Stefan Krastanov: Go ahead
Zhuangzhuang Chen: to the do. I have to start At first
Zhuangzhuang Chen: I will introduce a powerful tool that is really useful in the condom simulation, such as the quantum, circuit, analysis and quantum error correction. So maybe some of you may have participated in the winter school lecture. So if you have participated that you might be familiar with the style of my science.
Zhuangzhuang Chen: I just to use very few basic concepts in those slides. So, in order to make all my slides to be uniform. I utilize the tablet of
Zhuangzhuang Chen: the the size of this winter school that.
Okay, so let's get start t0 0ur main topic today.
Zhuangzhuang Chen: So for the quantum simulation quantum simulation is an important research areas for scientists, because it can provide insight into a wide range of physical systems that are difficult for us to study using the classical computers. One.
Zhuangzhuang Chen: of course.
Zhuangzhuang Chen: a are characterized by some complex behaviors that are hard for us to build models in classical physics, however, the condo simulation For why us? A useful tool, and
Zhuangzhuang Chen: the because it can enable the researchers to simulate the behaviors of quantum systems by using quantum computers.
Zhuangzhuang Chen: There are some traditional quantum simulators that a physicists usually use, such as the quantum Monte Carlo and the quantum information discern diagrams.
Zhuangzhuang Chen: However, those traditional counter simulators are not that efficient?
Zhuangzhuang Chen: The issue here is that the running time grows exponentially in the number of cubes. So here is the example. The quantum information decision diagram package takes more than 22 h to simulate the algorithm of 40 cubes.
Zhuangzhuang Chen: Therefore, simulating a 100 0r so and 2 base is out of the question for us.
Zhuangzhuang Chen: But fortunately there we there is a kind of circuit that can be simulated in the class of computer in polynomial time, and this kind of circuit we call this stabilized circuit
Zhuangzhuang Chen: the, and it is very useful in the quant of error fraction. Also. The Gosman near the room says that a stabilized circuit
Zhuangzhuang Chen: quantum circuit, that consisting of C naught
Zhuangzhuang Chen: and the tableaux algorithm, or the STEM cloud geometry, is really a powerful tool to implement this on the classical computer, and the later. I will show you how this simpler
Zhuangzhuang Chen: before we reach out to the So in the
Zhuangzhuang Chen: quantum competition we have cat, 0 and cat one, as the computational basis states.
Zhuangzhuang Chen: and the did not direct Notation Cat 0 represents a column matrix, 1 0 and the cat. One represents a column, matrix, 0, and one.
Zhuangzhuang Chen: and the single qubit state can be written as 5. It goes to r 4 times cat 0, plus beta top times, cat one.
Zhuangzhuang Chen: So since we only deal with the State 0 and a State one. So the probability of observing.
Zhuangzhuang Chen: since today t0 0 and state one master you must add up t0 0ne. Therefore, we have the amplitude of Alpha Square, plus the amplitude of Beta Square you close t0 0ne.
Zhuangzhuang Chen: and in the quantum computation.
Zhuangzhuang Chen: the the these false poly matrices can be seen everywhere, and they are, I'm. Did x gate zig it and wag it.
Zhuangzhuang Chen: and the different poly matrices have different meanings.
Zhuangzhuang Chen: So for the for the identity.
Zhuangzhuang Chen: so it well it it is, call it, and to make it identity operators, because if identity of return of X on the Conference
Zhuangzhuang Chen: status, cat, 0, and okay one, nothing will change.
Zhuangzhuang Chen: And as the X get. and what we call a bit flip
Zhuangzhuang Chen: good, because if there is X get X on the state t0 0, it becomes the to what state one, and if it acts on a state one, it will become 0
Zhuangzhuang Chen: also. We have the Z it, and we call this this flight to Kate. and when the zig it acts on 0 it remains the same. But if it x
Zhuangzhuang Chen: on the State one, it'll be council minus one. So the phase has changed.
Zhuangzhuang Chen: and for the
Zhuangzhuang Chen: so the white get it goes to the imagine. Imagine the United Times, the X Gate, and the Zikit. Therefore we call this bit of face fleet to get.
Zhuangzhuang Chen: There are some releases among those of 4 poly matrices. Here, for example, the X square you call square, you close to this squares you call to the identity.
Zhuangzhuang Chen: Also we have excel times away you close to minus y x y times z equals to minus the y and z times x you close to minus the x z
Zhuangzhuang Chen: erez agmoni. Therefore, from these relationships among poly matrices we can see that the X and Y and the operators are at. They can build with each other 150.
Zhuangzhuang Chen: But the polymerase, so it is. Self commute with commute is with itself, and it's identity.
Zhuangzhuang Chen: and this
Zhuangzhuang Chen: the result is very useful when we are dealing with the symptoms of the metrics, because in the same 5 matrix we usually have to consider the combativity of those quality of reach.
Zhuangzhuang Chen: Now let's review some some basic concepts
Zhuangzhuang Chen: in in of computation. So here is the Poly group.
Zhuangzhuang Chen: so polygroup for the in Cupid. The Poly group is defined to consist of all the tensor product or poly matrices, together with the multiplicative factors, plus minus one and the plus and minus I.
Zhuangzhuang Chen: So the reason we include these multiplicative factors, because we need to form a valid group.
Zhuangzhuang Chen: and in abstract algebra, in order to form the value group, the closer associativity, identity, and the universe property must satisfy.
Zhuangzhuang Chen: So here is a formula about the polygroup, and it is a tensor product of polling matrices.
Zhuangzhuang Chen: and the steps stabilize
Zhuangzhuang Chen: the stable that this concept of stabilizer is.
Zhuangzhuang Chen: is the forms of quantum error, correction. So a stabilizer is defined as an alien subgroup, as of for polygroup generated by commuting permission. Matrices.
Zhuangzhuang Chen: therefore, in order to form or stabilize a group, it must be the subgroup of a poly group.
Zhuangzhuang Chen: Therefore this this element is stabilizer must come from the Poly group. Also they need to the elements in the simple. As a group master can build with each other, so P. I. Times Pj. Equals to Pj. Transpare
Zhuangzhuang Chen: the they must commute with each other.
Zhuangzhuang Chen: and for the stabilizer code stabilize the code is
Zhuangzhuang Chen: defined to be the vector space we as stabilized by a subgroup as of the Poly group.
Zhuangzhuang Chen: So here is a notation of auto strip stabilizer code.
Zhuangzhuang Chen: So here the file is the stabilizer state.
Zhuangzhuang Chen: and the P. Is a.
Zhuangzhuang Chen: From the simple as a group that a fix the staple as a state.
Zhuangzhuang Chen: and also in the linear archive perspective.
Zhuangzhuang Chen: we can observe that
Zhuangzhuang Chen: the 5 is the I, and plus one is the
Zhuangzhuang Chen: therefore. In other words, the stabilizer code can be defined as the common plus one admin basis
Zhuangzhuang Chen: of a set of commuting policy matrices.
Stefan Krastanov: Hmm.
Zhuangzhuang Chen: Okay. We have already finished all of the basic concepts we need for our discussion. So here. now, let's focus on the simplistic stuff.
Zhuangzhuang Chen: So we observe that the whole initial polling matrix can, when cubic permission, poly matrices can be mapped
Zhuangzhuang Chen: to a pair of for the binary vectors. So how to
Zhuangzhuang Chen: this mapping works? How do we map the polymatr to binary vectors? So here is an example for the 3 cubic poly operators. So here we have 2
Zhuangzhuang Chen: X tens of Z. 10. So why we need to map down
Zhuangzhuang Chen: mean to the binary veterans eap. So here the a. Represents the X component on the B represents the Z component.
Zhuangzhuang Chen: the z component, then the since. So why you close to imagine the unit times X gate and zig it. So the X and the Z component together
Zhuangzhuang Chen: give us the wide component.
Zhuangzhuang Chen: therefore it to rise to the binary vectors. So here is the example. So that's tens of what? So the first qubit
Zhuangzhuang Chen: is
Zhuangzhuang Chen: first-commit is x. S0 0n the left hand side on the first position
Zhuangzhuang Chen: so it should be one.
Zhuangzhuang Chen: and on the right side. On the first to quit it should be 0,
Zhuangzhuang Chen: and to the circuit does qubit is the Zikit. They als0 0n the left side, as the second position is there on the right side as a set. No position. It is a what.
Zhuangzhuang Chen: and a third position. 30, two-bit is one
Zhuangzhuang Chen: the for
Zhuangzhuang Chen: the left side and right side to the third position. They are they all. They are both one.
Zhuangzhuang Chen: So that's how we
Zhuangzhuang Chen: So we write to the permission, polymetics as the pair of binary vectors.
Zhuangzhuang Chen: and for the Z. 10, so they transfer X. They are the they are similar. so the second, so first is 2 cubes are Z. Keys.
