Full ECC API (autogenerated)

QuantumClifford.ECC.CircuitCode — Type

CircuitCode is defined by a given encoding circuit circ.

n: qubit number
circ: the encoding circuit
encode_qubits: the qubits to be encoded

QuantumClifford.ECC.CommutationCheckECCSetup — Type

Configuration for ECC evaluator that does not simulate any ECC circuits, rather it simply checks the commutation of the parity check and the Pauli error.

This is much faster than any other simulation method, but it is incapable of noisy-circuit simulations and thus useless for fault-tolerance studies.

source

QuantumClifford.ECC.Concat — Type

Concat(c₁, c₂) is a code concatenation of two quantum codes (Knill and Laflamme, 1996).

The inner code c₁ and the outer code c₂. The construction is the following: replace each qubit in code c₂ with logical qubits encoded by code c₁. The resulting code will have n = n₁ × n₂ qubits and k = k₁ × k₂ logical qubits.

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QuantumClifford.ECC.DistanceMIPAlgorithm — Type

struct DistanceMIPAlgorithm <: QECCore.AbstractDistanceAlg

A Mixed Integer Programming (MIP) method for computing the code distance of CSS stabilizer codes by finding the minimum-weight non-trivial logical PauliOperator (either X-type or Z-type). Used with distance to select MIP as the method of finding the distance of a code.

Note

Requires a JuMP-compatible MIP solver (e.g., HiGHS, SCIP).
For some stabilizer CSS codes, the X-distance and Z-distance are equal.

logical_qubit: index of the logical qubit to compute code distance for (nothing means compute for all logical qubits)
logical_operator_type: type of logical operator to consider (:X or :Z, defaults to :minXZ).
solver: JuMP-compatible MIP solver (e.g., HiGHS, SCIP)
opt_summary: when true (default=false), prints the MIP solver's solution summary
time_limit: time limit (in seconds) for the MIP solver's execution (default=60.0)

source

QuantumClifford.ECC.NaiveSyndromeECCSetup — Type

Configuration for ECC evaluator that runs the simplest syndrome measurement circuit.

The circuit is being simulated (as opposed to doing only a quick commutation check). This circuit would give poor performance if there is non-zero gate noise.

source

QuantumClifford.ECC.ShorSyndromeECCSetup — Type

Configuration for ECC evaluators that simulate the Shor-style syndrome measurement (without a flag qubit).

The simulated circuit includes:

perfect noiseless encoding (encoding and its fault tolerance are not being studied here)
one round of "memory noise" after the encoding but before the syndrome measurement
perfect preparation of entangled ancillary qubits
noisy Shor-style syndrome measurement (only two-qubit gate noise)
noiseless "logical state measurement" (providing the comparison data when evaluating the decoder)

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QuantumClifford.ECC.TableDecoder — Type

A simple look-up table decoder for error correcting codes.

The lookup table contains only weight=1 errors, thus it is small, but at best it provides only for distance=3 decoding.

The size of the lookup table would grow exponentially quickly for higher distances.

source

QECCore.code_k — Method

The number of logical qubits in a code.

Note that when redundant rows exist in the parity check matrix, the number of logical qubits code_k(c) will be greater than code_n(c) - code_s(c), where the difference equals the redundancy.

source

QECCore.parity_matrix_x — Method

Parity check boolean matrix of a code (only the X entries in the tableau, i.e. the checks for Z errors).

Only CSS codes have this method.

Implemented in an extension requiring `Hecke.jl`

QuantumCliffordHeckeExt.LPCode — Type

struct LPCode <: QECCore.AbstractECC

Lifted product codes ((Panteleev and Kalachev, 2021), (Panteleev and Kalachev, Jun 2022))

A lifted product code is defined by the hypergraph product of a base matrices A and the conjugate of another base matrix B'. Here, the hypergraph product is taken over a group algebra, of which the base matrices are consisting.

The binary parity check matrices are obtained by applying A_repr and B_repr representation maps to each element of the base matrices. These linear transformations convert group algebra elements to their matrix representations while preserving the CSS orthogonality condition.

Mathematical Framework

Given classical parity-check matrices:

$A \in \mathbb{F}_q^{m_a \times n_a}$
$B \in \mathbb{F}_q^{m_b \times n_b}$

The lifted product construction produces quantum CSS codes with parity-check matrices:

\[\begin{aligned} H_X &= [A \otimes I_{m_b}, -I_{m_a} \otimes B] \\ H_Z &= [I_{n_a} \otimes B^*, A^* \otimes I_{n_b}] \end{aligned}\]

Commutative Group Algebra

When R is commutative, a single representation suffices since all elements naturally commute. Here $\rho(a) = \lambda(a)$ for all $a \in R$.

