Glossary

Quantum error correction, in plain language. Terms are listed alphabetically. Throughout the site, a dotted underline and a help cursor mean you can follow a term here for its definition.

Automorphic CNOTs

Given a CSS $n$-qubit code $\mathcal{C}$ and an automorphism $\pi \in S_n$ of $\mathcal{C}$, it's automorphic CNOT is a logical gate acting on two $\mathcal{C}$ code blocks $A$ and $B$ by: $\prod_i \CNOT_{Ai, B \pi(i)}$, where $\CNOT_{Ai,Bj}$ denotes the physical CNOT gate with control on the $i$-th qubit of block $A$ and the target qubit on the $j$-th qubit of block $B$. The filter has Automorphic CNOTs on QECdb filters codes with the following property: any logical circuits accross two $\mathcal{C}$ code-blocks $A$ and $B$ consisting of logical CNOTs with controls on $A$ and targets in $B$ can be expressed as a product of automorphic CNOTs. In the bipartite-CNOT generators sections of the pages of such codes, one can find a list of code automorphisms corresponding to the aforementioned generating set of automorphic CNOTs.

Automorphism gates

The term automorphism gate has no precise generally agreed upon definition. Roughly speaking, it is a gate obtained from a process of the following form: you give me an automorphism of a code or of some object derived from the code, I give you a logical gate in return. The term is used to refer to: SWAP + local Clifford gates, automorphic CNOTs and fold-transversal gates.

Automorphisms

Given an $n$-qubit code $\mathcal{C}$, a code automorphism is a permutation $\pi \in S_n$ such that a paulistring $P=P_1 \dots P_n$ is a stabilizer of $\mathcal{C}$ iff $P=P_{\pi(1)}\dots P_{\pi(n)}$ is also a stabilizer. Equivalently, it's a permutation of the qubits that leaves the codespace invariant - i.e. that is a logical gate.

CSS self-dual

A CSS code is said to be self-dual if for every subset of qubits $S \subseteq \{1 \dots n\}$, it holds that $\prod_{i \in S} X_i$ is a stabilizer of $\mathcal{C}$ iff $\prod_{i \in S} Z_i$ is also a stabilizer.

Destabilizers

Given a stabilizer code with stabilizers $s_1, \dots, s_t$ and logicals $x_1, \dots, x_k, z_1, \dots, z_k$, its $i$-th destabilizer ($i \in \{1, \dots, t\}$) is the unique Pauli operator anticommuting with $s_i$ and commuting with all other stabilizers and logical operators.

dx (X distance)

For a CSS code, dx is the smallest number of physical qubits on which an X-only logical operator acts nontrivially.

dz (Z distance)

For a CSS code, dz is the smallest number of physical qubits on which a Z-only logical operator acts nontrivially.

Isomorphic codes

Two $n$ qubit codes $\mathcal{C}_1, \mathcal{C}_2$ are isomorphic here when relabelling their physical qubits maps the stabilizer group of one exactly onto the other. More explicitly, that is - there exists some permutation $\pi \in S_n$ called an isomorphism such that a paulistring $P=P_1 \dots P_n$ is a stabilizer of $\mathcal{C}_1$ iff $P=P_{\pi(1)}\dots P_{\pi(n)}$ is a stabilizer of $\mathcal{C}_2$.

ZX-duality

Given a CSS code $\mathcal{C}$, we say that $\pi \in S_n$ is a ZX-duality if $S \subseteq \{1 \dots n\}$, it holds that $\prod_{i \in S} X_i$ is a stabilizer of $\mathcal{C}$ iff $\prod_{i \in S} Z_{\pi(i)}$ is also a stabilizer. Note that CSS self-dual codes have the ZX-duality $\pi=1$, which motivates calling codes with a ZX duality weakly self-dual codes.