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Advances in Operation Research and Production Management

Stabilizer Codes on a Cayley Graph

Yiqun Gui

Published Jul 9, 2025• 0 citations
Publisher

Abstract

The paper investigates quantum stabilizer codes and our innovative construction methodology. At first, we began with representing fundamental concepts and theories of quantum computing. In order to identify quantum error correcting codes, we extend the method of using polynomials to represent qudits on square lattices to accommodate more complicated situations in general Cayley graphs. The paper briefly reviews essential definitions and examples related to graphs, groups, and stabilizer codes, and later we propose the novel method and demonstrate its application by using the dihedral groupD4.

DOI
10.54254/3029-0880/2025.24624

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The paper investigates quantum stabilizer codes and our innovative construction methodology. At first, we began with representing fundamental concepts and theories of quantum computing. In order to identify quantum error correcting codes, we extend the method of using polynomials to represent qudits on square lattices to accommodate more complicated situations in general Cayley graphs. The paper briefly reviews essential definitions and examples related to graphs, groups, and stabilizer codes, and later we propose the novel method and demonstrate its application by using the dihedral groupD4.
Graph states are generalized from qubits to collections of nnn qudits of arbitrary dimension DDD, and simple graphical methods are used to construct both additive and nonadditive, as well as degenerate and nondegenerate, quantum-error-correcting codes. Codes of distance 2 saturating the quantum Singleton bound for arbitrarily large nnn and DDD are constructed using simple graphs, except when nnn is odd and DDD is even. Computer searches have produced a number of codes with distances 3 and 4, some previously known and some new. The concept of a stabilizer is extended to general DDD, and shown to provide a dual representation of an additive graph code.
Graph theory is important in information theory. We introduce a quantization process on graphs and apply the quantized graphs in quantum information. The quon language provides a mathematical theory to study such quantized graphs in a general framework. We give a new method to construct graphical quantum error correcting codes on quantized graphs and characterize all optimal ones. We establish a further connection to geometric group theory and construct quantum low-density parity-check stabilizer codes on the Cayley graphs of groups. Their logical qubits can be encoded by the ground states of newly constructed exactly solvable models with translation-invariant local Hamiltonians. Moreover, the Hamiltonian is gapped in the large limit when the underlying group is infinite.
Long distance communication of digital data, whether through a physical medium or a broadcast signal, is often subjected to noise. To deliver data reliably through noisy communication channels, one must use codes that can detect and correct the particular noise of the channel. For transmission of classical data, error correcting schemes can be as simple as the sending of replicates. For quantum data, and in tandem the development of machines that can process quantum data, quantum error correcting codes must be developed. In addition to a larger set of possible errors, quantum error correcting schemes must contend with other peculiarities of quantum mechanics, such as the no-cloning theorem which can prevent the sending of replicate messages. Stabilizer codes are one family of quantum error correcting codes which can protect and correct errors expressed in terms of the Pauli group, exploiting its group structure and utilizing classical codes and the corresponding duals. We develop and examine a family of quantum stabilizer codes which arise from reflexive stabilizers. Moreover, we provide a mapping from our reflexive stabilizer codes to the well-known CSS codes developed by Calderbank, Shor, and Steane. For the case of a 4-state system we show that these codes can obtain the minimal embedding for code which can correct any flip or phase error. We also provide heuristic algorithms for creating reflexive stabilizer codes starting from the noise of a quantum channel. Furthermore, we show that the problem can be posed in terms of finding maximal Cayley subgraphs with restrictions imposed by the set of potential errors.
We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes are verified in graph-theoretic terms as originally advocated by Schlingemann and Werner. Finally, possible generalizations of our scheme for the graphical construction of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum codes are briefly addressed.
We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes are verified in graph-theoretic terms as originally advocated by Schlingemann and Werner. Finally, possible generalizations of our scheme for the graphical construction of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum codes are briefly addressed.
Graphs are closely related to quantum error-correcting codes: every stabilizer code is locally equivalent to a graph code and every codeword stabilized code can be described by a graph and a classical code. For the construction of good quantum codes of relatively large block length, concatenated quantum codes and their generalizations play an important role. We develop a systematic method for constructing concatenated quantum codes based on “graph concatenation,” where graphs representing the inner and outer codes are concatenated via a simple graph operation called “generalized local complementation.” Our method applies to both binary and nonbinary concatenated quantum codes as well as their generalizations.