Merken
On the energy landscape of 3D spin Hamiltonians with topological order
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Erkannte Entitäten
Sprachtranskript
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about dysthymia but so of putting much for it right and in this 1 as an the that is all I want to present some results on threedimensional our components Vista potable point order and this is a trend book this garden when Khot was here so are in so
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our final will here is stored on the mute the quantum so they would like to store that's the Deligne would like to realtime device but then I start quantum states in the presence of similarly center are small imperfections like hardware imperfections and as we all know that a quantum state could put it in another city device Stukeley clear hot debate on this year on some form of it and so I would like to focus on I still couldn't cameras and this is a system and which are better potential use imitated by natural lowtemperature than the use of such memory at our would meet on the call active and put it on there and once the final don't step but as the storage art itself does not require any actions from the user and I know I missed it
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myself to a about what the memories with some of the political stability ports this a famous example is the twodimensional when a court by the end of the political member is part of our polite always provide us some protection ideas against local imperfections such as small all the relations also on it seems that you need to worry about is our protection against the thermal noise which is would but unfortunately at this hour villages strongly believe led on a twodimensional the political numbers are not over protection local optimal studied it the and has become something of a book exhibitions of twodimensional art that knowledgeable models are always stay neurons and as a program is wet wet optimal corporations can easily reach their stopping us from the ground state and even worse suppurations can they're not move any arms and this thing on most it leaves the details that will get us the and after some constant time there are no accumulated at you exceeds their official and I like or the the tools our input it state and for example this intuition has been rigorously from them by and is defined as the current is given and showed that memory time all the twodimensional liquid is a constant independent of the lattice sites and so a conclusion but we can draw from this our results it is but we need some a mechanism to in this these are my beauty of anions another focus on quantum so effective and far so I will talk about our I said on the threedimensional I know that threedimensional potable and in in particular I will be interested in that Our threedimensional quote which urgent work I showed us in the morning session and this book has a nice property that are and we might be killed in your answers Institute animated dramatically so when local area stand on a more pronounced sample from distance only and so the question that you would need for I would like to address is but this this temple models curves are something properties and and if so how long can be stored on quantum information infection the they can
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go the so the user of anonymity must be
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able to write store and beat information and I use need to spend some time explaining your how exactly be a model on the steps and and sort of start from input in and I I don't see that as system defined from some ddimensional Our latest of units I Eitel the most interested in the policy and so given a mystical I support but we can define our American eldonian stage and it is essentially just minus some of us that either generators for the transport so the ground state subspace fault inimical bone coincide to this was wanted a subspace of our port and on I had always assumed when the ground state energy is is your and everyone we it's the stabilize the costs of what Leonard on but they go and so did not so but it's not anthropological on this slide but no I have to explain what I mean by political courts and I knew of both units 1st of all this this stabilize incinerators must bear a geometrical so each generator that's if you do this located mostly other and the 2nd there I mean that the distance of this court must I grow as our sample to give our power of lake effects and so of most example within all of these things actually most of the matter is sort of sites the and as example until their most
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important for me is if it there at you the court is modeled by and 1 and just safely from 1 session that the forward again so that that's our topic you that's all I can recite all the letters on stabilizer generators the form of our Union cubes on the generate the site is and generate subtype of that's on and on this is 1 book at some are logical to this selected must have building Monday conditions on x y and z axes and a nice property of this report is wet I will 1st of all it is the political parties in Eastern's grows or at least the unit is a work safe but I remarkable property is that underscore there's no logical stink like operators but to live important for for Michael the know now with that described are