Non-Commutative Group Algebra

When R is non-commutative, distinct representations are essential:

A_repr implements the right regular representation: $\rho(a)x = xa$
B_repr implements the left regular representation: $\lambda(b)x = bx$

These ensure the critical commutation relation:

\[\begin{aligned} \rho(a)\lambda(b) = \lambda(b)\rho(a) \end{aligned}\]

which follows from the associative property:

\[\begin{aligned} \rho(a)\lambda(b)(x) = b(xa) = (bx)a = \lambda(b)\rho(a)(x) \end{aligned}\]

Constructors

Multiple constructors are available:

Two base matrices of group algebra elements.
Two lifted codes, whose base matrices are for quantum code construction.
Two base matrices of group elements, where each group element will be considered as a group algebra element by assigning a unit coefficient.
Two base matrices of integers, where each integer represent the shift of a cyclic permutation. The order of the cyclic permutation should be specified.

Below is a list of all constructors:

LPCode(A, B; GA, A_repr, B_repr, repr)

defined at /home/runner/work/QuantumClifford.jl/QuantumClifford.jl/ext/QuantumCliffordHeckeExt/lifted_product.jl:158.

LPCode(c₁, c₂; GA, A_repr, B_repr, repr)

defined at /home/runner/work/QuantumClifford.jl/QuantumClifford.jl/ext/QuantumCliffordHeckeExt/lifted_product.jl:175.

LPCode(A, B; GA)

defined at /home/runner/work/QuantumClifford.jl/QuantumClifford.jl/ext/QuantumCliffordHeckeExt/lifted_product.jl:195.

LPCode(group_elem_array1, group_elem_array2; GA)

defined at /home/runner/work/QuantumClifford.jl/QuantumClifford.jl/ext/QuantumCliffordHeckeExt/lifted_product.jl:203.

LPCode(shift_array1, shift_array2, l; GA)

defined at /home/runner/work/QuantumClifford.jl/QuantumClifford.jl/ext/QuantumCliffordHeckeExt/lifted_product.jl:211.

Examples

A [[882, 24, d ≤ 24]] code from Appendix B of (Roffe et al., 2023). We use the 1st constructor to generate the code and check its length and dimension. During the construction, we do arithmetic operations to get the group algebra elements in base matrices A and B. Here x is the generator of the group algebra, i.e., offset-1 cyclic permutation, and GA(1) is the unit element.

julia> import Hecke: group_algebra, GF, abelian_group, gens; import LinearAlgebra: diagind; using QuantumClifford.ECC;

julia> l = 63; GA = group_algebra(GF(2), abelian_group(l)); x = gens(GA)[];

julia> A = zeros(GA, 7, 7);

julia> A[diagind(A)] .= x^27;

julia> A[diagind(A, -1)] .= x^54;

julia> A[diagind(A, 6)] .= x^54;

julia> A[diagind(A, -2)] .= GA(1);

julia> A[diagind(A, 5)] .= GA(1);

julia> B = reshape([1 + x + x^6], (1, 1));

julia> c1 = LPCode(A, B);

julia> code_n(c1), code_k(c1)
(882, 24)

A [[175, 19, d ≤ 10]] code from Eq. (18) in Appendix A of (Raveendran et al., 2022), following the 4th constructor.

julia> import Hecke; using QuantumClifford.ECC;

julia> base_matrix = [0 0 0 0; 0 1 2 5; 0 6 3 1]; l = 7;

julia> c2 = LPCode(base_matrix, l .- base_matrix', l);

julia> code_n(c2), code_k(c2)
(175, 19)

Code subfamilies and convenience constructors for them

When the base matrices of the LPCode are 1×1, the code is called a two-block group-algebra code two_block_group_algebra_codes.
When the base matrices of the LPCode are 1×1 and their elements are sums of cyclic permutations, the code is called a generalized bicycle code generalized_bicycle_codes.
When the two matrices are adjoint to each other, the code is called a bicycle code bicycle_codes.

The representation function

We use the default representation function Hecke.representation_matrix to convert a GF(2)-group algebra element to a binary matrix. The default representation, provided by Hecke, is the permutation representation.

We also accept a custom representation function as detailed in LiftedCode.

