Understanding Noise in Near-Term Quantum Computers with Haimeng Zhang: Qiskit Summer School 2024

hello everyone I'm hon I'm an engineer at IBM Quantum today I'm here to talk about Quantum noise and how to characterize it uh on actual Hardware I want to start by talking about a Quantum circuit so this is an object that we usually work with um when we want to do any Quantum computation on our Quantum device so this Quantum circuit could come from any Quantum algorithms you are interested in but usually uh it has a certain structure um so over here it is really a simple 4 Cubit toy model uh we can see that

at the first step it initialized the cubit in ground state and then it apply um a bunch of quantum operations over here those are quantum Gates uh usually described by unitary matrices and on our Hardware there are um single Cubit and two cubic Gates that are native to our Hardware that we can Implement and at the end of the circuit we measure in the computational basis so this is the Quantum circuit that we usually work with um but bear in mind when we actually run this Quantum circuit on our Hardware um all the information is

processed in the physical world and in practice those Quantum operations we talked about are not ideal um that's why we care about um this topic uh Hardware noise so um We Now understand the performance and the power of our Quantum processors are still largely limited by noise that's why we want to understand what's the limitation of our Quantum Hardware as a result so we can make the best use of it so in this lecture we will go over different sources of noise and um sub we'll go into details of incoherent noise coherent noise and state

preparation and measurement noise and we want to show you where you can read it um from our IBM Quantum platform where we report those noise related metrics on a daily basis and in the end we'll end by talking about how those metrics are experimentally characterized and followed by this lecture you can also join the lab session to try it out uh by actually doing uh such an experiment using kiss kit runtime so let's come back to our circuit example and start by talking about what ideal operations should look like so as to give some examples

um so first thing we noticed is there are single Cuba Gates so one example is hotmart gate which put a Cuba state from ground state into super position State and something we can also notice is that there are rotation Gates over here which are the those RX Gates um over some angle so those gates are implemented typically by turning on certain Honan and the huttonian over here for rotation Gates around x axis has the sigma X form so those honi are what we physically engineer and control in real hardware and over here this is a

c not gate um so those are Gates we use to generate two Cubit entanglement so here we be we can briefly see that with a HMA gate and a c not gate we can put a 2 Cubit from ground state to a maximally in tangle state so those are a few examples of how Quantum operations are um are going to do at the end the circuit the measurement operator what do we mean mean by that it is a projective measurement in the cubid computational basis so at the end with measurement we will get um the

B Rings In classical form of zero and ones so what can go wrong in all the ideal operations we just talked about um so roughly speaking all the noise can be um characterized in different categories so the first one we are going to look at is associated with State preparation and measurement so we call them spam errors the SE second type of Noise We are going to talk about today is incoherent errors so those in general terms they correspond to loss of quantum information in the form of superposition and entanglement usually due to uh interactions

between the cubits and their environment the third type of noise that we're going to talk about today are coherent Aras usually they um are associated with imperfect gate implementations um so this gives an overview of the lecture today so before we jump into all the types of noise we're going to talk about let's first um quickly recap how do we represent a Quantum State mathematically so take the simplest case of a single cubid we know a single cubid Quantum State can be represented by a state Vector in the two-dimensional h space and the state factors

here uh We've WR 10 zero State and one state forms a aoral complete basis for that space so those are the two states we are going to seeing a lot uh later on what's special about the quantum state is that um besides ground state and excited state it can also be in a coherent superposition state of the two so how do we write a superposition State we use what we call uh probability amplitudes Alpha and betas those can be complex numbers and they represent um roughly speaking probability of the state being in ground or excited

state um we can as you can see that um we have a way to visualize those Quantum States on a sphere that is what we call block sphere so uh if we have to talk about all available uh states that can be represented on this block sphere we have to talk about there is a distin a difference between mixed state and pure state so a pure state is what we write in this outer product form um but if we want to talk about classical mixture of pure States that's where we have to write in this