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called the model is the storage of information and for that we need to set up some very mixed let our diets assistant what's the look at a molecule on state and to make our life easier we'll assume but this dynamics is call them it is well known but the most janoma program on Linux that Assistant this quantum mechanics is this at the but equation because you knew that when some of the seminal known on and we have some are you look generator when describes participation dissipation of energy and this this look to narrator involves a bunch of quantum jump operators at all this month and start of fan people your the sum of the ground state are holds an American plan and his ground state is important statement that 1 to to and I will impose some major look radiant has constraints so all all quantum jump operators are much below 0 and we must monitor and or suppose no way our memory system internet this there are some general bot we just get that some constant temperature good FIL inching considered used an dynamics of the memory system and I will not be my for them is but all know it can so that we can they'll approximate this minimax by michael than 1 used so there is that make up thing it's too hard and so this stuff from our but choose a bunch of local operators in the face of the from each of our memories system can couple articles was about for example this could this would be a the set of all singleton which about operates in are and then we can use this problem cooperatives' before I was there not dissipated part of the name but equation using this rule so occupancy rate of quantum jump operators there are to Unisys at the end on and in this case tells us are basically beach up and cooperators mesial each interaction may price this quantum jump operator and in this on a graph of DOS us how much more energy here than inference built from the system with about and by the use of wikis and the article made beaches family circuit is there a spectral spectral density of the most I know that but that but of OK and on the about this spectral density at must the detailed last condition and I think it's all here so basically it says that are populations increase in our time populations in peace and energy of the system are expansionist optical imaging this didn't bother conditions this is on there I was on their place in our model that depends on on the bottom which the you I suppose we have of all of our system and according to this thing with equation for some time and now we want to achieve important information and releases of the decoded step and only since some of the of the single measurement and then the you edit and and a antigens just some algorithm but faces in producing on health of speech but in our operator the need of on this of OK so and it was not all DOS at all altitudes this action of the for the the core a the down delta spent several months trying different heuristics and finally on something that what's reasonably so the title explain you call our of most the so they call it
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nations decoder of course I got this or that it is this idea of waste our determination efforts to test is the simple the so conceptually it's a similar to the production of the by the end of this year is although technically it's our time in different so here we are the next wave of more FIL so this start from our method seemed and I know it didn't seem to be just that like that so this makes it can go and our of which of these again are useful features of water dimensional germs and this a less indicate publications and reach artist analyzes have been sweet as a result of the police obligations that you release on your of so on OK so so the 1st step of the algorithm is field I find out connected to other classes of this the text the and so this the initial and give it is not so important at this 1 but we it at the same cluster will be text is not connected the kind of all across clock the result and something 1 unit of them at the time for example here the best and quite often in the last at this is a simple question these are all the questions then I of for each of these questions we must use of a subroutine 1st are fined as a minimum a candle on books that imposes this that holders of most of the European and national and by this this and then the police last in the title of an EU latent by applying some of the data somewhat inside of this article symbols but if you succeed you what is the relation of the of so we apply this procedure to every request and part of this assumed just for the sake of argument that be successfully mediated the but they cannot immediately that means what look at at this in this tower unit plants but I think you're will and I am going to this so now our initial syndrome like this talked our universe plants are now to so so it means that these new gets would qualify as a connected plot and the famous for this so the company to benefit questions in finally of motives and for the in fact will only last but it can by inside articles and most and how are you supposed the successful any later also different the and so since they are tiny
