\([[144,12,12]]\) gross code

\([[144,12,12]]\) gross code[1]

Alternative Names: \((3,3)\) BB6 code.

Description

A BB code which requires less physical and ancilla qubits (for syndrome extraction) than the surface code with the same number of logical qubits and distance. The gross code is equivalent to 8 copies of the surface code via a constant-depth Clifford circuit, and is an element of a larger family of 2D stabilizer codes [2]. The name stems from the fact that a gross is a dozen dozen.

The polynomials \(A=x^3+y+y^2\) and \(B=y^3+x+x^2\) of the gross code on \(\ell=m=12\) yield an indecomposable \([[288,16,12]]\) code [3; Table III]. A different BB QLDPC code with the same parameters was introduced in [4].

Protection

Up to equivalence, the gross code is the only qubit Abelian 2BGA code with check weight six and \(n\leq 200\) that reaches \(kd^2/n=12\) [5]. It has the largest \(kd^2/n\) among all qubit 2BGA codes with check weight six and \(n\leq 191\) [5].

Rate

An ancilla-added rate of \(1/24\). In contrast, the distance-13 surface code has ancilla-added rate \(1/338\).

Transversal and Permutation-Based Gates

Logical Pauli operators and fold-transversal gates studied in Ref. [6].Two-fold transversal gates, i.e., depth-one circuits of two-local code-preserving physical gates, realize a logical group of order at least \(460800\), short of the full logical Clifford group [7].

Gates

Clifford gates [8].

Decoding

The GDG sliding-window decoder [9], with a realization achieving a worst-case decoding latency of 3ms per window.AC decoder is faster than ordinary BP-OSD with no reduction of fidelity [10].Transformer-based neural-network decoder [11].Frontier decoder, a pruned dynamic-programming decoder, retains on average fewer than 100 states under circuit-level noise at \(p=10^{-3}\) [12].

Fault Tolerance

At physical error rate \(p=10^{-3}\) under circuit-level noise, the code suppresses the logical error rate to about \(2\times 10^{-7}\), enough to preserve \(12\) logical qubits for nearly one million syndrome cycles using \(288\) total physical qubits [1].Fault-tolerant modular quantum computing framework [13].

Code Capacity Threshold

Bit-flip noise: pseudo-threshold of \(\approx 5\%\) for the block logical error rate under BP-OSD decoding [3].

Threshold

Admits a pseudo-threshold of \(\approx 0.7\%\) for the circuit-based noise model [1].

Realizations

An FPGA implementation of the Relay-BP decoder [14].

Cousins

Kitaev surface code— The gross code requires less physical and ancilla qubits (for syndrome extraction) than the surface code with the same number of logical qubits and distance. The gross code is equivalent to 8 copies of the surface code via a constant-depth Clifford circuit, and is an element of a larger family of 2D stabilizer codes [2]. An architecture combining the surface and gross codes was proposed in [15].
Perturbed bivariate bicycle (PBB) code— The \([[144,12,12]]\) PBB code perturbs the gross-code polynomials \(A=x^3+y+y^2\) and \(B=y^3+x+x^2\) by \(C=y+x^3y\) and \(D=y^3+x^3y^3\), with mixed stabilizer generators of weight eight [3; Table II].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeQubit Stabilizer Hamiltonian-based QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeQLDPC Qubit CSS Generalized homological-product Stabilizer Hamiltonian-based QECC Quantum

Bivariate bicycle (BB) codeQLDPC Qubit CSS Generalized homological-product Lattice stabilizer Stabilizer Hamiltonian-based QECC Quantum

Parents

Bivariate bicycle (BB) code

The gross code is the BB code with \(A=1+x+x^{-1}y^3\) and \(B=1+y+x^3y^{-1}\) on the torus with lattice vectors \((0,12)\) and \((6,0)\) [16; Exam. 3 and Table II].