density Matrix representation where it is is simply a sum of all possible pure States in this outer product form uh and then um weighted by this probability uh in the front so this describes an ensemble of pure States so all density Matrix um can be written in this block Vector representation so for single cubic case this is how we write it in Block vector representation we simply write the density Matrix as a sum of all poly operators identity Matrix and poly XYZ and the block Vector components would correspond to the um components the coefficient in

front of those poly matrices so to give a few examples um if we have a plus state or minus State those are the maximally superposition State uh those States corresponds to the x x poles of the block sphere so those if we we write the block Vector they would look like um 1 0 0 as a block vector or minus one 0 0 uh over here if we have ground state or excited States I think that's pretty obvious they are um on the Z poles of the block sphere there's also a very special example example

of a maximally mixed Aid which is um you have equal probability uh being in ground and exi stays and that corresponds to the center of the block sphere so over here it's a quick recap of what do we mean by mix State how do we represent them in the form of density matrices and how to visualize them on block sphere um later on we'll see the fact of noise can be very well capsulated in this picture where we have a noise Channel and what does it do in general a noise Channel brings a pure State

into a mixed state so in the block Vector representation a pure state would have a vector length of one wellas a mix State uh after as an output of the noise channel would have a vector lens smaller than one in the ideal world we have unitary operations to describe our Quantum Gates so another consequence of noise is that all now the mapping from the inut state to Output state would have to be described uh by a noise Channel and we will see uh specifically what does this channel do to our input states with that uh

we can move on to talk about uh not different types of noise so the first type of Noise We encounter here is what we call spam errors State preparation and measurement errors so a quick example of a mixed state here uh is say we have imperfect State preparation um that is the case where we want to prepare a um state which is initialized in ground state but in practice um we don't have the full information of the state then there are some population of the state ending up in excit state so that is a uh

type of error that needs to be captured in the model for spam errors but in practice um spam errors and readout errors are usually very hard to be separate from each other so we use this um spam error model to capture them together so a consequence of State preparation and measurement errors is that after measurement we will get um noisy uh bit string probability distributions what do I mean by that so suppose we have an ideal uh GHz state for 4 cubits so the grain distributions here corresponds to the ideal case but due to perfect

State preparation and measurement errors uh we can get bit strings that are away from this ideal distribution so mathematically the noisy distribution and ideal distribution can be related in this form over here and the effect of spam errors can be captured by this a matrix so um in this probability vector here in index um describes the probability of measuring bit string I that corresponds to the I element of this probability Vector P whereas each index each element in this a matrix for example the I row J's column corresponds to the probability of observing State J

given the state is being prepared in I so experimentally uh we can try to characterize this a matrix which would give us an idea of how much spam errors there are uh over here so over here on the uh bottom right we can see an example of such an a matrix um usually it is very challenging to characterize this a matrix at scale because we can see that it has 2 to the N by 2 to the N elements where n is number of cubits um that's why um there has been ongoing research to try

to figure out how to characterize spam errors at scale but on our back end we report um this property this um what is the probability of measuring St state zero if the state is prepared in one this property we report them uh Cubit by Cubit on our back end and you can check it out um on our um IBM Quantum platform but in principle uh bear in mind that those readout errors and state preparation errors could also be correlated so with that um this this is we can see from the back end that the typical

readout error is on the order of 10 to minus 2 so that is the readout error we have on our back end and moving forward uh I would like to talk about incoherent areas which are captured uh by this noise this picture of noise channel so over here what we see is we have an imp row and an output and the output state after the noise channel is Den noted by uh row Prime with this map abson acting on row so mathematically uh we have a bunch of options to represent this noise Channel and one

of them is what we call uh cross representation so this is the mathematical form of it basically we will have a set of cross operators um to act on state row um the effective noise what does the noise do to the state would be captured by those operators and of course there are some constraints on how this map should be like some properties include that this map should be linear should be complete positive and choice preserving basically those properties enforce that um this map maps a physically relevant density Matrix to a physical density M uh