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later also because it means the accumulated undulation I will they have I use of consistent and here so how do we that those issues we speculated that emulation operator as when it can operate at like the light now there are 2 ways in which this decoder for playing 1st of all people that that in our our union plans become unbearable is let we still have some effects there so the canonical in this case you just because of that and and other people for the thing is that not we have successfully EU lakes of weakness in Mexico as that when it input it on that's not on the system to their regional state so in this case the had histological status all them and I also think it's also important that all steps can be implemented on the fusion
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the so there so our mind will use to to analyze this 1 digit I will start so we start from an arbitrary initial ground state of an amendment the bullet for I mean the they of in this example the form the I would and then there of Comp where you at the initial state that our final so you want state so if the memory what storage there must be a little and so it is not only the so we use an upperbound on this storage here for there are the dimension of the core of what it is and you can see what are the most lenient this is not a by but what is important is that I think the story chair in the case of phenomena of the selected sites this it can be used for sufficiently small temperature people and that it is in the temperature of about so I know it shows that this book and you have some something and what it is it's being the solid the size of the the is it becomes more and more reliable but potent needed in the latest spot size that too much so these are some of the maybe some people in the size of we just the financial and then the temperature and this month wise for me the I people sites all because you have the latest to start shareable bond of the memory of say set some but there also some small they're excellent but the a and there as a memory and the most popular man in his lattice sites the degree of 1 of no you know if you keep some some temperature you can ask what is the maximum memory and can obtain by the 2 nodes of optimal later said it's they almost we should try to choose 1 of size of of like this and then there are the maximum memory time will be it financial in but mostly so it's of infinite and 1 thing but it's it grows in the past the original that I know this
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so this theorem on the test a barred on the memory and the initial question is this belong to start and it's something like for exactly 4 pieces of quotations in order the and in that the performance by numerical simulations so I mean I I posted you cannot compute this astonishing chair made it but what we can do is to use a MonteCarlo method is to make these quantities view of the this is a is a probability all successful you wouldn't be here the system of key the that is so would get something in the next lecture in this 1 week case of that financial this and there is something we did not understand foul each time the have this our numerical estimates that the you and so these are our results so on the sample that is is mean a size of the lattice and people it is is many time in a lot of space and so here I was created for our part since you can and I think they just so that's it so this is where our largest intelligence the smallest that this is the smallest of images largest that and occupancy when you hear about its temperature started the in size the number of times grows up because we have built but in that situation it's the sample remotely size such that you can go home as a memory can actually becomes loss so it's known as consistent with what our all of you it and so on this the ISO file is there a median of this funnel no depends on this you can see that it's not clear idea of what it is also consistent this all of the this on the science and art are within the power of the optimal religious sites are depends on that the modified because of any of the optimal weight size so the the consistent of use of conditions of started when so in the the we can also test we can use this data to test our 2nd each that the maximum maximum limit climatic expenditure rose financial this this that less I wasn't can only explored the idea with a small range of but are associated what an inverse temperature meters a lot of that's maximum limited time and now the policy units but it in this field must be in a way that yeah it's that say from this the look as it is in connected by so if you use so useful but the but they get some of the heat and you know it and you can see that the use of the negative the to systematic there phonetic people's of my that so it's also up there are consistent with all of you and I
26:11
