Quantum error-correcting codes aim to protect information in quantum systems to enable fault-tolerant quantum computations. The most prevalent method, stabilizer codes, has been well developed for many varieties of systems, however, largely the case of composite number of levels in the system has been avoided. This relative absence is due to the underlying ring theoretic tools required for analyzing such systems. Here we explore composite local-dimension quantum stabilizer codes, providing a pair of constructions for transforming known stabilizer codes into valid ones for composite local-dimensions. In addition remarks on logical encodings and the counts possible are discussed. This work lays out central methods for working with composite dimensional systems, enabling full use of the computational space of some systems, and expanding the understanding of the symplectic spaces involved.Quantum error-correcting codes aim to protect information in quantum systems to enable fault-tolerant quantum computations. The most prevalent method, stabilizer codes, has been well developed for many varieties of systems, however, largely the case of composite number of levels in the system has been avoided. This relative absence is due to the underlying ring theoretic tools required for analyzing such systems. Here we explore composite local-dimension quantum stabilizer codes, providing a pair of constructions for transforming known stabilizer codes into valid ones for composite local-dimensions. In addition remarks on logical encodings and the counts possible are discussed. This work lays out central methods for working with composite dimensional systems, enabling full use of the computational space of some systems, and expanding the understanding of the symplectic spaces involved.

arXiv:2501.04888Jan 8, 20250 citations

Local-dimension-invariant qudit stabilizer codes

Lane G. Gunderman

Protection of quantum information from noise is a massive challenge. One avenue people have begun to explore is reducing the number of particles needing to be protected from noise and instead use systems with more states, so-called qudit quantum computers. These systems will require codes which utilize the full computational space. Many prior qudit codes are very restrictive on relations between the parameters of number of qudits, number of logical qudits, distance of the code, and number of bases. In this paper we show that codes for these systems can be derived from already known codes, often relaxing the constraints somewhat, a result which could prove to be very useful for fault-tolerant qudit quantum computers.Protection of quantum information from noise is a massive challenge. One avenue people have begun to explore is reducing the number of particles needing to be protected from noise and instead use systems with more states, so-called qudit quantum computers. These systems will require codes which utilize the full computational space. Many prior qudit codes are very restrictive on relations between the parameters of number of qudits, number of logical qudits, distance of the code, and number of bases. In this paper we show that codes for these systems can be derived from already known codes, often relaxing the constraints somewhat, a result which could prove to be very useful for fault-tolerant qudit quantum computers.

arXiv:1910.08122Physical Review AMay 28, 202010 citations

Stabilizer Codes with Exotic Local-dimensions

Lane G. Gunderman

Traditional stabilizer codes operate over prime power local-dimensions. In this work we extend the stabilizer formalism using the local-dimension-invariant setting to import stabilizer codes from these standard local-dimensions to other cases. In particular, we show that any traditional stabilizer code can be used for analog continuous-variable codes, and consider restrictions in phase space and discretized phase space. This puts this framework on an equivalent footing as traditional stabilizer codes. Following this, using extensions of prior ideas, we show that a stabilizer code originally designed with a finite field local-dimension can be transformed into a code with the same n , k , and d parameters for any integral domain. This is of theoretical interest and can be of use for systems whose local-dimension is better described by mathematical rings, which permits the use of traditional stabilizer codes for protecting their information as well.Traditional stabilizer codes operate over prime power local-dimensions. In this work we extend the stabilizer formalism using the local-dimension-invariant setting to import stabilizer codes from these standard local-dimensions to other cases. In particular, we show that any traditional stabilizer code can be used for analog continuous-variable codes, and consider restrictions in phase space and discretized phase space. This puts this framework on an equivalent footing as traditional stabilizer codes. Following this, using extensions of prior ideas, we show that a stabilizer code originally designed with a finite field local-dimension can be transformed into a code with the same n , k , and d parameters for any integral domain. This is of theoretical interest and can be of use for systems whose local-dimension is better described by mathematical rings, which permits the use of traditional stabilizer codes for protecting their information as well.