Matrix tricks if you're curious to learn more about um noise channels rigorously I would refer you to this um lecture on Quantum channels by John Watchers which is available on IBM Quantum learning platform um now we can see um several examples of incoherent areas and we will see later those examples are actually quite relevant when we want to understand noise that are actually going on on our Quantum Hardware so the first example is um the energy relaxation process so this is a process which brings um as the name suggests corresponding to the decay of this

the excit state to ground state this process is quite relevant for superc conducting cubits which is uh the hot wheare we were talking about here because we the cubits we are talking about operate at very low temperature typically around 20 Melvin and our cubid frequency is typically between 4 to six gahz given that um it's quite intuitive to say that there's a much stronger Decay from excited state to ground state compared to the opposite process um from ground state to excited state so this time scale of energy relaxing from excited state to ground state is

characterized by T1 that is what we refer as the relaxation time and that relaxation time can also be associated with a probability of a relaxation error uh for some Evolution time T so if we want to mathematically describe this energy relaxation process we can use what we call a damp amplitude damping Channel and over here we can see the cross operator form of this channel but basically what this channel does is that it takes an exited state to ground state with probability P whereas it leaves the ground state unchanged if we visualize the action of

this amplitude damping channel on the Block we can see what it does is that it brings all the state and contrive it to the ground state so at the end the study state of this um channel would be the ground state so the picture we're getting here is that ground state is actually a long lived state that is largely unaffected by this type of relaxation areas other state would be suffer from this T1 limit another type of noise we'll be talking about is Def phasing noise so that is the process where superposition States be become

classical mixture and a coherent State lose its phas information so similarly to the picture we have before for T1 relaxation the defacing time scale here is characterized by what we call T2 and T2 can also be related uh to a defacing error rate um by this equation here um we can see that both relaxation and defacing era has a consequence of turning Quantum information to classical information so this defacing channel this defacing process can also be mathematically described by a defacing channel and over here we can see the cross operator form of it the action

of Def phasing or pH stamping channel on the Block sphere is that it shrink the size of the X and Y component of the block Vector uh hence we will see that now the quantum States in this block Vector representation would have a un a vector lens of smaller than one and it has um less and less information about its face so those two process the energy relaxation and Def phasing are both two physical relevant um decoherence process on our Hardware another mathematically motivated uh noise model is what we call poly noise model um we'll

see that it has a form like this for single Cubit case what it does is that it uh apply the single Cubit poly operators to the Cub State uh at each given um probability so this is also a physically motivated uh noise model and it has actually been used to model our noisy Quantum processors in practice um given by this literature here and this paper showed that one can use use certain tomography or learning technique to learn the parameters uh associated with poly noise channel for a given uh device a special case of poly noise

model is what we call depolarizing Channel that is the case when the probability of XYZ poly operators are the same so this channel can also be written in the following form over here so what the deor rizing channel does is that with probability P it map the state to the maximumly mixed state and with probability 1 minus P it leaves the state and changed um so this is a formula directly suggesting uh deising channel is a special case of poly Channel given XY the poly operators have the same uh probabilities so poly channels we can

see um what's special about those channels are um the stady state of both poly noise Channel or deing Channel as a special case is that its steady state is the maximally mixed Aid that's why we say those channels are unal meaning that a map an identity to an identity in Max in maximally mixed Aid to itself after talking about different types of incoherent noise now let's move over to talk about coherent errors so as we mentioned Cohan errors are usually associated with imperfect gate implementations so in our Hardware gates are usually generated by turning on

certain Honan that we can control and engineer so let's look at a single Cubit case where we have um a single Cubit rotation gate around x axis suppose we want to implement an X gate and that corresponds to rotation around x axis of angle Pi what we do in practice is we would turn on this Honan of Sigma X form with amplitude V and we calibrate the rotation axis and the rotation angle corresponding to the desired angle but what if we engineered the wrong heronian suppose we have incorrect phase specified in the microwave generator that

