Stefan Krastanov: Yeah, I guess you can see the screen
Stefan Krastanov: and
Stefan Krastanov: go ahead, Marianne. Could you look at the chat because night around neither? I or
Narayanan Rengaswamy: Yeah, that sounds good. I'll keep an eye on the top.
Stefan Krastanov: Okay.
Stefan Krastanov: cool. Well, yeah, hi, everyone. My name is Matthews for you doing If we don't know me. I'm in. That's for Kelsey, and today my talk is going to be about one America action with surface codes, and to me very interesting that I generated slides. You should see Powerpoint
Stefan Krastanov: designer feature. And apparently this is what a surface oneameric correction with certain school to me, to it. Very nice.
Stefan Krastanov: Okay. So to give you a brief outline of the talk today. I'm gonna give you a very brief introduction accounting for some of the historical development of service goals. We're going to go over the definition of service calls themselves.
Stefan Krastanov: and learn how we can visualize single cubic errors in this code, learn how we can implement single cubic Logical operators.
Stefan Krastanov: See somehow how the cubits are actually implemented in a certain this gold
Stefan Krastanov: discuss a little bit about thresholds for discount. and finally, just i'm going to give you a brief summary of everything as well.
Stefan Krastanov: so that you know this presentation is based on the article shown there in this slide, and I'm. Using its notation throughout.
Stefan Krastanov: Just not. This is a bit different with some of the notation in the published literature. Recipes go, but I have a table in the end that will try to answer some of the questions you might have if you have encountered that this codes in the future.
Stefan Krastanov: So let's start with the overview. Well, initially, service calls are proposed as Tory codes by and if you don't know what a tourist is, it's just something like a tone, and imagine that in the service of the doughnut you put a bunch of qubits and the qubits can interact
Stefan Krastanov: if they are neighbors to each other, newspapers to each other in the in the stores.
Stefan Krastanov: and later on it was discovered that the oric structure can be waved off, and you can think of it as just a planner structure, a to designer structure.
Stefan Krastanov: This codes are interesting because they are resilient to local errors. In the qubits
Stefan Krastanov: there was a there. There is an initial implementation proposed. I would tolerate 3% of physical error rate per cycle of decoding of your code. This will be.
Stefan Krastanov: and we were going to send this later on.
Stefan Krastanov: There is some not Well, this initial interpretation assumes that you have a way to interact with the qubits that is not restricted to their locality in the in the tourist or in the surface.
Stefan Krastanov: and a more realistic implementation that would only take into account single qubit and 2 cubic gates was proposed later on, again giving you a point, 75 of a of a tolerance against a heirs.
Stefan Krastanov: You can compare this with the 2D's team and make them sure Code, and you see that they require a much more stringent value for your probabilities, so it can be successful in your fault tolerance upon computation. So you see, it's 2 to the time set to the minus 5 against point, 75%
Stefan Krastanov: cool, and as any
Stefan Krastanov: stabilize our code service calls are very interesting because you can keep track of the errors classically, and this is a very interesting feature, because you don't need to be performing a quantum operations to keep track of what's going on
Stefan Krastanov: somehow.
Stefan Krastanov: I mean you can represent the errors with the outputs of measurements which are classical, as we've seen before with other calls as well.
Stefan Krastanov: Okay. So to understand how some of these go to actually operate.
Stefan Krastanov: we have to talk about really the model of computation that is proposed with service goals. And there's any models on computation you need to be able to define. A qubit need to be able to perform a universal gate set, and you need to be able to perform measurements in a particular basis so that it can red out your data.
Stefan Krastanov: Well, this is a very interesting way of talking about things, but I won't be able to go in depth in all of them, because
Stefan Krastanov: the literature is quite sensitive, and there's a lot of things happening in. This goes for you to restrict to these 3, but only introductory a material on all of them.
Stefan Krastanov: So yeah, you won't be able to for know how you actually implement, let's say, universal. It set. But you know how you can implement single and qubit operators. And then.
Stefan Krastanov: hopefully, this can help you understanding. If you'd like how you'd implement this more complicated case.
Stefan Krastanov: single cup. It's not
like
Stefan Krastanov: logical cubic. This is one. Okay, this is. This is for the logical one. But I
Stefan Krastanov: the physical, is going to be something we are used to very much
Stefan Krastanov: cool. So what are some of our schools.
Stefan Krastanov: You see that lattice, that lattice, is it for scope?
Stefan Krastanov: You have 2 types of qubits in this class
Stefan Krastanov: you have data qubits which are in open circles, and you have measurement cubic, which are the closed surface. The idea is that you use the measurement qubits
Stefan Krastanov: to perform non-destructed stabilizer measurements in your data queues, and this is something new to us
Stefan Krastanov: from our discussions. We've been discussing about performing stabilizer management, but we never actually saw how to implement this in practice. And here this is a not a time to learn. You need this
Stefan Krastanov: additional qubits that you're going to use perform measurements so that you do not destroy that one information stored in your data. Cool. This is just one example of a lattice that you can use to define it for this gold. And this example we're going to use throughout. And again we're following the approach from the paper that I showed you in the outline slide.
so that you know each one of these green
Stefan Krastanov: Ross's, and is representing a stabilizer, and the yellow classes of process are also representing a stabilizer. And guess what the Green is telling you about a Z operator stabilizer, and the yellow one is telling you about an X operator stabilizer.
Stefan Krastanov: Oh, quick measurement cubes right.
Stefan Krastanov: This is new to us, and you can essentially thought about them as
Stefan Krastanov: of plus one minus one value that are gonna keep computing at every step of your computation. And this plus minus one value, was actually give you the ability to get some information
Stefan Krastanov: from the state of your data qubits.
Stefan Krastanov: You'll see that the circuit that performs the stabilizer measurement. We measure this data qubits in the Z basis, and they are actually what enable you to perform
Stefan Krastanov: error, detection. And it's interesting because in this idea of stabilize your goals. If you can detect the error, you can correct the error.
Stefan Krastanov: Okay, so that that last statement is not true. There are errors that you can there. Will they take that You You You can not perform the correct. Yeah, yeah, I yeah, I mean, yeah, if you're going to attack which error it was you could you would be able to. But yes, yeah, yeah, be careful with that pricing, because the thing here means that they that any error has happened. Yes.
Stefan Krastanov: thank you. All right.
Stefan Krastanov: That okay.
Stefan Krastanov: Yeah. So what do we need for stabilizing goals? We need
Stefan Krastanov: commutation relations essentially.
Stefan Krastanov: And what this lives here telling you is, if you look at the stabilizers, and they are drawn in the figure, you will see that they actually
Stefan Krastanov: respect the communication relation you expect them to respect.
Stefan Krastanov: So
Stefan Krastanov: it's very trivial to chat that whenever the stabilizers do not
Stefan Krastanov: overlap, which are here representing by the green circles
Stefan Krastanov: they will commute simply because they are acting in different data queues.
Stefan Krastanov: Now, when they are overlapping again.
Stefan Krastanov: if you're looking at
Stefan Krastanov: X
Stefan Krastanov: stabilizers is also trivial, because they have this from the same poly matrix. But when they are not when they are together in accidents, you will see that each data qubit
Stefan Krastanov: as
Stefan Krastanov: to X. Stabilize your neighbors
Stefan Krastanov: and 2 Z. Stabilize your neighbors
Stefan Krastanov: and we this gives you. Is you gonna get 2 minus signs when you compute the commutators
Stefan Krastanov: that would allow you to find that they actually commute between themselves.
Stefan Krastanov: So yeah, that's all you need to the the property we need for say.
Stefan Krastanov: and how do we measure them to stop on the previous picture, and just make sure that they are on the same page, because this is something that will come up in the next stock as well. This probably one of the really important features.
Stefan Krastanov: So
Stefan Krastanov: and this is something that you can kind of do for a classical both to it's just a different way to write down
Stefan Krastanov: a geometric structure for your parity, Jake.
Stefan Krastanov: so that it was already explained. The white qubits, the the the the actual physical qubits. So forget about the black ones for them.
Stefan Krastanov: I forget that there are queues. We just put a bunch of information storage physical objects instead of writing down the parity check matrix. Just
Stefan Krastanov: that's a matrix. We pull some geometric structure to it. We both, that
Stefan Krastanov: what is the neighbor of this lattice will belong to the same parity.
Stefan Krastanov: So, for instance, if we are to rewrite this structure again, where we care only about the wide circles, the black circles are just
Stefan Krastanov: notation keeping convention for us to know that there is some something measured there.
Stefan Krastanov: The white ones identity direct to the party checkmakers.
Stefan Krastanov: because we see every single white one. It wants to.
Stefan Krastanov: How many Shakespeare it's it's it's it's, it's, it's, it's it's it's it's, it's, it's, it's, it's it's it's, it's, it's, it's it's, it's it's, it's it's, it's, it's, it's, it's it's, it's, it's, it's, it's, it's, it's, it's, it's, it's, it'
Stefan Krastanov: for no your relevance. and every call I'm. Looking at the geometric structure we'll have again for.
Stefan Krastanov: and there will be 2 of them will be for a zip. Check through them will be from an exchange for 2 different.
Stefan Krastanov: So is this sort of a diagram representation of the matrix.
Stefan Krastanov: All right.
Stefan Krastanov: this not yeah, not really. This is representing actually the 2 bits. So the these are Well, yeah, these are all Cubans, both the the the black ones and white ones. But I guess we'll step. This point was, You can.
Stefan Krastanov: if you imagine that the black ones are not 2 goods at all. They are just giving you this classical. You know a a representation that you can check using fairy-check matrices for the logging, which I see ancillary cubits. Yeah, yeah, let me say something almost wrong.
Stefan Krastanov: Do you know how drafts Can you represent the graph on paper? That's a pretty picture?
Stefan Krastanov: Well, this is the geometric draft representation of our call
Stefan Krastanov: the parity-check matrix. Is it's not the but the parents check matrices against sound matrix representation of the signal information.
Stefan Krastanov: And there is another question
Stefan Krastanov: just yeah, I was wondering. So
Stefan Krastanov: can I describe the I guess, to add on check for an error quickly, and then just to make sure I understand the whole concept. So the white qubits they all the
Stefan Krastanov: one of the
Stefan Krastanov: So let's say I wanted to measure one
Stefan Krastanov: check one way you did.