arXiv:2303.17000QuantumFeb 12, 20243 citations

Intrinsic locality dimension of quantum codes

Yimin Lu, Esther Xiaozhen Fu, Zi-Wen Liu

Quantum error-correcting codes are a cornerstone of quantum computing, with broad and profound connections to physics and mathematics. In this work, we introduce the notion of intrinsic locality dimension of stabilizer codes, which is independent of the underlying geometry of quantum codes and naturally extends to non-integer values. Drawing on mathematical tools from fractal geometry and geometric measure theory, the intrinsic locality dimension accommodates flexible architectures and provides a quantitative measure of code connectivity, encompassing both topological codes and algebraic constructions such as bivariate-bicycle-type codes. We show how the intrinsic dimension serves as a fundamental organizing parameter that unifies code properties. In particular, we prove general limitations on code parameters and compatible fault-tolerant logical gates induced by the intrinsic dimension, generalizing the Bravyi--Poulin--Terhal and Bravyi--König bounds for regular topological codes, respectively. Furthermore, we consider implications on thermal properties: toward fully characterizing the geometry requirement for self-correcting quantum memories (SCQMs), we present a conditional no-go result for SCQMs in dimension $3-ε$ and take stock of existing results on low-dimensional SCQMs. Our theory provides a unifying mathematical framework for understanding the fundamental capabilities and geometric implementations of quantum error correction and fault tolerance.Quantum error-correcting codes are a cornerstone of quantum computing, with broad and profound connections to physics and mathematics. In this work, we introduce the notion of intrinsic locality dimension of stabilizer codes, which is independent of the underlying geometry of quantum codes and naturally extends to non-integer values. Drawing on mathematical tools from fractal geometry and geometric measure theory, the intrinsic locality dimension accommodates flexible architectures and provides a quantitative measure of code connectivity, encompassing both topological codes and algebraic constructions such as bivariate-bicycle-type codes. We show how the intrinsic dimension serves as a fundamental organizing parameter that unifies code properties. In particular, we prove general limitations on code parameters and compatible fault-tolerant logical gates induced by the intrinsic dimension, generalizing the Bravyi--Poulin--Terhal and Bravyi--König bounds for regular topological codes, respectively. Furthermore, we consider implications on thermal properties: toward fully characterizing the geometry requirement for self-correcting quantum memories (SCQMs), we present a conditional no-go result for SCQMs in dimension $3-ε$ and take stock of existing results on low-dimensional SCQMs. Our theory provides a unifying mathematical framework for understanding the fundamental capabilities and geometric implementations of quantum error correction and fault tolerance.

quant-pharXiv:2605.31441Jun 1, 20260 citations

Mixed-register Stabilizer Codes: A Coding-theoretic Perspective

Himanshu Dongre, Lane G. Gunderman

Protecting information in systems that have more than two basis states (qudits) not only offers a promising route for reducing the number of individual quantum locations that must be protected, while more accurately reflecting the structure of realistic quantum hardware, but also has some possibly enticing foundational strengths. While work in the past has largely focused on protecting information in quantum devices with locations that are some consistent local structure, this work considers coding-theoretic constraints on devices constructed from locations which may vary in their local structures -- these are mixed-register quantum devices. In this work we provide some general results for mixed-register Pauli operators, then identify some stabilizer encoded information forms that are forbidden. Building on these insights, we construct coding-theoretically optimal mixed-register stabilizer codes from sets of codes defined on coprime local-dimensions. The construction of such codes results in codes with logical subspaces that do not directly correspond to any of the constituent local-dimensions.Protecting information in systems that have more than two basis states (qudits) not only offers a promising route for reducing the number of individual quantum locations that must be protected, while more accurately reflecting the structure of realistic quantum hardware, but also has some possibly enticing foundational strengths. While work in the past has largely focused on protecting information in quantum devices with locations that are some consistent local structure, this work considers coding-theoretic constraints on devices constructed from locations which may vary in their local structures -- these are mixed-register quantum devices. In this work we provide some general results for mixed-register Pauli operators, then identify some stabilizer encoded information forms that are forbidden. Building on these insights, we construct coding-theoretically optimal mixed-register stabilizer codes from sets of codes defined on coprime local-dimensions. The construction of such codes results in codes with logical subspaces that do not directly correspond to any of the constituent local-dimensions.

