill, Knockholt, Kent, TN14 7NN, Tel: 01959 532269Registered in England No: 1534529
Website designed and developed by Website engine and hosting by Corby+Fellas


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Why use Rubbish Clearance Ltd?
- No Item Restrictions for Ardleigh or Blackmore End
- Labour Charges Included
- Save Money Now
- No Skip Lighting Costs
- Our rates are based on the percentage of the vehicle filled and you only pay for the space used. We do all the loading and all disposal fees are included.
- No Weight Restrictions
Live in CO7? We can move single bulky item or a full house clearance! Call: 0800 138 9100


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Vantec NexStar SATA/IDE to USB 2.0 Universal Storage Adapter (Page 1)
They say, no one likes a know-it-all, but what about a do-it-all? - As enthusiasts we have a lot of toys and many of these devices store data on different forms of flash media. To add insult to injury, the people around us tend to think we are their personal tier one tech support and expect us to be able to work with all of their devices as well. All of that connectivity can be a pain in the bum if you are trying to carry separate adapters.
I really couldn't believe that it was possible to get a single device that has the ability to connect to 66 different types of media. I mean come on, 66? that is SIXTY-SIX! I had no idea that there were even 66 different types of storage mediums in use today or at least new enough that anyone would want to still use them. Just before I start to write every review, I have a peek at Newegg and feel out the other products on the market and how the cost breakdown looks. Tonight I found out that the Vantec NexStar Universal Storage Adapter 66-in-1 is available for less than 30 Dollars. That is less than 50 Cents for each format that the USA is able to work with. Come on, pinch me, because I have to be dreaming.
Let's take a look at all of those media formats on the specifications page and see what the entire unit consists of.
Last updated: Nov 26, 2019 at 07:04 pm CST
PRICING: You can find products similar to this one for sale below.
United States: Find other tech and computer products like this over at Amazon.com
United Kingdom: Find other tech and computer products like this over at Amazon.co.uk
Australia: Find other tech and computer products like this over at Amazon.com.au
Canada: Find other tech and computer products like this over at Amazon.ca
Deutschland: Finde andere Technik- und Computerprodukte wie dieses auf Amazon.de


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This boxed calendar has a unique feature of five double sided colored pencils attached to the back stand. This will make it easy to color yourself calm all year long! Our lives become busier with each passing day, and as technology escalates, so does our access to work, obligations, and stress. Constant stimulation and expectation have left us burnt out and distanced from the present moment. Color Me Calm All Year Long offers up 365 coloring templates all designed to help you get coloring and get relaxed throughout the year.
- Desk Calendar
- 365 Day Calendar
- Stapled Binding
- Medium Square Grid
- Observes Major Holidays
Brand: Calendars.com


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Ivory Coast: What to expect from upcoming elections in the Ivory Coast | Ventures Africa
Ivory Coast presidential elections will take place on the 25th of October, and barely 2 weeks to the set date, there are already whispers of a 'rigged' election. Ivory Coast's President, Allasane Quattara has been accused of colluding with the Electoral commission of the Ivory Coast to 'sabotage' the elections. Quattara, who is contesting for a second term in office through his party Rally of the Republicans, is receiving backlashes from opposition candidates who have called for dialogue with him. They reportedly want the dissolution of the Electoral Commission in the country. October's election marks the first election held in this West African Country since 2010. The President of Ivory Coast in 2010, Laurent Gbagbo, refused to cede the Presidency to Allasane Quattara who had been declared the winner. This triggered a civil war that lasted 5 months and killed about 3,000 people. Allasane Quattara was later installed as the President through the combined efforts of the African Union and French forces. Laurent Gbagbo is currently awaiting trial at the International Criminal Court (ICC).Full Article: What to expect from upcoming elections in the Ivory Coast - Ventures Africa.


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Everyone knows that for every occasion gifting plays a vital role and thats exactly what we specialize in here at our E-Shop. At our Online Shopping Store you will find great and amazing Gifts for Online Delivery to Jamshedpur with the facility of Free Home Delivery. Our E-tail brings to you our special selection of Gifts for various occasions like Birthday, Wedding, Christmas, New Year etc. When you Send Gift Basket to Jamshedpur through our E-shop, you are sure to surprise them with the content of the Gift Baskets. On our website, you can also buy & send Mothers Day Gifts to Jamshedpur.
Our Website Portal offers an extensive collection of Quality Guaranteed Gifts like Personal Care Items, Watches, Perfumes and Showpieces etc for all kind of Indian Festivals. Get Hassle Free Delivery from our E-shop and Send Gift to Jamshedpur as per taste and style of your loved ones. When your selected Online Gift reaches your dear ones, it will show how you feel about the loved one youre buying it for. So, order online to Send Designer Gifts to Jamshedpur for all occasions and our 24x7 Customer Service shall have it delivered the very same day. Also, shop & send Mothers Day Gifts to Jamshedpur today.
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I Don't Want to Go to the Hospital!
From the Series Little Princess
The Little Princess doesn't want to go to the hospital, and she does everything she can to avoid it. But when she is finally forced to go, she finds that she really rather likes it. Why? Because they treat her like a princess there!
|Your Price
|Interest Level
|Preschool - Grade 3
|Reading Level
|Grade 2
|Fiction, Picture Books
|Andersen Press USA
|Andersen Press USA
|Publication Date
Library Media Connection
"This is sure to be a big hit with young readers….The colorful illustrations are humorous, young children will love reading about the Little Princess." —Library Media Connection
Booklist Online
"Hospitals made not so scary." —Booklist Online
School Library Journal
"The large, expressive watercolors have lots of amusing detail, like the cat having a fish dinner on the roof….Sure to be popular with fans of the series." —School Library Journal
Ross's popular little princess with the yellow crown needs to go to the hospital to have a lump removed from her nose. This look at a hospital stay will comfort any child facing the same fate, yet it avoids being purposeful or didactic due to the humorous tone. The watercolor cartoon illustrations, framed and set on white backgrounds with just one sentence per page, make this appropriate for both group sharing and one-on-one. Libraries may want to purchase if the other books in the series are popular (I Want to Win, 2012; I Want a Party! 2011, etc.). First published in Great Britain in 2000.
Author, Illustrator: Tony Ross
Tony Ross has illustrated hundreds of children's books including the beloved Amber Brown series and his own Little Princess books for Andersen Press. He has chalked up countless awards in his fifty-year career and his books have been published in more than fifty languages.


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Good to know
|Distance to city center
|0.3 mi
|Location rating
|Nearest Airport
|Funchal Madeira
|Distance to airport
|9.2 mi
Standard Rooms
Deluxe Rooms
Traveler rating
Time of year
Traveler type
Show reviews that mention
ExcellentDec 20199.0
The room was simple but comfortable and cosy. The cleaning was fine and the staff was helpful. The spa at casino hotel was relaxing and good.