Stefan Krastanov: for whether or not there's there I would have to conduct 4 measurements through the stabilizing code. See what the output of that is. So this is kind of like a
Stefan Krastanov: So it's a crucially important feature that you never measure a single data qubit.
Stefan Krastanov: You always measure either 4 green things or for the little things. Okay, so every time I do one measurement it measures all 4.
Stefan Krastanov: It measures something about all 4, you know.
Stefan Krastanov: and this just a quick this is going to be, I I hope, Clear in the butter slides. We're gonna see exactly how to detect the area
Stefan Krastanov: one. I guess the measurement queue that you always initialize it in the 0 state.
Stefan Krastanov: You'll see okay, You see, okay, here's the other thing. Is this is error
Stefan Krastanov: correctly be coded. What happens when i'm down there location.
Stefan Krastanov: you'll that you're also gonna see
Stefan Krastanov: is, yeah, if it's about the the the the measurements happen to will shortly explain that I I I's about the boundary, right? Yes, yeah. So the boundary is just going to work the same way, but you don't have all the for it. You have less.
Yeah.
Stefan Krastanov: This I don't know. If you previously or
Stefan Krastanov: E. Each data queue is not a logical to No, no, no, no, you know you're You're You're you're gonna get hit. Exactly. I don't think that was gonna be us.
Stefan Krastanov: Okay, yeah, I mean, if it's okay. Do you think the pace is good because you can see it? Thanks. I don't know. And please like. Let's
Stefan Krastanov: let's
Stefan Krastanov: try to give him a the a bunch of slides on a topic. But
Stefan Krastanov: let's make big bosses. You haven't. You haven't that I out way to the All right, so let's go cool. So how are your back, your stabilizers? Well.
Stefan Krastanov: the idea is that by
Stefan Krastanov: first my bedroom, disabled as you are projecting qubits into the eigenstates of the same lesson. Right so, or in that in space of this generated by this stabilizer.
Stefan Krastanov: and the X measurement is going to be controlled by data qubit. You can see the the circuit that we're going to use to perform these measurements here in the bottom.
Stefan Krastanov: and the Z are going to be controlled by measurements. So if you look at these 2 circuits, they look X
Stefan Krastanov: very similar to each other. One of them has control lines from the data
Stefan Krastanov: to the measurement. The other one has controls from the measurement to the data. And if you also look carefully done that your question comes here. Now, if you see this line.
Stefan Krastanov: the management qubit is in Nicholas as takes 0,
Stefan Krastanov: but then he goes under some transformations, and he gets measured in the Z basis.
Stefan Krastanov: and then
Stefan Krastanov: you started a 0. But you might have a 0 or one later on, and, in fact, depending on your measurement outcome right?
Stefan Krastanov: If you had a 0, you're actually projecting into a plus one eigenvalue.
Stefan Krastanov: If you had a one you're projecting to the minus one eigenvalue of your stabilizer
Stefan Krastanov: subspace. These are some of
Stefan Krastanov: Is there any? It's a chest to fill up space to match the amount of.
Stefan Krastanov: If you compare these 2 again, You see, there are 2 identity operators here, and these are simply weight operations in your on computer, so that the circuits have exactly the same down.
Stefan Krastanov: If you want to measure on the the Z stabilizer here?
Stefan Krastanov: Okay. So I just figured that this is swapped
Stefan Krastanov: because these is the axe measurement, right? And these are the same measurements.
Stefan Krastanov: So these circuits should be exchanged.
Stefan Krastanov: Which? Yeah. Okay.
Stefan Krastanov: So
Stefan Krastanov: this one's for the X right? And you see that you have some hot of art operation. You're applied right after your store. G: because you want to start your measurement qubit in a state that is an eigenstate of X.
Stefan Krastanov: So yeah, these were swapped. And it's important to also know that the order that you're performing the C knots matter? Okay, you need to do this sort of zigzagging pattern. You do a, B, C, D. And if you change this order you lose
Stefan Krastanov: the the the computation property we have.
Stefan Krastanov: So there are more orders you could use, but not every order will give you the structure. So I don't think you lose any of the commutation properly. That's fine. You can totally permute the
Stefan Krastanov: 3 4 5, and 6; but if you permute them it will be a little bit more difficult to do everything with parallel, because you want to
Stefan Krastanov: the the checks on the line. So some this is just an engineering question of how do you make the 2 cubic gates not flash with each other. So, if you permute them, it will be a little bit more. And knowing to make sure that
Stefan Krastanov: while you're doing the z-checks, you're not in logging the qubits that are listed. They's checks, I see. But if you just do all the checks, one after the other, then the order. Okay, okay.
Stefan Krastanov: You mean the order of DC. Nuts. Yeah, yeah. And another thing about the flipping of the scenes control and the target. The intuition behind that is that the going into the eggs basis basically flips control and target.
Stefan Krastanov: You can't see on the Z basis. If you go on the X basis.
Stefan Krastanov: basically you have flipped the control on target of the with respect to the new basis. So to an extent, this occasionally a confusing name here.
Stefan Krastanov: because it's not like the control controls.
It depends on the basis choice, I see.
Stefan Krastanov: Yeah. And then again, all the is in greed are are going to be measured for each step. So you're going to perform magnets for all. Oh, well, yellow and green.
Stefan Krastanov: A measurement cycle is a fine to me is just
Stefan Krastanov: one stabilizer measurement. Okay? So the Z cycles are going to allow you to identify X
Stefan Krastanov: errors. The
Stefan Krastanov: back cycles are going to help you identify the errors, and you can identify why errors by combining your results from both
Stefan Krastanov: measurements.
Stefan Krastanov: And again, one stat of our our error. Correction is essentially
Stefan Krastanov: measuring all of the stabilizers in the bridge, so in the paper. They mentioned that you need to to have this sort of lockstep, so you need to measure all of them once, and then you do all of them again.
Stefan Krastanov: so on, so forth.
Stefan Krastanov: So you couldn't do. Let's say i'm gonna measure this stabilizer, then this, then come back to this. No, you should go over all of them and then restart it process once all of the
Stefan Krastanov: Okay. So what are the Save Lives States from this measurements Again, we know they are going to have either a plus one or a one value, showing up for each one of the measurement qubits we have in our grid, and ideally
Stefan Krastanov: these are signaling to into which eigen subspace of your stabilizer you are into.
Stefan Krastanov: and as from similar as what we've seen before the stage that we get out of performing all the measurements is a state that is simultaneously stabilized by all of your
okay.
Stefan Krastanov: which is a minor, notational issue. Here it's any implied that you have the same X chicks and Z shakes for the same 4 qubit. Oh, yeah, this is just a exposition. This is just an example.
Stefan Krastanov: You you won't have something like this in the actual, the the overlap halfway. No, there the 2 days like this of chicks and the Z lasts are all set for one another.
Stefan Krastanov: I'm. Curious, based on the measurement results. Does that tell you something about the the quality
Stefan Krastanov: that I don't know? Let's discuss that during give a second.
That might be a good place to talk about. I guess I might have some. My, okay.
Stefan Krastanov: let's go. All right.
Stefan Krastanov: Okay, yeah. So this leads us to the introduction of Chiassin States, and it's interesting because when I talk to Stephen about this, he asked me, what are by us and States to begin with, and this is just their terminology to talk about your cold states. Okay, meaning a a quiet and state is just
Stefan Krastanov: the state of your data qubits all together after you perform the join stabilizer measure. And this relates to what you asked.
Stefan Krastanov: Okay. in the
Okay.
Stefan Krastanov: yeah, the the that you're encoding your your information into the entire
Stefan Krastanov: again. There are vectors that are stabilized by all of your stabilizers. You can in neutralize the State
Stefan Krastanov: by performing one measurement cycle.
Stefan Krastanov: Because, imagine you can I give you this rule or this grid of qubits? They can be in any safe. But after you perform the measurements you did all your projections. And then, after one cycle of measurement, you're going to have a State.
Stefan Krastanov: and this date is specified by your plus one and minus one values. Okay, now, there are 2 questions that are going to ask on the next slide. We're going to answer the next slide, but
Stefan Krastanov: it's. Maybe if you want to take a shot there. How many python states you have, if you have end data qubits, and what is the size of the stabilized subspace when you actually have K stabilizers.
Stefan Krastanov: anyone wants to answer this.
Stefan Krastanov: Okay. So
Stefan Krastanov: no
Stefan Krastanov: 2 to the end for a first. Okay, that's correct. for the second is to to the K. Is this the size of your subspace?
Stefan Krastanov: Okay. in this case 10 is 39 in the case. 38
Stefan Krastanov: can yourself here for for a second. Why is in Turkey 9 and K. 38?
Stefan Krastanov: Because well. yeah, the idea is
Stefan Krastanov: you have, you know, here it's just you have the cubits. Okay, yeah, you are. The Us. Numbers come from when you look at the lattice. Oh, yeah, you just count the the
Stefan Krastanov: open circles. and you count the black circles.
Stefan Krastanov: and so the open circles are Well, if you do the math, I think it's like, what is it? There's 3 there's 3 outs, each one has 6, 18, and then 20
Stefan Krastanov: for the greens.
Stefan Krastanov: and then 38 for for for most of these for practical purposes we'll probably look at the mostly squerish lattices.
Stefan Krastanov: and there will be a lot of things will be expressed in terms of the size of the levels.
Stefan Krastanov: Yeah, go ahead
Stefan Krastanov: that it should be easy, and X's. You said, or the to the the white and black.
Stefan Krastanov: So can you read that you said those?
Stefan Krastanov: It's basically a number of things in the lattice? What it the number of X's and D's. Or was it
Stefan Krastanov: number of okay.
Stefan Krastanov: those circles? So actually, that's a very great question. You have as many close circles as you have stabilizers. Okay.
Stefan Krastanov: Sorry. You said that the like subspace.
Stefan Krastanov: Yeah. But it's 2 to the Yes, yes, yes, yes, yes. So yes, yes. And my
Stefan Krastanov: you're great. Yeah to to down landscape. Yeah.
Stefan Krastanov: Question of the statements. So are there differences between asymmetric that
Stefan Krastanov: that might be something we'll be discussing in the next in the next number.
Stefan Krastanov: But yes.
yeah.