Zhuangzhuang Chen: Therefore on the right side it is 1 0ne, and the circuit is X. On the left side it should be one, and on the right side it should be 0.
Zhuangzhuang Chen: And also we can also use this binary representation to check whether these 2,
Zhuangzhuang Chen: where these 2 poly arbitrators can mute or not. So here we use simplic in nerve product To calculate. This simple in your product is defined
Zhuangzhuang Chen: as a times d transpose plus b times, c. Transpose so last. Let's do this example.
Zhuangzhuang Chen: So first we need to cap. So if
Zhuangzhuang Chen: if was a simpler, you know, product is 0,
Zhuangzhuang Chen: they can mute, and if simply, you know, product is one, the anti commute.
Zhuangzhuang Chen: So here
Zhuangzhuang Chen: now, let's first calculate the 8 times d transpose.
Zhuangzhuang Chen: So here this time times this one
Zhuangzhuang Chen: one times one is 1 0 times one is 0, and one times 0 is 0. Therefore 8 times the transpose equals t0 0ne.
Zhuangzhuang Chen: and for the B Times, C transpose
Zhuangzhuang Chen: 0 times 0 0ne times 0 and one times one.
Zhuangzhuang Chen: So the B Times, C transpose you close t0 0ne.
Zhuangzhuang Chen: therefore one plus one and you close t0 0, and we can conclude that this 2 poly operators commute with each other.
Zhuangzhuang Chen: So you might wonder why they simplified in the product could relate to the commutativity properties of
Zhuangzhuang Chen: of the party operators. So the proof is.
Zhuangzhuang Chen: it's straightforward. Yeah. So
Zhuangzhuang Chen: So so. So, as we already discussed the previous. We see that as the X of Y and the operators they are, and to commute with each other, but they can mute with themselves.
Zhuangzhuang Chen: 0 4. Is there a chat?
Stefan Krastanov: Is there a typo in the talk thing that B should be 0 1 1?
Yeah. Did you hear the question? The the the top wise panel we have, I think. There I agree. I think there might be a type wouldn't be.
Stefan Krastanov: I was trying to make sense of it
Narayanan Rengaswamy: just above your
Zhuangzhuang Chen: 0 1 0 1 1.
Narayanan Rengaswamy: Yeah. So in the top. So zhang wrong in the top
Narayanan Rengaswamy: box for it and
go ahead. Yeah, it's good.
Zhuangzhuang Chen: Okay.
Zhuangzhuang Chen: So.
Zhuangzhuang Chen: So just as we've discussed the previous, the X. Y. And Z operators enter can be able to with each other, but they commute with themselves.
Zhuangzhuang Chen: Therefore the simplicity in a product is to calculate how many times this anti commute happens.
Zhuangzhuang Chen: So, if it happens at an even time.
Zhuangzhuang Chen: they will commute; and it if it happens at all the time they under commute. So that's how they simulate. You know. Product relates to the
Zhuangzhuang Chen: competitiative property of poly operators.
Zhuangzhuang Chen: No, let's focus on the same type of meetings.
Zhuangzhuang Chen: So the and Qubit poly operators can be mapped into the laws of Austin.
Zhuangzhuang Chen: So the last column of the matrix is to check the so plus sign or minus sign, off with the quality operator. So when we're doing the calculation, we we usually ignore this, and we just focus on this part
Zhuangzhuang Chen: from X t0 1 0ne to Z one, and this pump.
Zhuangzhuang Chen: therefore this and platform matrix is a size of 2 n times 2,
Zhuangzhuang Chen: which sized in 2 n times 2, and metrics, and then represents the qubit.
Zhuangzhuang Chen: And when we observe this simple as a matrix, it is partitioned int0 4 parts, so that the part represents the X component. and the right part represents. The Z component
Zhuangzhuang Chen: also is X and the z component together for most of the web component, just as we discussed previous support to the binary representation or polling operators.
Zhuangzhuang Chen: and also the simplicity matrix is partitioned into the upper part and the lower part.
Zhuangzhuang Chen: so the other part and the lower part that shows the combativity properties of those party operators. For example, the logical x operator, master, anti-com build with the logical Z operators.
Zhuangzhuang Chen: But the the stabilizers must commute with each other
Zhuangzhuang Chen: also in order to form a
Zhuangzhuang Chen: So first
Zhuangzhuang Chen: the lowest from the first row to the
Zhuangzhuang Chen: they master commute with each other.
Zhuangzhuang Chen: and also the second. Poverty says that for all it belongs t0 0ne to the
Zhuangzhuang Chen: each slope and a commute with h plus angel.
Zhuangzhuang Chen: that is, for example, that is to say. if for the first row on the upper part, it must anti-commute with the first row in the lower part.
Zhuangzhuang Chen: and the third property tells us that for all I and H
Zhuangzhuang Chen: belongs t0 0ne to and such that I is not equal to h I
Zhuangzhuang Chen: the I stroke me with h plus, and those so that tell us that so again.
Zhuangzhuang Chen: for the first of all on the upper part. It's the meals with every those in the lower part. It accepts the first lower in the lower part, because in the second forward we have already that this
Zhuangzhuang Chen: they'll enter it with this show.
Zhuangzhuang Chen: And simple as a matrix is very
Zhuangzhuang Chen: useful in the counter, You know, quantum error cracks in circuit, and this
Zhuangzhuang Chen: can can be useful in this clue can be useful
Zhuangzhuang Chen: in the So the cliff is
Zhuangzhuang Chen: is defined as the unitary operator. You that maps poly operators to policy operators under this conjugation rules
Zhuangzhuang Chen: U. Times, E. Times U. Dagger equals to the E. Prime and the E and the E dagger
Zhuangzhuang Chen: are all forms of Poly Group.
Zhuangzhuang Chen: So there are several Cl. For kids, such as the heart of market, fiscal
Zhuangzhuang Chen: and the single good.
Zhuangzhuang Chen: So all
Zhuangzhuang Chen: all of these kids have their different properties.
Zhuangzhuang Chen: and we can, we can, since the part, since the cliff focus.
Zhuangzhuang Chen: That's the poly operators to another poly operators. We can regard this clearly for gate as the simplification of that X on the binary.
Zhuangzhuang Chen: And this this simplicity matrix can maps a binary operator of a policy matrix to another binary of which binary representation of the poly
Zhuangzhuang Chen: operators
Zhuangzhuang Chen: also the Cosmos Theorem says that the Clifford Gate can be efficiently simulated in the castle computer. and we can just
Zhuangzhuang Chen: but I simply track in the stabilizers of the input State through the circuit. Later, I will show you an example how this works
Zhuangzhuang Chen: for this.
Zhuangzhuang Chen: Oh.
Zhuangzhuang Chen: here is the example of what is the blackman matrix of the harder market?
Zhuangzhuang Chen: So the hard market is one of the cliff. Forget.
Zhuangzhuang Chen: and how to market to the matrix form can be written as this.
Zhuangzhuang Chen: S0 0ne over square root of 2 1 1 0ne, and the minus one here.
Zhuangzhuang Chen: Now, also from the previous slides. We know the
Zhuangzhuang Chen: the definitely we know the definition of this account location rules.
Zhuangzhuang Chen: So here it's the communication rules for the Harder.
Zhuangzhuang Chen: H. Times H. Dagger equals to the and h times the diagram it goes to X, and that is to say, when the X operator
Zhuangzhuang Chen: to pass through this.
Zhuangzhuang Chen: there's a hard market, it will be mapped into the Zeit and the Zig, it will be met into the X Gate.
Zhuangzhuang Chen: So here is a simpler form of a harder molecule in all the cubists. and the platform is like this.
Zhuangzhuang Chen: The I represents the identity.
Zhuangzhuang Chen: The M is the is the number of cubes.
Zhuangzhuang Chen: S0 0kay, we use the previous example to to the calculus to do the calculation. So to show how the simplicity matrix and that one poly operators to another point operators.
Zhuangzhuang Chen: So
Zhuangzhuang Chen: we can just combine this
Zhuangzhuang Chen: pair wires to binary binary vectors, together with form or matrix.
Zhuangzhuang Chen: another, a simple to form of the harder market acts on those 2 bits.
Zhuangzhuang Chen: and it is partition to into the 4 parts here.
Zhuangzhuang Chen: and as after that we can get a new
Zhuangzhuang Chen: binary representation. so we can usually observe that 1 0 1 0 1 1,
Zhuangzhuang Chen: it comes t0 1 0 0 0 1 0. So that is to say, is the X component and the z component they exchange with each other.
Zhuangzhuang Chen: and the result is the same with the communication rules. X
Zhuangzhuang Chen: maps to the Z and the Z. Maps to x.
Zhuangzhuang Chen: So that's how this implemented matrix works, and also the the fiscal at the scene, or they they both have the They are unique, simple to make it so. But since they are some time matrix.
