Phases of Quantum Matter from a Computer-Science Perspective | Victor V. Albert

foreign hi everyone thanks for joining the kiss kit seminar series we're on the 127th episode today of the series I am Maria villaris I'm your host for today's talk by Victor Albert I work part-time with the kiss kit Community team alongside my PhD which I do at the University of Oxford now while we wait a few minutes for everyone to arrive I'd love to know where you are all joining us from today so please use the live YouTube chat to let us know where you're joining from the live chat is open for the whole of the seminar so feel free to drop in questions and comments there for me or for the speaker Victor at any time and yeah I'd love to know where you're from normally I'm hosting and living in Oxford where um it would be late afternoon now but this time I'm uh here in New York visiting the um the IBM Quantum officers that's where where I'm joining from where I'm I've been filming some new videos and just uh yeah we have people joining us from Portland Oregon and from Delhi from Israel from Mexico City from I won't be able to pronounce that correctly prior Garage in India ecuadim cool loads of places Chile Southern California great Saudi Arabia San Diego great to see so many people from so many different places joining us today and from Berlin in Germany as well cool yeah thank you all for for joining from always different places cool so now I'll start introducing today's speaker so we can get started with the seminar so in case you have just joined us I'm your host Marie villaris from the kiss kick Community team at the University of Oxford this seminar Happens Live every week at 12 pm eastern time all of the episodes are recorded so you can find them on the kiss kit YouTube channel but you can only ask your questions and interact live in the comments if you join us live at 12 Eastern Time every week so I'm here with today's speaker Victor Albert Victor Albert is a physicist at the National Institute of Standards and technology and an adjunct assistant professor in the department of physics and the institute for Advanced Computer studies he enjoys pursuing the broad area of quantum science technology engineering and Mathematics with topics ranging from superconducting circuits to molecular physics Victor received his PhD from Yale in 2017 and he was a postdoc at Caltech prior to joining quicks so I will now hand over to Victor to tell us about his work on Quantum simulation so over to you Victor hi Maria yeah thank you uh everybody at IBM and uh particularly zlaco who actually we used to be grad students with we were the same year you know um I know he he runs these things uh sometimes so so I guess uh zaka was thought it might be interesting for people to hear about uh rigorous characterization of quantum phases of matter on Quantum devices um I'm not you know 100 full-time Quantum simulation researcher but I was interested in a particular question about how to Black Box characterize a state of matter um whether it's in a material or a device I'll be focusing on the device case so and and that led me to sort of this type of engineering or computer science mindset that uh is I believe required really to to to be able to to to truly you know verify that you've made what you wanted to make on your device you know IBM or whatever Google Etc so um why were you interested in doing this well it's because it's something we we will be able to do uh as we gain access to these more uh uh with these more complex and larger devices made out of qubits uh made out of you know Joseph's injunctions and stuff like IBM has uh in some sense we already have access and people already have started putting out papers and realizing this and that uh topological phase some effect like braiding uh associated with uh with these uh phases and uh so in some sense they will be able to do more complicated phases than the ones readily available in materials uh simply because the the materials side of things is just a lot harder so a phase you would expect naturally to describe some material which has you know like moles of atoms and uh and some interactions and we can't track every single atom so we have to kind of do some sort of course grading statistical physics Etc but uh now on the other hand that we have the ability to sort of build uh in some sense like atom by atom or qubit by qubit devices we'll be able to um map those phases onto those devices where we don't have you know 6 times 10 to the 23rd uh you know qubits yet but uh we'll have a large enough amount of qubits where we can have a notion of locality and uh be able to sort of simulate or realize and there's lots of any synonyms to this that people argue about but you know such that we can engineer this device such that it admits a topological phase okay and uh and this is really fun because there's a whole zoo of topological phases uh the classification is not 100 understood uh these topological phases admit very many exotic effects these effects have potential applications and intimate ties to Quant robust Quantum computing and uh and it's proven hard to do in real materials so we you know we have these simulators might as well just try to do this so I'll be talking about I'll be introducing this sort of notion of a phase very gently and and and sort of outlining what they are and uh giving you a few examples of where of of where you can actually of how you can rigorously characterize the topological phase I would appreciate questions uh you know during the talk uh feel free to interrupt with with I guess texting uh Maria and uh uh because I don't really know whether you're receiving the information uh because I'm just seeing my computer screen in front of me uh so uh feel free to sort of help me uh get and get this talk to be more Interactive so um right so first to to sort of get into the notion of what a phase is it's not a argument of a complex number I mean like solid liquid and gas so um so before we we Define what a what a phase is we have to talk about uh where we're defining this phase on okay and we this is assumed and usually not stated that we're defining this phase on some sort of local system A system that has things that are far away from where you are an assistant and and that are close so the system that has an ability to distinguish things that are far versus things that are close so conventionally it's called a notion of locality and I just want to talk a little bit about this notion of locality because when you get into Quantum which is a large field intersecting between computer science and physics locality is a very loaded word okay and I just want to really clarify it because I think it's important for everybody on both sides in the physics and the computer side to not be confused so uh this type of thing that goes without saying in physics I'm actually going to say it so what do we mean by locality it's defined with respect to a metric on some set of qubits so on the IBM chips it's going to be a 2d euclidean you know uh sort of manifold you're placing your qubits as points and there's qubits that are close and there's qubits that are farmed uh you can try to do this on a tour so the rydberg people are saying that they can in their traps they can do some sort of more more weirder periodic boundary conditions but generally the the chip people have basically a 2d kind of layout um maybe two layers of a 2d layout which is exciting but more or less all basically a 2d layout so when we speak of local operators we mean uh we have to consider two cases so in one case is the formal mathematical case which we require to even Define phases it's called the taking the thermodynamic limit where we have our number of qubits n or subsystems and generally n literally be Infinity so there