They could have some parking for their guests, we paid a lot of parking during our stay.
ExcellentJul 20198.3
Very clean, nice and helpful staff. Pool area really nice and not at all busy. Room held high standard. Breakfast at Pestana Casino Park extremely good!
The beds was simply to hard. Laying on the back or stomach was ok but the side. Also a bit noisy as our balcony was facing the main road. .
ExcellentMay 201910.0
Very friendly and helpful staff. Nice comfortable large sized room. Around a 5-10 minute walk from Funchal centre. Lots of nice bars/restaurants close to the hotel. Can recommend Casal de Penha restaurant down the road from the hotel, lovely food and staff.
GoodApr 20197.5
Location Very clean
Noisy roads all around Pestana apartment good but hotel casino staff not very warm & welcoming & spa was covered in flies which we reported to front desk as spa unattended
ExcellentDec 20188.8
Room size as expected. Big comfortable bed. Good location. Very small kitchen which could have been better equipped - not even a bottle opener. Good breakfast which was taken in the Pestana Casino Hotel over the road. Very friendly and helpful staff.
The chairs were not very comfortable for sitting in in the evening, they were more like dining chairs. Poorly equipped kitchen. loads of road noise. The Pestana Casino Hotel may have been 5 star at one time, but now it is just tatty! I felt like rubbing out the 5th star and a bit of the 4th one. Pool sun beds are old and faded. Paint is flaking off everywhere. The lockers in the indoor pool area are falling apart. Many of those little rubber stoppers under the toilet seats are missing or broken. It just needs a thorough refurb.
Located in Funchal (Se), Pestana Casino Studios is within a 15-minute walk of CR7 Museum and Lido. Featured amenities include dry cleaning/laundry services and a 24-hour front desk.
9.2 Excellent
based on 156 reviews
Things to Know
- New rooms
- Clean rooms
- Modern Hotel
Check in
Check out
Policies vary by room type and provider.
FAQs when booking at Pestana Casino Studios
Where is Pestana Casino Studios located?
Pestana Casino Studios is located at Av. do Infante, nº9 in Sao Martinho, 0.3 miles from the center of Funchal. Madeira Casino is the closest landmark to Pestana Casino Studios.
When is check-in time and check-out time at Pestana Casino Studios?
Check-in time is 3:00 PM and check-out time is 12:00 PM at Pestana Casino Studios.
Does Pestana Casino Studios offer free Wi-Fi?
Yes, Pestana Casino Studios offers free Wi-Fi.
Does Pestana Casino Studios offer free parking?
No, Pestana Casino Studios does not offer free parking.
Does Pestana Casino Studios offer free airport shuttle service?
No, Pestana Casino Studios does not offer free airport shuttle service.
How far is Pestana Casino Studios from the airport?
Pestana Casino Studios is 9.2 miles from Funchal Madeira. Pestana Casino Studios is 44.4 miles from Porto Santo.
Can you bring pets to Pestana Casino Studios?
No, pets are not allowed at Pestana Casino Studios.
Does Pestana Casino Studios have a pool?
Yes, Pestana Casino Studios has a pool onsite.
Reviews about Pestana Casino Studios
Most Popular Landmarks
- Madeira Casino 273 yd
- Museum of Natural History 0.6 mi
- Santa Clara Monastery 0.6 mi
- Frederico de Freitas Museum 0.6 mi
- Quinta das Cruzes Museum 0.6 mi
- Cathedral Se 0.7 mi
Getting Around
|Funchal Madeira
|9.2 mi
|Porto Santo
|44.4 mi
|Funchal Centro
|0.3 mi


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Gopher Gold Slots
Gopher Gold is another fantastic, gold-themed online slot from Microgaming. This is a video slot, with gold graphics to help you keep your eyes on the prize.
With 5 paylines, this is smaller than a lot of other video slots, but there is plenty on offer if you fancy trying your luck at the jackpot.
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Play Gopher Gold Slots Online
Just like the gophers in the game, you will have plenty to pick away at, and hopefully, take home a share of the gold.
How to Play Gopher Gold Slots
Gopher Gold is a little different in the fact that it uses 5 paylines. On each payline, you can bet one coin, but the value of the coin is set by you, just like it is in most other online slots.
Once you have set your coin size, you will bet your five coins, and spin to see if you can win the jackpot.
The aim, as always, is to match up the symbols. Gopher Gold uses standard J, Q, K and A symbols for its reels. These are complemented with little gophers, which act as wilds, plus carts and axes. Wilds can match any symbol.
Depending on which symbols you match, you can win big, and if you're lucky enough to land on the Gopher Gold logo symbols you'll get 12,000 times the bet for the payline.
Gopher Gold Slot Free Spins No Deposit
There are no free spins available with the Gopher Gold video slot, but it contains plenty of extra bonus features for you to take advantage of and help you win. With 5 paylines, plus wilds and scatters, there's plenty of ways you can win some serious money through Gopher Gold in April 2020.
Interactive Features
Gopher Gold doesn't use many interactive features, and it is a reasonably easy game once you have adjusted to the five paylines.
It does use wilds. These are the little gopher icons you will find on the reels, and these will substitute for any symbol you need in order to get a winning payline.
Paylines and Stakes
The Gopher Gold video slot uses 5 paylines, with 5 reels. This is not unusual for video slots, and you can select which paylines you wish to play on. For each payline, you can only bet one coin. While this may not seem like much, you can vary the size of that coin from $0.25 to $5. This means there is some serious money available, especially if you play on all 5 paylines.
Gopher Gold Slot on Mobile Phone
If you like to play your online slots on the go, you'll be pleased to know that you can play Gopher Gold on your mobile phone, your tablet, or whatever your preferred device might be. This game is available in all Microgaming casinos, so you shouldn't struggle to find it either.
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As a video slot, Gopher Gold is expected to have high-quality graphic design in order for it to function well and let you play at a natural pace. It certainly achieves this, but it may come at the cost of rather basic and old-fashioned imaging.
The Gopher Gold theme is certainly not lost in the design, as microgaming have made the bling background very clear, but it does look a bit vintage. Some may enjoy this, and it certainly doesn't affect the quality of the game, but it isn't the most aesthetically pleasing online slot game available.
Overall, it doesn't take away from the gameplay much.
About Gopher Gold Slots
Gopher Gold is a classic gold themed April 2020 video slot where the aim is to find your fortune.
The little gopher, axe and cart icons stick to the theme of the game, and there are standard J, Q, K and A icons too. There's also the Gopher Gold logo icon, which will pay out 12,000 times the bet for that line, which could be a lot of money.