Stefan Krastanov: Okay. So
Stefan Krastanov: this is the space we have for queues.
Stefan Krastanov: Why? Because we know that we.
Stefan Krastanov: as we're performing our measurements we're gonna have plus one values for each one of these stabilizer maximum qubit, whatever you'd like to call them. This introduces you constraints right, because, as we said, we'll know. Okay, this subspace of my
Stefan Krastanov: entire equipment space is actually one of the eigenvectors of my stabilizer.
Stefan Krastanov: and you can imagine that you have to to the end a space of qubits with 2 to the gate constraints, and this would give you a logic space that would be. I think there's a mind. This is to be N. Minus K. 2: yeah, this is their
Stefan Krastanov: There is your lunch to the spaces, 2 to the end, minus K. In the setting, which means.
Stefan Krastanov: however, constraints you have orchestrating your your the variability you can see at in your quies and state to keep because you
Stefan Krastanov: in there. Okay. So again, this is restricted to class and states, because.
Stefan Krastanov: as we said, they're the ones where we're projecting, there are. There are the output of our projection. So sorry I don't know. Restricted through. I I mean this cannot be any sort of state you would like to come up with.
Stefan Krastanov: Well, you have. What is it? A and cubic? You? You can have infinitely many states, but you cannot find every single state you'd like. They have to be stabilized by your stabilizers. That's the I.
Stefan Krastanov: This is the your call, somehow your code space.
Yeah.
Stefan Krastanov: So in in the paper where they introduced this term, not a
Stefan Krastanov: this is, for
Stefan Krastanov: do they restrict to pure states, or are you from peer-
Stefan Krastanov: your support?
Stefan Krastanov: Why, it Well, it just feels a little bit like
Stefan Krastanov: before you do like your direction stuff to stay in
direction.
Stefan Krastanov: I know it seems a little bit
Stefan Krastanov: so. The idea is just to say that
Stefan Krastanov: when you imagine you have a class in state. When you do correction, you also have a class in state. Your precedencies are changing.
Yeah.
Stefan Krastanov: it's just one of these kind of things also here like possible error corrections. And I I would just see the terminology from there. So
Stefan Krastanov: it sounds cool, but it's it's it's a little bit. Yeah, we can talk about. Yeah.
Stefan Krastanov: just this. Maybe it's a good question. But is there any significance to the etymology of the
Stefan Krastanov: I don't know.
of course.
Stefan Krastanov: Okay.
Stefan Krastanov: Okay, yeah. So here in this case again, you have a 39
Stefan Krastanov: data queue. It's 38 physical qubits. And you can.
Stefan Krastanov: Yeah.
Stefan Krastanov: Sorry. 38 measurement to physical springs which means you can extract one qubit out of this.
Stefan Krastanov: All right. I didn't tell you anything and how you're going to do this, but you can expect one. And this he said, that one logical queue, this whole thing here is one logical qubit.
Stefan Krastanov: and this is forgiving that.
Stefan Krastanov: And this is bad. Yeah, like these are bad. But if anything, this is this optimistic.
Stefan Krastanov: So we have more work to do. And there are. There are issues with this number being actually useful in practical. Yeah, okay.
Stefan Krastanov: I mean, you would be very impressive. You could do anything with 38 physical humans. Right?
Yeah.
Stefan Krastanov: Okay. All right.
Stefan Krastanov: Is there any reason why you want to go to that
Stefan Krastanov: to this large yeah.
Stefan Krastanov: Why not like 5 like Yes. So in the do you see the rates? I showed you before like any, since it's already. It's sooner we what the rate is you on the screen? Oh, no, no, no, no, I mean the not the rate. Sorry the the the
Stefan Krastanov: so you have 10 to the minus 5 in one of them before, and then. But okay, but
Stefan Krastanov: it it's just that. I mean they're raised the independent phone system size, right?
I guess.
Stefan Krastanov: So. I'm very happy that they, because there is a question there is an answer in that direction. So for sure, the torque is already much better than all the other examples we have worked with.
Stefan Krastanov: because it has a better threshold that the area of which it becomes remotely feasible to maybe considered using.
Stefan Krastanov: It's already yours, mind you, better than than in other situations.
Stefan Krastanov: So that's one of the
Stefan Krastanov: the geometric interpretation also gives us a way to do for multi-corders, which is not obvious. We have any polar issue that which is a very big deal we discussing in the second time. But why do you size of the the Tory calls? Well, if you have a small Tory calls
Stefan Krastanov: that you the the the, the
Stefan Krastanov: the line representing
Stefan Krastanov: logical error rates versus physical air rates is not particularly the scheme.
Stefan Krastanov: So, passing the threshold, the years to you got a little bit less here, but not by much become bigger and bigger. The system is
Stefan Krastanov: the more upgrade the thresholds are disappearing, the more it starts, looking like a phase transition between completely rent information and completely preserved information.
Stefan Krastanov: As the system side increasing, you get to observe our faith transition between perfect and
Stefan Krastanov: absolute gender.
Stefan Krastanov: Yeah.
Stefan Krastanov: I also.
Stefan Krastanov: So the whole thing where we have one qubit. Is that mean? Like if we want you? If we had a theoretical out of that prior, 3, 2 bits.
Stefan Krastanov: We need 3 of these lattices.
Stefan Krastanov: Yeah, you will. That is a very good question. And you're gonna see in a couple of slides how you can create
Stefan Krastanov: more than one cubic inside a lattice. Okay?
Stefan Krastanov: And the answer. But I think you can also do what you have multiple lattices together, right? Using a
Stefan Krastanov: Oh, I have 10 slides. Okay, I wish okay. You just say no to questions.
Stefan Krastanov: Okay. Aw: from meet you? Yeah, All right. So yeah, let's see how you detect errors. This should be, I guess, the what you're asking before. I imagine that you have this cubic a here in the center of these
Stefan Krastanov: grid, and it goes through the apps flip all right. So you know that x a anti commute with your zac, b, c, c, z, d, and with the a/C easy app. G.
Stefan Krastanov: This means that if you have a flip and you perform a management after you got the flip, you're gonna see a difference in the signs for your measured qubits
Stefan Krastanov: right after you performed a measurement.
Stefan Krastanov: This means that if you in one cycle you had something right. The 2 values, let's say minus one and plus one after the cycle you're gonna have plus one and minus one a flip in both, and that is a key
Stefan Krastanov: idea. You have a flip in both measured qubits. There are neighboring your data qubit, and this is also true for your acts measurements right? So if you had a Z. Flip, You're gonna look at the X
Stefan Krastanov: columns in this case, and if you see a joint flip between 2 Cubans.
Stefan Krastanov: then you figure it out an error. Okay?
Stefan Krastanov: Well, again, why gives you a flip in the forward?
Stefan Krastanov: Because it's just a flip and Z and a flip, it.
Stefan Krastanov: Yeah. So going back to the question, how about see on the boundary? So if you got it, okay, so I crop this image, or it wouldn't be on the boundary.
Stefan Krastanov: Yeah, that's it's drum. It looks like the bounder. So it's your others. Yeah. Yeah. So you you just gonna see a flip in here
Stefan Krastanov: because this is the only at stable z stabilizer. That is a non-trivial that is actually on seen on triple.
Stefan Krastanov: So you'd see a flip in here in the boundary. but you still figured out 2 flips from for Yes, in this case.
Stefan Krastanov: But so if you're going across the lines, and you're measuring across like that, are you going across the black measurements or across somebody.
Stefan Krastanov: So if you, if you have an error here, you've got a flip in these 2,
Stefan Krastanov: that if you had an error here
Stefan Krastanov: last year if you had an error. Here you go this you
Stefan Krastanov: and if I look joined you got a joint.
Stefan Krastanov: They're in, both of them with zoom. You wouldn't see a flip in this one, but you see a flip in the left one, and then the right one. It's useful to kind of forget about the black. It doesn't matter how the measurement is done in the examples he gave. Yes, a measurement was done with one black you within the center.
Stefan Krastanov: for some reasons that we'll discuss later on. That's actually a bad way to it. Doesn't really matter. How would you?
Stefan Krastanov: The the dreaming blog is wrong measurement. The yellow blob is an automation. the results on the blob as a call, and that blog
Stefan Krastanov: the one on the left. It touches a number of qubits depending on how many of them flip that blob as a whole, we have a trivial or an Ontario measurement result.
Stefan Krastanov: Yeah.
Stefan Krastanov: do you need all the data you could super, and with which you started with, Can they be like? Are they called over and towards loss? Yes, okay.
Stefan Krastanov: How much time do you have to let's say that. Okay, how do you read out errors? Well, very cool.
Stefan Krastanov: And I guess. Okay. The way you detect it is, if you perform a measurement here, and you only see one sign Change. then. That's an readout error in this measurement qubit.
Stefan Krastanov: Okay. But for now that you can only have one keep it going wrong
Stefan Krastanov: at a staff, because it's easy this way. You can identify what what happened right as we were discussing in the beginning. But yeah.
Stefan Krastanov: the 10 of them by every out there in one of them. And again, this is true for all, every measurement that you have. All right. Let's put all this together. They have this very beautiful picture in the from the
Stefan Krastanov: the
Stefan Krastanov: the paper.
Stefan Krastanov: Remember that detecting errors, meaning the tanting, the changes in the signs that you have
Stefan Krastanov: in your measurement measurement qubits. This is
Stefan Krastanov: somewhat wrong, I mean, if you can, of course identify for sure which error you have, that you can correct it if you see here the C. Is just showing you a flip in the in 2 cubits that are in the, in the, in the column direction, the axis showing a flip
Stefan Krastanov: in the role direction, the Y showing you of a 4 flip right in all the corners. So the X Y and Z here there. So
Stefan Krastanov: the axis, the the axis of is time, the time step, and if you all these squares represent winter and error has happened on that time Exactly. And then the
Stefan Krastanov: red lines.
Stefan Krastanov: the red lines are showing you the flip. Okay. So here you have a minus it a plus, and they become a plus and a minus. Okay. So on and so forth. And for the measurements you only see one red line.
Stefan Krastanov: Essentially you.
Stefan Krastanov: Of course, if you have.
Stefan Krastanov: Yeah, yeah.