Zhuangzhuang Chen: the form of similar matrix is a little bit complicated, complicated. So I just give you example about the hardware supply matrix of the hardware gate
Narayanan Rengaswamy: just to add a comment here. Yeah. So so what you said is exactly right. But just for clarity. So
Narayanan Rengaswamy: give me 1 s.
Narayanan Rengaswamy: Yeah. So that there's there should basically be a negative sign. So because at Y h dagger is minus y. So in your example. So let me see if I can annotate
Narayanan Rengaswamy: It's like camp.
Narayanan Rengaswamy: They did think no, but anyway. So if you yeah, so for the first one, so x tensor z tens or y, if you apply haramad to each one, so
Narayanan Rengaswamy: 3 changes to x.
Narayanan Rengaswamy: But why changes to minus y.
Zhuangzhuang Chen: So it we just add as a minus sign on the binary. Vector
No, no. S0 0n the right hand side after the arrow.
Narayanan Rengaswamy: Yeah. So after that, on the top line before the Z, you should put a minus sign.
Narayanan Rengaswamy: But of course, when you map it to the binary, vector you drop all the signs. So the binary vector, is correct.
Narayanan Rengaswamy: But just that, just for everybody's clarity. H. Y dagger. H yi x tiger is minus y. So you actually pick up a minus sign there, which which gets lost in the binary mapping.
Zhuangzhuang Chen: Yes.
Zhuangzhuang Chen: So
so yeah. So I just wanted to mention that. But but yeah, go ahead.
Stefan Krastanov: I wanted to ask something else here to all the folks, maybe especially from physics. This might look like
Stefan Krastanov: all really complicated like. Why would we ever boulder introducing it to you
Stefan Krastanov: when we already have the matrix represented the the there's a you intermittent. It's like how to use you intermittent. This is, you know we know how to use kids and vectors.
Stefan Krastanov: We. It is more difficult to learn how to rewrite this in terms of simplicity, matrices or stabilizers of roles, or what not. The value here is that this we're writing. Let's express a lot of things in
polynomial space and time instead of exponential space and time, even though it looks
Stefan Krastanov: more complicated to write down right now, for this one had a mark.
Stefan Krastanov: If you. this will be tractable while it won't, be tractable for the
Stefan Krastanov: yeah, you ready to go to go ahead.
Gabe Richardson: I have a question.
Gabe Richardson: Okay. So
Gabe Richardson: the the matrix that's on the bottom right corner. This one
Gabe Richardson: would there be, if I understand correctly, the way that we track the signs. Should there be an extra column at the end, and that's where we would know
Gabe Richardson: what the the sign is.
Zhuangzhuang Chen: So I
Zhuangzhuang Chen: I haven't consider that problem because I usually just
Zhuangzhuang Chen: when we we are doing so primarily the maximum as we just as just even all that time.
Zhuangzhuang Chen: So I have not that sure. Maybe we can add an extra column just like this part. We'll extra extra column to represents the science offers of poly operators.
Narayanan Rengaswamy: Yeah, Good. Good question. So yeah. So what John's Rob said is is correct. So so typically this homomorphism, like formally in mathematics. But so the this mapping that we have from this unitary space to this
Narayanan Rengaswamy: simpler takes space, that that that you know, ignores the any kind of science or scalars. But yeah, so if you are doing like a circuit simulation, as if the the science are important for your simulation. If they if they make a difference, then you should have this extra column, so you'll have 2 n plus one columns.
Narayanan Rengaswamy: and then you will carefully track the science at every step. But for for the application. The sign is not too important. Then you can just work with the 2 in columns.
Gabe Richardson: Okay, so it's it's application specific.
Narayanan Rengaswamy: Yes.
Gabe Richardson: And is there a like, would you say? Generally, we don't care about the signs, or is it? Is there not really a general rule from
Narayanan Rengaswamy: Yeah, I mean. So I know of some examples in my own research, where the sign does make a difference. But but I mean, maybe Stefan calls can also comment on this. Typically.
Narayanan Rengaswamy: the way the sign changes. Things is not
Narayanan Rengaswamy: super important. But in some cases you just want to be precise. So it's important. That's how I would put it.
Gabe Richardson: Okay, great. Thank you.
Stefan Krastanov: Change can frequently be just called a change of basis that you can beat up and not care about the library. I maintain, for instance. all the functions have an option to skip tracking the sign, because things become
Stefan Krastanov: 3 times faster. If you contract the sign, the the sign computation is actually way more complicated than the everything risks everything else here. Sure, it's just some small polynomial that you need to compute. But, unlike the rest of the computations. It's a couple of
Stefan Krastanov: complicated operations that that instead of just a binary x or so, there are a lot of reasons you might want to skip, computing that, especially if you're pushing for extremely high performance in your computations.
Narayanan Rengaswamy: Yeah, I completely agree. So
Narayanan Rengaswamy: I can like talk. We can talk offline, and they can give you one example where I couldn't avoid this, and, like Stefan said, it's just a pick pain
Narayanan Rengaswamy: to to track the science, because the binary mapping doesn't do it for you. You have to have, like more complicated in inner product formulas to track the sign, or you have to actually conjugate the matrices in the unitary space or do stuff, but but in neither of those basis it's nice.
Narayanan Rengaswamy: But in some cases it does make a difference. For example, when you are
Narayanan Rengaswamy: trying to figure out like how a particular circuit is acting on a code. And if you want to understand the logical operator, those kind of analysis the sign actually makes a difference, but otherwise I I typically in my simulation, set up also. I award it.
Stefan Krastanov: I just wanna make sure that I understand that. So what happens is here. You just have zeros and ones. But once you introduce the sign that's 0, plus one and minus one.
Stefan Krastanov: if that's what
Stefan Krastanov: it's 3 times more expensive. It's you have yours, and once for the
Stefan Krastanov: and internally the sign is just an extra bit, so everything is still just bits. You just have an extra call room of bits that track a sign per row.
Stefan Krastanov: right? But why is it a 3 times more expensive? Oh, because just because of coincidence of how CPU's work Today you have a lot of nice bit twice operations you can implement.
Stefan Krastanov: So most of this you see on this screen is be twice X or one single computer instruction. One single clock cycle on your CPU can d0 200 something queues at the same time.
Stefan Krastanov: If you want to track the the face in front of it.
just the expression, for that is a little bit more difficult to express in terms of bullying and algebra.
Stefan Krastanov: and it's that kind of boring, engineering reason that makes it forward
Stefan Krastanov: or the way in the regular mission. You need to track a inner product that's a little bit nasty to write down.
Stefan Krastanov: It's just that the the rule for the face is inspired. It comes from the Poly algebra, s0 0n, but it's unknowing to track down it's a slightly more complicated expression.
Stefan Krastanov: Is there any of the reason why we did that? X tensor the tensor of y? And then the tensor, the tensor X. It was just a nice example that our covers. So most of the
And so i'll prepare online cloud environment where you guys can just go in and claim some of these expressions. Simple, the
Stefan Krastanov: later expression and the bit twice expression behind it.
Is there a way where you can
Stefan Krastanov: somehow delay the keeping track of the signs and simplify things like.
Stefan Krastanov: I think not. Really. Okay. The depends on what you mean by that. But it Won't: Give you performance both.
Stefan Krastanov: Okay.
Stefan Krastanov: Yeah. Feel free to continue.
Zhuangzhuang Chen: Okay. So the
Zhuangzhuang Chen: seen on and the fiscal they all have their sympathy for. But they are more complicated with that of the part of our gate. But this
Zhuangzhuang Chen: what is this congregation rules? And here i'll show you this.
Zhuangzhuang Chen: this communication to about so as gate and the scene of the gate. So for the first gate we will map x to y. and that's the Z to z.
Zhuangzhuang Chen: So if we're not so x to y. So in the
Zhuangzhuang Chen: so in the same pattern matrix, we just add the X component into the Z component.
Zhuangzhuang Chen: and that's how we do the mapping, the simpler
Zhuangzhuang Chen: and the lower part is for this communication rules about the C. North. K.
Zhuangzhuang Chen: So see here means the the first qubit is the control qubit, and the second Cupid is a target qubit. So also, seeing our okay to will
Zhuangzhuang Chen: follows this communication rules and we will use this communication rule to do the poly tracking, you know, counter circuit.
Zhuangzhuang Chen: So here is the one also corners of the case, and we we want to do the poly tracking here.
Zhuangzhuang Chen: so
Zhuangzhuang Chen: will it act on the first a little bit what we will get in the final state. So here there is the party tracking works under this communication rules.
Zhuangzhuang Chen: So, just as we mentioned before, the access, the H.
Zhuangzhuang Chen: Z. Times H. Dagger equals to X,
Zhuangzhuang Chen: the up to the they pass through the harder market it it becomes X.
Zhuangzhuang Chen: Under this this X-ray is add to the the control qubit of a single gate. So it will be this scenario.
Zhuangzhuang Chen: It will be X X.
Zhuangzhuang Chen: Also excellent.