All fields:

A::Union{LinearAlgebra.Adjoint{<:Hecke.GroupAlgebraElem, <:Matrix{<:Hecke.GroupAlgebraElem}}, Matrix{<:Hecke.GroupAlgebraElem}}: the first base matrix of the code, whose elements are in a group algebra.
B::Union{LinearAlgebra.Adjoint{<:Hecke.GroupAlgebraElem, <:Matrix{<:Hecke.GroupAlgebraElem}}, Matrix{<:Hecke.GroupAlgebraElem}}: the second base matrix of the code, whose elements are in the same group algebra as A.
GA::Hecke.GroupAlgebra: the group algebra for which elements in A and B are from.
A_repr::Function: a function that converts a group algebra element to a binary matrix for A; default to be the right regular representation for GF(2)-algebra.
B_repr::Function: a function that converts a group algebra element to a binary matrix for B; default to be the left regular representation for GF(2)-algebra.

source

QuantumCliffordHeckeExt.LaCross — Type

struct LaCross <: QECCore.AbstractCSSCode

The La-cross code is a quantum LDPC code constructed using the hypergraph product of two classical seed LDPC codes. It is characterized by its parity check matrix H, which is derived from circulant matrices with specific properties. These codes were introduced in (Pecorari et al., 2025).

The La-cross code has two families, distinguished by their boundary conditions:

Periodic boundary (full_rank = false):

\[\begin{aligned} \left[\left[2n^2, 2k^2, d \right]\right] \end{aligned}\]

Open boundary (full_rank = true):

\[\begin{aligned} \left[\left[(n - k)^2 + n^2, k^2, d \right]\right] \end{aligned}\]

Note

When H is square and circulant (full_rank=false), classical checks connect opposite endpoints of the length-n classical code and give rise to a quantum code with stabilizers connecting opposite array boundaries, i.e. with periodic boundary conditions. On the contrary, rectangular parity-check matrices in $\mathbb{F}_2^{(n−k)×n}$ give rise to a quantum code with stabilizers stretching up to the array extent, i.e. with open boundary conditions.

Cyclic Codes and Circulant Matrices

A cyclic code is a linear code in which codewords remain valid under cyclic shifts. A circulant matrix is a square matrix where each row is a cyclic shift of the first row. When the parity-check matrix H is circulant, the code is fully determined by its first row:

\[\begin{aligned} H = \text{circ}(c_0, c_1, \dots, c_k, 0, \dots, 0) \in \mathbb{F}_2^{n \times n}. \end{aligned}\]

The elements $c_i$ (for $i = 0, 1, \dots, k$) correspond to the coefficients of a $degree-k$ polynomial:

\[\begin{aligned} h(x) = 1 + \sum_{i=1}^{k} c_i x^i \in \mathbb{F}_2[x]/(x^n - 1). \end{aligned}\]

This establishes a mapping between $\mathbb{F}_2^n$ and the quotient ring $\mathbb{F}_2[x]/(x^n - 1)$, where cyclic shifts in $\mathbb{F}_2^n$ correspond to multiplications by x in the polynomial ring. Since multiplication by x preserves the ideal structure of $\mathbb{F}_2[x]/(x^n - 1)$, cyclic codes correspond to ideals in this ring. These ideals are in one-to-one correspondence with unitary $mod-2$ divisors of $x^n - 1$ with a leading coefficient of 1. Consequently, the fundamental building blocks of cyclic codes correspond to the factorization of $x^n - 1$.

Note

For k = 1, the generator polynomial $h(x) = 1 + x$ defines the repetition code.

Polynomial Representation

The first row of a circulant matrix $H = \text{circ}(c_0, c_1, c_2, \dots, c_{n-1})$ can be mapped to the coefficients of a polynomial $h(x)$. For instance, if the first row is $[1, 1, 0, 1]$, the polynomial is: $h(x) = 1 + x + x^3$. This polynomial-based representation aids in the analysis and design of cyclic codes. For our implementation of La-cross codes, we leverage Hecke.polynomial_ring to work directly with polynomial rings rather than manipulating coefficient arrays explicitly.

Note

The next-to-next-to-nearest neighbor connectivity implies the use of a degree-3 seed polynomial $h(x) = 1 + x + x^2 + x^3$ in the ring $\mathbb{F}_2[x]/(x^n - 1)$ for a specific code length n. Additionally, the condition of low stabilizer weight requires the polynomial $1 + x + x^3$.

[[2n², 2k², d]] La-cross Code

Here is [[98, 18, 4]] La-cross code from with $h(x) = 1 + x + x^3$, n = 7, and k = 3 from Appendix A of (Pecorari et al., 2025).

julia> using QuantumClifford; using QuantumClifford.ECC; # hide

julia> import Hecke: GF, polynomial_ring;

julia> n = 7; k = 3; F = GF(2);

julia> R, x = polynomial_ring(F, "x");

julia> h = 1 + x + x^k;

julia> c = LaCross(n, h, false);

julia> import HiGHS;

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=HiGHS))
(98, 18, 4)

[[(n - k)² + n², k², d]] La-cross Code

Here is [[65, 9, 4]] La-cross code from with $h(x) = 1 + x + x^3$, n = 7, k = 3 and full rank seed rectangular circulant matrix from Appendix A of (Pecorari et al., 2025).

julia> c = LaCross(n, h, true);

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=HiGHS))
(65, 9, 4)

Here is [[400, 16, 8]] La-cross code from with $h(x) = 1 + x + x^4$, n = 16, k = 4 from (Pecorari et al., 2025).

julia> n = 16; k = 4;

julia> R, x = polynomial_ring(F, "x");

julia> h = 1 + x + x^k;

julia> full_rank = true;

julia> c = LaCross(n, h, full_rank);

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=HiGHS))
(400, 16, 8)

The ECC Zoo has an entry for this family.

