Questions tagged [css]
The css tag has no usage guidance.
12
questions
3
votes
1
answer
66
views
Proof for how logical operators generated systematically will satisfy Pauli commutation
Let me assume a CSS code.
Suppose I have $n$ qubits and have $(n-k)$ stabilizers which we label by the set $S = \{S_1, S_2, .. S_{n-k}\}$. Let me then find $2k$ Pauli operators $L_i$ that are
Not ...
- 552
1
vote
2
answers
100
views
Why parity-check matrices not reduced when working with Bivariate Bicycle codes?
Bivariate Bicycle (BB) codes were introduced in [1]. These families of QEC codes are defined via parity check matrices containing a two block structure of the form $H_{X} = (A|B)$ and $H_{Z} = (B^{T}|...
2
votes
1
answer
43
views
How are stabilizers defined with monomial labellings for Bivariate Bicycle codes?
I have been recently looking at getting familiarized with Bivariate Bicycle (BB) codes [1], and in general, Multivariate Bicycle codes [2]. I need some help figuring out the usage of the monomial ...
2
votes
3
answers
107
views
Intuition behind constructions of good qLDPC codes
A good $[[n, k, d]]$ qLDPC code is one where $\lim_{n \rightarrow \infty} \frac{k}{n} > 0$ and $k = O(n)$ and $\lim_{n \rightarrow \infty} \frac{d}{n} > 0$ and $d = O(n)$. A CSS code is one ...
- 3,137
0
votes
0
answers
23
views
How to find the matrices which generate the physical realizations of logical Pauli operators for the $[[6,4]]$ stabilizer code?
Question surrounds example given on page 8 of this paper.
The rows of the following matrix determine the physical states of the $[[6,4]]$ code:
$$G= \begin{bmatrix}
1 & 1 & 0 & 0 & 0 &...
- 705
1
vote
1
answer
57
views
Variant of Quantum Hamming bound for CSS codes
The arguments leading to the Quantum Hamming Bound for non-degenerate $[[n,k,d = 2t+1]]$ codes
$$\sum_{j=0}^t 3^j \binom{n}{j} ≤ 2^{n-k}$$
can be manipulated to find a stronger related bound for non-...
- 113
0
votes
1
answer
35
views
A question on dimensions of the basis vectors for the $[[6,4]]$ code
I am working from page $8$ of this paper.
The generator for the coset representatives of $C^{\perp}$ in $C$ for the $[6,5,2]$ classical code is given by $$G_{C/C^{\perp}} = \begin{bmatrix} 1 & 1 &...
- 705
0
votes
0
answers
48
views
Smallest distance 9 self dual code CSS code
Very similar to a previous question at Smallest distance 9 self-dual CSS code? .
I'm wondering whats the smallest distance 9 self dual CSS code. There is a 23 qubit distance 7 Golay code and a 47 ...
- 43
0
votes
1
answer
53
views
Implementing CNOT logical gate for the Steane code
I am looking at the transversal logical gates for the Steane code.
I am having trouble trying to implement the $CNOT$ logical gate for this code.
I understand that we can transversally implement a ...
- 705
5
votes
2
answers
118
views
Transversal CNOTs on CSS codes with multiple logical qubits
I am interested in the theory of implementing logical gates on quantum error correcting codes. From a practical view, transversal gates are very attractive. I have a question about transversal gates.
...
- 311
5
votes
1
answer
198
views
A better name for "weakly self-dual CSS codes"
Does anyone know a better name than "weakly self-dual"/"self-orthogonal" for CSS codes where $H_X=H_Z$, for example the Steane code, and the color codes?
Details
In the discussion ...
- 1,083
6
votes
1
answer
216
views
Smallest distance 9 self-dual CSS code?
The level-2 concatenated [[7,1,3]] Steane code, and the 4.8.8 color code are both self-dual [[49,1,9]] codes from the CSS family. Is there a distance 9 self-dual CSS code that has less than 49 qubits?
- 1,083