All Codes Available in this Benchmark Database

The 6.6.6 Honeycomb color code, defined on a hexagonal lattice. Each hexagon has an X and Z check on all the qubits on its vertices. A code of odd distance 𝑑 has either (3𝑑² + 1)/4 or (3𝑑 - 1)²/4 physical qubits.

My friend Nithin made this one. It is here as an example placeholder as we built out the page for this code family.

One of the earliest proof-of-concept error correcting codes. The smallest code that can protect against any single-qubit error. Not a CSS code.

The famous toric code, the first topological code. Terrible rate, ok-ish distance, awesome locality – a tradeoff that will turn out to be fundamental to codes with only 2D connectivity.

The [[8,3,3]] code from Cleve and Gottesman (1997), a convenient pedagogical example when studying how to construct encoding circuits, as it is one of the smallest codes with more than one logical qubit.

An open-boundary version of the famous toric code, the first topological code. Terrible rate, ok-ish distance, awesome locality – a tradeoff that will turn out to be fundamental to codes with only 2D connectivity.

The generalized bicycle codes (GBCs) extend the original bicycle codes by using two commuting square n × n binary matrices A and B, satisfying AB + BA = 0. The code is defined using the generator matrices: GX = (A, B), GZ = (Bᵀ, Aᵀ). See Table I in Lin and Pryadko (2023) for the subscripts.

One of the earliest proof-of-concept error correcting codes, a concatenation of a 3-bit classical repetition code dedicated to protecting against bit-flips, and a 3-bit repetition code dedicated to protecting against phase-flips.

One of the earliest proof-of-concept error correcting codes.

The two-block group algebra (2BGA) codes extend the generalized bicycle (GB) codes by replacing the cyclic group with a general finite group, which can be non-abelian. The stabilizer generator matrices are defined using commuting square matrices derived from elements of a group algebra: HX = (A, B), HZ^T = [B; -A] where A and B are commuting ℓ × ℓ matrices, ensuring the CSS orthogonality condition.

The [[2ʲ, 2ʲ - j - 2, 3]] family of codes, the quantum equivalent of the Hamming codes, capable of correcting any single-qubit error.

The 4.8.8 Square-Octagon color code, defined on a lattice where each qubit sits on a vertex shared by two octagons and a square. Each shape has an X and Z check on all the qubits on its vertices. A code of odd distance 𝑑 has (𝑑² - 1)/2 + 𝑑 physical qubits.