know I would like to shall there 2 of the bonds but I I don't just make that if some in origin and so on so basically because we can think about this document as and the standard error model and we would like to characterize our you meant to understand speech errors are likely to be generated in this animal and you can talk about I want you to think about it is that energy their own so it depends on how well they being we consider are all possible all all possible error costs that can what did you ladies operator and then proper that the 2nd right single cubic errors the show if you apply this was initial nonstate have been a set of intermediate signals and Beach said the following it because of that in Table of the cats just of island in light in and out of the image of a policy maker this the the we within each dual implemented and integrated by themselves similar to the parents the time so this file locally and then into the area most of because that's 1 of the parts but the question that will be and in this means things because this operator itself states that the new text from the the to you know basic information is left a
28:16
similar is all of them on degenerate there's this are sufficiently small and and so so these are the areas that are already wouldn't be illegal where and our interest is an engineer only consider confuse legal there might be expected such errors are very unlikely that the in the group this month so we call it the to solve a lot care we assume but our decoded and all areas are this image of data smaller than the in Japan so them and that Apple wanted to use on the various because this of most honored and so the and the I know it's find and the small in and a couple it's that supports our initial intuition but are beautiful of this and that that I think that does not depend on the energetical and it grows show that the this so that would be in this and the the example of the number of possible error crops that and even and was in the so in order to get the book this
29:53
and have the effect of the they're they're forced to acquire the on FidoNet multiscale for example the are and so this land this 1 now this and because there is just a couple bounded by a constant the that this so 1 of the most of the because of this at the proportional to the system size and this month is only useful if our energy that this on the risk of a trend in the system that so what we need is to let our decoder of terrorists whole there's the and into their smaller than some was sometimes will the and in this will have to use this
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will be given but that the court does not have been quite a lot of thought so that would be a monkey of the emission of these operators the so so a logical asking segmented just about operator such that but please to are built separated after so you get from the null space the and the regions suppose this I told impacts the the I know you will say
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that a logical thing segment steel so inspection on the ground state and there are different views by a local operators which are localized on in some 1000 people put and regions you know think that statements and often the constructionist in the middle and so dealing with
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lost cancel it says that Our whole distinct segments which are sufficiently stage I must be in the field the time later on quantify how speech has been sent and this after careful it's the distance between an original divided by by the size of the region we demand that the sum of all the such that analytical Scheme segment is artificial because of what be now as the focus this morning there and what are the issues are most useful the and of the decimal but most conventional stabilizer port affected by this property so for the 2nd
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result so it is to consider what they need the political unless a court on some new dimensional like this if you caught obviously spirals so then is there are will connect in service engineer at most a few cans of lot and the responsibilities of the depends only on on the social dimension and the constraint of the most people and yeah we can combine these are our results together because of from the storage area but in this the InfoWorld and we can't use cell and you can feel it you can look at so we get this this often on the case I notice that in fact means located each the group of these here the selection of the most the most difficult result the this now it was in the city I
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will consider I did the right lobe of the antigen video interfaces I was a logical operators have small engine barrier that would be a there but it was not recorded in the the and note part that group attention it involves 2 ideas these are the questions that I know will imply optimization of so suppose that that our honesty will to move to the center of the implements on some logical operators as opposed to some these this theme of their some to some of them even though so because some initial thing on this are not human right and because some long chain of single bit errors that goes from and each of these Tuesday along now on the accumulated there along this path how could they and then nonlocal suppose color that whole intermediates inference in this 2nd hour sparks and I was not something that the distance between any where all the sets if you want to the constant powerful and the most useful then no 1 would would increase the than asking so it's the same our local areas cannot move from individual defects they distance that more than all so it means that if you look at the center of the scene around on the spot obviously that stand up and when there are a little bit away from the initial location that we can use this optimization I began localize this accumulated error by the thing that you can multiplied by several at work modified there is supported on the some small neighborhood is the initial and the final signal and that the size disabled insulated of so if the Intermediate
