All Codes Available in this Benchmark Database

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 [[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.

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 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 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.

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

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.

Concatenated codes recursively encode the logical qubits of an outer code using an inner code. For outer code [[n₁, k₁, d₁]] and inner code [[n₂, k₂, d₂]], the result is [[n₁n₂, k₁k₂, d₁d₂]].

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

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

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 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.

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.