Fields

n::Int64: The block length of the classical seed code.
h::Nemo.FqPolyRingElem: The seed vector is represented with a degree-n polynomial of the form $h(x) = \sum_{i=0}^{n-1} c_i x^i$.
full_rank::Bool: A flag indicating whether to use the full-rank rectangular matrix (true) or the original circulant matrix (false).

source

QuantumCliffordHeckeExt.LiftedCode — Type

struct LiftedCode <: QECCore.AbstractCECC

Classical codes lifted over a group algebra, used for lifted product code construction ((Panteleev and Kalachev, 2021), (Panteleev and Kalachev, Jun 2022))

The parity-check matrix is constructed by applying repr to each element of A, which is mathematically a linear map from a group algebra element to a binary matrix. The size of the parity check matrix will enlarged with each element of A being inflated into a matrix. The procedure is called a lift (Panteleev and Kalachev, Jun 2022).

Constructors

A lifted code can be constructed via the following approaches:

A matrix of group algebra elements.
A matrix of group elements, where a group element will be considered as a group algebra element by assigning a unit coefficient.
A matrix of integers, where each integer represent the shift of a cyclic permutation. The order of the cyclic permutation should be specified.

The default GA is the group algebra of A[1, 1], the default representation repr is the permutation representation.

Below is a list of all constructors:

LiftedCode(A; GA, repr)

defined at /home/runner/work/QuantumClifford.jl/QuantumClifford.jl/ext/QuantumCliffordHeckeExt/lifted.jl:58.

LiftedCode(A; GA)

defined at /home/runner/work/QuantumClifford.jl/QuantumClifford.jl/ext/QuantumCliffordHeckeExt/lifted.jl:68.

LiftedCode(group_elem_array; GA, repr)

defined at /home/runner/work/QuantumClifford.jl/QuantumClifford.jl/ext/QuantumCliffordHeckeExt/lifted.jl:74.

LiftedCode(shift_array, l; GA, repr)

defined at /home/runner/work/QuantumClifford.jl/QuantumClifford.jl/ext/QuantumCliffordHeckeExt/lifted.jl:87.

The representation function repr

We also accept a custom representation function (the repr field of the constructor). Whatever the representation, the matrix elements need to be convertible to Integers (e.g. permit lift(ZZ, ...)). Such a customization would be useful to reduce the number of bits required by the code construction.

For example, if we use a D4 group for lifting, our default representation will be 8×8 permutation matrices, where 8 is the group's order. However, we can find a 4×4 matrix representation for the group, e.g. by using the typical 2×2 representation and converting it into binary representation by replacing "1" with the Pauli I, and "-1" with the Pauli X matrix.

Implemented in an extension requiring `Oscar.jl`

QuantumCliffordOscarExt.DDimensionalSurfaceCode — Type

struct DDimensionalSurfaceCode <: QuantumCliffordOscarExt.DDimensionalCode

Constructs the D-dimensional surface code using chain complexes and $\mathbb{F}_2$-homology.

Homological Algebra Foundations of Quantum Error Correction

The theory of chain complexes over $\mathbb{F}_2$ provides a unified framework for understanding error-correcting codes, where classical $[n, k, d]$ codes correspond to 2-term complexes and quantum CSS codes arise naturally as 3-term complexes satisfying the commutativity condition $H_Z^T H_X = 0$. The homological framework reveals that:

Quantum CSS codes arise from chain complexes where boundary operators correspond to parity checks.
Logical operators correspond to homology classes.
Higher-dimensional codes can be constructed through products of complexes.