You get to choose the value of each coin, from $0.25 to $5, and then you can bet one coin for each payline if you choose too. With the help of some wilds and a decent stake, the payouts can be quite substantial.
Gopher Gold Slots Software
Gopher Gold is an online slot game powered by Microgaming. Microgaming is a hugely successful software company that creates a wide range of online slot games with various themes and gameplay styles.
Gopher Gold is actually a video slot, but it maintains its high quality and can function at a normal gameplay rate. Unlike some lesser quality video slots, this should function at a normal speed while you try your luck to win the jackpot.
Gopher Gold Slots RTP Review
Gopher Gold has a Return To Player (RTP) rating of 96.3%, meaning that for every $100 bet in the game, $93.60 will, at some point, be dished out in rewards to lucky players. This is a fairly high RTP rating, although some online slots can be found with better returns than this.
It's also important to remember that this doesn't guarantee you'll win anything, it is simply a sign that most of the money can be won in April 2020.
Related Microgaming Pages
Check out more from Microgaming below to see other offers in April 2020.
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text:
Posted by Larry Hoover on July 20, 2004, at 19:42:55
In reply to Re: Surmontil/trimipramine for sleep, ibs » Larry Hoover, posted by yeltom on July 20, 2004, at 17:53:40
> Thanks for respoding. I actually had already read that abstract. How much are you taking?
25 mg, one hour before sleep.
> Are you taking it for sleep, ibs, to augment the zoloft?
The intent was to help with insomnia, which Zoloft made worse. I didn't know about the IBS link until today, and though I have IBS, I don't think my dose is large enough to affect that. Zoloft, initially, and at each dose increase, provoked extraordinary diarrhea that lasted days. My IBS med, Dicetel, fixed that problem up.
> How is it working?
It helps initiation of sleep. I also use 30 mg temazepam. The effect of the trimipramine seems to be diminishing over time. I expect I'll have to increase the dose. For primary insomnia, the dose is 100 mg, so I've got a ways to go, yet.
> How bad are the side effects?
Bit of dry mouth, but that passed after a week or so. Also, tended to sleep without moving for extended periods, and waking crampy. That passed, too.
> Let me know, will you. Thanks
You're welcome.
poster:Larry Hoover
URL: http://www.dr-bob.org/babble/20040719/msgs/368384.html


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Home Search Collections Journals About Contact us My IOPscience Logical error rate scaling of the toric code
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New Journal of Physics
The open access journal at the forefront of physics
Deutsche PhysikalischeGeseUschaft DPG IOP Institute Of PhySjCS
Logical error rate scaling of the toric code
Fern H E Watson and Sean D Barrett1
Department of Physics, Imperial College London, Prince Consort Road, London SW7 2AZ, UK E-mail: firstname.lastname@example.com
Received 24 June 2014, revised 1 August 2014 Accepted for publication 6 August 2014 Published 25 September 2014
New Journal of Physics 16 (2014) 093045
To date, a great deal of attention has focused on characterizing the performance of quantum error correcting codes via their thresholds, the maximum correctable physical error rate for a given noise model and decoding strategy. Practical quantum computers will necessarily operate below these thresholds meaning that other performance indicators become important. In this work we consider the scaling of the logical error rate of the toric code and demonstrate how, in turn, this may be used to calculate a key performance indicator. We use a perfect matching decoding algorithm to find the scaling of the logical error rate and find two distinct operating regimes. The first regime admits a universal scaling analysis due to a mapping to a statistical physics model. The second regime characterizes the behaviour in the limit of small physical error rate and can be understood by counting the error configurations leading to the failure of the decoder. We present a conjecture for the ranges of validity of these two regimes and use them to quantify the overhead—the total number of physical qubits required to perform error correction.
Keywords: overhead, toric code, topological quantum error correction
1. Introduction
Quantum computers are sensitive to the effects of noise due to unwanted interactions with the environment. To overcome this, fault-tolerant protocols that utilize error correction codes have
1 Deceased 19 October 2012.
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal
citation and DOI.
New Journal of Physics 16 (2014) 093045 1367-2630/14/093045+20$33.00
©2014 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
been developed. These schemes allow arbitrary quantum gates to be performed in spite of the noise that is ubiquitous in current models of quantum computing.
Recent progress has been made towards experimental implementations of quantum error correcting codes using small numbers of qubits realized using photonic systems, trapped ions and NMR techniques [1-5]. Superconducting qubits are another promising experimental technique for scalable fault-tolerant quantum computing [6-8], including surface code architectures [9].
The surface code [10, 11] is one of a family of topological codes, and is the basis for an approach to fault-tolerant quantum computing for which high thresholds have been reported [12-15]. The toric code [16] is among the most extensively studied of this family of codes, revealing much insight into related topologically ordered systems. A great deal of work has concentrated on calculating thresholds for various error models [17-19], and on the discovery and implementation of new classical decoding algorithms [19-25]. The toric code performs well, with high thresholds for some commonly studied noise models.
A high threshold is a very desirable property of an error correcting code since for all error rates below the threshold, increasing the number of physical qubits encoding the quantum information reduces the logical error rate. In a realistic setting the code must be operating at an error rate below the threshold. Other quantities then become important to characterize the performance of a quantum computer, for example the code overhead, the number of physical qubits comprising the code that are required to adequately protect the encoded quantum information. This is an important consideration for the practical implementation of fault-tolerant quantum computation and has recently begun to draw some attention [26-28].
The logical failure rate of the error correction, denoted here as Pfail, is a key metric of the performance of a code, since it describes the likelihood of failing to protect the encoded quantum information. In this work we seek the logical failure rate of the toric code for fixed code distance and physical error rate, p. The code distance is the minimum length of a stringlike operator that has a non-trivial effect on the code space, and in the case of the toric code such operators have a length equal to the lattice size L.
The toric code is a simple model that is closely related to other, more physically realistic systems. We expect therefore that results for the logical error rate scaling of the toric code could be applied in a range of other physical systems—most obviously the planar code (with open, rather than periodic, boundary conditions) and with noisy syndrome measurements. The techniques to determine the scaling of the logical error rate should be analogous although the numerics would be expected to differ from the toric code case [29]. Furthermore, once the scaling has been determined it can be used to calculate the fault-tolerant overhead for the planar code using the methods presented in this paper.
Below the threshold, the logical failure rate of a topological code is expected to reduce exponentially as we increase the code distance [16]. Although the code performance improves rapidly with increasing L, in the lattice of the toric code the total number of physical qubits scales as O (L2). Manufacturing, storing, and manipulating resources with such a scaling is a non-trivial task with technology available at present. We should then ask not simply how large we can make the code, but how many physical qubits are required to achieve a desired error correction performance.