Stefan Krastanov: And the M. Here is also an error. And and well, what this bullet is trying to convey is the following: If you have this scenario all right.
Stefan Krastanov: You could assume somehow that these were actually the errors you got, and then you could correct for all of them.
Stefan Krastanov: If the red lines you can then use yeah, X, Y. And Z. And exactly Exactly. So. This is some graphical representation of the lattice, but also for what's happening here measure. Yeah, yeah, this shows you the years pretty much. The
Stefan Krastanov: circles are every single measurement you perform, and then layered the go through performance multiple
Stefan Krastanov: And guess what. If you can do faster measurements.
Stefan Krastanov: you can improve your detection the faster can't matter right. The more more likely you are to not skip. Let's say 2 errors, or I don't know how many errors that you'd like to do this, of course, instantaneously if you don't have measurement there. Yeah.
Stefan Krastanov: there we go
Stefan Krastanov: all right. So lots of operations. This is good. I have a tool to to to read, slide the way you do this is you literally perform
Stefan Krastanov: axe operators in your data qubits when you're connecting through a line
Stefan Krastanov: that connects to boundaries of your surface code.
Stefan Krastanov: So in this figure here you see that this surface scope has
Stefan Krastanov: a Z boundary on the top, a Z boundary at the bottom, an X
Stefan Krastanov: boundary on the left and ax boundary in the right, and if you connect
Stefan Krastanov: the data queue is in here by performing and by connect, I mean performing an X operator in each one of them. The physical X operator. What you get is a logical.
Stefan Krastanov: This is how we implement a logical operator.
Stefan Krastanov: If you do this in the
Stefan Krastanov: so, this direction with Z. Flips, you're gonna get a logical Z operator
Stefan Krastanov: all right. If you use both strategies. you're gonna get a logical y operator.
Stefan Krastanov: And finally, it's interesting because any found any path that connects these boundaries. If you follow these rules, will act for you as a logical flip.
Stefan Krastanov: So what i'm showing here is the smallest
Stefan Krastanov: sequence of operations you need to perform in order to implement a logical Z and a logical X. And the way any path works is because of the stabilizer structure.
Stefan Krastanov: all right. So if you want to know more about these logical operators
Stefan Krastanov: well.
Stefan Krastanov: I think we won't have time to. Yeah, we're not far. Yeah. So too fast just to to say 2 things quickly in here. They commute to the stabilizers, which means they will map a code States to a code state. If I ask them state to rest and state whatever term logical to use, and they the important part is that they change that state
Stefan Krastanov: non-trivially right. So after you perform a logical Z in your cold state.
Stefan Krastanov: you're gonna get something that is not proportional to it. Does not trivially related to it. Okay, it's different. And what this actually gives you is that
Stefan Krastanov: this gives you all you need to implement a qubit somehow. Why? Because if you want something to implement a qubit. You need to have this commutation relation with some policy operators, and these are how the operators are going to give you these competition relations.
Stefan Krastanov: All right. Let's fast forward. So yeah, here's your qubit, right? You're gonna get all these.
Stefan Krastanov: You can think that it can somehow implement a qubit in this.
Stefan Krastanov: for instance in this. A lattice here, and just one thing that
Stefan Krastanov: I think it's interesting is.
Stefan Krastanov: and recall Correct me if I am wrong. Weeks. This d here mean being 5.
Stefan Krastanov: Give you some intuition on what is the minimum number of errors you need to have to flip a logical state without being able to detect.
Stefan Krastanov: if you have 5 lips
Stefan Krastanov: in either direction, any sequence of flips that connects to boundary such. But it this way. You're gonna get a logical error, and you won't be able to figure it out.
Stefan Krastanov: So the larger the the just, the the the farther the boundaries are between to other, the more is zoom if you are to these lunch orders. So just to rephrase some of this.
Stefan Krastanov: So remember now, when you go back to the the example of one error with the 2 stabilizers on the 2 sides, this one yeah, so say on a, we have an error that is dictable by the Z Stabilizers, one single p that experiencing an error
Stefan Krastanov: leads to 2 measurements. So being right, right? 2 measures.
Stefan Krastanov: because that quit is.
Stefan Krastanov: say, 2 neighbouring Cubans were experiencing this
Stefan Krastanov: error. Well, then, we will have an end measurements in the
Stefan Krastanov: right right.
Stefan Krastanov: but just just the extreme ones, not the one in the same way, because the one in the center. No. Belongs to 2 cubs. That is experience.
Stefan Krastanov: So a chain of errors
Stefan Krastanov: we will look like only 2 bright red.
Stefan Krastanov: You don't get to know about the errors in between. There's inside you get to the date only the the edges of the chain of pairs.
Stefan Krastanov: and that's why it matters how big your code is in your office is because your if you want this is big enough.
Stefan Krastanov: It's very difficult for a random chain to connect the 2 sides of the lattice and perform an undedictable logical operation, because the moment you reach the edges
Stefan Krastanov: they you don't have anything to to let you know that there was an error happening. Those 2 blocks just disappeared. No, this is sufficient to just have
Stefan Krastanov: I want to.
Stefan Krastanov: If you have only one red block, exactly one red line.
Stefan Krastanov: I think that means that you must have had a measurement here.
Stefan Krastanov: I see.
Stefan Krastanov: and that's why there there there was the tradition of the treaty lattice, where the third dimension was the time stairs, because then the same logic about connecting strings of
Stefan Krastanov: driving involves but in treaty, where the third dimension is time. The same logic leads to also being the able to take measurement here.
Stefan Krastanov: So it means that you can take it to the 3 lies, and then the same idea would: yeah, yeah. But that's why that's why the size is related to the distance of the call. Because the more difficult is to make a chain that connects the sides. The more difficult it is to
Stefan Krastanov: to
Stefan Krastanov: this is the exact same reason why it doesn't matter which, if you want to play a logical X-ray it doesn't matter which
Stefan Krastanov: That's a really important point. Imagine that the errors for the closed chain in the center, not touching the ages.
Stefan Krastanov: Well.
Stefan Krastanov: a closed loop doesn't have
Stefan Krastanov: doesn't have It's
Stefan Krastanov: So those 2 blocks will be will disappear. Yeah, that dial of disappearing blocks is fine. That's so. We kind of can't discuss it. It's about the generate goals. This some operation that happens that doesn't change the logical state. It's not in the technical area. It's what we launched. Operation. It's fine.
Stefan Krastanov: If they reached all the edges and disappearing engages instead of cancelling each other out.
Stefan Krastanov: then that so?
Stefan Krastanov: Well, either if you do it on purpose. It's a logical iteration, or if the errors did it without you knowing it, happen it's a logical error.
Stefan Krastanov: Yes, maybe this is
Stefan Krastanov: we're running out of time.
Stefan Krastanov: One should be a bit careful with saying, okay, like you have to add the loops off that that that's fine, because that's all. If you do this on a target, you would run into the trouble.
Stefan Krastanov: So it depends also on
the underlying.
Stefan Krastanov: It's announced that it will be minimally well need to.
Stefan Krastanov: Yeah. So the one there will be.
Stefan Krastanov: You really. Do you want to give her a distance of your
Stefan Krastanov: Okay? So just quickly, you can imagine that, based on everything you said, I can imagine that your States, with all your data qubits, has this form somehow, where you have a subspace of a sub, vector your state that is stabilized.
Stefan Krastanov: and whatever, when you perform stabilizers, you're not messing up with the logical part of your state, and you have a part of your say that you only access through logical operators.
Stefan Krastanov: Pretty cool. Okay, there is a way you can construct qubits inside large lasks.
Stefan Krastanov: Let's skip that one and a part of this is bad, because this is what I'm gonna tell you how you can initialize a qubit and how you can measure a qubit. But if you look at the paper you should be somehow. I guess
Stefan Krastanov: this is also really the last mention of what Kenneth said about. Be careful when you say calls loops, if go around the call, then there's
yeah.
Stefan Krastanov: And yeah, if you just just one sent this on this, if you want to know what the whole is. Is you shut down your measure, your stabilizer measurement for one cycle, and that will create a whole.
Stefan Krastanov: because when you do that you are not. It's restricting any constraint in your state space
Stefan Krastanov: for that stabilizer, so it's a to remove this. This is the thresholds. This is a quick figure for the threshold. They perform a simulation of the gold right with Varian deep for errors.
Stefan Krastanov: They did a very broad sort of error. Investigation, considering all these errors here, and I give an error probability, and to be fair, I don't know if they did an an average of all of these.
Stefan Krastanov: but they report. This particular.
Stefan Krastanov: Yeah, if you want to see the logical errors is here in the Y Axis, the error physical error age is on the x-axis, and you see how the threshold changes as you use your distance. Right? The threshold. Yeah. Sorry. Yes, yes, yeah. You're going to basically
Stefan Krastanov: So it becomes much more the improvement in the quality of the logical qubit becomes drastically better. Many orders of mind you better if you increase the
Stefan Krastanov: and that's why you might want to have such large what I mean.
Stefan Krastanov: I mean kind of thing. But I mean we would also have to analyze how you approach this like
Stefan Krastanov: actual business. Maybe they might have some some trade off between. You know how much you want to increase, and also like
Stefan Krastanov: you know how much she gets from
Stefan Krastanov: from the I mean, it's not good for you, but
Stefan Krastanov: me going today
Stefan Krastanov: 15 to 25. I don't see that. Much.
Stefan Krastanov: Yeah, yeah, that that's fair. And also you need to consider how expensive it is to need cold when it's bigger.
Stefan Krastanov: And just another thing this feature exists for basically
Stefan Krastanov: all
Stefan Krastanov: where you have some size parameter for the calls. You have some form, face transition and a pressure like.
Yeah.
Stefan Krastanov: Okay, yeah, You can see more of the there's extra in the paper, and this so for a summer you can skip if you want to take a look at this table. There is in a paper, too, and i'm gonna
Stefan Krastanov: give you the slides. It gives you a comparison between the terminology they introduce. And that's because terminology right? Just remember that they are stabilizer Goals surface schools are a simple, as they are called in a lattice of qubits. Yeah, you use
Stefan Krastanov: additional kip is to perform the instructive stabilizer measurements. Your code rate will depend on the lattice, of course. and the cube is selected here. Means.
Stefan Krastanov: if you're creating qubits, you can create more than one qubit for a given lattice. This would improve your rate.