Chain Complex Structure

A chain complex $C$ of length $n$ is a sequence of finite-dimensional vector spaces $C_j$ over $\mathbb{F}_2$ connected by boundary operators that are linear transformations $\partial_j \colon C_j \to C_{j-1}$:

\[\begin{aligned} C : \{0\} \longrightarrow C_n \xrightarrow{\partial_n} C_{n-1} \xrightarrow{\partial_{n-1}} \cdots \xrightarrow{\partial_2} C_1 \xrightarrow{\partial_1} C_0 \longrightarrow \{0\} \end{aligned}\]

with boundary operators satisfying $\partial_i \circ \partial_{i+1} = 0$ (equivalently, $\text{im}\, \partial_{j+1} \subseteq \ker \partial_j)$.We define:

i-chains: Elements of $C_i$
i-cycles: $Z_i(C) := \ker \partial_i$
i-boundaries: $B_i(C) := \mathrm{im} \partial_{i+1}$
i-th homology: $H_i(C) := Z_i(C)/B_i(C)$

Cohomology of the Dual Complex

Given a chain complex $C$ with boundary operators $\partial_j$, its cochain complex $\widetilde{C}$ is the dual sequence with coboundary maps $\delta^i = \partial_{i+1}^T$:

\[\begin{aligned} \widetilde{C} : \{0\} \longleftarrow C_0^* \xleftarrow{\partial_1^T} C_1^* \xleftarrow{\partial_2^T} \cdots \xleftarrow{\partial_n^T} C_n^* \longleftarrow \{0\} \end{aligned}\]

where

i-cocycles: $Z^i(\widetilde{C}) := \ker \partial_{i+1}^T$
i-coboundaries: $B^i(\widetilde{C}) := \mathrm{im} \partial_i^T$
i-th cohomology: $H^i(\widetilde{C}) := Z^i(\widetilde{C})/B^i(\widetilde{C})$

Note

The cohomology group $H^i(\widetilde{C}) = H(\partial_{i+1}^T, \partial_i^T)$ has the same rank as the homology group $H_i(C)$ and corresponds to $X$-type logical operators in the CSS code $CSS(\partial_i, \partial_{i+1}^T)$, while $H_i(C)$ corresponds to $Z$-type operators.

Classical Codes via Chain Complexes and $\mathbb{F_2}$ Homology

An $[n,k,d]$ classical code corresponds to a 2-term complex:

\[\begin{aligned} \{0\} \longrightarrow C_1 \xrightarrow{\partial_1 = H} C_0 \longrightarrow \{0\} \end{aligned}\]

where

$C_1 = \mathbb{F}_2^n$ (codeword space)
$C_0 = \mathbb{F}_2^{n-k}$ (syndrome space)
$H$ is the parity check matrix

Quantum CSS Codes via Chain Complexes and $\mathbb{F_2}$ Homology

Quantum CSS codes extend this to 3-term complexes:

\[\begin{aligned} \{0\} \longrightarrow C_2 \xrightarrow{\partial_2 = H_Z^T} C_1 \xrightarrow{\partial_1 = H_X} C_0 \longrightarrow \{0\} \end{aligned}\]

where

$C_1 = \mathbb{F}_2^n$ (physical qubits)
$C_2 = \mathbb{F}_2^{m_Z}$ (Z-stabilizers)
$C_0 = \mathbb{F}_2^{m_X}$ (X-stabilizers)

with the condition $\partial_1 \partial_2 = H_Z^TH_X = 0$ ensuring that CSS orthogonality is satisfied.

For any chain complex, selecting two consecutive boundary operators defines a valid CSS code. When qubits are identified with the space $C_i$, the code parameters are:

number of physical qubits: $n = \dim C_i$
number of logical qubits: $k = \dim H_i(C) = \dim H^i(C)$
code distance: $d = \min\{\text{wt}(v) | v \in (H_i(C) \cup H^i(C))\backslash\{0\}\}$

Note

Quantum error-correcting codes, which are represented as 3-term chain complexes, can be constructed by applying the homological or hypergraph product to two 2-term chain complexes.

For a detailed explanation, see the ECC Zoo's writeup on the CSS-to-homology correspondence.

D-dimensional Surface Code ((Berthusen et al., 2024), (Zeng and Pryadko, 2019))

We provide an explicit construction of the D-dimensional surface code within the framework of chain complexes and homology over $\mathbb{F_2}$.

The quantum code is obtained by applying the homological product (or hypergraph product) to two 2-term chain complexes. Our construction relies on taking the hypergraph product of these complexes.

Double Complex

Given chain complexes C and D, we construct a double complex derived from the tensor product of two 2-term chain complexes:

\[\begin{aligned} C \boxtimes D \quad \text{with} \quad \partial_i^v = \partial_i^C \otimes I_{D_i} \quad \text{and} \quad \partial_i^h = I_{C_i} \otimes \partial_i^D \end{aligned}\]

Total Complex

The total complex is derived from a double complex by taking the direct sum of vector spaces and boundary maps that share the same dimension:

\[\begin{aligned} \text{Tot}(C \boxtimes D)_i = \bigoplus_{i=j+k} C_j \otimes D_k = E_i \end{aligned}\]

with boundary maps:

\[\begin{aligned} \partial_i^E = \bigoplus_{i=j+k} \partial_j^v \oplus \partial_k^h \end{aligned}\]

The resulting chain complex, called the tensor product of $C$ and $D$, $C ⊗ D$, enables the construction of a CSS code when selecting any three consecutive terms in its sequence.