In order to answer this question, we examine the behaviour of the toric code in the presence of uncorrelated bit-flip and phase-flip noise. We numerically simulate the error correction procedure and use this to find the failure rate as a function of the input parameters L
and p and find two operating regimes. The first of these, which we will call the universal scaling hypothesis, extends ideas by Wang et al [30] and uses rescaling arguments based on a mapping to a well-studied model in statistical physics (the two-dimensional random-bond Ising model, or RBIM). This approach provides a good estimate for Pfan when the error weight (the number of qubits an operator acts on non-trivially) is high and code distance is large.
Rescaling arguments apply in the thermodynamic limit, and close to criticality, where the correlation length of the RBIM diverges and the appropriate length scale is the ratio of the lattice size to the correlation length, LIAs p decreases there is a point at which finite-size effects begin to dominate and we no longer expect the universal scaling hypothesis to apply. This limit corresponds to low physical error rates, as well as small lattices.
The second approach extends ideas by Raussendorf et al [12] and Fowler et al [15] to find an analytic expression for Pfan in the limit p ^ 0. When the error weight is low and the code distance is small this expression gives a good estimate of the logical failure rate. We will refer to this as the low p expression.
Although we know the limits in which each of these approaches is valid, we would like to make some quantitative statements about the range of parameters for which each is applicable. We shall present a heuristic argument for the range of L and p for which each regime gives a good approximation to the numerical data.
The structure of the paper is as follows. In section 2 we review the toric code and its properties. Readers familiar with this material may wish to skip to section 3 which discusses the universal scaling regime, in which rescaling arguments are used to estimate the logical failure rate. Section 4 describes the regime in which finite-size effects dominate the logical failure rate and the failure rate is dominated by spanning errors. In section 5 we present our conjectures regarding the ranges of validity of each of the two regimes described. In section 6 we use these results to demonstrate techniques to determine the overhead as a function of the single qubit error rate and the logical error rate. We conclude in section 7.
2. The toric code
2.1. Background
In the toric code, physical qubits reside on the edges of an L x L square lattice, as shown in figure 1. There are n = 2L2 physical qubits comprising the code. Periodic boundary conditions are imposed and the lattice can be imagined to be embedded on the surface of a torus.
The toric code is described by a set of two types of commuting stabilizer generators—the so-called vertex, Av, and plaquette, Bp, operators, defined as
Ay = 6v Xi, Bp = ®iGp Zi, (1)
where X and Z are the conventional single-qubit Pauli operators, v indicates a vertex and p a plaquette of the lattice. The Av operators therefore act on the four qubits surrounding a vertex of the lattice, and the Bp operators act on the four qubits surrounding a plaquette, see figure 1. These four-body measurements can be decomposed into four two-qubit CNOT gates with the addition of an ancilla [16].
Figure 1. Representation of stabilizer generators on an L = 5 toric code lattice. Qubits, shown as yellow circles, are placed on the links of the lattice. Note that the periodic boundaries are indicated by the dashed lines. The dual lattice is shown using grey lines. Top: a vertex operator on the primal lattice (left) and the dual lattice (right). Bottom: a plaquette operator on the primal lattice.
We denote the logical encoded state of the toric code by | toric. In the absence of noise, measuring any element of S = { Av, Bp } on this state will yield a +1 eigenvalue
Si \w)toric = + \w)toric, (2)
where Si G S. The stabilizer group is generated by S with multiplication being the group action. All elements of the stabilizer group act trivially on the code space. The code space of the toric code is four-dimensional and hence can encode two logical qubits. This is independent of n, hence the toric code protects a constant number of logical qubits regardless of its lattice size.
The symmetry between the primal lattice and the dual lattice (constructed by replacing plaquettes of the primal lattice by vertices and vice versa) shown in figure 1, reveals a useful symmetry in the stabilizers of the toric code. On the dual lattice the Av operators act on the qubits surrounding a plaquette, as shown in figure 1. By considering both the primal and dual lattices we can view all stabilizers as closed loops, meaning that all plaquette-type operators on the primal lattice have an analogous vertex-type operator on the dual lattice. It follows that all results calculated for either bit-flip or phase-flip errors are interchangeable with results for the other type.
In the language of algebraic topology, all of the stabilizers correspond to homologically trivial cycles. In figure 2 we show an example of a homologically trivial cycle that is generated by multiplying two adjacent stabilizer generators together. We see that all homologically trivial cycles act trivially on the code space.
The logical operators are also represented by cycles of Pauli operators. However, these cycles wrap around the torus and are not homologically equivalent to stabilizers. The logical operators correspond to homologically non-trivial cycles and have a non-trivial effect on the code space. The minimum weight of a logical operator is L.
There are two sets of Z and XX logical operators addressing the two encoded qubits (overbar indicates a logical operation). One of these, labeled Zi, is shown in figure 2 spanning the lattice vertically. The corresponding XX1 is also shown, and forms a closed horizontal loop on the dual lattice. By multiplying a logical operator by a subset of stabilizers we can continuously deform the minimum-weight cycle Z1 into any other operator spanning the lattice vertically. The
Figure 2. Left: Zi is a minimum-weight homologically non-trivial cycle, equivalent to a logical operator acting on the encoded information. Top: the H1 operator, drawn as a cycle on the dual lattice (lattice not shown). The H1 logical operator shares a single physical qubit with Zi and hence they anticommute. Right: an example of a homologically trivial cycle generated by multiplication of two adjacent plaquette operators.
Figure 3. A string of X errors is shown as a dashed line on the dual lattice. Measuring the two Bp generators indicated yields —1 eigenvalues because the stabilizer and error chain anticommute at these locations. Note that if the X error chain forms a cycle then it will not be detectable.
set of operators that are equivalent up to stabilizer operations belong to the same homology class [31].
Errors are detectable if they anticommute with at least one element of the set of stabilizer generators S. In this work we assume that stabilizers are measured perfectly. It follows that if any non-trivial eigenvalues are observed, this indicates the presence of errors with certainty. The pattern of stabilizers that anticommute with a given error reveals some information about the location and most likely type of error, although it cannot uniquely identify the error. This ambiguity is due to the code degeneracy.
The set of all errors on the lattice is called a chain, E. We use notation from algebraic topology to indicate the boundary of the chain of errors as dE (a good introduction to algebraic topology can be found in many textbooks, for example see ref. [32]). The errors commute with the stabilizers except at the boundary of the chain where the measured eigenvalues are non-
trivial. The full set of stabilizer eigenvalues is called the syndrome. Figure 3 shows a string of X errors and the two plaquette operators that anticommute with it.