there are crazy things that happen in this limit and one of the things that happened is we can actually have a definition of what a phase is and and strictly in this limit a local operator is just something that has support and a finite number of qubits because when you have an infinitely large table everything that doesn't go off to Infinity is is basically local because you could zoom out far enough and that thing will be small whatever that thing is living on whatever number of qubits it's acting on it will be small because you can keep infinitely zooming out because you have an infinitely large table now if you have a finite table uh you know of a finite ship then you have to define local as something that uh is not growing with the size of the table the size of the chip and is something that's not covering the whole table so there you have to have a sort of a notion of how much you can zoom in and out because you're limited by the size of your of your of your of your geometry or your table or your chip I'm using those three words synonymously um and uh this is how computer scientists what what computer scientists use for the word local okay but physicists use the word local in a different way I'm going to use the the word geometrically local to describe what physicists usually call as local and it just means that you're supported on a finite amount of sites fine automatic qubits but that finite amount is contained in some ball or in some circle you can draw a circle around all the qubits that this is some operator a is acting on uh and that the radius of that circle is independent of the size of the chip okay as the or as the size the number of qubits in general as the number of qubits grows the ball stays the same so um right and and as you in a similar way as you can define local operators which again to quantify phases we need to be a little bit more rigorous and have to think about these things you can also Define processes or maps that map operators to operators uh in uh and you can call them Quantum operations and you can you can you can look at a family of quantum operations that uh preserves the locality of an operator so Maps local operators to local operators and I like to call those Maps causal uh in in sort of the relativity uh flavor that uh they they don't cause some operator to all of a sudden jump over there that's really really far away right so uh sort of having defined what what we're really doing what we're dealing with that we're putting qubits on some space now we can talk a little bit about phases so this this definition here was proposed by my Nick Reed which is not this is a pretty good definition as far as I'm concerned uh that even applies to things like you're familiar with solids liquids and gases uh for a collection of similar particles a phase is a region in some parameter space in which thermal equilibrium States possess some properties in common that can be distinguished from those of other phases so phase is some large region in in whatever space of parameters your system or hamiltonian has that is that is United by some uh sort of coarse grained properties that distinguish it from other uh regions in the parameter space and in between and the boundaries of these phases uh you you have these things called phase transitions that cause sudden jumps in the face so for example you know uh uh water boils or something or water melts right so um just to Define uh here for us what what we mean by Quantum phase of matter we just mean uh for a family of of event cubic hamiltonians where each term in the hamiltonian is geometrically local and we know what that is now uh is that that maintains its Gap the hamiltonians maintain this Gap as we scale up the number of qubits so we have to have a hamiltonian for each uh number of qubits n or at least for some sequence of numbers of qubits and I with m i going to Infinity and as long as the hamiltonian remains gapped meaning that there is an energy difference between the ground state and the excited state uh then you can call this a gapped phase some people call it a topological phase but it doesn't have to be it doesn't have to have sort of non non-trivial topological properties it's just more or less just a phase and then there's the class of phases I won't discuss called gapless where the gap dies off in the thermodynamic limit and those those are you know substantially different from the ones that I'll be discussing okay and these Gap phases are what what I'll be talking about there's a huge two of them and those are the ones that'll be interesting I think and very doable and already doable to realize on Quantum devices so uh before I go off into what the types of phases are I wanted to tell you that besides being united by some core screened features in the in the parameter phase phases are also robust in the following sense so if you have some hamiltonian that's lying in this parameter space where where there's a particular phase say you you're engineering this hamiltonian in your chip but she didn't quite get it right you got you know the terms more or less right but you're off you know a bit you're on you have some you know uncertainty for each of the terms okay and um that's okay because this is a large region in parameter space and even if you didn't quite get the right thing uh the the nearby points in this parameter space will also be this phase unless you're super close to a phase transition which you don't want to be but if you're deep inside the phase you don't have to get it perfectly right so with Quantum simulators and with engineering phases you don't you're not going to uh you know necessarily need this full-blown overhead of quantum error correction you should be able to engineer this and if you get it good enough uh you should be able to see some features of the phase okay and there's lots of people already doing experiments on this where they're claiming that they have seen features in the face so this is something that is actively being done now um now on the theory side there are also uh mathematical ways to try to prove stability of a phase maybe basically you perturb the hamiltonian with some with some small change you make small changes of the hamiltonian you take the thermodynamic limit and you see if the Gap Remains the Same or the Gap doesn't close and and if the features are the same this is difficult to do and requires some math uh it has been shown for uh sort of this this table or this chip geometry that I'm talking about we have two Dimensions uh but uh there's lots of other exciting work uh for uh for Mirror correction coming in where you don't have that type of geometry you don't have a even geometrical local uh uh uh objects on your geometry you don't have geometric locality okay you have these crazy hyperbolic geometries or even crazier expander graphs which come from these qldpc codes and uh if you want to Define phases of matter or more exotic geometries you really kind of want to make sure it's stable and uh we have some work that we're trying to do that for for some of these other um other manifolds um and uh despite it being you know very kind of technical work I think it's important to do this because without it we're really not certain if putting a hamiltonian on a particular you know set of qubits arranged in some particular uh geometry is going to yield a stable phase of matter okay so this is I guess a first example of where I think sort of as physicists we should be a little bit more rigorous about showing some of these things because um we really can't guarantee it unless we're rigorous one and two we we can't guarantee it on on in in theory you know uh unless we're we're as rigorous as we can be and uh we want to be able to certify as best we can that we've made the right phase on our