Quantum Computing Stack Exchange
Quantum Computing Stack Exchange

Questions tagged [css]

13 questions
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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 in the comments of Example CSS codes and the properties "doubly even"…
Balint Pato
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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?
Balint Pato
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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. Background Here on stack exchange, I find many…
Vincent
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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 products of the stabilizers Not products of each…
JRT
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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 labels for Pauli operator definitions. Preliminaries…
Ezequiel Rodriguez Chiacchio
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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 where we take two classical codes $C_1$ and $C_2$ with…
user1936752
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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-degenerate CSS codes, $$\left(\sum_{j=0}^t…
The Vee
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How does Steane error correction for fault tolerance work?

I am interested in Steane error correction (not to be confused with the Steane code). This is described in Fig. 1 of this paper. The idea is to create a $\vert +\rangle_L$ state with as many registers as used by $\vert\psi\rangle_L$ and connecting…
JRT
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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 $H_{Z} = (B^{T}|A^{T})$. The matrices $A$ and $B$ are both…
Ezequiel Rodriguez Chiacchio
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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 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0…
am567
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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 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0…
am567
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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 qubit distance 11 Code but is there a distance 9 code…
mathNoob
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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 logical $S$ gate by: $$U = S \otimes S^{\dagger}…
am567
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