Bounds on achievable rates of sparse quantum codes used over the quantum erasure channel
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Transkript: Englisch(automatisch erzeugt)
00:00
and he'll talk about bounds on achievable rates of sparse quantum codes over the quantum erasure channel. I work with Gilles Emore on quantum LDPC codes or sparse quantum codes, and I will talk about the performance of this family of codes.
00:25
To begin, why do we study these kind of codes? We have a quantum channel which introduces errors, and then we use error correcting codes
00:44
to transmit information. The central equation in information theory is the determination of the capacity of the channel. It is the highest amount of information that we can transmit, vanishing error probability.
01:02
Therefore, we want a high rate, and as we saw this morning, we want an efficient decoding and algorithm, an efficient decoding and encoding algorithm. Otherwise, error accumulate, and we cannot compute if error accumulate.
01:20
To have this fast decoding, we use sparse codes. They are defined by matrix of flow rate, by equation of flow rate, and therefore, this gives us an efficient decoding algorithm. But in compensation, we know for classical sparse codes that we are a bit below the capacity of the channel,
01:42
and we will prove that this is the same thing for the quantum case. In the quantum setting, we cannot achieve the capacity with quantum sparse codes or quantum LPC codes. Another advantage of this kind of codes have small stabilizer,
02:03
stabilizer which are the identity almost everywhere. Here, when we have a typical error in a class, in a degeneracy class, we have several typical errors, and therefore, we use degeneracy. This is necessary when we want to obtain
02:21
good quantum codes for the decreasing channel, for example. I will begin by a proof of the capacity of the quantum eraser channel. Only for stabilizer codes, we will prove, using combinatorial arguments,
02:40
that we have at most achievable rate, one minus two P for the quantum eraser channel of probability P. After, we'll obtain using this new bound, that the sparse quantum code don't achieve capacity, and finally, I will talk about it in percolation theory.
03:01
Using quantum information, we obtain an upper bound and the critical probability in percolation theory for particular families of graphs. What do we know about the quantum eraser channel? Its capacity is one minus two P,
03:20
that was proved in 97 by Bennett, Devinsen, Zu, and Smolin, and the proof is based on the no cloning theorem. Therefore, we cannot improve this upper bound for a particular family of codes, for sparse codes, because it doesn't depend on the properties of the codes.
03:45
Now, to begin, what is stabilizer codes? We saw that we need a stabilizer group defined by air-independent generators, and the code will be the set of fixed point
04:03
of the stabilizer group. The rate of the code will be N minus air over N, and we know that we have a natural measurement associated with stabilizer codes. It is the syndrome. It is the vector of F2 to the air,
04:21
which is one, which is zero, if and only if the error commute with the high stabilizer, the high generator. Using this definition, we can see that we can measure the syndrome, and if two errors differ in the stabilizer,
04:41
then they have the same effect on the quantum code, and therefore, we can correct them in the same way. The quantum eraser channel, it has several equivalent definition. Here, we will use, which is well-adapted to the stabilizer formalism.
05:02
It is based on poly operator. We consider that the qubit is erased independently with probability P, and an erased qubit is subjected to a random poly error, I, X, Y, or Z, with probability of Carter,
05:21
and the raised position is known. Therefore, on n qubits, we denote by a vector of F2 to the n, the erased position. This vector is one, if and only if the ice qubit is erased, and we know this erased vector, this erased position,
05:42
and we know that, in this case, the state, the quantum state, is subjected to random poly error, which is included in the erased position. We can yet measure the syndrome of the error hetero-cure,
06:01
and the question is, with the knowledge of the erased position, and the syndrome, what is the error hetero-cure? What is the most probable error? We will see this on an example. We consider this stabilizer matrix. We have three independent generators.
06:21
It defines a quantum code of parameters five, two, and we take a random, we take an erasure. Here, the second and the third qubit are erased. How many errors can occur, and how many of them can we correct?
06:41
We have two erased components, and on each component, we have four different possible errors, I, X, Y, or Z. That give us four square different possible errors. How many of them counted? To create an error, we measure,
07:01
we know the erased position. We measure the syndrome, and given the syndrome, we associate a class of errors. Therefore, how many different syndromes are there? We are interested in syndrome of errors
07:23
which are included in the erased position. Therefore, only the blue matrix, H is erased matrix, is necessary, and when we look at this matrix, we can see that the third row is exactly the product of the two first row.
07:41
That means that the syndrome in the third component of the syndrome will depend on the two first components. For the two first components, we have two choices. The syndrome will be zero or one. Therefore, we have two to the two syndromes of error included in this erased position.
08:02
We have the number of syndromes, and given the syndrome, how many errors can we correct? How many errors are in the same degeneracy class and included in the eraser? To see this, we look at the red part.