Zhuangzhuang Chen: will become the up to the hotel mode, and this is the get
Zhuangzhuang Chen: is absolutely target power. Cupid of the C or G. So, which corresponds to this one.
Zhuangzhuang Chen: and it will become easy.
Zhuangzhuang Chen: and X becomes the of the harder market. and we will.
Zhuangzhuang Chen: And this part is a little bit tricky, because we both have Zig. We have zig it in both. The
Zhuangzhuang Chen: Therefore first we just even know this one.
Zhuangzhuang Chen: and then fit it as the identity.
Zhuangzhuang Chen: so we can first consider the zik. It adds the target to the
Zhuangzhuang Chen: It's easy.
Zhuangzhuang Chen: No. When this Z gets pass through this scene, or nothing will change.
Zhuangzhuang Chen: So just like this scenario.
Zhuangzhuang Chen: So that's how why we get a Z. Times Z. Here. The Times, c. Is the identity.
Zhuangzhuang Chen: So finally we will get a Z. As a final state. Therefore, from this small example we can see how powerful this poly tracking
Zhuangzhuang Chen: in the
Zhuangzhuang Chen: But if we just to calculate directly on each cube is, the calculation might be very complex. So you can see that as this poly tracking is very powerful, and we can use this tracking rule to simulate on the class of computer in
Zhuangzhuang Chen: hundreds of Cubans.
Stefan Krastanov: But the last part of your calculation I just also came from the reversibility of the C. Not right. And so I can just
Zhuangzhuang Chen: this one.
Stefan Krastanov: Yeah. And I can take it then, from the picture about
Zhuangzhuang Chen: Yeah. So first we just even know this. You know this one and the trips it as an identity. Right?
Zhuangzhuang Chen: So this is it is at the target.
Stefan Krastanov: So those are 2 different reasonable ways to do it. So Don't was asking Sl something like different. he was saying.
Stefan Krastanov: Wait, if I look up there, and I can just use the last diagram you have
Stefan Krastanov: who in the left you to this on the right.
Stefan Krastanov: If you use that in reverse, you get the same results you just described, and almost suggesting that that's
Stefan Krastanov: that. That's how it can be done. You were suggesting a different way to do it. You were saying that
Stefan Krastanov: Z. On the top and on the bottom.
Stefan Krastanov: Yeah.
Zhuangzhuang Chen: yeah, Can you consider the separately?
Stefan Krastanov: Yes, you were. You were composing them in Z on the top and I on the bottom.
Yep, go ahead.
Narayanan Rengaswamy: But the 3 quibits start in this in this like 0 0 0 state. So like, he was saying, if you directly do like matrix went to multiplication to simulate the output of the circuit to find site. Then you have this. You have all the Cds and how to month. So you effectively have, like 8 by 8 metrices
Narayanan Rengaswamy: that you have to multiply and then act on the input State to find out what size.
Narayanan Rengaswamy: But however, what this allos is. But how you can use this poly tracking is that the input, is what we call a stabilizer state it. It just means that it has stabilizers that are polymatr. So, for example, the reason John John started with Z on the first qubit
Narayanan Rengaswamy: was because z acting on 0 gives you back 0. So Z is a stabilizer of 0,
Narayanan Rengaswamy: and similarly for the other to quit. So basically the group generated by Z. On each of the 3 qubits of the input you get like a stabilize a group of with like 8 elements, which is basically all products of
Narayanan Rengaswamy: these policies and for identities. And so that is a full stabilizer group that uniquely identifies the input State 0 0 0.
Narayanan Rengaswamy: And similarly, what you're doing is you can take each of those stabilizers see on the first cubic Z on the second, C, which is on the third qubit, and if you track them like Zhang Zhong did then you get some outputs at the end, and those will form the stabilizer generators for the output State sign.
Narayanan Rengaswamy: and it is just an equal and representation of site. They uniquely identify site up to a global face. And this is why this kind of polytracking is very efficient for these kind of settings.
Stefan Krastanov: I have a question. So in the matrix of presentation you do not need to keep a dictionary, Webinar, in the stabilize the foundation that you would have to keep a dictionary for how.
Stefan Krastanov: and it is indeed a dictionary, but it's a dictionary for 3 letters. The rest is efficiently implemented out of that related dictionary.
Stefan Krastanov: The you know gate itself can be represented in the system like kick matrix that app. So it's. In fact, you addresses. So, indeed, there is a low, level multiplication in that needs to be.
There is some some dictionary.
Stefan Krastanov: but that's no different from saying you need a dictionary for for multiplication between the real numbers or integers. This is this will be a very simple.
Stefan Krastanov: large cooperation, that on which you'll build the rest of the
Stefan Krastanov: go ahead.
Zhuangzhuang Chen: Okay, sure
Zhuangzhuang Chen: is, we can use this poly tracking rules to find those to find the stabilizers of encoding circuit.
Zhuangzhuang Chen: So here I just use the example of the 3 1 3 code is. It is a well-known piece flip code.
Zhuangzhuang Chen: So we recall that the 0 you close t0 0 theta 0. So there's 0 stage 0, is the stabilizer.
Zhuangzhuang Chen: So Therefore, if we want to find a stabilizer, we just that to the it. X on the second cube it so we can do the conjugation. Same communications on the single gate.
Zhuangzhuang Chen: Yeah, they is at the target. So it becomes easy. and that at this time Z is add to the control of qubit. Nothing will change. So we will get to this as a one of those stabilizer. For this
Zhuangzhuang Chen: I will correct some code.
Zhuangzhuang Chen: and also, if we just act as a zip it. On the third qubit we can find another stipleizer, and the overall stabilizer is our Ii Zi iz, and z I. Z.
Zhuangzhuang Chen: Totally. There are 4 elements in this stabilizer group generated by Z. I. And I.
Zhuangzhuang Chen: So this is the concept conceptually correct of
Zhuangzhuang Chen: for choosing
Zhuangzhuang Chen: using this rule to
Zhuangzhuang Chen: to find the stabilizers of the encoding circuits. But in practical we
Zhuangzhuang Chen: we we have stabilizers, and to find and to to design the encoding circuits, so we are doing, the reverse.
Zhuangzhuang Chen: But still this poly tracking home location rules is really useful when we are doing the quantum I/O correction. So, for example, if we some bit of
Zhuangzhuang Chen: bit free. For all happens. We can also do some execution of the circuit, and then we can do the poly tracking rule to see how this arrow propagates in those
Zhuangzhuang Chen: in the in those quantum circuits. and therefore we can see that it's probably tracking through. It's a very powerful in the a quantum error correction, and also this communication will be
Zhuangzhuang Chen: can be used in other scenarios. But to
Zhuangzhuang Chen: quantum error corruption is is a is one of zoom.
Zhuangzhuang Chen: Okay. Now let me summarize what I have covered today. So the first I just introduced the concept of a polypit poly group and stabilized a stabilize the code and the cliff gets successful.
Zhuangzhuang Chen: and also we can see that Hermitian poly matrices can be measured into pail by spinning reptiles, and also the Clifford. The case can be represented as 2 n times 2 in binary. Simple, natural matrices acting on the poly operators.
Zhuangzhuang Chen: and the Clifford is is defined to map. To the
Zhuangzhuang Chen: maps are a poly operator to another poly operator. Also we can
Zhuangzhuang Chen: see that the the confluation, poly attraction is really a powerful tool in the quantum error creation, and it can reduce the complexity
Zhuangzhuang Chen: from directly using the New Zealand matrices multiplication to do the calculation on each cup
Zhuangzhuang Chen: and then. Now we can conclude that the quantum circuit, consisting solely of See not how much. And a fisc gate can be simulated efficiently on a classical computer. And that's what it goes to. My cell room tells us
Zhuangzhuang Chen: so. Here's my my reference, and my first reference is the paper I'm. Presenting today.
Zhuangzhuang Chen: and the second reference is a textbook recent from quantum computation and no control information, and this textbook gave us some of only mental background, mathematical backgrounds about to the quantum competition.
Zhuangzhuang Chen: and they also introduced the concept of for quantum circuit, and no quantum error correction.
Zhuangzhuang Chen: And the third reference is is a paper from by.
Zhuangzhuang Chen: and the this paper shows us in. They did in detail about the the synthetic
Zhuangzhuang Chen: detail about the same fashion, which is how how this, how this
Zhuangzhuang Chen: it's used in the Clifford case, and how we use the simplicity matrix to Max a logical circuit to a physical circuit.
Zhuangzhuang Chen: So if you are interested in the the simplified geometry, I strongly recommend you to read this paper to get a better view of that.
Zhuangzhuang Chen: Okay, that's all for my presentations and thanks for listening.
Stefan Krastanov: Thank you.
Stefan Krastanov: I will suggest we start with the second presentation right now before questions, because the second presentation will provide, we can
Stefan Krastanov: the same topics in a slightly different language. I think you're very valuable to to have both of those languages discussed a little bit before we go into the morning. That questions
Stefan Krastanov: try to share, and if it's not working, I think I have it now.
