[Paper Review] Quantum Cyclic Code

Sagarmoy Dutta, Piyush P. Kurur|arXiv (Cornell University)|Jul 10, 2010

Quantum Computing Algorithms and Architecture11 references3 citations

TL;DR

This paper introduces quantum cyclic codes as a generalization of classical cyclic codes to the quantum setting, beyond the standard CSS construction. It presents new non-CSS quantum cyclic codes with parameters [[5,1,3]], [[17,1,7]], and [[17,9,3]], and develops a polynomial-time quantum decoding algorithm for stabilizer codes of length $4^m+1$ using a BCH distance framework, enabling correction of up to $\lfloor (d-1)/2 \rfloor$ errors.

ABSTRACT

In this paper, we define and study \emph{quantum cyclic codes}, a generalisation of cyclic codes to the quantum setting. Previously studied examples of quantum cyclic codes were all quantum codes obtained from classical cyclic codes via the CSS construction. However, the codes that we study are much more general. In particular, we construct cyclic stabiliser codes with parameters $[[5,1,3]]$, $[[17,1,7]]$ and $[[17,9,3]]$, all of which are \emph{not} CSS. The $[[5,1,3]]$ code is the well known Laflamme code and to the best of our knowledge the other two are new examples. Our definition of cyclicity applies to non-stabiliser codes as well; in fact we show that the $((5,6,2))$ nonstabiliser first constructed by Rains\etal~ cite{rains97nonadditive} and latter by Arvind \etal~\cite{arvind:2004:nonstabilizer} is cyclic. We also study stabiliser codes of length $4^m +1$ over $\mathbb{F}_2$ for which we define a notation of BCH distance. Much like the Berlekamp decoding algorithm for classical BCH codes, we give efficient quantum algorithms to correct up to $\floor{\frac{d-1}{2}}$ errors when the BCH distance is $d$.

Motivation & Objective

• To define and characterize quantum cyclic codes as a natural generalization of classical cyclic codes in the quantum domain.
• To construct new quantum stabilizer codes that are not of the CSS type, including [[5,1,3]], [[17,1,7]], and [[17,9,3]] codes.
• To extend the concept of cyclicity to non-stabilizer codes, showing that the ((5,6,2)) code is cyclic under the new definition.
• To define a notion of BCH distance for stabilizer codes of length $4^m+1$ over $\mathbb{F}_2$, enabling efficient decoding.
• To develop a polynomial-time quantum decoding algorithm for such codes within the BCH error-correcting limit.

Proposed method

• Define quantum cyclic codes via the action of the shift operator on the underlying stabilizer group, generalizing classical cyclic codes to the quantum setting.
• Use the Gottesman-Knill framework to represent quantum codes via Gottesman subgroups and their associated totally isotropic subspaces in $\mathbb{F}_p^{2n}$.
• Introduce a generating pair $(g,h)$ for the underlying ideal in $\mathbb{F}_4[X]/(X^n - 1)$ to describe the code structure and enable efficient recovery of error polynomials.
• Leverage phase estimation and quantum Fourier sampling to compute inner products that recover coefficients of error polynomials modulo the generator.
• Apply the Berlekamp algorithm to decode errors up to $\lfloor (d-1)/2 \rfloor$ when the BCH distance $d$ is defined for codes of length $n = 4^m + 1$.
• Extend the framework to pseudo-stabilizer codes by showing that if the underlying Gottesman subgroup is cyclic, then the resulting code is also cyclic.

Experimental results

Research questions

• RQ1 Can the concept of cyclicity in classical codes be meaningfully extended to general quantum codes beyond the CSS construction?
• RQ2 What are the structural and parameter characteristics of quantum cyclic codes that are not of the CSS type?
• RQ3 Can a notion of BCH distance be defined for quantum stabilizer codes of length $4^m + 1$, analogous to classical BCH codes?
• RQ4 Is there a polynomial-time quantum decoding algorithm for such codes within the BCH error-correcting limit?
• RQ5 Are non-stabilizer quantum codes, such as the ((5,6,2)) code, also cyclic under the new definition?

Key findings

• The paper constructs a new [[17,1,7]] quantum cyclic code that is not of the CSS type, representing a novel code with high distance.
• A [[17,9,3]] quantum cyclic code is presented, also outside the CSS framework, demonstrating the generality of the construction.
• The well-known [[5,1,3]] Laflamme code is recovered as a special case of the proposed quantum cyclic code framework.
• The ((5,6,2)) non-stabilizer code is shown to be cyclic under the new definition, establishing the first known example of a non-stabilizer cyclic quantum code.
• For quantum stabilizer codes of length $n = 4^m + 1$ with BCH distance $d = 2t + 1$, a polynomial-time quantum decoding algorithm is constructed that corrects up to $t$ errors.
• The decoding algorithm relies on efficient quantum phase estimation to recover error polynomial coefficients modulo the generator polynomial, followed by the Berlekamp algorithm for error correction.