Once the syndrome has been established we employ a classical algorithm called a decoder to decide which correction chain, E', to apply. The goal of the decoder is to pair the non-trivial syndromes such that the total operator C = E + E' has the highest probability of being a homologically trivial cycle and thus a member of the stabilizer group. Failure of the decoding algorithm corresponds to the creation of a homologically non-trivial cycle. The decoder used in this work, the minimum-weight perfect matching algorithm (MWPMA), is described in the next section.
2.2. Error correction
The optimal threshold for the independent noise model that we consider here has been calculated using numerical techniques to be pc = 0.1093 [33-36]. However, there are no known efficient decoding algorithms that can obtain this threshold for the independent noise model on the toric code.
Several classes of sub-optimal efficient decoding algorithm exist [19, 22, 25, 37]. The one used in this work is a version of Edmonds' MWPMA [38, 39]. This algorithm pairs the nontrivial syndromes via a correction chain that has the least weight possible while satisfying the condition that its boundary matches the error chain boundary, i.e. dE = dE'. This ensures that the total operator, C = E + E', is a cycle. We denote the threshold for the MWPMA by pc0. Numerical simulations suggest that pc0 = 0.1031 ± 0.0001 [30].
Although this algorithm gives a high threshold [30], we shall consider a heuristic modification described in detail by Stace and Barrett [40], that includes the effects of the degeneracy of E' and can give thresholds up to pc0 « 0.106. Degeneracy counts the number of possible paths that the chain can take, given that its boundary and weight are fixed. Matchings with higher degeneracy have a higher probability of arising so they may be a priori more likely than some matchings with a lower weight.
The degeneracy itself is simple to calculate for a given (minimum-weight) matching. For instance, for a path m between two non-trivial syndromes, a and b, the degeneracy of that path Dm is given by the number of different combinations of the links in the matching. The product of all individual Dm is the total degeneracy of the matching, DM.
To take degeneracy into account we compute the matching using the MWPMA, where the edge weights dab are modified by the effect of the degeneracy of that path. Then the weight passed to the algorithm becomes dab — t ln Dm. Here t is a weighting that we assign to the degeneracy term. The degeneracy is added in such a way due to entropic considerations, see ref. [40] for details. The decoding algorithm minimizes this quantity globally and this has been shown to lead to an improved threshold [40]. We refer to this enhanced version of the minimum-weight perfect matching simply as the PMA decoder.
2.3. Simulating noise and error correction
An important tool in this work is the numerical simulation of the detection and correction of errors on a toric code. Repeating random trials allows us to examine the failure probability of the code over a wide range of parameters. As stated earlier, we consider uncorrelated bit-flip and phase-flip errors arising at a rate p. It suffices to perform simulations for only one of these types of error since the results will be equivalent for the other.
The behaviour of the toric code is simulated by placing an error with probability p on each individual qubit of the toric code lattice of linear dimension L, giving rise to a (usually disjoint) error chain E. The syndromes are measured and the PMA decoder is used to determine the correction chain E'. These correction chains are added, modulo 2, to E and a parity check with each of the appropriate logical operators is used to determine the homology class of the total operator C. The result of this random sample indicates whether the error correction succeeds or fails.
The outcome of the Bernoulli trial (a single simulation of error correction) is assigned the value nf = 0 if C is in the trivial homology class and nf = 1 if it is in any of the non-trivial homology classes. To gather statistics we repeat this procedure N times for the same input parameters (L, p). Of these N trials, Nf = 2f= 1nf,i will have failed to perform error correction successfully. We therefore estimate the error correction failure probability as ff-^ = Nf/N and the variance of such a distribution is a2 = ffaii(l — Paii)/N. The resulting data ffaii(L, p) characterize the toric code performance.
3. The universal scaling hypothesis
In [30], Wang et al used ideas from the theory of critical phenomena in finite-sized systems to show that there is a critical point in the failure probability of the toric code. To do this, they used the two-dimensional RBIM which is a model of ferromagnetism in which antiferromagnetic couplings arise at random. The probability distribution of antiferromagnetic couplings in this model matches the probability distribution of errors in the toric code, hence a mapping between the two models can be constructed [16, 21, 30]. The RBIM has been extensively studied and it is known to undergo a phase transition from an ordered to a disordered phase as the concentration of antiferromagnetic bonds is increased. This implies a phase transition in the corresponding quantity of the toric code: its logical failure rate.
Wang et al demonstrated that for the regime where L » where £ = (p — pc0)~Vo is the RBIM correlation length, we expect scale-invariant behaviour. This argument leads to the conjecture that in this regime the failure probability of the toric code is a function only of L/£ [30].
Below the threshold the failure rate is expected to depend exponentially on the system size [12, 16], and also more generally in the fault-tolerant case [41]
in ffaii a — L. (3)
Numerical evidence for this will be provided later, in figure 4.
Together, the exponential dependence on L and the scaling hypothesis fix the functional form of ffaii
Pfaii = Ae—a (L/(4)
= Ae—a (p —pco))°L. (5)
In this expression A and a are constants that can be determined using numerical fitting techniques, see section 5.1 and appendix A.
In practice the toric code will be operating in the correctable (p < pc0) regime so we use the rescaled variable x = (L/£)VV0 (alternatively this may be written as x = (p — pc0)L1 Vo) and
Figure 4. Dependence of the logical failure rate Pail on the size of the lattice. Each data point represents N = 107 runs. The data are plotted on a logarithmic scale and linear fits to a selected set of the data between p = 3.5% and p = 8% are shown. The four data sets shown in in the lower part of the plot (dashed lines) are examples of data with p < 3.5% for which linear fits could not be identified. In the grey region the linear relationship is expected to break down according to our validity conjecture, see section 5.
We determine the values of A, pc0 and v0 from a fit to data close to the threshold. In the remainder of this section we give evidence that the numerical data meet the two conditions required for the universal scaling hypothesis, namely an exponential decay of the failure rate as L increases and scale invariance.
3.1. Evidence for the universal scaling hypothesis
To observe the dependence of Pai on L and p we have generated a set of Monte Carlo data for 0.01 ^ p ^ 0.08 and odd lattice sizes in the range 5 ^ L ^ 23. We use the simulation method outlined in section 2.3 with each simulation repeated N = 107 times using Kolmogorov's Blossom V MWPMA implementation [42]. We pass modified weights to the algorithm to account for degeneracy as described in section 2.2.