Stefan Krastanov: I have a keynote file. I assume it's from for my only
Stefan Krastanov: i'm sorry.
Stefan Krastanov: Oh, were you able to share, or
Stefan Krastanov: it looks like that, You guys see? Yeah, You see this morning. Okay? Great.
So
Stefan Krastanov: I guess Start at your convenience. Please do introduce yourself. And oh, we should
turn off. Yeah.
Stefan Krastanov: Okay.
Stefan Krastanov: And you guys here
Stefan Krastanov: folks online. Could you hear from here? Well.
Narayanan Rengaswamy: yes, yes.
Stefan Krastanov: okay, cool. Yeah. S0 0kay, my name is Savier and I'm. Grange. This is a Phd students P. Rms.
Stefan Krastanov: Working on something, Anybody physics. So my background is not on the computer. But
Stefan Krastanov: yeah, I guess I am now enroll in the course we're glad to have you.
Stefan Krastanov: So yeah, I guess i'll be talking today about the how to do bundle computing in the testing board picture, which I think
Stefan Krastanov: the reviews with the presenter has over
Stefan Krastanov: much, much better. So it is my it is. I got behind my desk. That's a not much. I guess it's it's it's manager to me. I guess I know that
Stefan Krastanov: so very much. But but
Stefan Krastanov: essentially yeah. So motivation for this is that.
Stefan Krastanov: Do you think about the mechanics? We consider the state of evolution under the showing our picture?
Stefan Krastanov: This is just to be, you know, evolving at gets States under some
Stefan Krastanov: utilization which we think of some exponential, some.
Stefan Krastanov: and in the end this.
Stefan Krastanov: you know. my physics background. We are usually just in measuring some observable for some time.
Stefan Krastanov: and this is essentially taking the racket, or like the sandwiches between the all the State and some of it. And
there
Stefan Krastanov: this letter. Yep, Thank you. Sure.
Yeah. So
Stefan Krastanov: yeah, I said in the end, yeah, we're interested in some observable
Stefan Krastanov: just by looking at this.
Stefan Krastanov: I think, instead of evolving the state. You can just think of the whole thing that they're doing, just and measuring.
Stefan Krastanov: And this gives us that this gives rise to the pizza for a picture of time evolution essentially, that you guys do evolve over there within the bracket with the sandwich as as a seer, Shawn.
Stefan Krastanov: And there are some
Stefan Krastanov: different problems in the settings where we might be dressed with any of all. We never the operator instead of the state I can think of
Stefan Krastanov: first. That's how it's on the simulation where we adjusted, and measuring
Stefan Krastanov: time, or that it correlators as some here. This is like the expectation body of some two-point function of all but it. There is themselves
Stefan Krastanov: where the expedition body is given in terms of bikes on trace, with some financial, some comments on it, some temperature. In our
Stefan Krastanov: you got the information also. It's important. For instance, there are similar quantities to this time, or their with so much physical
Stefan Krastanov: different meeting this out of time, or the perilers, which measure is probably of information which but I don't have much time to talk about. But I guess, for for the present presentation the most important motivation is in one of the operations specifically in the
Stefan Krastanov: that we've discussed already.
Stefan Krastanov: And I would, I guess, show why it is so. So yeah, some basic. So the Heisenberg presentation.
I've already
Stefan Krastanov: discuss what it is, but it essentially you can think of. Apply some operator and onto a get State.
Stefan Krastanov: and then
Stefan Krastanov: we can assume that we evolved this by by by you, By you we get this new state.
Stefan Krastanov: which is, and i'm you times to get, and this and broad, essentially Isn't evolved over here that Mycember picture
Stefan Krastanov: s0 0ne
Stefan Krastanov: for what concerns it.
Stefan Krastanov: This Heisenberg time evolution, as well as so. If the initial state is an eigenstate of the a very third with some I good body Alpha.
Stefan Krastanov: that the evolved
Stefan Krastanov: State will also be, I can say. And you again. I you I get saved, but with the same I can value of the bulk of it here, so this will be very useful in some of the that's your format.
Stefan Krastanov: And so among all the classes of moderators that we can consider, we will be dressed in the those that those that belong to the body group, and
Stefan Krastanov: a ex operators acting on each cub it as the operators.
Stefan Krastanov: And this is.
Stefan Krastanov: and this is because essentially the remaining operators
Stefan Krastanov: I can be observed by by, by, by, by, by by multiplying X and Z.
Stefan Krastanov: And so yeah, i'm. On the portion port to the end of the 20 group, only to an operators are needed to to start with the rest.
Stefan Krastanov: Yeah. So within the quality growth, the most important group that we need for for our question is a a different group, and this is the fine as the
Stefan Krastanov: And then this sometimes is known as the centralizer of the group, because of this property here that the involved
Stefan Krastanov: open it, or must
Stefan Krastanov: about, or other education.
Stefan Krastanov: And then, yeah, the set of of editors that belong to this super group are, it's part of our objects which I think right. Ts. Are because of the the paper I guess, for is sorry about resonance. They
Stefan Krastanov: it was referred to as far. But this is the
Stefan Krastanov: matches, and then
Stefan Krastanov: this an output. And
Stefan Krastanov: yeah, this is a remark. But anybody magic's kind of you're doing so with the how them are going to them, please.
Stefan Krastanov: Yeah. So why? Why is this paper
useful? Well, it's because essentially
Stefan Krastanov: we can create a dictionary of the operations.
Stefan Krastanov: That's that's the
this is
Stefan Krastanov: essentially yeah. This is this here.
Stefan Krastanov: So essentially what I'm saying is that ready button circuit?
Stefan Krastanov: I' to look at the table, and I see what the result this. And so this is very efficient.
Stefan Krastanov: Yeah. So I'm: on. Yeah, the best of the the the of it there is, you know. It's the funny group. We will be interested in those that have us input states. That's why I invictors.
Stefan Krastanov: as I will show later. But
Stefan Krastanov: the this defines the
Stefan Krastanov: police that have us. I guess this the second value, the plus one I get value for all inputs. So the inputs will be.
Stefan Krastanov: So this is: tell you this about that; for if for any set of kits there will be a different 7 0r subset.
Stefan Krastanov: So let's consider this: this: this this example of 2 cubits, where the second one is fixed t0 0.
Stefan Krastanov: So it's not just for this circuit our identity magic on the
it' be on the first and the on the second.
Stefan Krastanov: and so this can see. There. Now, look at what happens when we will. It's not be nice, or I 10, c.
Stefan Krastanov: So this will give us.
Stefan Krastanov: Look up, table of the previous side.
Stefan Krastanov: This this circuit encodes one logical qubit on the first register, and that all you go through it is
Stefan Krastanov: that's essentially. We only have to keep track of the logic of it, or X, and then all you can operate there, and Z. I think, on the first do it.
Stefan Krastanov: And so, after Percy, not the Ix operator, evolves to X after the second one gives us I. X. So exactly on the second register.
Stefan Krastanov: That was for for the
Stefan Krastanov: I am to your folks a little bit more time on this slide, because it probably involves a lot of the more fundamental forations that are being discussed
Stefan Krastanov: on the very right. We have the stabilizers, which is how you represent the state you have these operators. He was referring to.
Stefan Krastanov: That's the
Stefan Krastanov: the state we care about is at the same time an eigenstate with all these operators.
Stefan Krastanov: The state is difficult to write down. These operators easy to write down, and everything else down in these circles is just looking through the look of table, of how these operators change when
Stefan Krastanov: when the gate is applied, and the important feature here is that this lookup table. or given the previous start, this look up table is just a small simpletic matrix, not an explanation to large.
Stefan Krastanov: Okay. Okay.
Stefan Krastanov: Okay. So how to find someidizers? We can argue that this
Stefan Krastanov: considering to.
Stefan Krastanov: I think M. And N on on side, will get us side, but also, I think, by end, and we also plus sign. Essentially this is already us the but I mean i'll be again
Stefan Krastanov: Nature of the side as a group. And then.
Stefan Krastanov: yeah, the other thing is that the only possible to see our bus might as well as we're discussing in the Us.
Stefan Krastanov: Our presentation. And this essentially tells us that the Stabilizer group is I. It's a more fit to the Z 2, so that they we're.
Stefan Krastanov: And so it's actually the the the full
Stefan Krastanov: operators sorry of moving and stuff that I thought about buying some of the
Stefan Krastanov: Okay. So let's consider. This is like an example to to to have a better invitation of how
Stefan Krastanov: and so by the view slide we have to to it's not. It is for for the circuit our
entity. Xx: Minus?
Stefan Krastanov: Why, why
Stefan Krastanov: and
Stefan Krastanov: yeah, what I show here on the is essentially the
Stefan Krastanov: the evolved.
