New method clarifies quantum code logical operators and distances
Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes
Information Theory
Summary
Quantum codes help protect information in quantum computers, but understanding their most reliable parts can be tricky. The authors studied a certain family of codes called bivariate bicycle codes and found a clean way to describe their logical operators and minimum error protection levels. They broke down these logical parts into simpler components and proved formulas to measure how far apart errors must be to affect the code. This helps both to count and understand the code’s key properties more clearly.
What this means in practice
- •For quantum hardware engineers: Improve design validation of quantum memory by better understanding minimum-weight logical errors in bivariate bicycle codes.
- •For quantum software developers: Enhance quantum error-correcting code simulators with explicit logical operator characterizations for more accurate distance checks.
Authors
Mohammad Rowshan, Simon Devitt
Abstract
Bivariate bicycle (BB) quantum codes are a prominent finite-length family of quantum low-density parity-check codes, but their minimum distance is usually established numerically rather than read from the defining polynomials. We study the $Z$-logical quotient $K/S$ over $\mathbb F_2[x,y]/(x^\ell-1,y^m-1)$ and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity $d_Z=\min(d_{\mathcal A},d_{\mathcal C})$. Using the Frobenius structure of the finite group algebra, we prove $r_{\mathcal A}=r_{\mathcal C}=k/2$ for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances $4,6,10,10,12,18$ of the six standard BB codes of lengths $18$ to $288$ and enumerates every minimum-weight logical operator. The resulting census shows that $[[108,8,10]]$ is the only one of the six whose distance is attained in a single component, with $d_{\mathcal C}=10$ and $d_{\mathcal A}=12$.