In figure 4 we plot the logical failure rate on a logarithmic scale, as a function of the lattice size. The shaded portion of the figure indicates the region where this exponential relationship is not expected to hold according to a conjecture that will be explained in section 5. Each set of data in figure 4 is fitted using a quadratic ansatz in L
For data in the range 0.035 ^ p ^ 0.08 and 5 ^ L ^ 23 the quadratic coefficient y is typically 2-3 orders of magnitude smaller than the linear coefficient This is strong evidence for a linear fit to the (logarithmic) data, suggesting a fit of the form p^ a e-L, matching equation (3). For data with values of p < 0.035 the quadratic coefficient was comparable in magnitude to the linear coefficient. A selection of this data is also shown in figure 4, demonstrating that the behaviour of the data for these values of physical error rate is ambiguous. Nevertheless, figure 4 establishes an exponential dependence of the logical failure probability on L for a wide range of the data.
ln Pail = a + pL + yL2.
Figure 5. Data obtained from numerical simulations of the toric code failure rate close to threshold, rescaled using x = (p — pc0)Ll/Vo. Each data point represents N = 106 runs. The finite-size correction is subtracted from pail. All of the data collapse to
a single curve and the threshold can be extracted as a fit parameter. Inset: the data prior to rescaling.
The universal scaling hypothesis in equation (4) also requires the system to be scale invariant which implies that the behaviour of Pai should depend only on the length scale L/£. This is demonstrated in figure 5 which shows the results of numerical simulations of the toric code failure rate close to threshold. The plot will be explained in detail in appendix A but now we simply note that rescaling the numerical data using the variable x = (LIleads to data collapse. This phenomenon describes the situation when data generated in different systems, in this case different lattice sizes, fall onto the same curve after an appropriate rescaling has been applied.
4. The low single qubit error rate regime
The universal scaling hypothesis is a good model for the logical failure rate when the lattice size is large and when there are sufficiently many errors. For a fixed lattice size, as p is reduced the universal scaling behaviour should not be expected to hold indefinitely. Indeed the numerical evidence suggests that when p becomes sufficiently small the scaling hypothesis fails. In the p ^ 0 limit the behaviour is given by the low p analytic approximation
Pail = -—-pL2]. (8)
This is justified by considering the uncorrectable error configurations in the p ^ 0 limit and calculating Pail directly. Restricting ourselves to low single qubit error rates we consider the minimum number of errors that can cause the error correction to fail, \L/2]. To cause the error correction to fail these errors must lie along a single minimum-weight homologically non-trivial cycle of the toric code. If they fall in this way the PMA will certainly apply the remaining LL/2J single qubit operators required to ensure C = E + E' is a logical operator. Figure 6 shows a sketch of how this happens.
Thus the expression in equation (8) for the failure rate is constructed via a counting argument. The first factor, 2L, is the number of minimum-weight homologically non-trivial
Figure 6. One way in which [L/2] errors lying along a minimum weight homologically non-trivial cycle will result in a logical error. The PMA decoder applies a correction chain that results in a non-trivial cycle, causing a logical failure. (a) The errors are distributed arbitrarily along one minimum-weight homologically non-trivial cycle of the lattice. (b) The syndromes that arise as a result of the error configuration are shown. (c) The minimum-weight perfect matching returns the correction chain E' with certainty. (d) The resultant cycle C = E + E' is homologically non-trivial, which means that the error correction has failed.
cycles of the code that exist. The second is the binomial coefficient which counts the possible combinations of \L/2] errors along a cycle of weight L. Finally we include a factor that accounts for the likelihood of exactly \L/2] errors occurring on a lattice constructed from 2L2 qubits, which is p[L/2] (1 — p)2L2-[L/2]. The single qubit error rate is small so we can neglect the final factor of (1
— p)2L2—[L/2| to obtain equation (8). In the low p limit the L dependence is Pal a e—[L/2] and we see that it is quantitatively different to the universal scaling regime, Pail a e—L.
5. The validity of the two regimes
The range of parameters we consider in our numerical simulations encompasses both the small p limit and the universal scaling limit. For small single qubit error rates the weight of the errors is typically much smaller than the code distance and the low p analytic expression is applicable. Conversely, for large L the number of errors can be much larger than the code distance and we expect a universal scaling hypothesis to apply. These regimes are distinct, as we see from their differing dependence on the code distance. Each of the two regimes will provide a good
approximation to the numerical data over some region of parameter space. We shall now make a heuristic argument to quantify those regions.
In order to make a conjecture about the validity of the regimes we consider the distribution of the number of errors that arise on a lattice of fixed size, at a known physical error rate. We will relate this distribution to \L/2], half the code distance. This number is significant to the PMA decoder because if the weight of the error chain, |E |, is less than this number then the error is certainly correctable. In the case when |E | ^ \L/2] a subset of the possible error configurations will lead to an incorrect pairing of syndromes, causing a logical failure. These are the spanning errors illustrated in figure 6.
The typical weight of errors on the lattice can be shown to be 2L2p. If 2L2p < \L/2] then the expected number of errors is less than half the code distance and logical errors are dominated by spanning chains, see figure 6. For a fixed p, as L increases this inequality is violated. When the number of errors is much greater than L but they are typically correctable, this is the universal scaling limit.
Requiring p » 1/L (up to a numerical factor) leads to a relationship between L and p that determines a minimum single qubit error rate for a given lattice size below which the universal scaling hypothesis breaks. We make the arbitrary but natural choice that the mean number of errors on the lattice must be two standard deviations above \L/2], leading to the expression
push «--. (9)
This expression, derived fully in appendix B, determines whether the behaviour can be considered to be within the universal scaling regime.
We can find an equivalent expression for p ^ 1/L, when the single qubit error rate above which the low p expression no longer provides a good approximation to the numerical data. This can be shown to be
Plp «-TT3-. (10)
When p ~ 1/L there is a 'crossover' region, in which the logical failure rate cannot be considered to be well approximated by either regime.
5.1. Testing the range of validity of the universal scaling hypothesis
Substituting pUSH given by equation (9) into the universal scaling hypothesis in equation (4) yields an expression for the minimum Pfail, for a fixed L, that belongs to the universal scaling regime. This expression is plotted as a grey line in figure 4 and hence the grey region indicates the region of parameter space where we do not expect the universal scaling hypothesis to hold. This supports the previous observation that most of the data we have obtained for p < 3.5% would lie outside the universal scaling region and therefore be poorly fit by equation (5).
We have fitted the universal scaling ansatz, equation (4) to the data that fall outside this grey region (the values of A, pc0 and v0 are all determined from the fit to the data around threshold). From the fit to the data in the universal scaling regime we find a = 32.31 ± 0.13. The data obeying the validity condition and the fit are shown in figure 7.
Let us now fix the code distance L and vary the single qubit error rate to see how the full set of data behaves in relation to the universal scaling limit. For each fixed L in figure 8,
Figure 7. Data satisfying the condition p > pUSH, plotted on a logarithmic scale and coloured according to lattice size. Each data point represents N = 107 runs. Also shown in black is the fit of the ansatz, equation (4) with all values taken from the threshold fit (see appendix A) except for a which was extracted using a fit to the data set shown.
