Classical coding theory contains several techniques to obtain new codes from other codes, including puncturing and shortening. Both of these techniques have been generalized to quantum codes. Restricting to stabilizer codes, this paper introduces more freedom in the choice of the encoded states after puncturing. Furthermore, we also give an explicit description of the stabilizers for the punctured code. The additional freedom in the procedure also opens up for new ways to construct new codes from old, and we present several ways to utilize this in the search for codes with good or even optimal parameters. In particular, we use the construction to obtain codes whose parameters exceed the best previously known and which are better than what general puncturing guarantees. Lastly, the freedom in our puncture procedure allowed us to generalize the proof of the Griesmer bound from the classical setting to stabilizer codes for qudits of prime dimension since the proof relies heavily on the puncturing technique.
Classical coding theory contains several techniques to obtain new codes from other codes, including puncturing and shortening. Both of these techniques have been generalized to quantum codes. Restricting to stabilizer codes, this paper introduces more freedom in the choice of the encoded states after puncturing. Furthermore, we also give an explicit description of the stabilizers for the punctured code. The additional freedom in the procedure also opens up for new ways to construct new codes from old, and we present several ways to utilize this in the search for codes with good or even optimal parameters. In particular, we use the construction to obtain codes whose parameters exceed the best previously known and which are better than what general puncturing guarantees. Lastly, the freedom in our puncture procedure allowed us to generalize the proof of the Griesmer bound from the classical setting to stabilizer codes for qudits of prime dimension since the proof relies heavily on the puncturing technique.
In this work, we introduce a technique for reducing the length of a quantum stabilizer code, and we call this deflation of the code. Deflation can be seen as a generalization of the well-known puncturing and shortening techniques in cases where more than a single qudit is removed. We show that the parameters of the deflated quantum code can be controlled, and argue that a similar approach is not as beneficial when applied to classical linear codes. Furthermore, it is shown that deflation introduces additional freedom compared to applying just puncturing and shortening consecutively. We exemplify that it is possible to obtain better parameters by deflating a code rather than consecutively using puncturing and shortening.
Quantum error-correcting codes are essential for reliable quantum communication. The achievable parameters can be improved by using entanglement-assisted stabilizers at the cost of consuming pre-shared Bell pairs. If the availability of such Bell pairs is intermittent, the ability to adapt the code to available resources is beneficial. That is, to pick a single mother code that can be adapted to different entanglement budgets. To achieve this, we extend an existing puncturing construction for standard/unassisted stabilizers to the entanglement-assisted setting through a three-step procedure: Interpreting the code as an unassisted stabilizer, puncturing a receiver-side qubit, and recovering an entanglement-assisted representation. This procedure reduces the number of required Bell pairs by one while preserving the number of transmitted and logical qubits. We derive a bound on the resulting distance loss and establish conditions for preserving the original code distance. Finally, we analyse distinct, randomly generated, entanglement-assisted stabilizers with five to eight transmitted qubits. Of these codes, 14.75% have a greater distance than their associated unassisted representations. Among this subset, 81.4% admit at least one puncturing choice that preserves the original entanglement-assisted distance. These results illustrate how puncturing can reduce entanglement requirements while maintaining code distance.
Recently, the field of quantum error-correcting codes has rapidly emerged as an important discipline. As quantum information is extremely sensitive to noise, it seems unlikely that any large scale quantum computation is feasible without quantum error-correction. In this paper we give a brief exposition of the theory of quantum stabilizer codes. We review the stabilizer formalism of quantum codes, establish the connection between classical codes and stabilizer codes and the main methods for constructing quantum codes from classical codes. In addition to the expository part, we include new results that cannot be found elsewhere. Specifically, after reviewing some important bounds for quantum codes, we prove the nonexistence of pure perfect quantum stabilizer codes with minimum distance greater than 3. Finally, we illustrate the general methods of constructing quantum codes from classical codes by explicitly constructing two new families of quantum codes and conclude by showing how to construct new quantum codes by shortening.
Abstract Recently, the field of quantum error-correcting codes has rapidly emerged as an important discipline. As quantum information is extremely sensitive to noise, it seems unlikely that any large scale quantum computation is feasible without quantum error-correction. In this paper we give a brief exposition of the theory of quantum stabilizer codes. We review the stabilizer formalism of quantum codes, establish the connection between classical codes and stabilizer codes and the main methods for constructing quantum codes from classical codes. In addition to the expository part, we include new results that cannot be found elsewhere. Specifically, after reviewing some important bounds for quantum codes, we prove the nonexistence of pure perfect quantum stabilizer codes with minimum distance greater than 3. Finally, we illustrate the general methods of constructing quantum codes from classical codes by explicitly constructing two new families of quantum codes and conclude by showing how to construct new quantum codes by shortening.
Introduction Most quantum error-correcting codes (QECCs) that are designed to correct local errors are stabilizer codes that correspond to additive classical codes. Because of this correspondence, stabilizer codes are also referred to as additive codes, while nonstabilizer codes are called nonadditive codes. Shortly after the invention of stabilizer codes by Gottesman [G96a] and independently Calderbank et al. [CRS+98], the nonadditive ((5, 6, 2)) code was found [RHS+97], which constitutes the first example of a nonadditive quantum code with a higher dimension than any stabilizer code of equal length and minimum distance. About ten years later, the first one-error-correcting ((9, 12, 3)) code and the optimal ((10, 24, 3)) code were found [YCL+09, YCO07], both better than any stabilizer code of the same length. Here we describe a framework that not only allows a joint description of these codes, but also enables the construction of new nonadditive codes with better parameters than the best known stabilizer codes. These codes are based on the classical nonlinear Goethals and Preparata codes, which themselves are better than any classical linear code. In Section 10.3, the framework will be presented from two different points of view, namely union stabilizer codes and codeword stabilized codes, each highlighting different aspects of the construction. To illustrate the relationship of these nonadditive codes to stabilizer codes we first recall the main aspects of the latter. Section 10.4 is devoted to methods for obtaining new codes, including the aforementioned quantum Goethals–Preparata codes.
In this paper, we prove how to extend a subset of quantum stabilizer codes into a qudit hybrid code storing log2p classical bits over a qudit space with dimension p, with p prime. Our proof also gives an explicit procedure for finding the entire collection of stabilizer algebras for all of the subcodes of the hybrid code. This allows extra classical information to be transmitted without having to arduously search for additional codes and their associated codewords, and also provides a first lower bound to the amount of classical information able to be transmitted in a qudit hybrid code, but unfortunately only allows for log2p classical bits to be decoded by a receiver.