Quantum CSS Codes: Generalizations & Applications

• Quantum CSS Codes are fundamental quantum error-correcting codes constructed from pairs of classical codes to protect against both bit and phase errors.
• They leverage subfield linearity and alternative inner products, such as trace forms, to expand design space and improve error correction parameters.
• Constructive methods including cyclic, group character, and BKLC constructions enable practical implementation and certification of high-performance asymmetric quantum codes.

Quantum CSS (Calderbank–Shor–Steane) Codes are a foundational class of quantum error-correcting codes constructed from pairs of classical codes, typically binary or $q$-ary linear codes. The original CSS framework relies on two nested classical codes and the use of standard inner products to define their mutual orthogonality, combining their classical protection against bit and phase errors into a quantum setting. The CSS paradigm has inspired a series of generalizations, including the use of subfield-linear codes, alternative bilinear forms, and asymmetric error models, enhancing the flexibility and performance of quantum codes for both symmetric and asymmetric noise channels.

1. Generalization of the CSS Construction: Subfield-Linearity and Inner Products

The standard CSS construction requires two $𝔽_q$-linear codes $C_1, C_2 \subseteq 𝔽_q^n$ satisfying $C_1^\perp \subseteq C_2$ with respect to the Euclidean (or, for certain fields, Hermitian) inner product. The code parameters are

$$[[n, k, d_z/d_x]]_q,\quad \text{where}\ k = \dim C_1 + \dim C_2 - n,\quad d_x = \min\{ \text{wt}(v) : v \in C_1 \setminus C_2^\perp \},\ d_z = \min\{ \text{wt}(v) : v \in C_2 \setminus C_1^\perp \}$$

The CSS-like approach (Ezerman et al., 2012) extends this structure in two essential ways:

• Subfield-linearity: Codes need only be linear over a subfield $𝔽_r \subset 𝔽_q$; thus, $C \leq 𝔽_r$-linear in $𝔽_q^n$. This allows the use of codes not available as $𝔽_q$-linear codes and is particularly potent for composite $q$ and extension fields.
• Alternative inner products: Rather than restricting to the standard Euclidean or Hermitian product, duality can be defined with respect to the trace Euclidean or trace Hermitian inner products. For example:
  • Trace Euclidean: $$\langle u, v \rangle_{Tr_{q/r}} = Tr_{q/r}\left( \sum_{i=1}^n u_i v_i \right)$$
  • Trace Hermitian (when $q = r^2$): $$\langle u, v \rangle_{trH} = Tr_{q/r}\left( \sum_{i=1}^n u_i \overline{v_i} \right)$$, where $\overline{v_i}$ denotes the Frobenius conjugate.

This dual freedom enables code designers to employ a broader array of classical codes and leverage the MacWilliams identity for weight enumerators with alternative bilinear forms:

$$W_{C^{\perp_*}}(X, Y) = \frac{1}{|C|} W_C(X + (q-1)Y,\ X-Y)$$

The result is a more general CSS-like construction that produces quantum codes with parameters

$[[n, \log_q(|C_2| / |C_1^{\perp_*}|),\ d_z/d_x]]_q$

where the dual $C_1^{\perp_*}$, distances $d_x, d_z$, and inclusion are with respect to the chosen inner product.

2. Impact on Code Performance: Removing Linearity and Inner Product Constraints

The relaxation to subfield-linearity unlocks a significantly larger design space for quantum codes. Many high-performance classical codes exist as $𝔽_r$-linear subspaces but not as $𝔽_q$-linear codes, particularly for $q = r^m$ and $m > 1$. This is especially evident in composite or extension fields, such as $q = 4, 8, 9$.

By selecting codes that are only $𝔽_r$-linear and using the trace inner products, it becomes possible to find nested pairs $C_1, C_2$ such that the quotient $|C_2| / |C_1^{\perp_*}|$ is maximized for a fixed $n$, leading to a higher code dimension $k$ and improved minimum distances for $X$ and $Z$ errors.

The flexibility in the choice of inner product is exploited as follows:

• For fully $𝔽_q$-linear codes, standard Euclidean (or Hermitian for quadratic extensions) inner products are used.
• For $𝔽_r$-linear codes, the trace Euclidean (or trace Hermitian) is essential, as the dual of an $𝔽_r$-linear code is again $𝔽_r$-linear only with respect to the trace form. The use of trace inner products preserves the validity of the code duality and MacWilliams identity, ensuring consistent weight enumerator relations and error counting.

These innovations have led to the explicit construction of AQCs that outperform standard CSS codes (those constrained to $𝔽_q$-linearity and standard inner products) in parameters such as block length $n$, code rate $k/n$, and achievable asymmetric distances $(d_z/d_x)$ (Ezerman et al., 2012).