Quantum device for people to believe us and so extra type of rigor uh you know that rules out other phases and shows and certifies that we've done this phase and not some other phase is something that's really badly needed I feel uh in physics we can try to show a signature of a phase but ruling out that all the other phases are something that we don't see is proving very difficult uh for some people right now and uh that's why I think it's important to have this sort of mindset of of really trying to develop Black Box algorithms saying like Okay give me your data give me your your measurements and with some uh guarantees that I get from you I can certify that you're like this phase and not this phase and maybe maybe this other phase okay so um this is I think an important topic and it's going to require a very intimate marriage between physics and computer science and engineering so um yeah I'm just talking at my computer screen happy to take any questions uh so far from people if in case there are any um yeah uh there's a question because you can have mixtures of phases also but uh yeah I think the question is asking about if you can have mixtures of phases as in can you you mean yeah well yeah um for the standard sort of closed system hamiltonian based Paradigm where you have a hamiltonian and a ground state or ground States uh I'm not aware of being able to have mixtures of phases there's a whole sort of extension of open of of of of phases of matter to open systems where now instead of a hamiltonian you have what's called a Markov Master equation or a linbladian okay or an open more generally an open system it's devolving dynamically in time but now instead of ground States this thing is evolving non-unitarily it has steady States and if it's a many body system again defined on some geometry uh okay it might have a phase of matter but because now you're evolving non-unitarily you have more power and you can break a lot of these phases that you have in hamiltonians you can have new phases or fewer phases actually because you can break a lot of the structure that you can't with with hamiltonian systems and there's lots of work in that direction and I wouldn't be able to do a due diligence because it's just so rapidly progressing there you can have mixed States um uh that are not necessarily even Gibbs ensembles or thermal States uh you can have all sorts of different uh interesting uh behaviors but I'll be talking about a little more about just like the basic conventional hamiltonian yeah yeah we've had a clarification on that question says that um he's saying that one can have mixed phases like for example in a whole system one can have a phase which is superconducting impulse and insulating in others um to determine which is which you need the kind of face certification that Victor is talking about that's another application if you have a large enough device you can have patches right where there's different phases going on correct um yeah yeah yeah we have a couple more questions um one is with the geometry of arranging the qubits depend on gauge group geometry of particles well I guess I I was a bit misleading here with using the word particles in Nick's definition uh I am really kind of specializing to qubits or cudits here uh as opposed to actual particles so I'm I'm mapping this definition that he was using he was defining using particles um physical particles into uh sort of the setting where you have an engineered device where you you have qubits designed out of I guess in some sense particles as well photons atoms ions Etc but there really it's the qubits that are sort of the fundamental ingredients and whatever the excitations and in those types of devices you really have just the bare the basic sort of uh setup there is going to be bosonic and there's not going to be any extra symmetries okay so so unlike dealing with intrinsically fermionic systems like some chunk of material where there's a bunch of electrons floating around here the engineering is done such that in the end you get basically like a system of bosonic they call them hardcore bosons you can call them that okay um it's basically a bunch of qubits uh and so we we if we have if we want to engineer any extra thing like a gauge Theory we have to introduce the gauge constraints you know manually okay by hand and uh that's where if we want to engineer a symmetry protected phase we need to introduce the Symmetry by hand if we want to simulate fermions we got to introduce fermions by hand we got to do Jordan vigner or some other uh fancy transformation brother so all of this stuff is extra overhead if you want to do things other than bosonic uh you know spin one half based lattice systems and there's a question could you increase problem complexity by running on multi-gpu Quantum simulators the setting I'm thinking about is just one big Quantum device like some IBM I don't know Eagle you guys have so many of them I've lost count uh you know some some big chunk of Chip uh and uh I see maybe you're saying like can we have multiple chips well if you can connect them in some way that maintains the geometric locality you can consider doing some sort of bilayer situation bilayer geometry and that would increase in principle complexity there are there is a type of phase called invertible phase where you have sort of a phase and it's partner that if you combine them together you can actually uh they kind of kill each other off and you can annihilate each other like particles and anti-particles and you get basically a trivial phase yeah that that would be kind of cool to do but connecting chips is real hard right so and because they're probably going to be in different fridges yeah cool um yeah thanks thanks for the questions everyone and uh yeah let you carry on yeah I'm happy to get the questions uh so phases uh right so how do we classify phases so again I'm gonna basically kind of specialize and I'm using the word specialize here very uh ironically because when you specialize you still get these huge zoo of phases so specializing to 2D bosonic systems uh the way you characterize them is how their excitations behave above the ground state so you basically have a hamiltonian a ground state you apply uh little operators that that perturb the ground state it's the ground state it's got the lowest energy but you so you perturb it with an operator meaning it'll excite the energy it'll it'll it'll be expressed as a superposition of some different higher lying excited States and uh if this excitation was caused by some geometrically local operator then you can think of it as kind of like a blob sitting in some region of space and you have another excitation and you can you can you can mess with them you can braid them you can bring them together and the behavior they exhibit is a way to classify the face different phases will have different excitation behaviors there's two places expectations can occur one I already mentioned they can occur in the bulk so you apply a local operator and you get some sort of excited state and um sorry hold on I'm getting a background music uh you mean yo you can stream yard song in my ear about having a great show stream yard background music okay it's faded away so actually I don't know what happened excitations can occur in the bulk of the manifold so like I said a local patch can be uh you can apply a little operator that perturbs your ground state in a local patch uh and uh this for for the class of phases these topological phases that I'm interested in these these create anions okay and you can think of them as these little blobs but but they can interact with each other okay and they can braid and they can be combined what's called fused together and the braiding patterns and the fusion that these these these