Subfamilies

[[L² + (L − 1)², 1, L]] 2D Surface Code

The 2D surface code is constructed using the hypergraph product of two repetition codes.Thus, we obtain a new 3-term chain complex:

\[\begin{aligned} E_2 \xrightarrow{\partial_2^E} E_1 \xrightarrow{\partial_1^E} E_0 \end{aligned}\]

Examples

julia> using Oscar; using QuantumClifford; using QuantumClifford.ECC;

julia> D = 2; L = 2;

julia> c = DDimensionalSurfaceCode(D, L);

julia> code = parity_checks(c)
+ X_X_X
+ _X_XX
+ ZZ__Z
+ __ZZZ

julia> import HiGHS; import JuMP;

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=HiGHS))
(5, 1, 2)

When L = 4, we get [[25,1, 4]] 2D surface code from (Berthusen et al., 2024).

julia> using Oscar; using QuantumClifford; using QuantumClifford.ECC;

julia> D = 2; L = 4;

julia> c = DDimensionalSurfaceCode(D, L);

julia> code = parity_checks(c)
+ X___X___________X________
+ _X___X__________XX_______
+ __X___X__________XX______
+ ___X___X__________X______
+ ____X___X__________X_____
+ _____X___X_________XX____
+ ______X___X_________XX___
+ _______X___X_________X___
+ ________X___X_________X__
+ _________X___X________XX_
+ __________X___X________XX
+ ___________X___X________X
+ ZZ______________Z________
+ _ZZ______________Z_______
+ __ZZ______________Z______
+ ____ZZ__________Z__Z_____
+ _____ZZ__________Z__Z____
+ ______ZZ__________Z__Z___
+ ________ZZ_________Z__Z__
+ _________ZZ_________Z__Z_
+ __________ZZ_________Z__Z
+ ____________ZZ________Z__
+ _____________ZZ________Z_
+ ______________ZZ________Z

julia> import HiGHS; import JuMP;

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=HiGHS))
(25, 1, 4)

Chain Complex

The construction is as follows:

\[\begin{aligned} C = \left( C_1 \xrightarrow{\partial} C_0 \right) \quad \text{and} \quad D = \left( D_1 \xrightarrow{\partial^T} D_0 \right) \end{aligned}\]

where $\partial$ is the $(L-1) \times L$ parity check matrix:

\[\begin{aligned} H = \begin{pmatrix} 1 & 1 & & \\ & 1 & \ddots & \\ & & \ddots & 1 \\ & & & 1 \end{pmatrix} \end{aligned}\]

[[L³ + 2L(L − 1)², 1, min(L, L²)]] 3D Surface Code

The 3D surface code is obtained by taking the hypergraph product of a 2D surface code with a repetition code. Thus, we obtain a new 4-term chain complex:

\[\begin{aligned} F_3 \xrightarrow{\partial_3^F} F_2 \xrightarrow{\partial_2^F} F_1 \xrightarrow{\partial_1^F} F_0 \end{aligned}\]

Metachecks

Z-type metachecks: $M_Z^T = \partial_3^F$

Example

Here is an example of [[12, 1, 2]] 3D Surface code with L = 2 from (Berthusen et al., 2024).

julia> using Oscar; using QuantumClifford; using QuantumClifford.ECC;

julia> D = 3; L = 2;

julia> c = DDimensionalSurfaceCode(D, L);

julia> code = parity_checks(c)
+ XX__X_____X_
+ __XXX______X
+ _____XX__XX_
+ _______XXX_X
+ Z_Z_Z_______
+ _Z_ZZ_______
+ _____Z_Z_Z__
+ ______Z_ZZ__
+ Z____Z____Z_
+ _Z____Z___Z_
+ __Z____Z___Z
+ ___Z____Z__Z
+ ____Z____ZZZ

julia> code_n(c), code_k(c)
(12, 1)

Note

For the 3D surface code, there is an asymmetry between the Z- and X-bases (Berthusen et al., 2024). Specifically, the Z-distance ($d_Z$) is 4, whereas the X-distance ($d_X$) is 2. As a result, the code has the parameters [[12, 1, 2]].

julia> import HiGHS; import JuMP;

julia> dz = distance(c, DistanceMIPAlgorithm(solver=HiGHS, logical_operator_type=:Z))
4

julia> dx = distance(c, DistanceMIPAlgorithm(solver=HiGHS, logical_operator_type=:X))
2