08:23
If two errors are in the same degeneracy class, two errors included in the erased position, they differ in a stabilizer which is included in the blue position, the erased position. That means that this stabilizer corresponds
08:41
to a relation between the row of the red matrix. In this case, we can see that the first and the third row are the same in the red part. This gives us the non-trivial relation, and this gives us two stabilizers,
09:02
the identity and this product of the two row, which are included in the erased position. Therefore, in each degeneracy class, we have two errors that we can correct in the same way. We can, therefore, correct two to the two times two errors
09:25
between, among the four square possible errors. Therefore, we cannot correct this erasure. And we can repeat this argument for a general erasure.
09:41
We obtain the number of correctable erasure in function of the rank of the sub-matrix He, the small sub-matrix, the matrix of the erased component and the sub-matrix of the non-erased component, the large sub-matrix H. Using this, we obtain a lower bound on the error probability
10:01
and therefore, we can see that the maximum achievable rate of a family of stabilizer code is below one minus two p minus g of p, whereas the function g depends on this difference of rank, of the large sub-matrix, a large random sub-matrix
10:24
and a small random sub-matrix. This gives us a bound which is below one minus two p. We recover the bound on the capacity, of the capacity of the quantum eraser channel, which comes from the cloning theorem.
10:43
But only from combinatorial arguments. And moreover, for general stabilizer codes, we cannot say something better, but for a sparse, if we have a family of sparse codes,
11:01
then we will prove that this function g of p can be bounded and we are here strictly below the capacity. Now we'll estimate this function for a sparse matrix. We take a random sub-matrix, He. Generally, we'll have pn columns
11:24
and when p grows, we arrive to a square matrix. In this position, for this p, we have a square matrix for a general matrix. This matrix He will have almost full rank
11:42
and we cannot say something for the bound. But for a particular matrix, a sparse matrix, we have rows of low weight, we have rows of weight three and in the other coordinate, it is the identity.
12:04
Therefore, we have a non-zero probability to have a new row in the matrix He, such in this matrix. And we can enumerate the number of new rows,
12:20
the expected number of new rows in the random sub-matrix. We obtain a function which is linear in the length. This gives us a non-trivial lower bound on the function g and therefore, this proves that LDPC codes, LDPC quantum codes don't achieve capacity.
12:43
And in fact, we can improve this bound if we consider different case. When we have in the same column two z, we are shown four rows of weight one in the sub-matrix.
13:01
We have a relation between the two rows. This tell us that the row of the sub-matrix will not be full. The rank of the sub-matrix will not be full. And therefore, we obtain an improvement of the preceding case using new rows. And we can continue using the relation
13:20
included in two columns, three columns, et cetera. This give us a more accurate bound and this bound looks like to this. We cannot see anything but we obtain a bound on achievable rates of CSS codes of type two m.
13:42
That means that the columns are rate two and the rows are rate m. There are m non-identity coordinate on the stabilizer, on the generators. And if we have sufficiently large distance in the x part and the z part,
14:01
we obtain a bound on the, an upper bound on the rate to see what is this bound. For code of type two eight, we have the bound of the capacity of the quantum eraser channel in red.
14:23
With this kind of code, we know that the rate is at most, at least one minus two l over, one minus two times two over eight. This give us 0.5. Therefore, using the capacity, we can say that the maximum eraser threshold is 0.25.
14:45
And using the blue curve, which is a curve when we consider only the new rows in the present, we obtain something a bit better. And in the black curve, we consider the relation
15:01
included in one, two, three till 30 columns. This give us a bound which is something like 0.2, which is clearly better than 0.25 for the tolerable eraser threshold.
15:23
I will pass the application in percolation theory. To conclude, we obtain the same thing for CSS code of type LM. We use hypergraphs, property of hypergraphs
15:40
to obtain this, and we can do the same thing for stabilizer code. To open question, can we do the same thing for the depolarizing channel? Can we have a new way to obtain, to approach the capacity of the depolarizing channel?
16:01
And the bound is better for CSS code than stabilizer code. Does that mean that we can construct better stabilizer code for the eraser threshold? Thank you for your attention.
16:27
I like questions.
16:45
So where in your proof did you use the fact that age should satisfy the symplectic scalar product? That age that? Yeah, like basically that your stabilizer operator should commute with each other, so age should satisfy the symplectic scalar product.
17:05
We don't need the commutation relation for submatrix here. We don't have the commutation relation between the row, it's right. I think it is a question.
17:22
But we can define the rank of the submatrix. It will be the rank of the row space without the commutation relation. And we don't need the commutation relation to obtain an information about the number of syndromes and the number of error included in the eraser.
17:50
One last question, or was this the last question? Looks like that, I guess everybody's hungry. There is a final announcement from the organizers,
18:02
and thanks again for all the speakers. Thanks.