corresponds to some component of rotation around y- Axis or suppose we have poorly calibrated our PSE duration and our pulse angle is not corresponding to um angle Pi so in those two cases the coherent errors can be modeled by unwanted unitary gates in the circuit in the case of misalignment a noisy um X gate with coh errors can be decomposed into an ideal X gate together with a Y rotation gate that corresponds to rotation around y AIS with some um small angle in the case of miscalibrated post durations the noisy X gate can be DEC

composed into the ideal X gate together with a x rotation gate with an over or under rotation angle so that's the case where we can model coherent errors in the case of two Cubit coherent erors that is the case where we usually deal with what we call cross talk Aras and this has something this has something to do with how two cubic gates are implemented take the case of Eagle processors where our native two Cub gates are what we call echoed cross resonance Gates and they are equivalent to a c not gate up to some

single Cubit rotations so what does ECR Gates look like so in practice what we implement is a Honan of the ZX form so those are the interactions we use to generate um entanglement uh between two cubits so in the echoed cross resonance gate we Implement rzx rotations with opposite amplitude pi over4 and minus Pi over4 back to back with some single cubid uh RX rotation gate in between so that whole sequence Implement an ECR gate but we can see that U the full Honan describing this um gate evolution is actually has some unwanted terms so

those unwanted terms would correspond to coherent errors if the calibration part is not done perfectly so special attention is usually paid to this form where we have ZZ um coupling between two cubits and in our device um specific to egole processors those ZZ coupling are actually resulted from static capacitive coupling that cannot be completely turned off even when the gates are not implemented so those type of coupling can result in unwanted cross talk between the target cubits to the nearest neighboring connected cubits so those are uh a source of cross talk errors that we have

to deal with also bear in mind that the cross residence Gates we're talking about here are usually way longer than single cubic Gates that means our two cubic Gates has a way longer usually hundreds of NCS of gate duration so that's why um usually two cubic gates are considered the main source of error um in the context of the whole circuit after talking about single Cubit and two Cubit coherent errors I want to quickly summarize by an observation that Cohan errors oftentimes result in oscillations in signals to see this I would refer you to an

example in our last year kkit Global summer school where um we look into a case in simulation uh to take account of single cubid over rotation error depolarizing error and spam errors so for this example showing on the right here the ideal expectation values for the observable single cube of Z would always be either one or minus one and we can see how under the simulated noise uh the signals uh diverge from that and we can see uh due to this single Cubit over rotation error we see oscillations in the signal so if you do

see signature of unwanted oscillations that could possibly indicate the presence of coherent errors uh in your experiments and from this example we can also see that coherent errors can build up much more rapidly than incoherent errors at least in short time um so they can be very undesirable and tricky to deal with there are a number of techniques that can deal with coherent errors for example dynamical decoupling and and twirling if you're interested in those techniques I refer you to uh the lecture from this year's summer school on execution on noisy Quantum hotwar fighting errors

before fought tolerance um by my colleague Pedro so now we have talked about both incoherent noise and coherent noise um with that let's put what we have talked about uh in practice and look look at the following example where we want to implement a six Cubit circuit on a chain of physically connected cubits and the information we have here is that first of all um suppose we know one of the cubits Cubit 2 would have really bad T1 T2 time has very um low coherence time and we also know um ECR Gates or two cubba

gates are not desirable to implement in par in parallel due to uh cross talk areas so those information can actually inform us to schedule our gates in a logical circuit um in a more um noise resilient way what do I mean by that suppose in this three examples we are implementing exactly same circuit in the ideal case but in order to avoid cross talk between Cubit 1 and Cubit 2 which are physically connected we can also try introduce a barrier in between those two two cubic Gates so that those two pairs of two cubic gates