quant-pharXiv:2603.28459Mar 31, 20260 citations

Quantum Error Correcting Codes and Fault-Tolerant Quantum Computation over Nice Rings

Sang-Jun Lee

Quantum error correcting codes play an essential role in protecting quantum information from the noise and the decoherence. Most quantum codes have been constructed based on the Pauli basis indexed by a finite field. With a newly introduced algebraic class called a nice ring, it is possible to construct the quantum codes such that their alphabet sizes are not restricted to powers of a prime.\n\nSubsystem codes are quantum error correcting schemes unifying stabilizer codes, decoherence free subspaces and noiseless subsystems. We show a generalization of subsystem codes over nice rings. Furthermore, we prove that free subsystem codes over a finite chain ring cannot outperform those over a finite field. We also generalize entanglement-assisted quantum error correcting codes to nice rings. With the help of the entanglement, any classical code can be used to derive the corresponding quantum codes, even if such codes are not self-orthogonal. We prove that an R-module with antisymmetric bicharacter can be decomposed as an orthogonal direct sum of hyperbolic pairs using symplectic geometry over rings. So, we can find hyperbolic pairs and commuting generators generating the check matrix of the entanglement-assisted quantum code.\n\nFault-tolerant quantum computation has been also studied over a finite field. Transversal operations are the simplest way to implement fault-tolerant quantum gates. We derive transversal Clifford operations for CSS codes over nice rings, including Fourier transforms, SUM gates, and phase gates. Since transversal operations alone cannot provide a computationally universal set of gates, we add fault-tolerant implementations of doubly-controlled Z gates for triorthogonal stabilizer codes over nice rings.\n\nFinally, we investigate optimal key exchange protocols for unconditionally secure key distribution schemes. We prove how many rounds are needed for the key exchange between any pair of the group on star networks, linear-chain networks, and general networks.Quantum error correcting codes play an essential role in protecting quantum information from the noise and the decoherence. Most quantum codes have been constructed based on the Pauli basis indexed by a finite field. With a newly introduced algebraic class called a nice ring, it is possible to construct the quantum codes such that their alphabet sizes are not restricted to powers of a prime.\n\nSubsystem codes are quantum error correcting schemes unifying stabilizer codes, decoherence free subspaces and noiseless subsystems. We show a generalization of subsystem codes over nice rings. Furthermore, we prove that free subsystem codes over a finite chain ring cannot outperform those over a finite field. We also generalize entanglement-assisted quantum error correcting codes to nice rings. With the help of the entanglement, any classical code can be used to derive the corresponding quantum codes, even if such codes are not self-orthogonal. We prove that an R-module with antisymmetric bicharacter can be decomposed as an orthogonal direct sum of hyperbolic pairs using symplectic geometry over rings. So, we can find hyperbolic pairs and commuting generators generating the check matrix of the entanglement-assisted quantum code.\n\nFault-tolerant quantum computation has been also studied over a finite field. Transversal operations are the simplest way to implement fault-tolerant quantum gates. We derive transversal Clifford operations for CSS codes over nice rings, including Fourier transforms, SUM gates, and phase gates. Since transversal operations alone cannot provide a computationally universal set of gates, we add fault-tolerant implementations of doubly-controlled Z gates for triorthogonal stabilizer codes over nice rings.\n\nFinally, we investigate optimal key exchange protocols for unconditionally secure key distribution schemes. We prove how many rounds are needed for the key exchange between any pair of the group on star networks, linear-chain networks, and general networks.

Jun 29, 20170 citations

3-ε

and take stock of existing results on low-dimensional SCQMs. Our theory provides a unifying mathematical framework for understanding the fundamental capabilities and geometric implementations of quantum error correction and fault tolerance.