excitations exhibit uh are are basically what classifies the phase um so you have basically the types of anions that you can create Pi you have the statistics or the phases you get when you per when you braid them you exchange them change statistics can be more than just bosons and fermions hence the name anions and when you bring two of them together you get you get another you know zoo of them potentially and those are and which ones you get when you combine two are called the fusion rules okay and together this this basically invites category Theory because these types of things all collectively make up what's called a particular category so it's very abstract but it's also very physically relevant and different types of categories basically correspond to different types of phases okay uh but that's not all the places where excitations can occur excitations can also occur on the edge of a system so now if you have a geometry with edges you're going to have extra topological properties going on in the edge potentially you may or may not so if you have excitations on the boundary these you want to think of as sort of moving and transporting uh either charge or current uh uh or exhibiting current and the current can can be sort of uh electrical current like in Quantum Hall physics or it can be thermal current Heat current so there's there's these systems where if you if you perturb them the right way they start uh conducting things on the edge but not in the bulk so the bulk is quiet but the edge is conducted and so now if because you have this 2D you know geometry you can you can move things clockwise and you can move things counterclockwise and um uh these this these currents are also kind of intrinsically protected and robust to small perturbations uh just like these anions are and uh the there's a sort of a number that quantifies the difference between for for sort of your conventional integer based uh uh thermal transport there's a there's a number that quantifies the difference between how many are moving things to the right versus to the left and that's called the chiral Central charge so here uh not All Phases can exhibit this so for example your everybody's favorite surface code doesn't have anything non-trivial on the boundary like this it has stuff in the bulk e m charges but it doesn't have anything on the boundary on the other hand um your Quantum hall effect uh is uh you know not interacting picture doesn't have anything in the bulk but does have something on the boundary and then there are other theories uh like uh I believe the Semion model which has something on the bulk and something on the boundary so you basically kind of have to take into account both of these things when it comes time for the boundary when you have a non-zero Central charge that's when you start talking about these things called chiral topological phases because you know the left and the right it has a just there's a difference between the two and that's Quantified mathematically in the in the fact in the idea that the the conjugate the complex conjugate of the density Matrix is not equal to the density Matrix okay so you have some some issues with changing the sign of the comp of I in the density Matrix and that'll yield a sort of the the partner to that state we have um a question about um can you get static excitations on the boundary excitations which don't move uh yeah so if you bring an anion so for the surface code there's this whole game you can play where you take these anions and you drag them out to the edge and they'll be half they'll either be you know remain excitations on the edge or not um they and then there's these this jargon that comes along with this so if they if they don't excite the the ground state on the edge then they condense if they excite they're said to be confined but that type of sort of behavior is not not really this intrinsic topological current property that I'm discussing um at the moment um so yeah you can you can play these games and people do this with if they want to like they think about Gates and there's different types of boundaries that you can have even for things that don't admit any Movers um the the things that only have stuff in the bulk that that don't admit uh these these these excitations on the edge uh then you can you can drag bulk excitations to the edge and see how they behave uh I forgot to mention thanks for the question that these uh Moving excitations On The Edge are are gapless meaning that the for a geometry with Edge the gap of the system actually will close uh okay and and the the things that close it are located on the edge and are precisely these types of Movers okay question says thanks but they understood the only example I agree yeah um so right and there's this beautiful relation called the gauss-sum rule that as far as I know it was just kind of like thrown out there by kataya in an appendix that maybe somebody can correct me but um this uh well I think it came it was done earlier I think Qatar I'm not sure if you called it the gaustum rule but later on people were able to relate it and generalize it uh maybe related to older work um that unifies these properties so there's there's intimate connections between the the things that you get in the bulk and the things that you get on the edge and you see here on the left hand side I have the chiral Central charge in this exponent that's going to be exhibiting itself module 08 and on the right hand side I have the the sum of the exchange statistics of all the anions and this is supposed to be an equality and it's a very beautiful formula that sort of tells you that there are some relations between and some restrictions on what you can have on the bulk given an agent vice versa okay so uh we are Engineers we want to make sure that if we're given some data we want to uh with some promises on the data we want to certify that the data tells us that we're not in this phase or in this other phase or maybe one of these two phases Etc and I'm going to go through a list of things that I've seen where this is at least in principally possible on paper okay so I'm going to try to list some of the rigorous examples that I've seen uh hopefully all of them and I'll also list some of the other ones that are more physically motivated and that are not quite but maybe around the corner from a rigorous proof um so the problem is given access to two cop to two copies of a 2d gapped face PSI either from numerix or from experiment determine the phase and uh sort of a vague preliminary idea that some of these things that I'm going to discuss are based on is that you basically try to probe locally the phase using expectation values of observables now if you're given access to just the state you can take expectation values with the state and that gives you a polynomial of degree one in the variables of the state and the coherences in the populations of the state however you will see that that's not necessarily going to be enough and you may need multiple copies of the state which correspond to polynomials of the state and its conjugate of higher degree okay and so in general there's a belief sort of heuristically that looking at polynomials on local patches of the state will more or less extract sort of the core screened properties of the state if you're given if you're given a state that is assumed to be in you know one phase okay um and uh this is kind of the flavor of things I I think would be useful to try to investigate uh here I'm also mentioning that we need the conjugate because if we're looking at chiral phases we may need to figure out whether there is a notion of chirality inside the state and for that we may need to look at polynomials if it's conjugate okay so uh here's a few instances where you can rigorously show