Stabilizer codes over qudits have been widely studied but their encoding procedures have not been investigated in detail by many. The encoding procedure proposed previously by Grassl et al. generates the stabilizer codes over qudits for prime d, while it may only generate a subspace of the codespace over qudits where d is a power of a prime. In this paper, we propose an encoding procedure to generate the stabilizer codes over qudits for d being a power of a prime. The procedure involves reducing the problem of encoding over qudits with d = pkto a problem similar to encoding over qudits with d = p.
Encoding of Quantum Stabilizer Codes Over Qudits with d=pk
Stabilizer codes over qudits have been widely studied but their encoding procedures have not been investigated in detail by many. The encoding procedure proposed previously by Grassl et al. generates the stabilizer codes over qudits for prime d, while it may only generate a subspace of the codespace over qudits where d is a power of a prime. In this paper, we propose an encoding procedure to generate the stabilizer codes over qudits for d being a power of a prime. The procedure involves reducing the problem of encoding over qudits with d = pkto a problem similar to encoding over qudits with d = p.
Quantum coding schemes over qudits using preshared entanglement between the encoder and decoder can provide better error correction capability than without it. In this article, we develop procedures for constructing encoding operators for entanglement-unassisted and entanglement-assisted qudit stabilizer codes over Fpk, with p prime and k ≥ 1 from first principles, generalizing prior works on qubit-based codes and codes that work over Fp. We also provide quantum encoding architectures based on the proposed encoding procedures using one and two qudit gates, useful toward realizing coded quantum computing and communication systems using qudits.
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.
In this work, we compute the number of [[n,k]]d stabilizer codes made up of d -dimensional qudits, for arbitrary positive integers d . In a seminal work by Gross \cite{Gross2006} the number of [[n,k]]d stabilizer codes was computed for the case when d is a prime (or the power of a prime, i.e., d=pm , but when the qudits are Galois-qudits). The proof in \cite{Gross2006} is inapplicable to the non-prime case. For our proof, we introduce a group structure to [[n,k]]d codes, and use this in conjunction with the Chinese remainder theorem to count the number of [[n,k]]d codes. Our work overlaps with \cite{Gross2006} when d is a prime and in this case our results match exactly, but the results differ for the more generic case. Despite that, the overall order of magnitude of the number of stabilizer codes scales agnostic of whether the dimension is prime or non-prime. This is surprising since the method employed to count the number of stabilizer states (or more generally stabilizer codes) depends on whether d is prime or not. The cardinality of stabilizer states, which was so far known only for the prime-dimensional case (and the Galois qudit prime-power dimensional case) plays an important role as a quantifier in many topics in quantum computing. Salient among these are the resource theory of magic, design theory, de Finetti theorem for stabilizer states, the study and optimisation of the classical simulability of Clifford circuits, the study of quantum contextuality of small-dimensional systems and the study of Wigner-functions. Our work makes available this quantifier for the generic case, and thus is an important step needed to place results for quantum computing with non-prime dimensional quantum systems on the same pedestal as prime-dimensional systems.
There is a connection between classical codes, highly entangled pure states (calledk-uniform or absolutely maximally entangled (AME) states), and quantum error correcting codes (QECCs). This leads to a systematic method to construct stabilizer QECCs by starting from ak-uniform state or the corresponding classical code and tracing out one party at each step. We provide explicit constructions for codewords, encoding procedure and stabilizer formalism of the QECCs by describing the changes that partial traces cause on the corresponding generator matrix of the classical codes. We then modify the method to produce another set of stabilizer QECCs that encode a logical qudit into a subspace spanned by AME states. This construction produces quantum codes starting from an AME state without tracing out any party. Therefore, quantum stabilizer codes with larger codespace can be constructed.
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.
Quantum computers are a revolutionary class of computational platforms with applications in combinatorial and global optimization, machine learning, and other domains involving computationally hard problems. While these machines typically operate on qubits—quantum information elements that can occupy superpositions of the basis∣0⟩and∣1⟩states—recent advances have demonstrated the practical implementation of higher-dimensional quantum systems (qudits) across various hardware platforms. In these hardware realizations, the higher-order states are less stable and thus remain coherent for a shorter duration than the basis∣0⟩and∣1⟩states. Moreover, formal methods for designing efficient encoder circuits for these systems remain underexplored. This limitation motivates the development of efficient circuit techniques for qudit systems (d-level quantum systems). Previous works have typically established generating gate sets for higher-dimensional codes by generalizing the methods used for qubits. In this work, we introduce a systematic framework for optimizing encoder circuits for prime-dimension stabilizer codes. This framework is based on novel generating gate sets whose elements map directly to efficient Clifford gate sequences. We demonstrate the effectiveness of this method on key codes, achieving a 13–44% reduction in encoder circuit gate count for the qutrit (d = 3)[[9,5,3]]3,[[5,1,3]]3, and[[7,1,3]]3codes, and a 9–21% reduction for the ququint (d = 5)[[10,6,3]]5code when compared to prior work. We also achieved circuit depth reductions up to 42%.