## Description

An \([[n,k,d]]\) stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Pauli strings. Codes can be defined from two classical codes and/or chain complexes over \(\mathbb{Z}_2\) per the qubit CSS-to-homology correspondence below. Strong CSS codes are codes for which there exists a set of \(X\) and \(Z\) stabilizer generators of equal weight.

The stabilizer generator matrix is of the form \begin{align} H=\begin{pmatrix}0 & H_{Z}\\ H_{X} & 0 \end{pmatrix} \label{eq:parity} \tag*{(1)}\end{align} such that the rows of the two blocks must be orthogonal \begin{align} H_X H_Z^T=0~. \label{eq:comm} \tag*{(2)}\end{align} The above condition guarantees that the \(X\)-stabilizer generators, defined in the symplectic representation as rows of \(H_X\), commute with the \(Z\)-stabilizer generators associated with \(H_Z\).

Encoding is based on two related binary linear codes, an \([n,k_X,d_X]\) code \(C_X\) and \([n,k_Z,d_Z]\) code \(C_Z\), satisfying \(C_X^\perp \subseteq C_Z\). The resulting CSS code has \(k=k_X+k_Z-n\) logical qubits and distance \(d\geq\min\{d_X,d_Z\}\). The \(H_X\) (\(H_Z\)) block of \(H\) (1) is the parity-check matrix of the code \(C_Z\) (\(C_X\)). The requirement \(C_X^\perp \subseteq C_Z\) guarantees (2) and also implies \(C_Z^\perp \subseteq C_X \). Basis states for the code are, for \(\gamma \in C_X\), \begin{align} |\gamma + C_Z^\perp \rangle = \frac{1}{\sqrt{|C_Z^\perp|}} \sum_{\eta \in C_Z^\perp} |\gamma + \eta\rangle. \tag*{(3)}\end{align}

A CSS code has stabilizer weight \(w\) if the highest weight of any stabilizer generator is \(w\), i.e., any row of \(H_X\) and \(H_Z\) has weight at most \(w\). In the context of comparing weight as well as of determining distances for noise models biased toward \(X\)- or \(Z\)-type errors, an extended notation for asymmetric CSS codes is \([[n,k,(d_X,d_Z),w]]\) or \([[n,k,d_X/d_Z,w]]\). The quantity \(\min\{d_X,d_Z\}\) is often called the worst-case minimum distance and is often less than the actual code distance due to degeneracy [4].

To find the minimum distance of degenerate CSS code, we have to first remove the codewords of the smaller codes as those codewords correspond to stabilizer generators instead of logical operators. Thus the general formulae for the minimum distances \(d, d_Z, d_X\) for an \([[n,k,d]]\) or \([[n,k,(d_X,d_Z)]]\) \(CSS(C_X, C_Z)\) code are: \begin{align} d_{X}&=\min\{ w_H(c) | c \in C_X \setminus C_Z^\perp \} \tag*{(4)}\\ d_{Z}&=\min\{ w_H(c) | c \in C_Z \setminus C_X^\perp \} \tag*{(5)}\\ d&=\min\{d_X,d_Z\}~, \tag*{(6)}\end{align} where \(w_H\) is the Hamming weight of a codeword.

### CSS-to-homology correspondence

Qubit CSS-to-homology correspondence: CSS codes and their properties can be formulated in terms of homology theory, yielding a powerful correspondence between codes and chain complexes, the primary homological structures. There exists a many-to-one mapping from size three chain complexes to CSS codes [5–8] that allows one to extract code properties from topological features of the complexes. Codes constructed in this manner are sometimes called homological CSS codes, but they are equivalent to CSS codes. This mapping of codes to manifolds allows the application of structures from topology to error correction, yielding various QLDPC codes with favorable properties.

A chain complex of size three is given by binary vector spaces \(A_2\), \(A_1\), \(A_0\) and binary matrices \(\partial_{i=1,2}\) (called boundary operators) \(A_i\) to \(A_{i-1}\) that satisfy \(\partial_1 \partial_2 = 0\). Such a complex is typically denoted as \begin{align} A_2 \xrightarrow{\partial_2} A_1 \xrightarrow{\partial_1} A_0~. \label{eq:chain} \tag*{(7)}\end{align} One constructs a CSS code by associating a physical qubit to every basis element of \(A_1\), and defining parity-check matrices \(H_X=\partial_1\) and \(H_Z=\partial_2^T\)). That way, the spaces \(A_0\) and \(A_2\) can be associated with \(X\)-type and \(Z\)-type Pauli operators, respectively, and boundary operators determine the Paulis making up the stabilizer generators. The requirement \(\partial_1 \partial_2 = 0\) guarantees that the \(X\)-stabilizer generators associated with \(H_X\) commute with the \(Z\)-stabilizer generators associated with \(H_Z\).