something about uh uh the the things you need or or or the things that are not enough to show that you're in a phase so I'll have this table here so on one hand I'll see and so this this first line is basically showing that any degree one polynomial namely an expectation value of any observable will not be enough to differentiate uh two types of phases a topological phase and a non-topological phase and I'm not talking about symmetry protect it's just purely topological like surface code single copy observables don't work so why not well let's assume that we have such an observable that whose expectation value is going to be close to let's say for convention plus one and all representatives of a trivial phase and -1 and all represent as the topological phase so what does this mean this means that for any member of the topological side we're going to be um you know uh greater than or less than zero okay we can this minus one will be smeared out but it'll be less than zero strictly so we have a boundary between any element of the trivial side which will be greater than zero okay but remember uh topologicals phases are robust to small deformations okay and so we can deform a given representative by some Quantum circuit whose depth does not scale with the system size whose depth is independent of the system size and in the thermodynamic limit that depth would just become a small blur of the state and won't change the phase one way to do that is just to even do apply a single qubit rotation on each qubit okay but that already smears the expectation value so much using properties of these single qubit unitaries that on average if we average over all possible single qubit unitaries we just get the trace of the observable left and we're left with no signature of the phase no uh you know signature of PSI at all it's a complete this single qubit unitary is completely washed out this observable and this is true for both sides because uh you can perturb both Islands the trivial and the topological islands in the same way and be guaranteed to stay in the same phase but this observable will be washed out on both sides and this leads to a contradiction because now both sides are going to be equal but I had assumed by contradiction that one side is always greater than the other so we can't use a single copy observable to distinguish between all elements of a topological phase from all elements of a trivial phase okay so well what's the good news Well when can we do this well we can do this for some phases in particular if we add symmetries so if we now restrict ourselves to only systems that admit a certain symmetry hamiltonians that are symmetric under certain symmetry uh then inside that class of systems we can distinguish between systems that are topologically trivial or non-trivial in particular there was a proof uh done by Hal tasaki using this Aflac Libra twist operator which is the the observable that we're going to use to distinguish between uh topologically trivial and non-trivial phases with O2 symmetry so let's see how this twist operator does this so this now now this thing's these things will work so uh this local twist acts on L sites so this is a one-dimensional system so chain and it acts on some segment of the chain of size l is going to not scale with the system size because we are assuming this is we need a local operator here we're going to show it's a local operator that can distinguish the phases and it twists it imparts a phase that depends on the on the the number of the site so in part sort of if you think of an the arrow is the phase that imparts it'll twist as it goes along the chain you see there's a k dependence here um and uh you can show that its expectation value is close to plus one and minus one respectively in the two phases uh now how why does this work so here's this sort of short outline of the proof amazingly if you take state PSI that's a ground state um either in either Island this twist operator and take the thermodynamic limit and look at the difference between the Twisted state so the state with the twist applied and the state without the twist you will see that actually they have the same energy and because the ground state is unique You can conclude that they're the same state up to a phase so even though this twist is an operator it winds up giving you something equivalent to a global phase in the thermodynamic limit and so if these really are just the same state then their energies you can show that their energies are are you know approaching the same value and with the assumption that or with the with the extra fact that that you have a unique State then you know that they're the same state which means that the overlap between the twisted and the original ground States is going to be close to one and scaling uh you know and approaching one with the same basically rate as the as the difference in energy and what is the overlap between the twisted and the untwisted ground state that's just the expectation value of the twist operator uh by symmetry we can we can this is where the Symmetry comes in and it's required all pieces of this O2 part are required to tell you this it tells you that the expectation of this twist has to be real meaning it's only plus one or minus one and um because also the twist is unitary right so yeah and now the the kicker is why can't why do we not need to blow up L the size of the twist as we go off in the thermodynamic limit well because if we say start with a uh nearly infinite you know an L that's almost the size of the chain and we shrink it down to something that's not too small but something that's still independent of n the value of the twist doesn't change because the twist is continuous and L so the way I've written it the way tasaki was writing it is this equation up at the top and the reason it's written in such a funny way is because if you let L go to a continuous variable you'll see that the twist is actually a continuous function as a matrix of L and so this means that we don't require L to scale the system size because we can just pick a sufficiently large L and uh up to some of these Corrections that are 1 over L will be close enough to 1 and minus one and we will be able to resolve the Symmetry protected and the trivial phases unlike in the previous case where we had the freedom to smear our state using local operators and thereby smear any operator in this case we're dealing with O2 symmetric States we don't have that freedom to smear the states and we have an operator that we can prove that can remain local and diagnose the two phases so as far as I know the only proof that you know of an operator that can actually do this all right so that's the first two rows the second the third row is something that uh is again another symmetry protected phase where there's been a lot of serious rigorous work on and here the operator we're going to use is called the hall conductivity um now in this example which was rigorously proven you need an extra ingredient you don't just need a state you need also a hamiltonian and a symmetry uh but on top of the Symmetry you also need the hamiltonian so we have to be given a hamiltonian and here's the hamiltonian it's got some geometrical pieces we have this hamiltonian we also need a symmetry that the hamiltonian respects in this case it's the charge can of currents without charge and this charge Q This Global charge you can also re relay using little pieces little local pieces the the full H commutes with the full Q uh and uh thereby the full Q commutes with little pieces of H as well however little pieces of cue don't commute with little pieces of age so globally you can have charge conservation but locally you can have a little flow of you know and and of charge going on in the system so when you have