Stefan Krastanov: Yeah. Aside from the I of the
Stefan Krastanov: what happens when we evolve through this the first circuit first of the the client market, then
Stefan Krastanov: bye, bye.
Stefan Krastanov: And s0 0ne interesting thing to remark here is that even though see what it industry. It is basically
Stefan Krastanov: very efficient through the uptable.
Stefan Krastanov: The
Stefan Krastanov: Yeah. I guess that my motivation for the subject matter is that it's it's very convenient. They've got this up to the air direction.
Stefan Krastanov: and the reason is because this allows us to be back
Stefan Krastanov: easily. The errors, and by the audiences correct them also easily.
Stefan Krastanov: And s0 0kay, for instance, this one. See that this this circuit here.
So the circuit and
Stefan Krastanov: the text one you can error. So it's not you guys, for this group are
Stefan Krastanov: shown here. Xxx. X.
Stefan Krastanov: And the errors are detective anytime we find a element of these hybridizer set that does not commute with the error or the enemy
Stefan Krastanov: question. So if we find to provide an operator on to the twist on, there is some one of them.
Stefan Krastanov: In this case this is okay, correct, correct. I wonder
Stefan Krastanov: that I can be seen by a blind
Stefan Krastanov: that's got to be seen by a blind
Stefan Krastanov: x, Y or Z, or what is he
Stefan Krastanov: onto that first 1 s? You guys.
Ali: Can I ask a question here?
Ali: Okay, when we're actually
Ali: trying to do quantum error correction, I'm. Assuming we start from the stabilizers starting from like.
Ali: maybe classically motivated code.
Ali: So is this directly useful, for I like. How do we then figure out how to construct that
stabilizer state.
Ali: Do you have to work backwards. try to come up with the circuit and work backwards, using the Clifford tracking to come up with the circuit or something.
Stefan Krastanov: Yeah, that's a that's actually a great question. I I also I mean, I I also have the same question. I I guess
Stefan Krastanov: I know how how this is like I I it's usually.
Stefan Krastanov: I think it's not trivial, for for the Jerry case. Find the stabilizer states that correspond to even set up operators.
Stefan Krastanov: But
Stefan Krastanov: yeah, maybe in what direction was the question, Was it? You have a cold
Stefan Krastanov: described with the set of stabilizers?
Stefan Krastanov: How do you actually make a code word in your computer? Was that the question? Or
Ali: that's the question
Stefan Krastanov: It so there is. There are a couple of different ways, but the easiest one is probably just start doing your creation.
Stefan Krastanov: You measure the state. You have some initial state. You don't know what to do with this you start doing the error correction. You measure the stabilizer Well, every single time you're measuring one of those you're forcing a constraint on your on your state.
Stefan Krastanov: You're steering the stage from whatever it was, forcing it into being. Your cold work.
Stefan Krastanov: You don't know which it can be a mixture of all the words.
Ali: so
Stefan Krastanov: the the logical operator, and so that will project it. That will project you in one of the logical queue called words.
Stefan Krastanov: And now that you know which one, if it's not the one you want, You just perform logical big flips up to with this the one you want.
Stefan Krastanov: Okay. What was I talking to? I don't know who was I talking to?
Ali: This is Ali from the University of Arizona.
Thank you. Just so. Whenever anyone is speaking of, because we're in 2 different spaces. If you can just say who's talking.
Stefan Krastanov: should we continue?
Stefan Krastanov: Sure, Go ahead.
Stefan Krastanov: Okay. So yeah, the review is circuit kind of next up to an error
Stefan Krastanov: 5 t0 0ne correct
Stefan Krastanov: errors. Well, it's a bit more difficult. But that, yeah is essentially that if we found one.
Stefan Krastanov: if we have 1 0ne error, and we know which operator response to them essentially, that you have to apply the same operator again, because that will give us the
Stefan Krastanov: so; for if you want to now correct one error up t0 0ne error, then we need 5 pivots instead of and
Stefan Krastanov: and so this is this modest.
Stefan Krastanov: What do we Operation Code? The correct of the one give it. And this is not as if I' to code
Stefan Krastanov: some guys are here. Are those for here and
Stefan Krastanov: this yeah, that would be goal to. Well, yeah, the the this is yeah, this one. But you both give it anization part. For this
Stefan Krastanov: code is as we're done, as here, as in terms of like double brackets. The first entry is a number of
Stefan Krastanov: 3 is a go distance, which I guess I haven't defined yet, but the go distance
Stefan Krastanov: for so this has to go distance 3. And then what is the number of what you call greater? This one.
Stefan Krastanov: Okay. So there are some certain errors that will give us, and I give you one. I'm going to start at your state, and these are errors that can be corrected, but they are. They are harmless. So you can
Stefan Krastanov: forget about that.
Stefan Krastanov: By what this means is essentially that the code is degenerate.
Stefan Krastanov: Otherwise it's all the yours, I think i'm here to, because this one it's not. It's not the
Stefan Krastanov: and I guess. Yeah, this is.
Stefan Krastanov: I guess the reason why it's not this. I
Stefan Krastanov: it's yes, that's very useful. You've got the very versions. Because.
Stefan Krastanov: yeah, what are the needs of? You know your resources and and physical resources. And this is captured by the by this year. And here.
Stefan Krastanov: yeah, and then
Stefan Krastanov: yeah, I guess I guess it's good.
Stefan Krastanov: So yeah.
Stefan Krastanov: yeah, we can go to the
Stefan Krastanov: essentially.
Stefan Krastanov: I only focus on evolution of the
Stefan Krastanov: and however this has it simulations in particular, we cannot.
Stefan Krastanov: There are some gates that are outside of the group, and and this
Stefan Krastanov: this doesn't provide a universal
Stefan Krastanov: on a computer. And
Stefan Krastanov: yeah, I guess the size. Yeah, that particular but particular
Stefan Krastanov: what I do about my integration here and coding on decoding.
Stefan Krastanov: Yeah.
Stefan Krastanov: bye, bye, Thank you.
Stefan Krastanov: Thank you. So let's open up the floor for questions to both presenters. If your question is specifically targeted t0 0ne or the other just mentioned that otherwise we'll do it more in a panel fashion.
and the presenters obviously.
Stefan Krastanov: if you, if you not certain about some other questions, not them back to us; and if we're uncertain, or at least try to find where to find the answers.
Stefan Krastanov: I I I have a question. S0 0ur poly's, and see what's universal for quantum mirror correction.
Stefan Krastanov: Well, then, you
Stefan Krastanov: you can
Stefan Krastanov: do. You know, no matter what kind of errors.
Stefan Krastanov: those that are on the body
Stefan Krastanov: meeting Facebook. I mean this error or
Stefan Krastanov: coming directed. So now
Stefan Krastanov: 2 could be dates that are not
Stefan Krastanov: got it. So it's
Narayanan Rengaswamy: I mean, typically so. For like designing codes and like doing analysis and stuff, we work with like poly channels like depolarizing or other channels. So we typically sample poly errors and measure stabilizers find the signs of the stabilizers, and then fix the eigenvalues to fix the errors.
Narayanan Rengaswamy: But of course, in practice we cannot always guarantee that the error will be a poly that could be some like over rotation when you shine lasers and like, do Gates, if you want to do like a pi over 2 rotation. You might do like pi over 2 plus epsilon or something, so that can be this kind of like coherent errors, and als0 0ther
Narayanan Rengaswamy: stochastic errors that are not necessary poly errors, but the powerful thing in a quantum that distinguishes itself, for example, from analog computing
Narayanan Rengaswamy: it's because, for example, an analog updating, if the error is like a continuous number, you have to exactly find the number and fix the error, whereas in quantum mechanics. You have this
Narayanan Rengaswamy: crucial principle called the Digitization of Errors, or like the discretization of errors.
Narayanan Rengaswamy: which basically says that when you measure stabilizers, because the quantum quantum mechanics postulates. Tell us that when you measure an operator you project the state t0 0ne of the eigenvalues, one of the icon spaces of the operator.
Narayanan Rengaswamy: So when you, when you, you know, do a see a sequence of central measurements, stabilize the measurements, then you're guaranteed that you have projected this state
Narayanan Rengaswamy: to the common L in space of those stabilizers, up to some changes in signs
Narayanan Rengaswamy: so effectively. Even if you had a non poly error to start with, because your measurement projects you t0 0ne of these eigen spaces. That is an effective poly error. That also represents that particular. The change from the original plus one. I can space to this new eigen space.
Narayanan Rengaswamy: so you can apply that most likely poly error and hope that everything is fixed.
Narayanan Rengaswamy: Now, when does this work is, there are caveats, meaning that there should not be too much too many correlations, and, like other conditions under those constraints, you can show that this process of measuring stabilizers and projecting the state, and then doing a poly
Narayanan Rengaswamy: correction actually fixes the error.
