Qubit CSS code[1–3] 

Also known as Qubit Euclidean code.

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}

Specializing to the case when \(C_Z=[n,k,d]\) is dual-containing yields an \([[n,2k-n,\geq d_Z]]\) self-dual qubit CSS code (a.k.a. weakly self-dual qubit CSS code) with \(C_X = C_Z^\perp\) and with \(n\) necessarily odd. Its \(X\)-type and \(Z\)-type stabilizers are identically supported.

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 qubit 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 \(H_Z\). See [9; Table 3.2] for a Rosetta stone comparing statistical mechanical models, CSS codes, and chain complexes.

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 [10–12]. 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.

CSS codes have a CSS lower bound against depolarizing noise because CSS decoding does not take into account correlations between \(X\)- and \(Z\)-type noise [13]. An upper bound is formulated in Ref. [14].

Steane enlargement: An \([[n,2k-n,d]]\) CSS code can be converted to a \([[n,k+k^{\prime}−n,\min(d,\left\lceil 3d^{\prime}/2\right\rceil )]]\) code for particular \(k^{\prime}\) and \(d^{\prime}\) via the Steane enlargement construction [15].

Using linear programming (LP) to solve a set of equations and inequalities on weight distribution of a classical self-orthogonal code \(C=(n, 2^n-k)\) and its dual, one can find a \(C\) such that the \([[n,k,d]]\) CSS code constructed using \(C\) and its dual would have rate and distance close to the Singleton bound [16].

Rate

For a depolarizing channel with probability \(p\), CSS codes allowing for arbitrarily accurate recovery exist with asymptotic rate \(1-2h(p)\), where \(h\) is the binary entropy function [17].

Magic

Various self-dual CSS codes yield magic-state distillation protocols whose yield parameter \(\gamma \to 1^{+}\) [19][18; Thms. 4.1 and 4.2].

Encoding

Stabilizer measurement [20].Clusterization, i.e., measurement of a particular cluster state [21].Entanglement purification [22].Reinforcement learning encoding circuits [23].ZX calculus provides a canonical form of an encoding circuit [24].Automated fault-tolerant encoding circuit synthesis [25].

Transversal Gates

Transversal Pauli gates since qubit CSS codes are qubit stabilizer codes.Transversal CNOT gates preserve the logical subspace, up to \(X\)-type Paulis, iff a qubit stabilizer code is CSS [4,26]. The Paulis are necessary for when the code is stabilized by stabilizers with a minus in front of them, e.g., \(-XXXX\) and \(ZZZZ\).Self-dual CSS codes admit a transversal Hadamard gate.Fold-transversal [27,28] Clifford gates are transversal gates combined with qubit permutations. Some of these can be obtained from automorphism groups of the underlying classical codes [29; Thms. 2-3].Necessary and sufficient conditions for a CSS code to yield a transversal gate in the Clifford hierarchy have been formulated [32][30; Thm. 9][31; Thm. 5]. There are routines that can determine what diagonal gates in the Clifford hierarchy are realized by a code [33,34].

Gates

LDPC CSS code symmetries called \(XZ\)-dualities allow for fold-transversal gates, i.e., transversal gates followed by qubit permutations [28].Generalized lattice surgery [35].

Decoding

Coherent decoders allow for measurement-free error correction [36]. One method is table/multi-control decoding [37], which scales exponentially with the number of ancillas used in syndrome measurement. A fault-tolerant measurement-free scheme for low-distance CSS codes is formulated in Ref. [38]. Another method, the Ising-based decoder, utilizes the mapping of the effect of the noise to a statistical mechanical model [17,39] such that the decoding problem maps to preparation of the ground state of an Ising model. See [9; Table 3.2] for a Rosetta stone comparing statistical mechanical models, CSS codes, and chain complexes.Decoders based on neural networks [40].MaxSAT decoder [41].

Fault Tolerance

Steane error correction [42], where fault-tolerance is ensured by preparing ancillary encoded states and extracting syndromes via \(CNOT\) gates.Fault-tolerant error correction and logical measurements using flag qubits for distance-three cyclic CSS codes [43]. Parallel syndrome extraction for distance-three codes can be done fault-tolerantly using one flag qubit [44]. Distance-preserving flag fault-tolerant error correction can be done using lookup tables for small codes [45]. Any self-dual CSS code with bounded-weight stabilizer generators admits flag fault-tolerant syndrome extraction [46].Homomorphic gadgets fault-tolerant measurement unify Steane and Shor error correction [47].A fault-tolerant error-correction protocol using \(O(d\log d)\) syndrome measurements can be applied to any CSS code with distance \(d \geq \Omega(n^{\alpha})\) for any \(\alpha > 0\) [48].Fault-tolerant measurement-free scheme for low-distance CSS codes [38].Automated fault-tolerant encoding circuit synthesis [25].Fault-tolerant homological measurement of logical Pauli operators [49].