are not implemented in parallel hence reduce cross talk errors as much as possible another information we had was if Cubit 2 has low coherence time what we can do is we can delay the implementation of ECR Gates to as late as possible so that cuber 2 stay in ground state for as long as possible before engaging in any active computation so this is a nice example where using our understanding of incoherent noise and coherent noise how we can come up with optimal ways to schedule our circuit and this is what we call as late as

possible scheduling strategy if you want to read more about that I would refer you to this uh paper here uh below so so far we have mostly talked about um local noise for single cubba and two Cuba case so in the following I want to briefly present uh the picture of how those noise can propagate in time and space so suppose in the picture of a layered circuit and over here what we seeing is that we are implementing layers of two cubic Gates and those layer circuit are actually quite relevant in applications of near time

Quantum Computing um so this particular case we are looking at Brick work random circuit on a 1D chain um so it has been derived theoretically that uh if we have random to CU Gates and all noise are local then the output of this noisy circuit would converge fast to maximum mixed Aid um and the noise model in the Deep circuit regime can actually be very well approximated by global depolarizing noise so although this is only true for random circuits but it can only it can also speaks to uh the limitation of running circuits on an

actual Hardware so um there's a limmit on how deep we can run circuits before a Quantum state becomes uh maximally mixed a hence uh purely classical in previous slide we are interested in what's the output state of a noisy Quantum circuit but in a lot of applications we are only going to measure the some observable and estimate is expectation values suppose the observables we are interested in are local meaning we are only going to measure a subset of cubits in this case uh we are only going to measure observable on a single cubid and it

has been our understanding over here is that the effect of noise can be more moderate than the case we presented uh before uh the intuition here is that only the noise in a specific region of the circuit is going to contribute to the era in the final expectation values um and this region is uh what we refer to as a l um this is over here we are drawing what we call a backward L con that is um if we start from the back of the circuit from the measurement and we back propagate the observable

um and shade it the region where the wires are physically connected to this cubid that is measured and we can um if we keep doing this um backwards we can shade the entire region and within this region those errors are going to contribute to the to the expectation values um this is pointed out in the paper reference blow and using this intuition we can argue that uh the noise or the errors in local observables is going to be less than a general um observable on all cubits so far we have looked into single cubid two

cub but incoherent and coherent noise and how they can propagate um over circuits um in a theoretical setup but um how do we figure out and characterize noise in practice is a different question so in the following we'll look at uh how noise can be experimentally characterized to understand that I want to first start by talking about how do we character an unknown Quantum State and then we'll talk about how do we characterize an unknown Quantum process so take the example of a single cubid we can expand the single cubid density Matrix in the poly

bases as the following so by a simple counting argument we can see that uh in order to measure all the possible all the coefficients corresponding to those po we need to perform um at least 4 to the N minus one measurement um where n is number of cubits minus one it is because we know choice of row is one so we can save one measurement so in order to in general to characterize an unknown Quantum State needs exponentially number of experiments um what about the case where we are interested in Cur izing an unknown Quantum

process it turns out the procedure of characterizing an unknown process is the following first we prepare for to the end input States and then we pass it to the quantum channel that we are we want to characterize and then we measure in all for to the end complete basis by the simple um Counting we can see that we need need on the order of 16 to n experiments to characterize an unknown Quantum process so that is even harder than doing State tomography so it has been um an ongoing research topic in making this um scalable

but um from this we have learned that um characterizing a process in general is quite um costly um but experimentally we have we can use the trick of randomization that is we use random gates in order to impose certain symmetry over the quantum Channel hence we do not need to do that many experiments so in practice that is what uh randomized benchmarking is trying to do so over here I'm showing a simple single Cubit two randomized benchmarking protocol where we we Implement layers of random cliffer Gates where um CI are single Cubit or two Cubit