that you can Define these little currents that talk about flow of charge from site J to site k once you have these ingredients you can Define these currents and that we're going to construct the operator the a that we need to distinguish trivial and topological phases using these currents we have to do one more thing to these we have to smear them by up uh basically multiplying them by the greens functions of the hamiltonian the kind of thing you do in perturbation Theory this is in other parts of the world it's called quasi-adiabatic Evolution there's a lot of history behind this operator you can do this using a local or a causal Quantum operation I don't necessarily need to get into that for the purposes of this talk but this is a perfectly well defined thing to do and once you've smeared out these currents you just take their commutator and you get a an observable whose expectation value will tell you the conductivity on the edge of this system okay and this thing is going to be even integer for bosonic 2D topological phases and you can use this to diagnose and so the the in principle the regions here you're defining x y y bar X bar or perp um they're big however because you're looking at the commutator between these currents the commutator is only on the overlap regions and that that is going to be a local object in general and that's one of the techniques you basically have to use to show that that the conductance remains uh you know doesn't blow up and remains a quantized observable in the thermodynamic limit okay I want to mention a brief uh connection to a related uh phase and a related operator which was uh recently derived to um to work as a diagnosis for these types of phases which have a U1 charge symmetry uh this is not a rigorous proof but it was based off of topological Quantum field Theory arguments uh you know these high-end physics theories that are Continuum that describe these phases without the need of a lattice um and uh the reason I want to mention is because this this this next local observable that uh does this actually has a lot of ingredients that are similar to the observables we've discussed already so I want to point out this connection because I think there might be more to work on there more to extract in more General cases so this observable uh we'll extract What's called the Churn number um and this chart number is by the way the same thing as the whole conductivity uh in the previous case um so okay so here's a sort of expression for it um involving all these different operators and I'm just going to introduce them so um the the first the W operator is basically the symmetry so you have you have again a symmetry and here's how they Define it and you just apply the Symmetry to the to some region r the V operator is similar to The Twist so this is what I wanted to point out um it has a sort of it's here it's in a different context but it does the same thing that the twist does okay it it um it applies a phase generated by the charge that's dependent on the position y see the Y there in the in the in the in the exponent exactly the same as the twist and the third part is something that uh basically makes the expectation value of this operator equivalent to a partition function on uh some weirdly connected surface this is really kind of the secret sauce here and I encourage you to uh you know read the paper because I don't have enough know-how to really explain this but this was really the the key Insight I think to be able to make this work so this is a third ingredient that again I think also has some room to be generalized to other systems and I think there's also ongoing work in this direction okay um it would be nice to have more rigorous backings of this besides numerix for this churn insulator case that's another interesting Direction um the other another case that I wanted to mention is uh briefly I guess without an example is more generally a one one-dimensional symmetry protected phase is protected by on-site symmetries or time reversal symmetries also have a series of different types of parameters or expect whose expectation values can be done in particular with matrix product States they can be used to classify the face given some symmetry okay and now I'll go kind of back up to where we started with and look at more General phases symmetry or not uh and top that the and consider uh functions of the density Matrix preferably local that can be used to diagnose whether it's in a topological phase or not and there's several different sort of key developments in this in this direction it's more general direction besides the first first row that I've discussed so the third from last rows is a tanglement spectrum it's just the eigenvalues of the reduced density Matrix you can think of them as functions of the density Matrix there's some polynomial the density Matrix and uh if you look at these in specific sub-regions of your geometry they give you insight into what the phase is another thing is entropic uh variables uh infantropic functions uh of the density Matrix such as this topological entanglementary which is just some some combination of of Von Neumann or sort of you know density Matrix entropies of the reduced the reduced density Matrix on different patches combined such that it extracts a little piece that quantifies in a coarse grained way the topological phase of the system extracting What's called the total Quantum dimension of all the anions and the third thing is something that we were that was developing with with the collaborators um called the modular commutator uh it's sort of a quadratic version of a topological entanglement entropy now that includes sort of uh two instances of the density Matrix on different regions and it looks at their commutator sorry log of the density Matrix on different regions and looks at their commutator so that part I wanted to mention and I wanted to mention that we conjecture that this modular commutator extracts our good old friend the chiral Central charge from the beginning of this talk so this case uh let's let's look at how you would in the in the original kataya formulation extract the chiro Central charge you give you would be given a hamiltonian and then instead of being given a charge now that you're dealing with a with a system with no particular symmetry you define a different type of current a local energy current so the hamiltonian commutes with itself energy is preserved but the little bits and pieces of the hamiltonian may not so again you can have flow of energy in the little bits and pieces of the system and you can define energy currents and uh then you can go from there and kind of do a similar type of thing as you did with the with the hall conductivity however what we were thinking about is well what if we're not given a hamiltonian can we still uh extract the Cairo Central charge only from properties of the state and uh well if we don't have hamiltonian let's just make one up from the state and the best way to make one up is just take the log of the state any reduced density Matrix can be expressed as an exponential of some hamiltonian and we just take the log and we get the hamiltonian and we call it the mod let's call it the modular hamiltonian that's how it was called in high energy Theory and we can similarly Define currents uh for the bits and pieces of the modular hamiltonian and now the belief is that basically this modular hamiltonian will have the same type of phase characteristics as the whatever the original Hamilton system would have been and if we sort of subscribe to this belief then we can um turns out that if you look at commutators between the modular hamiltonian bits and pieces you cook up this observable you evaluate its expectation value and you can extract uh the the chiral Central