Stefan Krastanov: Finally, the distance of your errors will depend on how far the boundaries are from each other. and this also true. When you create Cupid's, too. By the way.
Stefan Krastanov: Okay, that's the end. Thank you.
Stefan Krastanov: Let's have the second speaker set up. So most of the questions probably will overlap for both.
Stefan Krastanov: And as you're doing that, I want to mention
Stefan Krastanov: about that again about the rate. And this is a disco. So we have a distance. That's If we have in Quebec on the square lattice.
Stefan Krastanov: the distance will be square root of N. Because the number of
Stefan Krastanov: the surface, If any cubits, then the side is so the the the distance in terms of the number of us. You have this word on that. and the relative distance
Stefan Krastanov: us, which means the distance over the total number of cubits it's Sc. To F over N. Which is one over the
Stefan Krastanov: which doesn't scale up the rate of and growth. If the relative business goes to here
Stefan Krastanov: they're raised
Stefan Krastanov: is one over N, and that even more obviously go through 0. So this goal is actually
Stefan Krastanov: painful.
Stefan Krastanov: It's it's not the
Stefan Krastanov: practical calls. The the the nearest neighbor. Constraint on the interactions is wonderful. Many codes require difference.
Stefan Krastanov: How your connectivity
Stefan Krastanov: So that's that's incredibly valuable to the nearest neighbor constraint. but
Stefan Krastanov: it leads to this terrible rate in a terrible distance. And so it was only recently, like
Stefan Krastanov: last summer. That sound proved that you can cash quantum goals that have both
Stefan Krastanov: cost on trade and cost and business.
Stefan Krastanov: Sorry. Someone constructed my controls that that
Stefan Krastanov: relative
Stefan Krastanov: Yes, sorry. Relatively. This is an important difference here relative to things versus the the distance of the historical draws their relative distance. The distance, with respect to how many queue that you have in total
Stefan Krastanov: goes down.
Stefan Krastanov: Okay, Should we have the next speaker set up?
Stefan Krastanov: Yes, sound check again. Could you confirm that you can hear us so we can hear you, too.
Evan Anderson: Yeah, I can hear you. Can you hear me?
Stefan Krastanov: Yes, yes.
Evan Anderson: okay.
Evan Anderson: Share my screen.
Evan Anderson: hey? Can you see my screen now?
Stefan Krastanov: Yes, but Excuse me, give me a second, because the device from which we're streaming is having issues.
Evan Anderson: Okay.
Stefan Krastanov: Okay, there were no issues besides me being stupid.
Stefan Krastanov: Okay.
Evan Anderson: Okay. So
Evan Anderson: just heard about torque code and surface codes. So
Evan Anderson: i'm gonna talk about how you might decode
Evan Anderson: these surface codes specifically with minimum weight. Perfect matching.
Evan Anderson: Yes.
Stefan Krastanov: Okay.
Evan Anderson: sorry. What was that?
Stefan Krastanov: So?
Evan Anderson: Oh, yeah, sorry.
Evan Anderson: So i'm Evan Anderson. I'm at the University of Arizona Fourth year.
Evan Anderson: Phd: students.
Evan Anderson: I've mostly been working on quantum information theory
Evan Anderson: work specifically as it relates to covert communication.
Evan Anderson: But i'm slowly transitioning into coding theory.
Evan Anderson: And
yeah, so
Evan Anderson: just introduce the talk. So there's going to be a a few concepts that are similar to what
Evan Anderson: Taylor's talked about. So is this idea of threshold.
Evan Anderson: So the threshold is a way to describe.
Evan Anderson: When a system can form fault, tolerant, con
Evan Anderson: compute, computation, or act sort of as a perfect memory, and there's the caveat here that
Evan Anderson: this occurs when your system, size, or your number of qubits goes to infinity.
Evan Anderson: The
Evan Anderson: obviously the threshold is going to differ system to system depending on what's being modeled.
Evan Anderson: What your noise, error, model is, or whether or not your including measurement in the system.
Evan Anderson: For
Evan Anderson: on computing, whereas you don't necessarily need that
Evan Anderson: through your memory.
Evan Anderson: there's also this idea of a break even point. So i'm not gonna talk about this necessarily, but it's it's somewhat related to this idea of threshold.
Evan Anderson: And so, if you're reading the literature. I
Evan Anderson: A break even point effectively is. if you have, say, a 100 different qubits.
Evan Anderson: and some are a little bit better than others, and you're able to identify, hey? This is my very best qubit.
Evan Anderson: physical qubit.
Evan Anderson: Now, can I create a logical qubit out of those 100 physical qubit that performs better than my best physical qubit. Because if you can't.
Evan Anderson: then why not? Just.
Evan Anderson: you know, do everything with your best physical qubit, instead of encoding everything
Evan Anderson: into a logical qubit.
Evan Anderson: and my understanding is that this this break, even point, at least for
Evan Anderson: superconducting systems.
Evan Anderson: has only been
Evan Anderson: observed.
Evan Anderson: Yeah, within like the last year, maybe even the less
Evan Anderson: 8 months or so.
Stefan Krastanov: Yale University. And
Evan Anderson: oh, really
Stefan Krastanov: they they! They have the paper on demonstrating it, you know, contrived manner
Stefan Krastanov: a lot.
Stefan Krastanov: I 6 years ago.
Evan Anderson: Oh, okay, I was. I was
Evan Anderson: referring to the work that Vladimir. It done with Michelle divers group. Or is it? Yeah.
Narayanan Rengaswamy: but I think you. But I think your point is for for i'm trucks, probably because I think continuum with this experiment last year.
Evan Anderson: Yeah.
Evan Anderson: right? And so again. This is
Evan Anderson: what I sort of just talked about.
Evan Anderson: So the threshold depends on the code itself. So right now we're looking at the surface code or the or specifically, maybe, the torque code.
Evan Anderson: And so
Evan Anderson: if our stabilizer measurements for each set of plaquettes or vertices look something like this, we can create a
Evan Anderson: circuit.
Evan Anderson: and then this circuit is going to have errors. And so maybe there's just a depolarizing error where
Evan Anderson: we're not performing any gates. But oh. a bit. An X Eric Kurz or Z. Eric, or why Erik is here.
Evan Anderson: When you actually perform the gates. there could be
Evan Anderson: get errors, so does it see not actually perform a C. Not, or does it just
Evan Anderson: maybe it just performs it, not gate.
Evan Anderson: And then also. there can be errors with your measurement
Evan Anderson: that you'll need to introduce when we're considering the threshold.
Evan Anderson: Likewise, we need to consider the decoding algorithm. When we're talking about the threshold. In this case we're going to talk about
Evan Anderson: minimum weight, perfect matching.
Evan Anderson: So this is a similar plot to what I think, Mteo said in his
Evan Anderson: talk. whereas here i'm. I took this from Oops got to add the reference here.
Evan Anderson: but Ibm is using a heavy hex surface code.
Evan Anderson: So this isn't quite the same as the Torque code. We we're seeing it's a
Evan Anderson: more complicated variation on it.
Evan Anderson: But as we see that as we increase the size of the surface code, we introduce more and more physical qubits. then the logical error rate is on the left hand side. Here
decreases. Yeah.
Evan Anderson: as the physical
Evan Anderson: physical error rate decreases as well. And so this this idea of the threshold is at what point does
Evan Anderson: continually increasing the size of the
Evan Anderson: system actually make our logical error rate decrease faster
Evan Anderson: and so ideally, as D. Goes to infinity. This pro this becomes a step function.
Evan Anderson: So for the surface of torque code, with perfect matching decoding
Evan Anderson: it's
Evan Anderson: around one numeric results show it's run one for 2 qubit
Evan Anderson: gates or see not errors.
Evan Anderson: and practically one for just the polarizing error in the
Evan Anderson: physical qubit itself.
Evan Anderson: And then this this equation is just basically your step function where here t is just your
Evan Anderson: point. 0 0 4 5.
Evan Anderson: Yeah. Any Any questions so far?
Evan Anderson: Pretty straightforward.
Evan Anderson: So now let's get on to decoding. So first we need to introduce
Evan Anderson: what is perfect matching. So let's say you have a graph like this on the
Evan Anderson: left hand side we have 8 vertices and edges between
Evan Anderson: a number of those vertices.
Evan Anderson: So a perfect matching graph is simply just the set of edges that
Evan Anderson: connect all
Evan Anderson: vertices, with exactly one other
Evan Anderson: where to see.
Evan Anderson: And so there's 1 2 3 4 5 6 7 8, 9 different sets of perfect matches for the specific graph.
Evan Anderson: There's 1 one thing. Suppose we had an extra vertex out here. It's there are some graphs where it's possible that you can't connect every single
Evan Anderson: edge to another edge, because we have a odd number of edges or a number of vertices. Then
Evan Anderson: obviously, one's gonna have to be back
Evan Anderson: all out by itself.
Evan Anderson: Right. So now, if we assigned weights
Evan Anderson: to each of these edges, so maybe this weight is just the distance it's not quite drawn to scale, but save
Evan Anderson: it says, the distance to this is the distance one. Then we can define
what the minimum weight perfect matching is.
Evan Anderson: Again, we've to find these distances. then, for this
Evan Anderson: graph or this perfect matching graph.
Evan Anderson: the wait. This is 5.7,
Evan Anderson: whereas on the other extreme the wait for this perfect matching graph is 6,
Evan Anderson: and everything else in between these 2
Evan Anderson: is in between these 2. So it's 5.8, 5, point 9, etc.
Stefan Krastanov: Sorry. Just
Stefan Krastanov: any question. I think I see why it is so for B. But there is one on the top, which is 2. The next one is one, then one, then 2, so 2 plus one plus one plus 2 makes
Stefan Krastanov: like 6. Why is it so for 8?
Stefan Krastanov: Where is this to come from? Like?
Evan Anderson: Yeah. So this the basically is the
Evan Anderson: oh, sorry it this calculation should be one.
Evan Anderson: Yeah, this should be a little bit smaller. Actually.
Evan Anderson: I guess we we do need to give a wait for the for the diagonal edges to
Evan Anderson: had this calculate the end
Stefan Krastanov: sounds good.
Evan Anderson: Yeah.
Evan Anderson: Okay.