random cliffer Gates sampled from a finite cliffer gay set the final layer of the randomized benchmarking sequence implements a operation an inverse operation where the total sequence in the ideal case corresponds to an identity operation and then um after the whole sequence we measure the probability of the state getting back to the ground state at the end of the sequence in the whole randomized benchmarking experiment we vary the sequence length M and we plot this state survival probability as a function of M in order to extract error rate per gate from RB experiments we fit

the survival probability to an exponential of the following form note that uh this exponential decaying form we're fitting to is actually derived under the assumption that the noise model follows the depolarizing noise model so after fitting and extract the parameter Alpha which corresponds to the error rate in a depolarizing noise model will report the average gate error of the following form and the average gate error can also be related to average gate Fidelity or the process Fidelity which has more mathematically well defined the RB experiment is actually how we characterize single Cubit gate error and

ECR gate eror uh for backends and those metrics are reported um on our IBM Quantum platform over here and you can check those numbers for different backends over here something to note that um is that the two Cubit ECR errors for um ego device in this case IBM sherberg is an order of magnitude larger than single Cubit gate erors which um is saying um the two cubic gate Aras are actually uh the main source of error when we want to uh Implement circuits a limitation of randomized benchmarking protocol is that it is insensitive to coherent

errors due to um randomization motivated by that we have been using a new metric what we call layer Fidelity or error per layer gate in order to capture the effect of cross talk errors so this new metric layer fidelity is defined over say unconnected cubits similar to uh what we shown on the figure here we are concerned with a set of connecting Gates defined over uh those cubits that form um an entangling Clifford layers and we ask what is the process fality of this layer of gates as we have um talked about those layer structure

are quite relevant um and they serve as building blocks for a lot of circuits we care about and this layer fality includes cross talk and can be related to some other metrics to infer the aror mitigation overhead um in doing experiments the layer Fidelity experiments actually expands on the randomized benchmarking protocol so we'll see how the circuits would look like if we want to do if we want to measure layer Fidelity so first um in layer Fidelity experiments we want to define a layer structure over some physically connected cubits in this case it is a

uh 1D chain of 10 cubits the next thing we do is we want to separate this layer structure into disjoint sets and measure with randomized benchmarking protocol individually for each disjoint set that is um what the steps three is for so for each layer we will do simultaneous direct randomized benchmarking experiment with barriers for each layer in those disjoint sets what the the property is that each Cubit would only have one to cubic gate being applied on at the end of this protocol uh we will get a bunch of fidelities from doing RB experiment and

the layer Fidelity reported is simply multiplying those fidelities together so this is um the expression for the eppg metric error per layer gate we report it is simply one minus layer fidelity up to 1 over n where N2 Q is number of two cubic Gates uh in the circuit so what's nice about this metric is that it allow us to compare eppg to the errow per gate measured in isolated RB experiments by comparing those two sets of data um over here um Sol dark blue for layered Fidelity experiment and Light Blue for isolated standard RB

experiment we can already see the uh effect of cross talk era so um the takeaway here is that when the eror per layer gate is characterized in the context of a layer structure the gate fality or the gate era um tends to be worse so we get a higher uh gate error in those layered um context whereas when they are um being measured separately um they have a less error rate which um agrees with our intuition that Gates two CU Gates should be run separately rather than in parallel um over here if we compare different

processors uh blue for ego processor and red for Heron processor we can see that the difference in comparing layered Fidelity experiments versus standard uh RB experiments is less that is because our Heron device suffers less from the cross talk area so the eppg metric is actually um a Masher we can use to capture continuous Improvement of our Hardware with that I would like to point you to our um backend property where this EPG metric is reported uh in this case for IBM Torino and um this is where you can check what's the um eppg for

each back end and in this case it is 8% if you're interested in characterize this met trick on your own for different devices you care about I would encourage you to join the live session where we would actually run a layer Fidelity experiment via kuit runtime led by my colleague Sensa so with that I would thank you for joining us today and hope to see you soon