charge on a sufficiently large region um again that's just the conjecture but we have much a lot of numerical evidence uh to back it up and some subsequent work uh one of the instances that singles out this specific observable from others is that it's chiral remember the chiral Central charge is something that tells you how different is rostar from Rome in some sense which means that if you do take the conjugate of rho you're supposed to get a different value for this charge namely minus the charge current central charge and this this the way this commutator structured you indeed do that so this sort of opens up the the direction of trying to be able to cook up sort of chiral observables that diagnose the chiral nature of the entanglement of a many body state that as far as I know have not really been present in sort of the information Theory literature uh that and the many body entanglement literature okay that sums it up uh there are many different other developments that I haven't highlighted and I'll get to in the convention in the conclusion but right now I'll mention sort of okay so what's the output so we have some mathematical tools and techniques and we have some case studies and instances where we can really understand what the phases of a given system sometimes we need just the state sometimes we need the hamiltonian sometimes we need a symmetry but we can we can get a grasp of it given enough ingredients what can we do with that well one of the things we were working on is is is is is is then plugging into a computer and asking the computer uh with some minimal guarantee asking the computer well what phase is is this state in and uh so one of the things you can use is just you know machine learning algorithms and not even the complicated ones just these things called support Vector machines so the first thing you need to do before you throw a state in the computer is uh have an efficient description of it and there was a method uh sort of a tomography protocol called classical Shadows that obtained efficient snapshots of a Quantum State I won't necessarily get into it but you can just say that these SFI are actually tensor product operators that on average when you add enough of them up give you the state okay once you have an efficient description of the state you can throw it into the machine learning algorithm and here for these support Vector machines what you do is the following the state itself is will be characterized let's say by some uh local functions of the density matrices some polynomials of arbitrary order but those are non-linear in row the quadratic cubic Etc we want to unroll those non-linear correlations and and and and features into a linear into a linear function a vector and the way that this was thought of by by Robert Huang was uh to map the state or the shadow of the state into another Hilbert space into a vector called a feature Vector in another Hilbert space that linearizes all all of these different combinatorial combinations of reduced density matrices namely all powers on all regions of all different sizes so this sounds really complicated but and it kind of is but you can actually write it down formally so you take a shadow s on a region r of the state PSI you you you do a a block vector stack all the different shadows of all on all regions of a fixed support W fixed number of qubits w take that and then do that for all w okay and then you want to do all powers of the reduced density Matrix then take that really long infinite vector and do its tensor Powers tensor tensor products of it with itself which gives you all the possible combinations of the regions and you do that for arbitrary number of tensor products so you get this huge double infinite vector okay and well this looks unwieldy what are you going to do with it well but before I tell you that it's not as unwieldy as you think first I want to tell you that we've succeeded in what we wanted to do we unrolled all the non-local non-linear pieces of the density Matrix into a linear vector now any function of the state can be written as uh as this inner product with some feature function and the vector everything's been linearized okay well now uh we have this huge Vector space we need to unders we need to gain a feeling for where things are in this Vector space in order for the machine learning algorithm the support Vector machine to distinguish between elements of different phases the idea is that the feature vectors for different phases will cluster in different regions of this feature Vector space because they are so different and their correlations have made the feature vectors be far away from each other if they're in different phases but close to each other if they're in the same phase okay and that'll be great if only we can figure out where the vectors are well it turns out because of the interplay of Shadows and the way this feature Vector is written down inner products between vectors that tell you how far they are can be calculated efficiently when you do the inner product between these you get a double power Series in p and w that because of the way the Shadows are structured they both get rolled up into an exponential so the inner products between these vectors of exponentials of exponential of a very simple quantity so they're efficiently to calculate efficiently calculable doing this you plug in this into a machine learning algorithm the support Vector machine calculates all the inner products spits out a hyperplane separating the two clusters of of feature vectors which will be corresponding to two different phases and the Machine learning algorithm can do this irrespective of whether you promise you you're giving it two different phases or not the reason being is because the feature Vector does all the job of enrolling the correlations and and helping the machine learning algorithm characterize the phases without you telling what the machine algorithm is expecting to see and the only guarantee we need to promise it is that there exists a local invariant a so here we need a local invariant so we can do this for us for symmetry protected phases for example then irrespective of whether we give the machine learning algorithm the invariant or not we don't need to give it the invariant it will find the corresponding hyper plane numerically uh We've you know tested this with with for example a symmetry protected phase um but numerically we see something even more interesting if we test this on a topological phase where we're not guaranteed to have or in fact there is no local observable that it isn't that can distinguish the phases the feature Vector is so expressive that it naturally clusters the two phases topological and trivial in separate corners of the of the feature Vector space and thereby the machine learning algorithm can still differentiate them without them us telling the algorithm what what to expect and it can do this as long as we don't wash out the states too much uh using a constant depth circuit you see here the the the distinguishability gets smeared out as we crank up the circuit depth so uh I've used up I guess all of my time and I'll just wrap up uh in I guess the current day stock is a bit outdated we have access to devices that allow us to simulate phases of matter not readily available in natural materials we need to design engineering protocols to verify and certify that a state created on a device is indeed a representative of the desired state this means that we need to convert physical condensed matter topological field Theory category Theory intuition into rigorous engineering protocols that take in some assumptions and spit out some guarantees on what the state is and when we do this of course there's going to be some back action namely that as we sort of gain a more uh uh