[[6L⁴ − 12L³ + 10L² − 4L + 1, 1, L²]] 4D Surface Code

The 4D surface code is constructed by taking the hypergraph product of a 3D surface code with a repetition code. Thus, we obtain a new 5-term chain complex:

\[\begin{aligned} G_4 \xrightarrow{\partial_4^G} G_3 \xrightarrow{\partial_3^G} G_2 \xrightarrow{\partial_2^G} G_1 \xrightarrow{\partial_1^G} G_0 \end{aligned}\]

[[33, 1, 4]]

Here is an example of [[33, 1, 4]] 4D Surface code with L = 2 from (Berthusen et al., 2024).

julia> using Oscar; using QuantumClifford; using QuantumClifford.ECC;

julia> D = 4; L = 2;

julia> c = DDimensionalSurfaceCode(D, L);

julia> import HiGHS; import JuMP;

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=HiGHS))
(33, 1, 4)

Metachecks

Both X and Z-type metachecks available:

$M_Z^T = \partial_4^G$
$M_X = \partial_1^G$

To obtain surface codes of greater dimensionality, we alternate between C and D and then form a product with the chain complex representing the DDimensionalSurfaceCode (Berthusen et al., 2024).

Note

The procedure described above for the DDimensionalSurfaceCode can alternatively be performed using an L × L repetition code and only the chain complex C. In this case, the result would be the DDimensionalToricCode.

Implemented in an extension requiring `JuMP.jl`

QECCore.distance — Method

distance(
    code::QECCore.AbstractECC,
    alg::QuantumClifford.ECC.DistanceMIPAlgorithm
) -> Any

Compute the distance of a code using mixed integer programming. See QuantumClifford.ECC.DistanceMIPAlgorithm for configuration options.

Specifically, it computes the minimum distance of a Calderbank-Shor-Steane code by solving two independent Mixed Integer Programs for X-type ($d_X$) and Z-type ($d_Z$) distances. The code distance is $d = \min(d_X, d_Z)$.

Background on Minimum Distance

For classical codes, the minimum distance, which measures a code's error-correcting capability, is equivalent to its minimum weight. This can be computed by generating all possible codewords from combinations of the generator matrix rows, calculating their weights, and finding the smallest. While accurate, this method takes exponential time. Vardy (Vardy, 1997) demonstrated that computing the minimum distance is NP-hard, and the corresponding decision problem is NP-complete, making polynomial-time algorithms unlikely.

For quantum codes, classical intuition does not always apply. The minimum distance is given by the minimum weight of a non-trivial logical operator. This is generally unrelated to the minimum distance of the corresponding stabilizer code when viewed as a classical, additive code. White and Grassl (White and Grassl, 2006) proposed mapping quantum codes to higher-dimensional classical linear codes. This mapping allows the minimum distance of the quantum additive code to be inferred from that of the classical linear code but increases parameters from n to 3n and d to 2d, adding complexity. Furthermore, once a minimal weight vector is identified, it is essential to verify whether it belongs to the Pauli group 𝒫ₙ over n qubits (Sabo, 2022).

Additionally, to illustrate how classical intuition can be misleading in this context, consider that the [[7, 1, 3]] Steane code has a minimum distance of three, despite all its elements having a weight of four. This discrepancy occurs because stabilizer codes are defined by parity-check matrices, while their minimum distances are determined by the dual (Sabo, 2022).

Note

The minimum distance problem for quantum codes is NP-hard, and this hardness extends to multiplicative and additive approximations, even when restricted to stabilizer or CSS codes, with the result established through a reduction from classical problems in the CWS framework using a 4-cycle free graph (Kapshikar and Kundu, 2023). Despite this, methods that improve on brute-force approaches are actively explored.

For a more in-depth background on minimum distance, see Chapter 3 of (Sabo, 2022).

Quantum Error Correction and Code Distance

The foundation of quantum error correction lies in protecting logical quantum information from physical errors by encoding it across multiple qubits. A quantum code's performance is fundamentally characterized by its distance (d), which quantifies the code's ability to detect and correct errors. Practically, the distance represents the minimum number of physical qubit errors required to cause an undetectable logical error - one that corrupts encoded information while evading the code's error detection mechanisms.

Fundamentals of Quantum Code Distance

The distance d of a quantum error-correcting code represents its robustness against physical errors and is defined as (Ekert et al., undated):

\[\begin{aligned} \begin{equation} d = \min_{P \in N(S)\setminus S} \mathrm{wt}(P) \end{equation} \end{aligned}\]

where:

$N(S)$ denotes the normalizer of the stabilizer group S.
$\mathrm{wt}(P)$ represents the weight of Pauli operator P.
The minimization is taken over all logical operators that are not stabilizers.

