Draft:Bivariate bicycle code

A bivariate bicycle code (BB code) is a family of quantum error-correcting codes introduced by Sergey Bravyi and collaborators. BB codes are CSS codes and quantum low-density parity-check (qLDPC) codes. They generalize earlier bicycle-code constructions by using polynomials in two commuting variables.

Construction

Let $x$ and $y$ denote cyclic shift operators of orders $\ell$ and $m$. Two sparse bivariate polynomials $a(x,y)$ and $b(x,y)$ over the binary field define commuting $\ell m\times \ell m$ matrices $A$ and $B$. A BB code has CSS check matrices

$$H_{X}={\begin{bmatrix}A&B\end{bmatrix}},\qquad H_{Z}={\begin{bmatrix}B^{\mathsf {T}}&A^{\mathsf {T}}\end{bmatrix}}.$$

Because $A$ and $B$ commute, the CSS orthogonality condition $H_{X}H_{Z}^{\mathsf {T}}=0$ is satisfied. The construction uses $n=2\ell m$ data qubits. Sparse choices of $a$ and $b$ give bounded-weight stabilizer checks and bounded qubit degree, which are the defining locality properties of qLDPC codes.

The code can be represented on a periodic two-dimensional grid. Some interactions are non-local in that grid, but the connectivity graph found by Bravyi and collaborators decomposes into two planar degree-three subgraphs. Their syndrome-extraction circuits use weight-six checks and require each physical qubit to interact with six others.

Error correction and implementation

The original work analyzed circuit-level noise and proposed a combination of iterative message passing and ordered-statistics decoding. It identified finite BB codes with lower qubit overhead than comparable surface-code constructions at target logical error rates, subject to additional connectivity requirements.

Later work studied BB codes on architectures limited to two-dimensional local gates. A bilayer proposal using local operations and classical communication found regimes in which BB codes used fewer physical qubits than surface codes while giving comparable simulated logical error rates.

Examples

The best-known finite example is the Gross code, a $\left[\!\left[144,12,12\right]\!\right]$ BB code: it encodes 12 logical qubits in 144 data qubits and has distance 12. Its name refers to a gross, a quantity of 144.

Other parameter sets reported in the original construction include BB codes of lengths 72, 90, 108, 144, 216 and 288, obtained by searches over sparse bivariate polynomials.

See also

• Quantum error correction
• Low-density parity-check code
• CSS code
• Toric code

Category:Coding theory