Narayanan Rengaswamy: But if you have more correlated errors or more nasty things, then this projection might already change the logical information in a non trivial way, which you might not be able to figure out just by measuring syndrome. Am I making sense?
Stefan Krastanov: Yes, I have a
Stefan Krastanov: sample to that. So Maria kind of mentioned that example. I just want to mention it explicitly. So you have one qubit. and you you want to just keep it. You don't want anything to be happening to that.
Stefan Krastanov: But a little bit of a rotation kind of straight. Now it's a superposition of the State 0 that it used to be, and a little bit epsilon of State one.
Stefan Krastanov: This is not a poly operation. This, if you don't have Cuba like that, and there's indignant, and s0 0n. Even even simulating that is, is prohibitively expensive. This is not something that fits the stabilizer for myself.
So sure. One way to correct that here would be to know for a fact that that was the rotation that happened. And then just backtrack.
Stefan Krastanov: We don't have that capability under the formalism we describe today. What we can do is
Stefan Krastanov: we measure the stabilizer, or in this case we just measure the qubit.
Stefan Krastanov: and there is some chance that it's 0. A very high chance. That is 0, and some low chance that it's one some low chance that effectively. There is an X operation, a only bit flip that has happened
into this measurement. As an already mentioned.
Stefan Krastanov: This
Stefan Krastanov: terms this coherent small rotation that can not be represented as a poly operation into something that is probabilistically a poly operation. Either. No, all the operation or some small chance for a polynomial flip.
Stefan Krastanov: So
Stefan Krastanov: what happens when your measurement introduces an error? It is not a all year. So so you're modeling a measurement as it would only give you a all year at the end right.
Stefan Krastanov: I mean, or even a perfect microphone. What happens when your actual apparatus induces a channel that is not described as a policy channel. So
Stefan Krastanov: couple of different things need to be specified here. But first you're the one that's making the measurement. You say that you're trying to measure the stabilizer. The stabilizer is a only operator. You're measuring
Stefan Krastanov: your your zoom room is a poly operator. That that's part of the call. So that
Stefan Krastanov: is what it is right.
Stefan Krastanov: However, maybe there is some arrow top of it. Your measurement is not very precise, so it's not that your measurement is not the poly operation at all. Sorry it's on the it's not measurement of a poly, observed. Well, at this you're attempting to.
Stefan Krastanov: but there might be some extraordinary on that that might be the measurement worked out fine. But there was a classical bit slipping the digital electronics of the measurement apparatus, and it gave you the wrong result.
Stefan Krastanov: The measurement was in a basis that was actually a little bit rotated. So sure you are making a poly measurement. But implicitly, there was a small rotation before the only measurement.
that access of measurement, actually
Stefan Krastanov: the drift Slowly.
All of this
Stefan Krastanov: is sources of errors that can in most cases be modeled as some mixture of all the operations before or after things have happened. And so sure this is more noise, so it's. You need to make sure that your colds can correct for that as well. But it is still a question of having enough redundancy to detect these errors. With that there was some major operation that happened before you start measuring the synchronizer.
Stefan Krastanov: or there is calls by measuring the stabilizer.
I see.
Ali: have a
Ali: so follow up on my initial question. So
Ali: this last point on this conclusion slide about encoding, plus decoding, being the
Ali: the consequence of.
Ali: or the attractive feature of, the Clifford circuits is that
Ali: they allow you in the office of this, so I understand that
Ali: to encode the State we can use the method. That stuff on the describe which is, start with the I'm Tree State and start doing qu American action.
Ali: and that will end up with an encoded state. But for recording and operator.
Ali: I now see that a circuit would be helpful.
But how do we construct the circuit?
Ali: If we know the
Ali: code we're using is my question.
Stefan Krastanov: I will say that this so? A large question, and I think the you for some for colleagues here will be presenting on this over the next few weeks.
Ali: 0 0kay.
Stefan Krastanov: And so, just to briefly mention, there are some straightforward ways to find what you do physically. If the operator you want to perform is a poly operation, logical audi0 0peration
Stefan Krastanov: a little bit more complicated if you want to perform a logical
Stefan Krastanov: is shown.
Stefan Krastanov: and it's actually really complicated question with an active area of research. If you want to perform a logical normally for the operation, and there are a lot of
Stefan Krastanov: a lot of caveats and difficulties when you're starting that
Ali: I see.
Stefan Krastanov: So I just wanted to mention that trying to to do what I was saying i'll be doing the previous question. We're answering was from the Ts. And I think he has another one. Yeah.
Stefan Krastanov: is there? This is within the this, just this question: is there a them that says that you can do this? Non, Clifford.
there is a German that says that there is no way to do an air protected. So not only for the gate, any of different gate.
Stefan Krastanov: If You're just doing the circuit directly, but there is some work around. So about around this noble Quran, and this noble cheering, and the workarounds will probably be topics for the second half of this this semester.
Narayanan Rengaswamy: So let me let me let me qualify what what Stefan said, just to be clear to everyone. So I mean, so fall tolerance is is is something we haven't discussed yet. In this course we will discuss that later. But yeah, doing a logical non Clifford gate fall tolerantly, is challenging.
Narayanan Rengaswamy: Now, the theorem that, Stefan pointed out the bravi connect theorem. Is that sorry? The Eastern kernel, theorem is that you cannot do a universal set of logical operations using transverse of gates. So transversal gates mean that
Narayanan Rengaswamy: you do not interact. So the physical circuit that you are implementing is is doing an independent unitary on each qubit.
Narayanan Rengaswamy: or more gently, an independent operation on a fixed partition of queues. So, for example, you might want to, if you have like 50 cubic. You can group them int0 2 0r 3, but you fix the partition, and with with respect to the fixed partition, the only operations, you allow.
Narayanan Rengaswamy: are unit trees that interact only qubits within the group.
Narayanan Rengaswamy: but not qubits across groups. So for those kind of transverse operations, you cannot get a universal set of logical operations.
Narayanan Rengaswamy: And and so you have to do some operations transversely. But other operations using other fault to run methods, and that's where things get challenging for some codes. We know that you can get a universe. You can get all logical. Clifford gets using transfers of operations like this team code which we'll hear about soon
Narayanan Rengaswamy: or but but you cannot. Yeah, but you cannot get like a logical T gate, or like a logical, not to for git
Narayanan Rengaswamy: also transversity on the same code. So you have to do something else, and there are different methods developed. Each one comes with its own resource overhead, and, like Stefan said, it's a massive field. I've done some work in my Phd. But it's it's. It's just a hard problem.
Narayanan Rengaswamy: So people are trying to figure out how to do how to do it.
Stefan Krastanov: One third question
Stefan Krastanov: Go ahead. Something else.
Stefan Krastanov: You're the one that's doing the
Stefan Krastanov: Okay. So I apologize for this. I'll let you read the question. But I want to make the point that, as we mentioned, this is a very wide area of research, and these questions are difficult. They deserve their own presentation. Yes.
Stefan Krastanov: so could you repeat your question now that I've in terms of you like that? Just so we know what you'll be talking about in a couple of weeks. Yeah. So my question was, Can is there a proof that it could actually do universal one computation?
Stefan Krastanov: All? Okay, yeah, I mean. So yeah, I mean, so whether we can do it or not, it's not the question that is solved like we we know of
Narayanan Rengaswamy: quite a few different ways and different codes, and on which we can do these things? The question is just like resource overhead, and like, is it practical? Are there better methods to do it? Things like that? So, for example, on the surface code, people have figured out how to do universal sort of kits.
Narayanan Rengaswamy: But the problem is, you're investing like 100 cubits per logical quid.
Narayanan Rengaswamy: Which means, if you want to do any interesting algorithm you are. You probably want like 50 t0 100 logical qubits, which means even a very, very optimistic calculation gives you 10,000 physical qubits to build
Narayanan Rengaswamy: with a lot of like moving things around and like grading, and all these things that are just like an engineering nightmare.
Narayanan Rengaswamy: So that so the question is just like, yeah, just coming up with better methods or new techniques to get universe in front of.
Stefan Krastanov: I want to bring back to the the particular it's of the stabilizer promoters
Stefan Krastanov: This the blow representations you like to matrix or whatever it can represent a State, but we also use it to represent a code, an airbreaking code.
Stefan Krastanov: And how come
Stefan Krastanov: that representation cover both of calendars, both of both of those applications.
Stefan Krastanov: If you would like to try to speak a little bit about that, and then we can feel in as well after that
Stefan Krastanov: to speak of college.
Stefan Krastanov: So in the sound examples, and they know from Arizona feel free to jump in as well in some of the examples we have at the blow list of stabilizers, and that list of stabilizers represent the known state. This was exactly the state I was working with.
Stefan Krastanov: If it is the state that's stabilized, that is the that is the I I can say of all these stabilizer operations. But we're using the set of stabilizers to refer to a State to a state of the system.
Stefan Krastanov: But also we saw that for all of the air correcting calls, we we define Directory in cold by a list of stabilizer measurements that need to be known.