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kingdoms as far as it means that accumulated a respect for the that's updated small the maybe at some of by but we know that our entire past implements some logical operators which this I mean actually because this is a little more so it means that and the something to be ashamed of which are not sparse and article that single on them I think that
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know sponsors the 2nd idea the Commission is that this no sting so is excellent Meyer so you he tells us what you can see that in plaster all defense of size such fostered cannot move and on the moon by local areas moment Eastern's out as part of the this so you know the idea is that this is the virus and the fine arts classes and of syndrome rumsoaked different optical phase and then we show that in order for the next accumulated there are the order on the inside the market is 1 that we did things wrong because there's Our almost every intermediates that the and finally the initial are you because a single images on them so that they're there I think that did I think that's the answer that is that all the elements that the not and things although FIL I don't have time
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to get from 1 so far that it just
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happened by going through this and all the fact FIL questions and then you'll decoding algorithm you say that you you draw a box around kind of clusters of syndromes and then you try and figure out what I will probably errors and a annihilates those of those syndromes do do that just by brute force within the box you just try all possible and how the operators do not books or the mention much but will of the because political scientist and a lot of what can use
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Fehlermeldung
Punkt
Hardware
Mengentheoretische Topologie
Gruppenoperation
Temperaturstrahlung
Physikalisches System
Störungstheorie
Quantisierung <Physik>
Bildschirmmaske
Twitter <Softwareplattform>
Festspeicher
Dezimalsystem
Energiedichte
Zustand
Quantisierung <Physik>
Zusammenhängender Graph
Ordnung <Mathematik>
Speicher <Informatik>
Hardware
01:29
Resultante
Mengentheoretische Topologie
Adressraum
Extrempunkt
Font
MIDI <Musikelektronik>
Kurvenanpassung
Dimension 2
Kraftfahrzeugmechatroniker
Kategorie <Mathematik>
Kardinalzahl
Stellenring
Speicher <Informatik>
Dynamisches RAM
Temperaturstrahlung
Flüssiger Zustand
EinAusgabe
Knotenmenge
Quantisierung <Physik>
Feuchteleitung
Verbandstheorie
Einheit <Mathematik>
Information
Simulation
Aggregatzustand
Kubischer Graph
Web Site
Stabilitätstheorie <Logik>
Wissensrepräsentation
Decodierung
Mathematische Logik
Zeitdilatation
Zahlenbereich
Geräusch
EMail
Nichtlinearer Operator
ROM <Informatik>
Informationsmodellierung
Stichprobenumfang
Quantisierung <Physik>
Abstand
Optimierung
Fehlermeldung
Relativitätstheorie
Fokalpunkt
Zeichenkette
Flächeninhalt
Mereologie
ROM <Informatik>
Energiedichte
Ruhmasse
04:47
Web Site
Stabilitätstheorie <Logik>
Decodierung
Schreiben <Datenverarbeitung>
ROM <Informatik>
Schwach besetzte Matrix
Gittermodell
Informationsmodellierung
Bildschirmmaske
Einheit <Mathematik>
Schwebung
Code
Stichprobenumfang
Abstand
Speicher <Informatik>
Leistung <Physik>
Soundverarbeitung
Nichtlinearer Operator
Fehlermeldung
Kategorie <Mathematik>
Gebäude <Mathematik>
Physikalisches System
EinAusgabe
QuickSort
Unterraum
Rechenschieber
Abstand
Energiedichte
Generator <Informatik>
Würfel
Konditionszahl
Information
Verkehrsinformation
Kubischer Graph
08:01
Dissipation
Gewichtete Summe
Inferenz <Künstliche Intelligenz>
Dichte <Physik>
Familie <Mathematik>
Gleichungssystem
Internetworking
Chatbot
Unterraum
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Algorithmus
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Minimum
NotepadComputer
Einflussgröße
Nichtlinearer Operator
Befehl <Informatik>
Güte der Anpassung
Stellenring
Heuristik
Speicher <Informatik>
Bitrate
Wiki
Gruppenoperation
Quantisierung <Physik>
Dichte <Physik>
Konstante
Eigenwert
Menge
Festspeicher
Konditionszahl
Punktspektrum
Messprozess
Information
Aggregatzustand
Nebenbedingung
Gruppenoperation
Automatische Handlungsplanung
Interaktives Fernsehen
Sprachsynthese
Nichtlinearer Operator
Quantenmechanik
Zustandsdichte
Physikalisches System
Informationsmodellierung
Fächer <Mathematik>
Quantisierung <Physik>
Speicher <Informatik>
Optimierung
Gleichungssystem
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Algorithmus
Videospiel
Fehlermeldung
Graph
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Schlussregel
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Beamer
Mereologie
ROM <Informatik>
Digitaltechnik
Energiedichte
Speicherabzug
Surjektivität
13:19
Resultante
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Extrempunkt
Wasserdampftafel
Wellenlehre
Klasse <Mathematik>
Statistische Hypothese
Extrempunkt
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Physikalisches System
Gittermodell
Algorithmus