technical understanding of these qualitative physical behaviors we'll be able to learn something about the phases on the way made a lot of progress I've highlighted a few examples that I thought were very interesting uh more systematic efforts I think would be useful because for example I've pointed out that this twist is present in many in at least two different cases here that are quite distinct in the spin one chains uh with O2 Symmetry and the 2D bosonic systems with the with the charge U1 charge conservation I wanted to mention some follow-up work that I did not get to but I apologize for missing um in particular sort of sort of things along similar lines have been uh developed uh for extracting not the total Quantum dimension of all the anions but individual Dimensions from a single wave function uh kataya had a formula for non-interacting fermions for the real space Churn number for fermionic phases I only covered bosonic cases there's been extensions of that modular commutator has also been investigated a bit more thoroughly uh machine learning algorithms can be used to find for example Wilson Loop operators which are who's who which you can you can use to characterize phases um uh certain phases and uh lots of work for specific examples but again I really wish there could be a more General sort of understanding of these behaviors um for the systems of Interest so with that I want to thank you for your attention I guess any questions great thanks a lot for the talk that was that was really good really interesting um we have a a couple of questions in the the comments um and yeah if anyone else has any can drop them there um sorry I guess I sorry I should mention that yeah machine learning work was spearheaded by Robert Huang um and uh um other other wonderful collaborators uh are listed here uh and the Cairo Central charge was spearheaded by Isaac and Bowen um and uh there's these two papers here that you want to consult so uh and I I work at Maryland I'm paid by nist I'm a physicist for the National Institute of Standards and Technology but I sit at this quick Center for University of Maryland College Park and we welcome anybody who wants to join us we have more than enough Quantum centers uh and uh any sort of area of quantum we have experts in the very exciting place to work okay thanks great so yeah check out those opportunities if if you're interested say we have one question is um can this phase classification be used to prepare a phase so come the class the can the classification tell how far we are from the truth since you mentioned about phases as parameter space say maybe we could adjust those yeah yeah I think it can for example if you if you cook up a hamiltonian you start trying to diagnose it the state that you made that is the ground state of this hamiltonian you start to see that this thing you know is nearly gapless and it's got some sort of like conformal invariance you know scale and variance starts to come in then you you start to get you can get a feel that the state is too close to a phase transition and then you want to move away from that right deep into the phase so that you have nice stability and rope and and robustness and you get uh you know the actual sort of more representative functions of the phase so yeah you can you can kind of maybe if you have enough knobs you can use it to travel along Traverse the parameter space the knobs are kind of defined by your physical system you know so here for these uh rydberg scenarios that this spt numerix was uh sort of simulating uh or no maybe sorry I'm maybe confusing this I mean okay so in this case if you have an engineered system uh you you the knobs are determined by what you can do in the lab and you have to be able to map the knobs to an actual parameter space of a phase if you can do that then the classification problem will help you Traverse the parameter space cool thanks and another question is um is the radius of bull r on the chip independent of the phases we are trying to classify what's the minimum good question I didn't get into things like correlation length so um right so this coarse grading of systems into uh sort of little local regions um requires you to make the local regions big enough so that you get rid of any sort of short range effects that might obscure your understanding of the Course Grand overall phase and uh these short range effects are are collectively you know sort of one of the things that quantifies them is this correlation length and uh it is expected um that many of these phases have sort of exponentially diminishing correlation link and you don't necessarily need that information these sort of local effects to diagnose the phase and so when you when you sort of think about how big of a region you need roughly to characterize the Fades using a local operator you need it to be big enough so that it kind of encompasses the uh that it that it deals with this Coral decaying correlation length um there are sort of special representatives of phases that have zero correlation length um whose expectation values of like operators on different sites uh doesn't Decay as constant as you as you put the operators on sites farther than farther away the surface code is one of them okay the quantum double model is another extreme Nets or the other so uh but but if you start to now apply a constant depth circuit to those phases then everything gets smeared out you start to gain more sort of connectedness that's that's not an extensive property of the phase that just kind of smears it out locally and you have to consider regions that are larger commensurate with the depth of the circuit that you use to smear off the face now formally even though numerically you'll be able to use a large enough region to diagnose you may still not have a proof that it is always sufficient that's another issue that's present in this topological entanglement entropy for example okay or that whether if you smear out one of these things that you put on one patch by a constant depth circuit is it going to be invariant that's another issue that people are are dealing with for these things I'm trying to prove we have a more of a comment uh which says great work Victor and team 2D topological system reminds me of skirmions as well thanks I don't know much about I can't comment anything intelligent about skirmions questions and for joining and thanks again Victor for giving a great talk um so just as we wrap up um uh yeah I just wanted to remind everyone to join us next week at again at 12 p.m Easter Daylight Time and um next week we'll have Christine mushick speaking about simulating one-dimensional Quantum chromodynamics on a quantum computer and just before um yeah so we can see lots of people saying in the comments thanks for the update thanks great I've enjoyed it a lot so yeah great and uh okay yeah do you have any final comments for our audience Victor that you'd like to share yeah I guess I guess I maybe do so the we really want to think about General characterization of these phases which is going to involve some sort of combination of adapting these high-end continuous field Theory physical techniques with uh you know digital computer science and engineering approaches and I don't think we're very close to a full-scale characterization of phases we do have specific instances where we can do this but I'd really love to work with people and like have see other people working on a more General approach cool or hopefully maybe that we have interest to some of the people that tuned in today all that watch this this recording later so yeah thanks everyone and we'll wrap up there so bye see you next time

Channel: Qiskit

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Length: 70min 50sec (4250 seconds)

Published: Sat Jul 01 2023