The distance reveals the essential property that it equals the smallest number of qubits that must be affected to produce a logical error undetectable by stabilizer measurements. The normalizer condition $P \in N(S)$ ensures the operator commutes with all stabilizers, while $P \notin S$ guarantees it performs a non-trivial logical operation (Ekert et al., undated).

Mixed Integer Programming

We compute the minimum code distance for CSS (Calderbank-Shor-Steane) codes by solving MIPs.

The distance is computed separately for X-type ($d_X$) and Z-type ($d_Z$) logical operators, then combined to give the true code distance: $d = \min(d_X, d_Z)$.

X-type and Z-type Distances

The X-type distance ($d_X$) and Z-type distance ($d_Z$`) are defined as the minimum number of errors required to implement a non-trivial logical operator of the opposite type, subject to the following constraints:

For $d_X$ (where $U = X$), the errors are Z-type (phase flips), and the constraints involve the X-stabilizer matrix $\mathbf{H_X}$ and X-logical operators $\mathbf{L_X}$. The error vector is denoted as \mathbf{e}_Z`.
For $d_Z$ (where $U = Z$), the errors are X-type (bit flips), with constraints given by the Z-stabilizer matrix $\mathbf{H_Z}$ and Z-logical operators $\mathbf{L_Z}$, and the error vector is $\mathbf{e}_X$.

\[\begin{aligned} \text{Minimize} \quad & \sum_{i=1}^n e_{V,i} \quad \text{(Hamming weight of errors)} \\ \text{Subject to} \quad & \mathbf{H_U} \cdot \mathbf{e}_V \equiv \mathbf{0} \pmod{2} \quad \text{(Commutes with U-stabilizers)} \\ & \mathbf{L_U} \cdot \mathbf{e}_V \equiv 1 \pmod{2} \quad \text{(Anti-commutes with U-logical)} \\ & e_{V,i} \in \{0,1\} \quad \text{(Binary error variables)} \end{aligned}\]

Example

A [[40, 8, 5]] 2BGA code with the minimum distance of 5 from Table 2 of (Lin and Pryadko, 2024).

julia> import Hecke: group_algebra, GF, abelian_group, gens; import HiGHS; import JuMP;

julia> using QuantumClifford, QuantumClifford.ECC

julia> l = 10; m = 2;

julia> GA = group_algebra(GF(2), abelian_group([l,m]));

julia> x, s = gens(GA);

julia> A = 1 + x^6;

julia> B = 1 + x^5 + s + x^6 + x + s*x^2;

julia> c = two_block_group_algebra_codes(A,B);

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=HiGHS))
(40, 8, 5)

A [[48, 6, 8]] GB code with the minimum distance of 8 from (A3) in Appendix B of (Panteleev and Kalachev, 2021).

julia> l = 24;

julia> c1 = generalized_bicycle_codes([0, 2, 8, 15], [0, 2, 12, 17], l);

julia> code_n(c1), code_k(c1), distance(c1, DistanceMIPAlgorithm(solver=HiGHS))
(48, 6, 8)

Applications

Mixed-integer programming (MIP) is applied in quantum error correction, notably for decoding and minimum distance computation. Some applications are as follows:

The first usecase of the MIP approach was the code capacity Most Likely Error (MLE) decoder for color codes introduced in (Landahl et al., 2011).
For all quantum LDPC codes presented in (Panteleev and Kalachev, 2021), the lower and upper bounds on the minimum distance was obtained by reduction to a mixed integer linear program and using the GNU Linear Programming Kit ((Makhorin, 2008)).
For all the Bivariate Bicycle (BB) codes presented in (Bravyi et al., 2024), the code distance was calculated using the mixed integer programming approach.
(Lacroix et al., 2024) developed a MLE decoder that finds the most likely chain of Pauli errors given the observed error syndrome by solving a mixed-integer program using HiGHS package ((Huangfu and Hall, 2018)).
(Cain et al., 2025) formulate maximum-likelihood decoding as a mixed-integer program maximizing $\prod_{j=1}^M p_j^{E_j}(1-p_j)^{1-E_j}$ (where binary variables $E_j \in {0,1}$ indicate error occurrence) subject to syndrome constraints, solved optimally via MIP solvers despite its NP-hard complexity.

source

Full ECC API (autogenerated)

Implemented in an extension requiring Hecke.jl

Implemented in an extension requiring Oscar.jl

Implemented in an extension requiring JuMP.jl

Implemented in an extension requiring `Hecke.jl`

Implemented in an extension requiring `Oscar.jl`

Implemented in an extension requiring `JuMP.jl`