Code Capacity Threshold

Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [50,51].

Realizations

Fully homomorphic encryption [52].Cryptographic applications stemming from the monogamy of entanglement of CSS code and error words [53].

Notes

See Refs. [4,54–56] for simple examples of CSS codes.Introduction to Qubit CSS-to-homology correspondence by M. Hastings; see also the book [9].Entanglement purification protocols with qubit CSS codes are related to quantum key distribution (QKD) [57].Qubit CSS codes can be used in quantum repeaters [58].

Parents

Children

Cousins

  • Qubit stabilizer code — Qubit CSS codes are qubit stabilizer codes whose stabilizer groups admit a generating set of pure-\(X\) and pure-\(Z\) Pauli strings. Transversal CNOT gates preserve the logical subspace iff a qubit stabilizer code is CSS [4,26]. Any \([[n,k,d]]\) stabilizer code can be mapped into a \([[4n,2k,2d]]\) self-dual CSS code via the Bravyi-Leemhuis-Terhal mapping (a.k.a. the Majorana mapping, named as such because it is done via an intermediate Majorana stabilizer code) [18][61; Corr. 1], which preserves geometric locality of a code up to a constant factor. Any \([[n,k,d]]\) stabilizer code can be mapped onto a \([[2n,2k,\geq d]]\) two-block CSS code via symplectic doubling, which preserves geometric locality of a code up to a constant factor. For any non-CSS qubit stabilizer code \(\mathsf{C}\), there exists a CSS code \(\mathsf{C}^{\prime}\) such that \(\mathsf{C} = DQ\mathsf{C}^{\prime}\), where \(D\) is a diagonal Clifford operator, and where \(Q\) is an element of an XP stabilizer group [34; Prop. B.3.1].'
  • Two-block CSS code — Any \([[n,k,d]]\) stabilizer code can be mapped onto a \([[2n,2k,\geq d]]\) two-block CSS code via symplectic doubling, which preserves geometric locality of a code up to a constant factor.
  • 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 or homogeneous [62] 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 GV bound [63].
  • Random stabilizer code — Random CSS codes asymptotically achieve linear distance with high probability, achieving the quantum GV bound [1].
  • Subsystem qubit stabilizer code — Qubit CSS "seed" codes can be used to produce subsystem stabilizer codes [64].
  • Cycle code — Cycle codes, including the Petersen cycle and Hoffman-Singleton cycle codes, feature in magic-state distillation protocols [18; Appx. A.2.1][19; Sec. VII.A].
  • Quantum spherical code (QSC) — CSS codes concatenated with two-component cat codes form QSCs which have a weight-based notion of distance.
  • Amplitude-damping (AD) code — An \([[n,k,d_Z=t+1,d_X=2t+1]]\) qubit CSS code protects against \(t\) AD errors [66][65; Sec. 8.7].
  • Constant-excitation (CE) code — Qubit CE codes are protected from coherent noise in the form of transversal \(Z\)-rotations because such rotations act identically on all codewords [67,68]. In the case of qubit CSS codes, all codes oblivious to such rotations are CE codes [67,68]. Any \([[n,k,d]]\) CSS code can be made into an \([[mn,k,>d]]\) CE code [67].
  • 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 [69; Fact 17].
  • EA qubit stabilizer code — As opposed to CSS codes, EA qubit stabilizer codes can be constructed from any linear binary code.
  • Quantum polar code — Quantum polar codes are CSS codes used in an entanglement generation scheme that generally requires entanglement assistance. They require assistance only to determine positions to store information which optimally protect against both bit and phase noise. Without this assistance, they are just CSS codes constructed out of polar codes. A variant of quantum polar codes exists that does not require entanglement assistance [70].
  • Majorana stabilizer code — Every \([[n,k,d]]_f\) Majorana stabilizer code is associated with a \([[2n,2k,d]]\) self-dual qubit CSS code [61; Lemma 2].
  • Union stabilizer (USt) code — An \([[n,2k-n,d]]\) CSS code can be converted to a \([[n,k+k^{\prime}−n,\min(d,\left\lceil 3d^{\prime}/2\right\rceil )]]\) code for particular \(k^{\prime}\) and \(d^{\prime}\) via Steane enlargement. This code can be treated as a union stabilizer code [71].
  • XP stabilizer code — Each XP-regular code can be mapped to a CSS code with a similar logical operator structure [72].
  • 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 [21].
  • 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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“Qubit CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/qubit_css
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