Figure 8. Logarithmic plot of all numerical data following the rescaling transformation (L, p) ^ x = (p - pc0)l}!1"0. The universal scaling fit is also shown in black. The data are plotted on a logarithmic scale and coloured according to lattice size L. For fixed L, decreasing x corresponds to reducing p. As we do this the universal scaling hypothesis breaks at a point predicted by equation (9). This is indicated for a single lattice size (L = 11) as a vertical line.
reducing x corresponds to reducing p. When p becomes sufficiently small the scaling hypothesis fails and as expected the failure rate deviates below the universal scaling law.
5.2. Testing the range of validity of the low error rate regime
We have proposed that, in the low p limit, spanning errors of the type illustrated in figure 6 dominate when 2L2p < \L/2]. This is the validity condition we use for the low p regime, see equation (10).
We can rewrite equation (8) in terms of L and the rescaling variable, x. Figure 9 shows this analytic expression plotted for some small values of L along with the numerical data. As the probability of errors decreases on a fixed lattice the mean number of errors will approach \L/2]. As expected, the low p expression gives a good approximation for small lattice sizes and low
' Pf.ii 10"1
Figure 9. The full set of renormalized data, coloured by lattice size. The low p analytic expression, equation (8) is shown for some small lattice sizes. As x decreases the analytic expression tends towards the data. This numerical evidence suggests that the analytic expression is an underestimate of the failure rate for this range of parameters.
physical qubit error rates. The data and low p analytic expression converge as x decreases, so for fixed lattice size as the physical error rate decreases the approximation improves.
6. Comparison of the overhead in the two regimes
So far we have concentrated on determining the logical error rate as a function of the lattice size and single qubit error rate. Now we wish to demonstrate that it is possible to invert these relationships to find the overhead, Q. This will be a function of the experimentally determined single qubit error rate, p, and maximum tolerable logical failure rate /£¿1.
In this work we demonstrate the calculation for the toric code with perfect stabilizer measurements. However the same techniques shown here will also be applicable to more physically realistic settings, for example a planar code with noisy stabilizer measurements. Although the numerics will differ from those presented here, the methods used are expected to be directly analogous.
The first step in calculating the overhead is to determine which of the two regimes (universal scaling or low p) the code is operating within. To do this we use the expression for Push in equation (9), to find the minimum error rate for which the universal scaling hypothesis holds. Similarly we find pLp, the maximum error rate for which the low p expression holds, using equation (10). In figure 10 we plot these two bounds, and the regions of validity that they indicate. Figure 10 therefore shows the region of (/fai1, p) parameter space for which each of the regimes is expected to give a good approximation to the logical error rate. Once the correct regime has been identified, the overhead can be calculated.
In the universal scaling region the logical failure rate is Pfai1 = Ae-a(P-Pco)i,°L. By using this to find the lattice size L as a function of Pfai1 and p, and recalling that there are 2L2 physical qubits comprising the toric code, we find the overhead in the universal scaling regime is given by
Figure 10. The range of validity of each of the regimes is indicated as a function of the independent variables p and Pfa;i. The uncoloured part of the plot is the crossover region between the two regimes.
Figure 11. A 3D plot of the overhead, on a logarithmic scale, in each of the two regimes for 0 < p < 8% and 10-7 < Pa;i < 10-3. This plot reveals the gap between the two regimes over the whole region of parameter space considered. It also reveals drop in overhead as the single qubit error rate is reduced, which is particularly striking for the low p regime.
where the constant a has been determined from fits to the data in this work, see section 5.1. The remaining parameters, A, pc0 and v0, can be determined from a fit to data generated close to threshold, see appendix A for this calculation and for their numerical values.
The analytic expression for the low p regime, equation (8), can be simplified by assuming that \L/2] = \L/2J = LI2 and using Stirling's approximation n ! = (n/e)nV2nn. Inverting this simplified expression we obtain a solution for L that uses the Lambert W function [43]. We can simplify this using the approximate form for the lower branch of the function [44]. It follows that an approximate expression for the overhead in this regime is given by
&Lp((a;l, p) = 2
in fil - ln( -in fi)
Figure 11 shows a 3D plot of the overhead as a function of Pfa;i and p. There is a significant gap between the two plots for most of parameter space (see figure 12) and an increase in overhead is seen as both p and Pfaii are increased. Allowing a higher logical failure rate will naturally reduce the overhead required, as will reducing the single qubit error probability.
Figure 12. (a) The overhead for the toric code calculated for a physical error rate p = 5% for desired fidelities 10-7 ^ Pfail ^ 10-2. (b) The overhead for logical failure rate Pfa;l = 10-7 and 0 < p < 8%. The plots can be considered to be practical bounds on the overhead for the parameters considered.
Figure 12 shows the difference between the required overhead in the two different regimes. For the range of parameters considered the low p expression always gives an estimate of the overhead that lies below the value given by the universal scaling hypothesis.
The low p expression tends to underestimate the logical failure rate for the range of numerical data simulated. Hence this may be considered to be a practical lower bound on the overhead required for those parameters. Conversely, the universal scaling hypothesis is an overestimate of the logical failure rate for most of the numerical data, and hence can be considered to be a practical upper bound to the resources required.
7. Conclusions
We have found two distinct operating regimes of the toric code. In one, the data can be rescaled and an ansatz based on this scaling and the exponential dependence of the failure rate on L can be used to find an empirical expression for p^. In the other, a counting argument gives rise to an analytic expression for the failure rate in the p ^ 0 limit. We propose, using the probability distribution of the error weight for fixed (L, p), heuristic conditions for the range of validity of each expression.
The expressions describing the two regimes have been inverted to calculate the system size required to achieve a desired logical success rate for a given single qubit error rate. We have used the expressions for the logical failure rate to demonstrate techniques to calculate the overhead, Q (Pfai1, p).
We expect that the techniques we have demonstrated in this work will be applicable in a wide range of settings. In particular, more physically realistic geometries such as the planar code, whose logical failure rate is expected to be higher than that of the toric code [29]. Furthermore, we expect that the methods we have demonstrated can be used to calculate the overhead of a fault-tolerant quantum memory, in which the stabilizer measurements are imperfect. Since all topological codes are based on similar principles the techniques outlined in this work can be expected to be directly applicable despite the fact that the numerics in these cases will differ from those presented here.
Based on the numerical evidence, we claim that for most practical purposes the two regimes bound the required overhead. The numerical results presented in this work are dependent on the choice of the decoder. Similar scaling relationships would be expected for other decoding algorithms, particularly renormalization group-based decoders such as [22, 37].