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Determiniertheit <Informatik>
Speicherabzug
Turm <Mathematik>
Biprodukt
Grundraum
Softwaretest
Soundverarbeitung
Parametersystem
Nichtlinearer Operator
Teilbarkeit
Relativitätstheorie
Wendepunkt
GibbsVerteilung
Plot <Graphische Darstellung>
Symboltabelle
Physikalisches System
EinAusgabe
Biprodukt
Algorithmische Programmiersprache
Datenfeld
Einheit <Mathematik>
Softwareschwachstelle
Mereologie
Emulator
Aggregatzustand
18:30
Resultante
Einfügungsdämpfung
Polynom
Extrempunkt
Minimierung
RaumZeit
Einheit <Mathematik>
Theorem
Meter
Metropolitan area network
Softwaretest
Sichtenkonzept
Inverse
Stichprobe
Speicher <Informatik>
Globale Optimierung
Sampler <Musikinstrument>
Dynamisches RAM
EFunktion
Datenfeld
Näherungsverfahren
Verbandstheorie
Festspeicher
Digitalisierer
Konditionszahl
Ordnung <Mathematik>
Simulation
Zentraleinheit
Aggregatzustand
Web Site
Theorem
Gewicht <Mathematik>
HausdorffDimension
Zahlenbereich
ROM <Informatik>
Bildschirmmaske
Diskrete Simulation
Stichprobenumfang
Schätzung
Inverser Limes
Speicher <Informatik>
Bildgebendes Verfahren
Leistung <Physik>
Schätzwert
Elektronische Publikation
Medianwert
Unendlichkeit
Minimalgrad
Mereologie
Zentrische Streckung
Speicherabzug
26:09
Gruppenkeim
Zahlenbereich
Sprachsynthese
Nichtlinearer Operator
Extrempunkt
Kubischer Graph
Stichprobenfehler
Informationsmodellierung
Ganze Zahl
Vererbungshierarchie
BoltzmannGleichung
Bildgebendes Verfahren
Folge <Mathematik>
Nichtlinearer Operator
Fehlermeldung
Teilbarkeit
Speicher <Informatik>
Einfache Genauigkeit
Elektronische Publikation
Gruppenoperation
Energiedichte
Feuchteleitung
Lemma <Logik>
Flächeninhalt
Menge
Einheit <Mathematik>
Rechter Winkel
Mereologie
Hochvakuum
Energiedichte
Information
Computerunterstützte Übersetzung
Ordnung <Mathematik>
Tabelle <Informatik>
Fehlermeldung
29:49
Soundverarbeitung
Nichtlinearer Operator
Fehlermeldung
Teilbarkeit
HausdorffRaum
Physikalisches System
Nichtlinearer Operator
Dialekt
RaumZeit
Gruppenoperation
Zeichenkette
Energiedichte
Feuchteleitung
Twitter <Softwareplattform>
Hochvakuum
Energiedichte
Klumpenstichprobe
31:24
Nichtlinearer Operator
Befehl <Informatik>
Stabilitätstheorie <Logik>
Sichtenkonzept
Gewichtete Summe
Kategorie <Mathematik>
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Sprachsynthese
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Fokalpunkt
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Code
Hochvakuum
Abstand
33:04
Resultante
Nachbarschaft <Mathematik>
Nebenbedingung
Stellenring
Theorem
Bit
Mathematische Logik
Bitfehlerhäufigkeit
Mengentheoretische Topologie
Inferenz <Künstliche Intelligenz>
HausdorffDimension
Minimierung
Gruppenkeim
Zellularer Automat
Videokonferenz
Nachbarschaft <Mathematik>
Spirale
Trennschärfe <Statistik>
Endogene Variable
HamiltonOperator
Abstand
Speicher <Informatik>
Internationalisierung <Programmierung>
Feuchteleitung
Nichtlinearer Operator
Fehlermeldung
Teilbarkeit
Cookie <Internet>
Einfache Genauigkeit
Zeichenkette
Feuchteleitung
Dienst <Informatik>
Flächeninhalt
Menge
Mereologie
Energiedichte
URL
Kantenfärbung
Beweistheorie
Kubischer Graph
Fehlermeldung
37:02
Nichtlinearer Operator
Stellenring
Fehlermeldung
Subtraktion
Computervirus
Momentenproblem
Dichte <Physik>
Klasse <Mathematik>
Stellenring
Einfache Genauigkeit
Element <Mathematik>
Technische Optik
Invariante
Nachbarschaft <Mathematik>
Zeichenkette
Schwach besetzte Matrix
Maßstab
Gewicht <Mathematik>
Flächeninhalt
Mereologie
Ordnung <Mathematik>
Streaming <Kommunikationstechnik>
Bildgebendes Verfahren
Phasenumwandlung
Internationalisierung <Programmierung>
38:46
Nichtlinearer Operator
Fehlermeldung
Polynom
Quader
Orthogonale Funktionen
Dichte <Physik>
Annulator
Globale Optimierung
Übergang
Nichtlinearer Operator
ROM <Informatik>
Quantisierung <Physik>
Datensatz
Gittermodell
Algorithmus
Lemma <Logik>
Gruppenkeim
Forcing
Code
HamiltonOperator
Qubit
Cluster <Rechnernetz>
Kubischer Graph
Demo <Programm>
Fehlermeldung
Metadaten
Formale Metadaten
Titel  On the energy landscape of 3D spin Hamiltonians with topological order 
Serientitel  Second International Conference on Quantum Error Correction (QEC11) 
Autor 
Bravyi, Sergey

Lizenz 
CCNamensnennung  keine kommerzielle Nutzung  keine Bearbeitung 3.0 Deutschland: Sie dürfen das Werk bzw. den Inhalt in unveränderter Form zu jedem legalen und nichtkommerziellen Zweck nutzen, vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. 
DOI  10.5446/35312 
Herausgeber  University of Southern California (USC) 
Erscheinungsjahr  2011 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Informatik, Mathematik, Physik 
Abstract  We explore feasibility of a quantum selfcorrecting memory based on 3D spin Hamiltonians with topological quantum order in which thermal diffusion of topological defects is suppressed by macroscopic energy barriers. To this end we characterize the energy landscape of stabilizer code Hamiltonians with local boundedstrength interactions which have a topologically ordered ground state but do not have stringlike logical operators. We prove that any sequence of local errors mapping a ground state of such Hamiltonian to an orthogonal ground state must cross an energy barrier growing at least as a logarithm of the lattice size. Our bound on the energy barrier is shown to be tight up to a constant factor for one particular 3D spin Hamiltonian. 