Evan Anderson: So now let's
Evan Anderson: go back to just a reminder of what logical gates are
Evan Anderson: on this sort of
Evan Anderson: Let us So if we have just a loop here.
Evan Anderson: then, if we were to measure
Evan Anderson: this loop, we'd get
Evan Anderson: effectively a plus one eigenvalue here.
Evan Anderson: because these 2 zs cancel each other, these 2 zs cancel each other, and you have one.
Stefan Krastanov: And just to compare this to the previous notation. Here Your circles
Stefan Krastanov: are the logical qubits, and you're not explicitly drawing the big green and other blobs. You're just saying right now. We are measuring something that we can correspond to one single blow or
Stefan Krastanov: like this is just Gates.
Evan Anderson: Yeah. yeah. yep. And so
Evan Anderson: if we just if essentially we said, hey, I want to
Evan Anderson: perform a Z gate on these for physical qubits that effectively does nothing to the overall state of the system
Evan Anderson: like what?
Stefan Krastanov: Oh, the the the blobs from the previous talk. They can only take a look right
Evan Anderson: right. Yep.
Evan Anderson: And similarly, if we again applied the gates to this set of
Evan Anderson: physical qubits. We're forming a loop
Evan Anderson: here
Evan Anderson: and again, this is
Evan Anderson: does nothing to the physical state, so it just acts as a logical identity. and
Evan Anderson: effectively One way you can think about this identity is, if you have
Evan Anderson: a loop that you can essentially collapse to something that looks like this
Evan Anderson: effectively, that collapses to a point, and
Evan Anderson: there's nothing on the State itself. Okay?
Evan Anderson: Whereas logical
Evan Anderson: operator is a loop around
Evan Anderson: the surface itself.
Evan Anderson: And so that's what
Evan Anderson: what we saw with the Torque code. You essentially cannot collapse this because it goes around sort of the
Evan Anderson: the whole of the donut, so to speak.
Evan Anderson: But you can deform it such that the path around that loop can be. It can take any path that it wants. It could go all the way around here. Come down.
Evan Anderson: go back up up here
Evan Anderson: and then down here. And so this is the effectively the least amount that you can
Evan Anderson: quote, unquote, compress. It is
Evan Anderson: this: this number of these.
Okay.
Evan Anderson: So now let's consider syndrome on this graph.
Evan Anderson: So what
Evan Anderson: what does this look like? So i'm gonna to mark this blue box with a
Evan Anderson: green circle as a stabilizer measurement with an eigenvalue of plus one.
Evan Anderson: And this
Evan Anderson: red box. Oh, sorry! This red circle, this blue box is a measurement
Evan Anderson: that has an eigenvalue of minus one, and
Evan Anderson: say, we've
Evan Anderson: the environment has introduced. This is the error at this physical qubit.
Evan Anderson: So what do we do?
Stefan Krastanov: They're also on the same thing.
Stefan Krastanov: These blue squares. They correspond to the blobs, the the
Stefan Krastanov: the blows that Matthews have between his logic, between his physical humans
Evan Anderson: correct.
Evan Anderson: And so
Evan Anderson: the idea is, you'll effectively measure every single square here. I'm. Just
Evan Anderson: showing an example of a handful of them.
Evan Anderson: and you measure these 4 physical qubits you get a plus one eigenvalue. That's fine.
Evan Anderson: Likewise we're over here.
Evan Anderson: We measure these 4 physical qubits get a plus one eigenvalue. We're good.
Evan Anderson: Right.
Evan Anderson: But now we measure these 4, and we have a z error here. And what does that mean? We have a minus one eigenvalue.
Evan Anderson: and that's not good. So now we have a a non-trivial syndrome at this. and so a qubit.
Evan Anderson: and likewise, if we do it on the other side of this error, we get.
Evan Anderson: we also get a syndrome. So this is
Evan Anderson: what our syndrome look like on the graph when we just have a single air.
Evan Anderson: Okay.
Evan Anderson: Now, what happens when there's 2 errors. Well. we do similar set of measurements
Evan Anderson: where syndrome shows up here.
Evan Anderson: But when we measure the
Evan Anderson: 4 physical qubits in between these 2 errors.
Evan Anderson: these 2 areas effectively cancel each other out because
Evan Anderson: Z times the identity.
Evan Anderson: and
Evan Anderson: and we get a plus one eigenvalue.
Evan Anderson: But on the other side of this is here
Evan Anderson: we we get a minus one. And so
Evan Anderson: our syndrome show up at these 2 points on the graph.
Okay.
Evan Anderson: and similarly for a string of Zeros.
Evan Anderson: the syndrome show up at the
Evan Anderson: end of the strings where there's no overlap, because if you were to measure this one. these 2 cancel each other out and get a plus one eigenvalue
Evan Anderson: here to measure these 4.
Evan Anderson: You, these 2 would cancel each other out plus eigenvalue.
Stefan Krastanov: Just one thing that for me is very confusing in this. In all these diagrams I always get tripped over it reading papers. Here.
Stefan Krastanov: Prince, happened only where solid lines intersect. Only there you Place X: measurement for qubit operation.
Evan Anderson: Yeah.
Stefan Krastanov: Onto the lattice of Z:
Evan Anderson: Yes. So yeah. And so
Evan Anderson: for the
Evan Anderson: X checks right. Your your
Evan Anderson: you'd have the you'd be measuring the vertices. these these 4,
Evan Anderson: as opposed to
Evan Anderson: these 4.
Evan Anderson: And so we're just doing one type of error in this talk, but it applies to
Evan Anderson: the other type as well.
Evan Anderson: Okay. So what if you had
Evan Anderson: a graph
Evan Anderson: that has right? We don't know where the errors are.
Evan Anderson: so we we only see the syndrome. So we have a syndrome here, and we have a syndrome here.
Evan Anderson: This could be caused by an error that happened here. or
Evan Anderson: it could be caused by a string of errors where
Evan Anderson: the string loops around to the other side. and
Evan Anderson: and then we have 2 z's here, and we get
Evan Anderson: again syndrome at the end of the strength.
So this is a particularly
Evan Anderson: bad situation, because
Evan Anderson: more than likely, almost every
Evan Anderson: decoding algorithm is gonna say your error occurred in the middle.
Evan Anderson: And so you're going to perform a correction in the middle. But what happens when you do that? Well, now you have a extra Z here, and you've formed a chain of z's which introduce a
Evan Anderson: logical operation on your qubit.
Evan Anderson: This all makes sense any questions.
Stefan Krastanov: There is a question from here.
Evan Anderson: Yes.
Stefan Krastanov: Could we measure the like?
Stefan Krastanov: And, Jason, I understand we're measuring it. I know you measure up, and
the bottom
Evan Anderson: or not as that.
Evan Anderson: So
Evan Anderson: do you mean like.
Evan Anderson: Do you mean measuring these 4? Or do you mean measuring like
Evan Anderson: this way?
Stefan Krastanov: So say
Stefan Krastanov: we. You have marked up the in red the measurements that that are non-trivial, but you perform every single measurement as well. It's just that
Stefan Krastanov: for these errors
Stefan Krastanov: these are the only measurements that will be non-trivial if you go one up and measure there. There are no Z flips there. There will be nothing to detect there
Stefan Krastanov: to measure. Where the dashed lines intersect. You will be measuring for the other types of errors. So again, you want to take those with years, and you can not
Stefan Krastanov: permit yourself to measure more things than that because then you're not left with any degrees of freedom.
Evan Anderson: Yeah. and sort of to
Evan Anderson: ruder at that point. If you
Evan Anderson: when you're doing this measurement. these measurements you're keeping track of
Evan Anderson: sort of where a syndrome might pop up.
Evan Anderson: And so
Evan Anderson: yeah. So if you measure these.
Evan Anderson: you're going to measure these, and there's no syndrome that pops up here. So
Evan Anderson: this is You're just
Evan Anderson: left with just these 2,
Evan Anderson: even if you measure everything else.
Evan Anderson: So now we saw
Evan Anderson: 2 syndrome on the graph. What happens if we have 4
Evan Anderson: or sorry 2 syndrome on the lettuce?
Evan Anderson: What happens if we have 4 syndrome? And can we convert this to a graph.
Evan Anderson: Oh, the answers, Yes, so
Evan Anderson: the process of converting the syndrome to a graph is effectively.
Evan Anderson: You choose all the syndrome on your lattice as your vertices.
Evan Anderson: and then connect all of those vertices
Evan Anderson: to every other.
Stefan Krastanov: You mentioned something here given what I said at the very beginning, I said that the lattice is kind of the graph representation of, and then you for the cold, and you also have a stabilizer matrix representation.
Stefan Krastanov: Here we're introducing yet another graph, and graph for that particular set of stabilizer measurements that's
Stefan Krastanov: emerges from the lattice and the given set of stabilizer measurements.
Stefan Krastanov: It's all the same. Drive.
Evan Anderson: Yep.
Evan Anderson: Yeah, so this this graph is dependent on
Evan Anderson: the results of your stabilizing measurements. This one on the right here, like like step on the same
Evan Anderson: right. So how do we add weights to this graph because this is a minimum weight. perfect, matching decoder.
Evan Anderson: Or we take the Manhattan distance as weights. Typically, there are
Evan Anderson: obviously other ways to take these weights, especially when you start introducing non-uniform
Evan Anderson: error across all of your physical qubits to say
Evan Anderson: Yeah, that say we know with you know 80% certainty that this physical qubit is going to introduce an error. Well.
Evan Anderson: then, the wait
Evan Anderson: to this one. This syndrome is going to be
Evan Anderson: much higher if we go this way than if we were to go
Evan Anderson: by the bottom path.
Evan Anderson: So
so we introduce these weights. So these the connections between
Evan Anderson: these
Evan Anderson: syndrome that I have here are pretty straightforward. So if 2, 3, and 3
Evan Anderson: so passes to through 2 physical qubits, that's another way to
Evan Anderson: look at it.
Evan Anderson: And then the minimum weights
Evan Anderson: between these other
Evan Anderson: edges. our loops around
Evan Anderson: the lattice itself. So
Evan Anderson: so this this syndrome to this syndrome is only 2 distance of 2, because we loop around.
Evan Anderson: passed through 2 physical qubits, whereas if we went this other direction.