3. Constructive Methods: Cyclic, Group Character, and BKLC Constructions

Three principal constructive methodologies are employed:

• Cyclic code construction: For $𝔽_r$-linear cyclic codes in $𝔽_q^n$, codes are described as $𝔽_r[x]$-modules with multiple generator polynomials. When $q = r^2$, every $𝔽_r$-linear cyclic code can be generated by a pair of polynomials (e.g., $$C = \langle a_0(x) + \omega a_1(x),\, a_1(x) \rangle$$ with $\omega$ generating the extension).
• Group character code construction: Explicitly constructed via group algebra techniques using characters of elementary abelian $t$-groups and organized by weight. The ordering induces naturally nested pairs suitable for the CSS(-like) construction.
• Best-known linear code (BKLC) construction: If a classical code contains a codeword of weight $n$, one applies a structural theorem yielding an AQC with parameters $[[n, k-1, d/2]]_q$.

Each construction is paired with linear programming bounds (applying Delsarte's method and MacWilliams identities) to optimize quantum code parameters and certify non-existence or optimality.

Extensive tabulated results in (Ezerman et al., 2012) document optimal and good AQCs for $q\in\{2,3,4,5,7,8,9\}$, including instances where non-$𝔽_q$-linear codes with trace inner products achieve strictly better parameters than any possible $𝔽_q$-linear symmetric code.

4. Mathematical Formulation and Examples

Key expressions for the generalized CSS(-like) construction are:

• Dual with generic inner product:

$$C^{\perp_*} = \{ u \in 𝔽_q^n : \langle u, v \rangle_* = 0\ \forall v\in C \}$$

• MacWilliams identity for $𝔽_r$-linear codes with bilinear form $*$:

$$W_{C^{\perp_*}}(X,Y) = \frac{1}{|C|}\, W_C(X + (q-1)Y,\,X-Y)$$

• Quantum code from $𝔽_r$-linear nested pair with a duality-type inclusion using the chosen inner product:

$$C_1^{\perp_*} \subseteq C_2 \implies [[n,\log_q(|C_2|/|C_1^{\perp_*}|), d_z/d_x]]_q$$

where

$$d_x = \min\{ \mathrm{wt}(v) : v\in C_1 \setminus C_2^{\perp_*} \},\quad d_z = \min\{ \mathrm{wt}(v) : v\in C_2 \setminus C_1^{\perp_*} \}$$

Explicit constructions:

• Binary example: $[[4,1,2/2]]_2$ from an ad hoc pair of codes generated by

$$\begin{pmatrix} 1 & 0 & 1 & 0\ 0 & 1 & 0 & 1 \end{pmatrix}$$

• Quaternary example: $[[12,2,5/4]]_4$ from $𝔽_2$-linear cyclic/group character constructions.

Tables in (Ezerman et al., 2012) compare parameters of $𝔽_q$-linear and subfield-linear AQCs for various $q$, with many optimal codes for $q \in \{4,8,9\}$ exceeding those obtainable with standard CSS constraints.

5. Comparative Perspective: Practical Performance and Applications

Removing the $𝔽_q$-linearity and standard inner product requirement categorically improves achievable code parameters for asymmetric error models, as it allows:

• Higher rates and dimensions: Subfield linearity enables larger $|C_2|/|C_1^{\perp_*}|$ quotients for fixed $n$, directly translating to higher encoded qubits per block.
• Greater flexibility in asymmetric scenarios: Quantum memories and communication systems can be tuned to the prevalence of $X$ vs. $Z$ errors, optimizing code distance for the dominant error mechanisms.
• Extension to non-trivial algebraic structures: The group character and cyclic module constructions generalize both classical code theory and quantum code design, supporting family-wide optimization.

By integrating the use of subfield-linear codes, alternative bilinear forms, and linear programming (for parameter bounding and infeasibility proofs), the CSS-like construction extends the boundaries of quantum error correction, making possible the systematic generation and certification of best-known and provably optimal AQCs for various alphabets and code lengths.

6. Summary and Outlook

The CSS-like generalization broadens the CSS error correction scheme by:

• Allowing pairs of nested subfield-linear codes in place of full-field linear codes,
• Enabling any suitable nondegenerate bilinear form (Euclidean, Hermitian, trace variants) to define duals and code inclusions,
• Utilizing cyclic, group character, and BKLC constructions to systematically produce high-performance asymmetric quantum codes,
• Leveraging MacWilliams relations and linear programming to certify optimality and non-existence.

These developments yield families of pure asymmetric quantum codes with parameters that surpass those of classical, symmetric, $𝔽_q$-linear CSS constructions. The approach unifies and extends previous methods, offering a comprehensive framework for the exploration and practical deployment of quantum error-correcting codes tailored for realistic, non-symmetric noise environments and for diverse field sizes (Ezerman et al., 2012).