Usually, the chain complex (7) used in the construction comes from the chain complex associated with a cellulation of a manifold. When the manifold is a two-dimensional surface, its entire chain is used. Higher-dimensional manifolds allow for longer chain complexes, and one can use the three largest non-trivial vector spaces in its chain.

CSS codes saturate a type of error correction uncertainty relation [2; Thm. 3], which is a special case of an entropic uncertainty relation between a pair of bases [9–11]. The code state \(\sum_{c\in C_{Z}}|c\rangle\) can be expressed in terms of either basis states labeled by the code \(C_{Z}\) or its dual, satisfying, with equality, the relation \begin{align} |C_{Z}||C_{Z}^{\perp}| \geq 2^{n}\,. \tag*{(8)}\end{align}

## Protection

Detects errors on \(d-1\) qubits, corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) qubits.

Using the relation to chain complexes, the number of encoded logical qubits is equal to the dimension of the first \(\mathbb{Z}_2\)-homology of the chain complex, \(H_1(\partial, \mathbb{Z}_2) = \frac{\text{Ker}(\partial_1)}{\text{Im}(\partial_2)}\).

The distance of the CSS code is equal to the minimum of the combinatorial (\(d-1\))-systole of the cellulated \(d\)-dimensional manifold and its dual.

## Rate

## Encoding

## Transversal Gates

## Gates

## Decoding

## Fault Tolerance

## Code Capacity Threshold

## Realizations

## Notes

## Parents

- CPC code — CSS codes are a subset of CPC codes [37], with the latter not requiring the two classical codes to be related.
- Movassagh-Ouyang Hamiltonian code — Movassagh-Ouyang codes stem from a prescription that converts an arbitrary classical code into a quantum code.
- Modular-qudit CSS code — Modular-qudit CSS codes for \(q=2\) are (qubit) CSS codes.
- Galois-qudit CSS code — Galois-qudit CSS codes for \(q=2\) are (qubit) CSS codes.

## Children

## Cousins

- Qubit stabilizer code — CSS codes are stabilizer codes whose stabilizer groups admit a generating set of pure-\(X\) and pure-\(Z\) Pauli strings. Additionally, any \([[n,k,d]]\) stabilizer code can be mapped onto a \([[2n,2k,\geq d]]\) CSS code, with the mapping preserving geometric locality of a code up to a constant factor [38] (see also [39]).
- Linear binary code — The CSS construction uses two related binary linear codes \(C_X\) and \(C_Z\).
- Dual linear code — CSS codes for which \(C_X=C_Z \equiv C\) are called self-orthogonal since \(C^{\perp} \subseteq C\). The stabilizer group of such codes is invariant under the Hadamard gate exchanging \(X\) and \(Z\).
- Alternant code — Alternant codes used in the CSS construction yield quantum codes that asymptotically achieve the quantum Gilbert-Varshamov bound [40].
- Random quantum code — Random CSS codes asymptotically achieve linear distance with high probability, achieving the quantum Gilbert-Varshamov bound [1].
- Algebraic-geometry (AG) code — Algebraic geometry codes can be plugged into the CSS construction to yield asymptotically good quantum codes [41].
- Quantum spherical code (QSC) — CSS codes concatenated with two-component cat codes form QSCs which have a weight-based notion of distance.
- Quantum locally testable code (QLTC) — A qubit CSS code defined by \(H_{Z}\) and \(H_{X}\) is glocally testable with some soundness iff the constituent codes \(\ker H_{Z}\) and \(\ker H_{X}\) are locally testable with the same soundness [42; Fact 17].
- Majorana stabilizer code — When constructing a Majorana stabilizer code from a self-orthogonal classical code with an odd number of bits and generator matrix \(G\), a more complex procedure must be applied to ensure that the fermion code has an even number of Majorana zero modes, and thus a physical Hilbert space [39,43]. Rather than taking \(G\) to be the stabilizer matrix as in the even case, we take \(G\oplus G\). This is a concatenation of classical codes as in the CSS construction and it yields a mapping \([2N-1,k,d]\rightarrow [[2N-1,2N-1-k,d^\perp]]_f\). This procedure may be further generalized by concatenating two different self-orthogonal classical codes with an odd number of bits, as is often done in the CSS construction.
- Cluster-state code — A resource cluster state can be constructed out of any qubit CSS code via foliation. Conversely, CSS codes can be constructed out of cluster states [14].
- XP stabilizer code — Each XP-regular code can be mapped to a CSS code with a similar logical operator structure [44].
- EA qubit stabilizer code — As opposed to CSS codes, EA qubit stabilizer codes can be constructed from any linear binary code.
- Qubit BCH code — Some qubit BCH codes are CSS.
- Subsystem CSS code — Subsystem qubit CSS codes reduce to (subspace) CSS qubit codes when there is no gauge subsystem.

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## Cite as:

“Qubit CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/qubit_css