Haah cubic code (CC)

Haah cubic code (CC) [1]

Description

A 3D lattice stabilizer code on a length-\(L\) cubic lattice with one or two qubits per site. Admits two types of stabilizer generators with support on each cube of the lattice. In the non-CSS case, these two are related by spatial inversion. For CSS codes, we require that the product of all corner operators is the identity. We lastly require that there are no non-trivial string operators, meaning that single-site operators are a phase, and any period one logical operator \(l \in \mathsf{S}^{\perp}\) is just a phase.

Haah showed in his original construction that there is exactly one non-CSS code of this form, and 17 CSS codes [1]. The non-CSS code is labeled code 0, and the rest are numbered from 1 - 17. Codes CC1-CC4, CC7, CC8, and CC10 do not have string logical operators [1,2]. The original cubic code in this family can be obtained by gauging [3–12] a fractal-symmetry Ising model [6; Sec. IV.B]; related fracton gauging constructions also appear in [5,6].

Under renormalization group flow [13], cubic code 1 fragments into itself and the Haah B-code (a.k.a. CC1B), which has four qubits per unit cell [14][2; Eq. (D2)]. In this context, cubic code 1 is sometimes called the Haah A-code or CC1A. Cubic codes 11-17 fragment into combinations of themselves, their corresponding B-codes, and stacks of surface codes [14; Table 1].

The Haah A-code can be written in a similar form as the Sierpinski prism model code [15]. The Haah B-code admits a topological defect network construction out of two copies of the 3D surface code [16].

Encodings using geometries with boundaries as well as lattice defects have been studied [17]. CC1A and CC1B have been generalized to manifolds more general than 3D lattices [18,19].

Protection

Cubic codes protect against simultaneous independent Pauli errors on different sites (not qubits, since there can be 2 qubits per site). Codes CC0-CC4 are known to have distance \(d \ge L\), meaning they can achieve macroscopic code distance as \(L\to\infty\).

Rate

The encoding rate depends on the code implemented, but code CC0 has been shown to have \(k \ge L\) on a periodic finite cubic lattice of side length \(L\). For codes CC1-CC4, only \(k \geq 2\) is guaranteed, and \(k\) depends sensitively on \(L\) [1]. For example, cubic code 1 has \(k = 4L-2\) when \(L = 2^p\) but \(k=2\) for odd \(L\) not divisible by 15 or 63, according to empirical formulas exact for \(L \leq 200\) [1].

Decoding

Hard-decision RG decoder [20].BP-OSD decoder [21].

Code Capacity Threshold

Cubic code 1 has a threshold of \(7.97(4)\%\) under BP-OSD decoding for code capacity bit-flip noise [22]. This estimate uses periodic lattices of odd side length on which the code encodes \(k=2\) logical qubits [22]. Under full postselection, the clean statistical mechanical model of cubic code 1 is the self-dual fractal Ising model [22]. This model has a unique transition at \(p=1/(2+\sqrt{2})\approx 0.2929\), the self-dual value shared by zero-rate PSD codes with a unique threshold [22].

Cousins

Kitaev surface code— Under renormalization group flow [13], cubic codes 11-17 fragment into combinations of themselves, their corresponding B-codes, and stacks of surface codes [14; Table 1].
3D surface code— The Haah B-code admits a topological defect network construction out of two copies of the 3D surface code [16].
3D color code— The 3D color and cubic code families both include 3D codes that do not admit string-like operators.
\((2,2)\) Loop toric code— Neither code has string-like logical operators [1,23], whose constant energy cost of creation precludes self-correction [24,25]. The membrane logical operators of the \((2,2)\) loop toric code give an energy barrier that grows with \(L\) and a self-correcting memory [23,26]. Cubic code 1 has only a logarithmic energy barrier [27] and thermalizes rapidly at every positive temperature under Davies dynamics [28].
Abelian two-block group-algebra code— CSS cubic codes are Abelian 2BGA codes over the group \(G=\mathbb{Z}_{L}^{\times 3}\) [29; Sec. III.A]; e.g., cubic code 1 corresponds to group-algebra elements \(a = 1+x+y+z\) and \(b = 1+xy+xz+yz\) [30].
Cluster-state code— A short-range entangled cluster-state model with fractal \(X\)-type symmetries on both sublattices can be built from the cubic-code gauging data. Gauging one sublattice yields, up to a local circuit, either the cubic code or its ungauged fractal-symmetry Ising model, while gauging both sublattices returns the cluster model up to local swaps and Hadamards [6; Secs. III.F, IV.C].
Lifted-product (LP) code— A lifted-product code constructed with coefficients in the ring \(R=\mathbb{F}_2[x,y,z]/(x^L-1,y^L-1,z^L-1)\) is a cubic code [31; Appx. B].
Sierpinski prism model code— The Haah A-code can be written in a similar form as the Sierpinski prism model code [15].
Permutationally self-dual (PSD) CSS code— CSS cubic codes are qubit Abelian 2BGA codes [29; Sec. III.A], so they are permutationally self-dual [32; Thm. 6]. For cubic code 1, transversal Hadamard composed with the spatial inversion \(\vec{r}\to-\vec{r}\) and a swap of the two qubits on each site exchanges the \(X\)- and \(Z\)-type stabilizers [33].
Fibonacci code— The Fibonacci code is designed to mimic the fractal properties of (quantum) Haah cubic code so that studying the former can help us toward the development of an efficient algorithm for the latter [34].
Self-correcting quantum code— Cubic code 1 has a logarithmic energy barrier [27] and is partially self-correcting below a temperature-dependent system-size cutoff [20]. Under Davies dynamics with single-qubit \(X\) and \(Z\) couplings on a periodic lattice, it thermalizes in time \(O(\log n)\) at every positive temperature [28]. Its memory time is therefore at most logarithmic in \(n\) for any decoder, so it is not self-correcting [28].
Bosonization code— Bosonization can be used to realize a Haah cubic code with an emergent fermion from a Majorana stabilizer code [35]. This code is shown to be distinct from the original code [36].

Member of code lists

Primary Hierarchy

References

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Cite as:

“Haah cubic code (CC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/haah_cubic, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/fracton/haah_cubic.yml.