Evan Anderson: we'd pass through 3 physical qubits, and that would be a weight of 3.
Evan Anderson: So we have a minimum weight graph now.
Evan Anderson: So what does perfect matching do? It
Evan Anderson: finds the minimum weights it's, find the perfect matching graph that has the minimum W. And so
Evan Anderson: in this case it's just the 2,
Evan Anderson: my 2.
Evan Anderson: And so yeah, this is the distance to from this syndrome to this syndrome is around the lattice
Evan Anderson: likewise
Evan Anderson: this: this syndrome for the syndrome is around the lattice top to bottom.
Evan Anderson: Okay. So this
Evan Anderson: So this this minimum weight perfect matching graph now tells us where the most oh, where are assumed? Errors should be. So that means
Evan Anderson: I assume we have 0 here.
Evan Anderson: and likewise here in here.
Evan Anderson: And of course, what is the recovery operation? Well, we just
Evan Anderson: apply a Z gate to wherever we have our assumed errors.
Evan Anderson: and
Evan Anderson: so it's web
Evan Anderson: our applied. The gates are in green.
Evan Anderson: and theoretically, when we
Evan Anderson: measure this again, this should
Evan Anderson: effectively have no syndrome, because we've applied these
Stefan Krastanov: just one here. Why is it valuable to assume that the gears we go, or the me, or wait once?
Evan Anderson: Yeah.
Evan Anderson: it's a good point. So
Evan Anderson: assume these are the minimum weight ones. Because
Evan Anderson: if we were to go this way, if we were to say
Evan Anderson: the errors that we got. we're.
Evan Anderson: or if we didn't do the minimum weight. and we went this direction. We'd have to have an error on 3 physical qubits. That's. P. Cubed
Evan Anderson: versus 2 cubic, 2 physical qubit. So P. Squared. So it's
Evan Anderson: has to do with like the probability of failure, of error, of the physical qubits themselves.
Evan Anderson: Is that what you were asking.
Stefan Krastanov: Yes, I it's exactly what you said. The the meanwhile way perfect matching
Stefan Krastanov: predicts the sake of chains of errors that are hopefully, maybe approximately.
Evan Anderson: we want the set of
Stefan Krastanov: possible chains that are the most probable to you have have, and this is at least on the perimeter of what is the most probable one, because it gives us the shortest ones that involve the fewest errors.
Evan Anderson: Yep. yeah. And i'll, i'll touch on.
Evan Anderson: Why, this isn't exactly the maximum likelihood and a couple of their 2 slides here.
Evan Anderson: So right so it's, giving us the shortest chains from one syndrome or set of chains from one syndrome to the next syndrome.
Evan Anderson: So our our our example is pretty trivial here. We could
Evan Anderson: understand it within.
We're doing 4
Evan Anderson: syndrome, so we only had 2 edges that we really had to worry about.
Evan Anderson: But even this example, what if our syndrome look like this, then.
Evan Anderson: you know, doing it by hand.
Evan Anderson: starts to become much more tedious. So
Evan Anderson: these algorithms are fairly efficient to do minimum weight perfect matching.
Evan Anderson: I believe. Worst case scenario.
Evan Anderson: The
Evan Anderson: it's an order of n cubed. Where n is your number of qubits or number of vertices rather.
Evan Anderson: and then average case is order, n it's essentially number syndrome.
Evan Anderson: So
Evan Anderson: what we were alluding to earlier was that minimum white. Perfect matching is not necessarily
Evan Anderson: the maximum likelihood decoder
Evan Anderson: maximum likelihood effectively takes into account
Evan Anderson: the class of the error. And so what that means i'll get to in a second. Let's say we have a smaller lattice. Now that looks like this. what can happen
Evan Anderson: if we have our minimum white perfect matching.
Evan Anderson: Let's just say we know this is distance 2
Evan Anderson: and say, we're always gonna perform our Z corrections here.
Evan Anderson: and if the real real errors happened
Evan Anderson: where we
Evan Anderson: performed our correction right. This is the perfect scenario we
Evan Anderson: We predicted exactly where the errors are happening. but
Evan Anderson: we have another scenario again. This is distance 2. But now
Evan Anderson: what happens if the real errors go from top and bottom? Well.
Evan Anderson: we form this loop, and
Evan Anderson: what we saw earlier was these loops effectively perform
the identity.
Evan Anderson: a logical identity
Evan Anderson: on your system.
Evan Anderson: So again, this is good. We're effectively corrected the error and
Evan Anderson: perform the identity operation on our logical state.
Evan Anderson: Okay. But now what happens if
we again
Evan Anderson: choose these 2 as the correction.
Evan Anderson: But the real errors happened at these 2 nodes. Oh.
Evan Anderson: is this also distance to the answer is, yes.
Evan Anderson: If we go this way you come around. come around the lattice and then go down. So it's
Evan Anderson: You're passing through 2 nodes, and so that's distance 2. But what you see here
Evan Anderson: is that now we've created a chain
Evan Anderson: of the errors where we've
Evan Anderson: looped around and down up. and this turns out to be a a logical operation on our
Evan Anderson: our logical state.
Evan Anderson: And so now we've introduced a logical error by performing our correction on these 2.
Evan Anderson: And so
Evan Anderson: what the maximum likelihood? I not actually, I didn't quite calculate it for
Evan Anderson: the specific scenario. But
Evan Anderson: what the maximum likelihood would do in this situation where you have multiple possibilities of
Evan Anderson: where your Zers could be with a distance to or wait to.
Evan Anderson: It looks at all of the possibilities, all the possible paths that would introduce either a logical identity or
Evan Anderson: a
Evan Anderson: logical Z.
Evan Anderson: And so that's you know it's traversing every single path on this on this lettuce.
Evan Anderson: and then
Evan Anderson: it
Evan Anderson: sums up all of the possible, the weight of all of the possible paths that
Evan Anderson: create a logical identity sums up all of the possible paths that would create a logical
Evan Anderson: z flip. and whichever one of those is
smaller.
Evan Anderson: then it assumes that that
Evan Anderson: is the where the er happen. So the maximum likelihood decoder. Could the in theory
Evan Anderson: choose accurately between
Evan Anderson: these 3 options
Evan Anderson: clear to everyone. You want me to explain it in a different way.
Stefan Krastanov: Let's try for. Yeah, there are a couple of questions. So how this is determined, the most likely error, based on, like the
Stefan Krastanov: pretty much so perfect maximum like it would be color which is exponentially expensive to run because it needs to brute force in your in your everything
Stefan Krastanov: we'll just look at all multiple paths.
Stefan Krastanov: the probability for everything to one of those Ps. And it will sum them up
Stefan Krastanov: so here it will go for longer than it will for 4 steps, 5 steps, and you know some funny topology, and they will have
Stefan Krastanov: only one short bat. but many, many long paths.
Stefan Krastanov: and if the total probability for something being a longer P is higher, that will be the correct, the the current calling it won't be the me on the left one
Stefan Krastanov: and the mean week. Perfect matching is
Stefan Krastanov: efficient
Stefan Krastanov: because it gets to look on the meal left one hoping that it's a good approximation
Stefan Krastanov: for the highest
Stefan Krastanov: and for people more into statistical physics, internal dynamics. This is also related to minimizing energy versus advising entropy. But that's
Stefan Krastanov: that it's been a few years since I've been able to do that.
Stefan Krastanov: Yeah. it's a bit too like when you're increasing the distance.
Stefan Krastanov: And by distance, I mean, that is the last. Yeah, there is. There is no statements about what the threshold is under maximum like you called.
Stefan Krastanov: and all the threshold is under, you know, with perfect matching and other decoders so frequently you'll be saying, i'm this much better than me, and like perfect matching with my fancy the caller
Stefan Krastanov: and i'm approaching efficiently the you know the maximum likelihood threshold.
Stefan Krastanov: And yeah, the offending. You can frequently drag about being a little bit better than Max, then, you know, wait for it, matching with some small smart that's in terms of threshold. There are other things that you might want to measure to.
Stefan Krastanov: Are there any
Stefan Krastanov: it's known, or
Stefan Krastanov: I I don't know
Stefan Krastanov: I don't.
Stefan Krastanov: Was there something else from here. I thought I mentioned the hands.
Stefan Krastanov: the any.
Stefan Krastanov: Okay, let's continue.
Evan Anderson: Okay. So just have a couple of more slides here.
Evan Anderson: So
Evan Anderson: you saw a similar
Evan Anderson: diagram like this in the taste, this talk. But
Evan Anderson: now, if we start to assume that our measurements could be faulty, that we have errors in our measurement.
Evan Anderson: Well, then, now we need to introduce this additional dimension to our 2D
Evan Anderson: surface code. where effectively we just continue to do repeated measurements for some set of time, or some number of rounds.
Evan Anderson: and as time increases in the vertical direction, syndrome can
Evan Anderson: pop up in the Time domain as well.
Evan Anderson: So now our our syndrome graph might become even more complicated.
Evan Anderson: And now we need to
Evan Anderson: do minimum weight, perfect matching on
Evan Anderson: this
Evan Anderson: 3D lattice or
Evan Anderson: send you a breath.
Yeah.
Evan Anderson: okay. And my last slide here is, if
Evan Anderson: head from it's Andreas Walrus group.
Evan Anderson: So this is what a physical
Evan Anderson: a superconducting chip might
Evan Anderson: look like. So here these stars are the qubits themselves, and the
Evan Anderson: blue lines connecting them are the
Evan Anderson: how you would perform your 2 qubit gates. So your c nuts.
I believe the
Evan Anderson: purple lines are the
Evan Anderson: read out for
Evan Anderson: measurement.
Evan Anderson: So you have, you know, a single wire, but multiple
Evan Anderson: branches that connect to the qubits themselves.
Evan Anderson: and then the
Evan Anderson: forget what the difference is between the pink and the green wire. One of them
Evan Anderson: is right just performs the single qubit
Evan Anderson: operations. Possibly I think the other. Yeah, I
Evan Anderson: So I think the other one what it does, is it? It modifies the energy levels. So you don't
Evan Anderson: get
Evan Anderson: crosstock when you're performing certain gates or measurements between your 2 nearest
Evan Anderson: or 3 nearest neighbors.