Stefan Krastanov: and that same object, the set of stabilizer measurements of stabilizers represents in one situation at a state
Stefan Krastanov: any other situation on error correcting cold? What is the relationship between those 2?
Stefan Krastanov: I'm sure plans to compute their question. But I guess I mean usually
Stefan Krastanov: you would define your stupidizers, and from their
you can define the there's some
Stefan Krastanov: that that's fair. That's a little bit about the camera as well. I was saying that
Stefan Krastanov: the definition of the State is the the stabilizers. I don't need.
Stefan Krastanov: I can't representation, sure, One firstly valid way to write down complete description of a State, and this completely define what stage you are in is to write it down as a. K.
Stefan Krastanov: But it's the perfectly valid approach to write down a bunch of stabilizers, and to never touch K. Notation, and that is a state it's a state represented in a different notation.
Stefan Krastanov: But they this still a perfectly well defined state. It is one single state.
Stefan Krastanov: What's your original as a kit or a set of stabilizers?
Stefan Krastanov: I mean, I guess. Yeah, I guess that's the reason why I what's more about the question. I mean, I guess it depends on what you're being
Stefan Krastanov: by by by by state for me the you know something that has to race one, and, like the usual definitions of up on home State that. But
Stefan Krastanov: you know, if I take the the trace of a polio, but it would you be. But no, you're not taking the trace of a polio bridge. You're taking the trace of the state defined by
Stefan Krastanov: the set of operators. It is an I can save up so sure the trace operation is much more
Stefan Krastanov: might be much more difficult to compute if your state is defined by some other representation. But it does this, this trace operations to exist, and I guess what I was going for is that
Stefan Krastanov: when you use the stabilize or notation, what you're doing is
Stefan Krastanov: you're pulling a bunch of constraints on what your state is permitted to be. The first stabilizer limits this the cubic space in which your State can exist. The second stabilizer limits it even more. The 2 stabilizer gets the size of the space in cash yet again, so the more stabilizers you have listed.
the more constraints you have.
Stefan Krastanov: and if you have as many constraints as number of qubits, you have perfectly defined your state. There is only one solution to that type of constraints.
Stefan Krastanov: If you remove one of the constraints
Stefan Krastanov: now you have a the blow that's not square. You have fewer rows than columns, fewer roles than the qubits. Now we have a degree of freedom left. Now we have a cold space to explore that if any state in that space is permitted
Stefan Krastanov: according to the the stabilizer constraints to your quote. And that is basically the definition of an you're correcting code, some subspace of the whole space that you're permitted to work with.
Stefan Krastanov: And that's where that's why we get to use the same representation for both things. Basically You can think of a state.
Stefan Krastanov: some error correcting code where you have as many constraints as qubits. It's just that you're not left with any degrees of freedom any more. You You convey that you can then call 0 degrees of 0 bits of information in in that call, because the cold is now perfectly constrained to all the State has to.
Stefan Krastanov: So there's a one t0 0ne map in between space and the
Stefan Krastanov: But there's a thing right, because you.
Stefan Krastanov: I don't think so. Any stabilizer state can be expressed that I can.
Stefan Krastanov: and that's easy
Stefan Krastanov: if the computation is straightforward.
Stefan Krastanov: and if let's say it's represented as a kid. It's a stabilizer state. We're already know that for a fact, someone an oracle gave us a State a kid in the kit representation in the vector representation and told us it's a stabilizer state
Stefan Krastanov: finding the stabilizers for that kid is an expensive operation that they're financially difficult to compute operation.
Narayanan Rengaswamy: and also just to just to complete the point and let tied the point that Stefan, what was making to this poly tracking thing that we saw
Narayanan Rengaswamy: so exactly so like, Stefan said. If you have, say, 3 qubits like in the example that John Wrong had, and if you have 3 independent stabilizers for the 3 cubic state. Then you have fully defined the State by its full stabilizer group.
Narayanan Rengaswamy: And this is why, when you start with the tensor product of 0 0 0. You can generate a stabilizer group independently using Z on the first qubit, which is a stabilizer, Z and the second qubit. And so the different combinations of the different T. Of products Of these 3 Z's will give you 8 possible.
Narayanan Rengaswamy: the I operators, and this is a full stabilizer group that uniquely identifies the input state.
Narayanan Rengaswamy: and instead of doing the kit and doing the unit tree on the kit and trying to figure out what the output state is from the circuit.
Narayanan Rengaswamy: What John John was doing was just tracking the stabilizers through the circuit, meaning doing this side of Heisenberg picture by evolving, evolving the operator rather than the State.
Narayanan Rengaswamy: And then what we get at the end. You will still get 8. If you, If you track all of the 8 input stabilizers, you will get 8 0utput stabilizers
Narayanan Rengaswamy: and those uniquely represent the output state.
Narayanan Rengaswamy: And so you have just used this efficient polynomial poly tracking to figure out what the circuit did to your State rather than having to do a large, exponentially complex matrix vector multiplication.
Stefan Krastanov: Or you said yet another thing. You now your constraints about, not how the State evolves, because the constraints uniquely define the state.
Ali: And here we're exploiting the the fact that
the complexity is reduced
Ali: by using the stabilizer. Formalism from just doing matrix multiplication is exploiting the fact that these matrices are very specific.
Ali: like there, poly are there. Clifford matrices?
Stefan Krastanov: Yeah.
Okay.
Stefan Krastanov: So to elaborate. Sorry I see there was some muting dock at first.
Narayanan Rengaswamy: And yeah, so the it's because the matrices are Clifford and the input. Is a stabilization.
Ali: I see, I see.
Stefan Krastanov: to reiterate on something
Stefan Krastanov: you have exponential remaining policies for which your stabilizer state is not even state.
Stefan Krastanov: If you were to track all the polling operations for which this is true, it's still exponentially expensive, and, like we do that in general front of my accounts all the time. It's just that our constraints are not
Stefan Krastanov: poly constraints. There are some general non poly observables, and most of the physics version of quantum mechanics is finding the 0 rules that provide you for with sufficiently many constraints to fully define a basis. Set.
Stefan Krastanov: The special thing about the poly operations is that also you still have exponentially many of them
Stefan Krastanov: all explanation. Many of them can be described by a linearly large generating set. The the set of all the observables has a group structure
Stefan Krastanov: that lets you efficiently write down all exponentially, many poly observables or no about all these, all the the exponential manufacturers
Ali: I see. I see
Stefan Krastanov: so a question that I asked about this one.
Stefan Krastanov: The the polys are, I guess, one of the views about is one of the goods you can group. Searches can come up with in the set of
Stefan Krastanov: that
Stefan Krastanov: from them, so that you could have this inscription to find a stabilizer, but not
Stefan Krastanov: they have their particular structure.
Narayanan Rengaswamy: Yeah. And and the nice thing about policies that they form an up. So if you yeah, if you use the L 2 norm for matrices, then the polymatr is form a basis for all matrices under the If i'm an orthogonal basis
Narayanan Rengaswamy: for all 2 to the end by 2 to the and matrices.
And so that's why you can represent, if, even if you have, like, a crazy quantum channel that you have, like cross operators
Narayanan Rengaswamy: for the Channel, you can actually take the cross operators and decompose them into sums of police so effectively. Your big cross representation can be mapped into something that just uses poly.
Narayanan Rengaswamy: And so this is why one of the important theorems proved early on, and error correction was that people showed that if you have a recovery operator.
Narayanan Rengaswamy: I think it's a specific recovery operator. But anyway, so if you have, if you can correct, you know an error channel with certain cross operators. Then they showed that, using that particular recovery operator. You can also correct any other quantum channel
Narayanan Rengaswamy: whose cross operators are linear combinations
Narayanan Rengaswamy: of the address that you can correct. And this is why we care about correcting poly errors, because we know that up to some caveats like, with not too many correlated errors, and s0 0n. We know that if we can correct poly errors, then we can also correct linear combinations of all errors on those 2,
Narayanan Rengaswamy: at least in principle.
Stefan Krastanov: I think we have been running out of time the last few minutes, so we should probably adjourn for today.
Stefan Krastanov: Okay, yeah. We have our assignments for next week
and later tonight we'll be asking for them on cheers for the week after that.
Stefan Krastanov: and next week we'll be looking into some of the historically first air correcting calls that you've been developed for prompting information, and then on step up to week after that to the screen. Break. Right? Perfect. So for we have more time.
Stefan Krastanov: Okay.
Stefan Krastanov: n0 3 weeks. So yeah, we are careful there. Yes, yeah. So we we do have things for next week, and we'll be asking for the week after that. And then there is, there is a break at least from our end.
Narayanan Rengaswamy: Yeah, I mean, 2 weeks from now is our break, and I think the week after is your break, so we'll figure out
Stefan Krastanov: perfect
Narayanan Rengaswamy: sounds. Good, thank you.
Stefan Krastanov: No.