Calderbank-Shor-Steane (CSS) stabilizer code

Calderbank-Shor-Steane (CSS) stabilizer code

Description

A stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type operators. The two sets of stabilizer generators can often be related to parts of a chain complex over the appropriate ring or field.

CSS codes can also be viewed as an instance of a two-step convex-geometric construction: one first chooses an intermediate subspace stabilized by one type of generator, and then imposes the other type of generator so that the remaining diagonal error slopes vanish via an application of the Tverberg theorem [1; Sec. 4.3].

Cousins

Rotor stabilizer code— A rotor stabilizer code admitting a set of generators such that each generator consists of either angular position or angular momentum operators is a CSS code.
Subsystem CSS code— Subsystem CSS codes reduce to CSS stabilizer codes when there are no gauge degrees of freedom.
Quantum lattice code— Quantum lattice codes defined on rectangular lattices are CSS codes. There is no known relation to chain complexes for such codes. More general lattices, obtained from rectangular lattices by Gaussian transformations, yield non-CSS codes.
Asymmetric quantum code (AQC)— 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 block quantum codes is \([[n,k,(d_X,d_Z),w]]\) or \([[n,k,d_X/d_Z,w]]\).

Member of code lists

Primary Hierarchy

Quantum Domain

Group quantum Kingdom

Group-based quantum codeQECC Quantum

Stabilizer codeHamiltonian-based QECC Quantum

Parents

Stabilizer code

Group GKP codeQECC Quantum

CSS codes are Abelian group GKP codes, i.e., group GKP codes constructed out of Pauli-type operators.

Calderbank-Shor-Steane (CSS) stabilizer code

Children

Homological rotor code

Homological rotor codes are rotor CSS codes constructed from chain complexes over the integers in an extension of the qubit CSS-to-homology correspondence to rotors. The homology group of the logical operators has a torsion component because the chain complexes are defined over the ring of integers, which yields codes with finite logical dimension. Products of chain complexes can also yield rotor codes.

Rotor GKP code

Rotor GKP code stabilizers are purely position and purely momentum rotor Pauli-type operators, making these codes CSS.

Generalized homological-product CSS codeHomological BB Kitaev surface

Bosonic CSS code

Modular-qudit CSS codeColor BB Homological Quantum RM code Kitaev surface