Evan Anderson: Yeah. So this is
Evan Anderson: a surface code on an actual chip itself.
Evan Anderson: That's the
Evan Anderson: and it might talk so
Evan Anderson: happy to answer any questions or
Evan Anderson: anyone might have.
Stefan Krastanov: Thank you.
Stefan Krastanov: Just before we start with the questions in terms of presentations.
Stefan Krastanov: future presentations please AIM to follow the style of the presentations today. Sure, today it was a little bit easier because we're talking about
Stefan Krastanov: topological calls which it's implicitly, as rich, sorry, explicitly, have a lot to do with
Stefan Krastanov: pretty pictures, but I do think all to. It was very difficult topic today. The value that was contributed by the a tone of graphics that bothers us at it was really really helpful. Please AIM to
Stefan Krastanov: cap at least as much diagrams and cafe little text on your slides.
Stefan Krastanov: Okay, so let's talk about your questions.
Stefan Krastanov: So can these be applied to like tramp architectures or energy center architectures or
Stefan Krastanov: under
through?
Stefan Krastanov: So for I am traps. You have better connectivity. So you don't need to necessarily be restricted to that.
Stefan Krastanov: I mean, in a sense you have. We're going to do it because the iron traps are. That's the
Stefan Krastanov: you, you. It means that for me to do something of all of them at the same time, or virtual person, or you can consider other classes, and with Irs you also can show full dance around. You can physically take one of your qubits and move it around which also brings more connectivity.
Stefan Krastanov: There's all head to that. But you you are not working on a too deep of circuits, which is what transpose. Are
Stefan Krastanov: so you get to have more than just the toy call. The turf is very natural for this.
Stefan Krastanov: with the envy centers the
Stefan Krastanov: very
Stefan Krastanov: complicated question like you, you know one way you have way. This connectivity you got with smaller registers. You cannot scale them up in another way. They are way better at actually long distance communication.
Stefan Krastanov: So the question there is
Stefan Krastanov: Then, over the last month we had a bunch of back into Canada. Here we were presenting on
Stefan Krastanov: spin cubits in in in classical silicon like that you can do at the foundry and those you can naturally do a couple of lines, and you can shuffle those lines between each other. So you have access to really interesting tree architectures.
Stefan Krastanov: So yet another topology that's very different. These are truly.
Stefan Krastanov: for the fish is that great, but they were interesting because they were an interesting canvas matter. Physics system that you can play with statistical physics, stores, problems with it.
Stefan Krastanov: It. Kind of
Stefan Krastanov: right business is also really good for transpose, because transmitter of 2D lattice, but also that's a reason to be scared about the future of
Stefan Krastanov: questions.
Stefan Krastanov: And so what happens when you have
Stefan Krastanov: it that are not?
Stefan Krastanov: It's the I think you mentioned it that some
Stefan Krastanov: the beginning, this this block where you have this.
Stefan Krastanov: How would you correct? Let it go? If something has, you know
theers
Stefan Krastanov: or any other kind of
Stefan Krastanov: sort of Not necessarily.
Stefan Krastanov: I don't know if you mean like.
Stefan Krastanov: I guess i'll give it a more, because I think I get that infinite to be many, several errors that we have.
Stefan Krastanov: I don't. I don't
Stefan Krastanov: so usually. Well, I I don't know. I don't know what happens when you for this particular example. But we usually what you do is you shake the
Stefan Krastanov: new offline condition the condition that we're listing a couple of times
Stefan Krastanov: already, Where you check with your
Stefan Krastanov: you can just check the condition itself. You want that your mission
Stefan Krastanov: means this: you're checking with your that errors after you work on your measurements, get
Stefan Krastanov: projected into all the years. Maybe that's not the case in which case. No, you cannot
Stefan Krastanov: here, and I don't really have any intuition here about the one we mentioned, but that's that's usually the hope that the measurements that you're performing.
Stefan Krastanov: Force your system to to move like something that have only 4 years. So if you see the paper that I refer to, he has a model. Where do you see? Not here?
Stefan Krastanov: This is the with the same problem. It's something like a deep providing zoom, but that paper probably has some. It's a different error, not a not having happened out of nowhere. But you see not. I want to use chatting soundly polarization here, which is going back to all years, which is.
Stefan Krastanov: i'm not correlated easy here.
Stefan Krastanov: So you're going to do something that it you don't want. Yeah, I mean, i'm just
Stefan Krastanov: most errors are not pretty. All the errors. The hope is that your measurement will turn them into a mixture of
Stefan Krastanov: for years, or let's hold it there.
Stefan Krastanov: That's what my math teacher is to go.
Stefan Krastanov: It was French.
Stefan Krastanov: Yes. yeah, I I guess
Stefan Krastanov: you mentioned this, but it seems like the
Stefan Krastanov: let me have your back.
Stefan Krastanov: Is it like.
Stefan Krastanov: If you have some specific Scripture, that there is you some approximation of maximum, which here the topological structure is, what for me to do, Catholic, I are more efficient. It is all your it does provide you.
Stefan Krastanov: and we'll work
right. It's a
Stefan Krastanov: that you know, whatever
that was efficient.
Stefan Krastanov: So you used to go when designing these things from one of the directions you start with, the structure that you know will give you a good dec folder, and you call it will also give you a good rate in this.
Stefan Krastanov: Here is that one you can be drinking.
Stefan Krastanov: It gives you way. or you start with an algebraic structure that for which you can prove that we have
Stefan Krastanov: good. The funny thing is that you can true good rates and distance. If you start with the travel conscriptions.
Stefan Krastanov: so that that's a a ton of family of theurms in terms of random colds. Have a ton of good properties.
Stefan Krastanov: but you hear around them. They don't that structure that can be exploited for you
Stefan Krastanov: Us. And it's
Stefan Krastanov: very.
Stefan Krastanov: It's very deep math research in in
Stefan Krastanov: multi graph theory, computer science about
Stefan Krastanov: finding
Stefan Krastanov: finding something that looks random, but has to. so that you can exploit these features. and we'll be discussing expander graphs, which is the for the typical example of that technique.
Stefan Krastanov: Trout. A lot of the
master.
No question
Stefan Krastanov: one. What's the maximum number?
Stefan Krastanov: Well, that would be the distance. I think so. It will be the oh, sorry, no, I apologize. That's
Stefan Krastanov: I don't know the answer is. but it wouldn't be all
Evan Anderson: kicked it
Evan Anderson: like if you had. See.
Evan Anderson: here's all across here, all of these would light up.
Evan Anderson: and you had zeros all across here
Evan Anderson: of these. Yeah.
Evan Anderson: I think you can theoretically have them everywhere.
Stefan Krastanov: If you're in that situation. You probably just
Stefan Krastanov: the second question. I guess this is not
Stefan Krastanov: it's coach, but presumably
Stefan Krastanov: the more red notes you can have is, let's say
Stefan Krastanov: the
Stefan Krastanov: the lower the
your logical here.
Stefan Krastanov: Yeah. And actually, that's something that's you could even make use of that to say, okay.
Stefan Krastanov: I should just cease my communication and start over. Yeah, and that's using concatenated calls. I think that Random or some of our secure and colleagues recently that will work on the company. It goes where through the bottom call was symbols on the code.
Stefan Krastanov: But you had some information about the certainty with which you were performing the correction, and that the information was used the higher of the
Stefan Krastanov: the chain.
Stefan Krastanov: Yeah, it's it's it's it's it. It would be with till my question is coming.
Stefan Krastanov: Discussions with it.
Stefan Krastanov: But yeah, concatenating that information and using the information about the certainty with which you're correcting the lower code can make the creation of the Ch. Your code better.
Stefan Krastanov: but I guess it has even more use them just there. Yeah. So
Stefan Krastanov: So I said that you have a computation going on at some point
Stefan Krastanov: you can just say
which is.
Stefan Krastanov: you can't believe the results are yet.
Stefan Krastanov: Yes.
Stefan Krastanov: I have a comment on. I guess something you were gonna ask where you asked about characterize the fidelity. I think what you can do is that you go to
Stefan Krastanov: just try to run your code without doing any operation.
Stefan Krastanov: and then, based on how often your measurements are changing as you progress.
Stefan Krastanov: which i'll give you an intuition like what's going on in each of the Cubans.
Stefan Krastanov: because you have this
Stefan Krastanov: it's telling you.
Stefan Krastanov: and you can say, oh, how often do do I get If you live in these to right it's so related
Stefan Krastanov: you, you ought to be able to.
Stefan Krastanov: It's important, even physical
it was.
Evan Anderson: Yes, and that's that's sort of what this
Evan Anderson: break, even point is about, so
Evan Anderson: they
Evan Anderson: effectively. They They have characterized all of their physical qubits, and they know with probability p this this qubit after
Evan Anderson: you know time t is gonna have a x-foot.
Evan Anderson: But this qubit right next to it has much lower p for that same sort of time period.
Evan Anderson: and so we choose the second one as our best physical qubit.
Evan Anderson: and
Evan Anderson: if
Evan Anderson: if we can do error, correction, where our logical
Evan Anderson: qubit
Evan Anderson: has a lower probability of error.
Evan Anderson: then our best physical qubit, then we've done.
Evan Anderson: then actually doing quantum air correction is useful
Stefan Krastanov: in the last few minutes we have left. I just want to bring up from next week.
Stefan Krastanov: One of the things will be talking about interesting bias version of the Service code.
Stefan Krastanov: and the other thing will be jumping into
Stefan Krastanov: more interesting positive calls
Stefan Krastanov: for. So we need one volunteer from the dark all side of things, the last article side of things, and one volunteer for classical Ldpc calls and believe propagation.
Stefan Krastanov: If there is someone that wants to discuss these, please message me, and tonight i'll just send
Stefan Krastanov: a message with a random draw. If not, is volunteered for.
Stefan Krastanov: And yeah, it will be out of the people that we presented. Yet we probably will certainly need you to have great beads.
Stefan Krastanov: but for the monetary stick to those that account for that yet, and so some people can volunteer for papers later on. They they have that long.
Stefan Krastanov: But again for next week please volunteer. If you're interested in my.
Stefan Krastanov: as
Stefan Krastanov: I guess we can adjourn.
Stefan Krastanov: Okay.
Stefan Krastanov: not really. I mean, someday, that's whichever one.