This work raises several open questions. It has been shown that the MWPMA decoder has a quadratically lower logical failure rate than the renormalization group algorithm [45]. However, we still believe that a comprehensive comparison of all existing decoders over the whole region of (relevant) parameter space would be interesting and worthwhile. A possible scenario is that the size of the topological code that can be realized will be fixed by technological limitations. In that case, a comparison of the analysis presented in this work for all known decoders below threshold would reveal which should be implemented to minimize the logical failure rate.
Decoders with high thresholds usually require a longer running time than those with more modest thresholds. We expect a tradeoff between time and space resources, suggesting that those decoders with longer running times may have smaller physical qubit overheads. This is interesting, because although a high threshold is desirable, for practical implementations the running time and physical overhead are also important constraints. Therefore it seems that a balance between these three figures of merit may be of interest for practical quantum computation.
Several of the limitations we faced have been addressed by Bravyi and Vargo in [26] during the preparation of this manuscript. The first of these addresses the crossover region between the two regimes we have identified. Bravyi and Vargo have constructed a heuristic ansatz that interpolates between the dependence on L of the low p regime, /ai1 a e-L2\ and the dependence expected for larger physical error rates, Pfai1 a e-L. These functional forms match the two regimes we have identified so the ansatz by Bravyi and Vargo could lead to a method for interpolating between them.
Another benefit of the technique by Bravyi and Vargo is that it provides a fit to the numerical data in the small and moderate p regimes. A significant limitation we faced was the availability of resources to run the Monte Carlo simulations of the error correction procedure. For example, it was impossible to obtain data for Pfai1 < 10-7 due to the running time of the decoder. Bravyi and Vargo have discovered a new technique for probing very low error rates on surface codes [26]. Obtaining data for very low logical error rates using this algorithm would help us to verify the conjecture of the range of validity of the low p expression, particularly for larger lattice sizes than we were able to test.
While heuristic approaches are very flexible, our universal scaling hypothesis has the following advantages. It addresses the large L limit and gives particularly good approximations to the numerical data for moderately large single qubit error rates. The functional form for the universal scaling hypothesis, given in equation (4) is derived from the phase transition of the RBIM, which is a model of statistical physics that the toric code error correction can be mapped to, meaning that it is not a heuristic expression. It is also easily invertible and its pre-factor, A, does not depend on the code distance.
Ultimately the implementation of universal quantum computing that is found will set the input parameters that determine which of the regimes it operates within.
I would like to thank T Stace for many valuable discussions in the early stages of this work and his idea of considering universal scaling in such an analysis, as well as for his careful reading of and comments on this manuscript. I would like to thank D Browne for his help in preparing this paper, and thank D Jennings and H Anwar for useful discussions and helpful comments on this manuscript. We acknowledge the Imperial College High Performance Computing Service for computational resources. FHEW was supported by EPSRC (Grant number: EP/G037043/1).
Appendix A. Determining the threshold
In section 3 we rescaled the numerical data using the variable x = (p — pc0) L1v0. In order to do this, we must first establish the values of the threshold, pc0, and critical exponent, v0. The universal scaling hypothesis, equation (4), also relies on knowing the failure rate at threshold in the large L limit. In this appendix we show how these quantities are obtained from a fit to data close to the threshold.
The threshold for the stand-alone MWPMA decoding has been calculated previously as 10.306 ± 0.008% [30]. Since we allow the degeneracy of the matching to affect the choice of correction chain, we repeat the calculation in this work to obtain the threshold for our enhanced PMA decoder.
To find the logical failure rate Pfail we numerically simulate the error correction protocol, enhanced minimum-weight perfect matching (PMA), using the same method described in section 3.1. We performed N = 106 simulations of the error correction procedure for p close to 10.3% and for odd lattice sizes in the range 5 ^ L ^ 25. This set of data was only used for the purpose of finding the threshold and critical exponent, and is not the main data set used in this work.
The lattice sizes we use are far from the large L limit, so following the method from Wang et al the fitting ansatz was constructed by taking a quadratic expansion in x around the threshold x = 0 and accounting for finite-size effects by adding a single non-universal term that is dependent on the lattice size [30]. The ansatz is
Pfail = A + Bx + Cx2 + DL~l/(A.1)
where A, B and C are expansion coefficients, D is the coefficient of the non-universal term, and
x = (p - pC0)L1V0. (A.2)
Here v0 is the critical exponent and pc0 is the threshold error rate for our PMA decoder.
Figure 5 shows the rescaled data with finite-size effects subtracted, and the fit to the data. The relevant parameters were found to be
pc0 = 0.1028 ± 0.0002,
The threshold for our modified decoding algorithms was found to be in agreement with the value found by Wang et al for the unmodified MWPMA [30]. This does not achieve the maximum threshold of pc0 ~ 10.6% that is possible when the degeneracy of the matching is included [40]. This is because in the simulations performed for this paper we allow only a weak dependence of the choice of matching on the degeneracy in our modified PMA decoder. This means that the choice of matching is only weakly dependent on the degeneracy of the matching and the effect on the threshold is small. The value of the critical exponent v0 found here is in agreement with the value found by Merz and Chalker when calculating the optimal threshold value [34], although it does not agree with value found by Wang et al for the MWPMA decoder.
The analysis presented in this appendix establishes the validity of the rescaling approach to the analysis for this choice of decoder by demonstrating that the scaling asatz, equation (A.1) provides a good fit to the collapsed data close to the threshold.
Appendix B. Deriving the validity conditions
In this appendix we outline the derivation of the validity condition for the universal scaling hypothesis, pUSH given in equation (9). The validity condition for the low p expression, pLp given in (10) is not explicitly shown, but can be reproduced using a similar argument.
The single qubit errors occur independently and at a rate p. The weight of the error that arises, ^ |, obeys a binomial distribution with a mean that coincides with the typical error weight
V = 2L 2p, (B.1)
and a variance of
^2 = 2L2p (1 - p). (B.2)
According to the central limit theorem the binomial distribution can be approximated by a normal distribution for large enough lattice size.
For the universal scaling hypothesis, the condition we have proposed is that ^, the mean of the probability distribution, is large with respect to [L/2]. This implies that the weight of the
error chain that results is larger than \L/2\ with high probability. We can write this as p » \L/2\, or
n— > 2
where n is the number of standard deviations above \L/2] we require the mean to lie. We have chosen n = 2 for both the universal scaling hypothesis and corresponding condition for the low p expression.
Substituting equations B.1 and B.2 into equation B.3 we obtain
2L 2p - s] 2L 2p (1 - p) >
Solving for p and taking only the highest order terms, we arrive at the expression for pUSH in equation (9).
The expression for pLp in equation (10) is obtained similarly, by requiring
p + a < L . (B.5)
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