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.
The protection of quantum information in systems leveraging more than two basis states (qudits) is an ongoing challenge. A tool for transforming the plethora of qubit stabilizer codes into qudit stabilizer codes is the local-dimension-invariant (LDI) extension to the stabilizer formalism. A drawback of the LDI method is that generally in order to ensure the distance of the resultant code is at least preserved the newly selected local-dimension will need to be any prime bigger than a large cutoff value. In this work we consider the repercussions of this large cutoff value, then by focusing on quantum low-density parity-check (qLDPC) codes we are able to bring down this cutoff value. Notably, we are able to reduce it in some cases to the initial local-dimension and for arbitrary CSS qubit-derived codes to singly exponential in d with no dependence on the number of logicals k. This reduces the size of the caveat in LDI forms in their distance preservation condition.
One hurdle to performing reliable quantum computations is overcoming noise. One possibility is to reduce the number of particles needing to be protected from noise and instead use systems with more states, so-called qudit quantum computers. In this paper we show that codes for these systems can be derived from already known codes, and, in particular, that degenerate stabilizer codes can have their distance also promised upon sufficiently large local dimension as well as an alternative bound on the local dimension required to preserve the distance of local-dimension-invariant codes, which is a result which could prove to be useful for error-corrected qudit quantum computers.
We develop a theory of stabilizer codes whose stabilizer groups consist of commuting qudit Paulis, oscillator displacements, and/or planar-rotor displacements. We consider codes whose stabilizer groups form generalized lattices in quantum phase space, a condition that guarantees a finite logical dimension. We construct oscillator-rotor, rotor-qudit, and oscillator-rotor-qubit codes that cannot be decomposed into separate subsystems by generalized Clifford transformations. We also introduce an oscillator-qubit code arising from a partially lifted Golay code, where Construction A is applied to half of the Golay code coordinates. We derive analytical forms of logical operators from the symplectic dual lattice, determine Clifford-Gaussian gates from lattice symmetries, show that the code dimension equals the covolume of the stabilizer lattice, and organize the syndrome subspaces into a vector bundle. Our technique builds on earlier non-commutative geometric results by Rieffel, but it can often be interpreted simply as concatenating a given stabilizer code with a Gottesman-Kitaev-Preskill (GKP) code, applying conventional lattice-theoretic results, and unconcatenating. We formulate distances, optimal decoding, and quantum weight enumerators, recovering the Pauli, qudit, and GKP versions together with known MacWilliams identities. We construct finite-energy codewords whose encoding is an isometry up to an error exponentially small in the inverse damping parameter.
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.
Abstract We address the task of verifying whether a quantum computer, designed to be protected by a specific stabilizer code, correctly encodes the corresponding logical qubits. To achieve this, we develop a general framework for subspace verification and explore several practically significant stabilizer code subspaces. First, we present two efficient verification strategies for general stabilizer code subspaces, using measurements of their stabilizer generators and stabilizer groups, respectively. Then, building on the observation that certain tests can be conducted in parallel due to specific structural properties of the subspace, we propose a coloring strategy tailored to graph code subspaces and an XZ strategy for Calderbank–Shor–Steane code subspaces. Compared to stabilizer-based strategies, these new strategies require significantly fewer measurement settings and consume fewer state copies, achieving near-global optimality. Notably, all the strategies employ a limited number of Pauli measurements, are non-adaptive, and work for mixed states, enabling efficient experimental certification of whether quantum states remain within the target code subspace in noisy quantum computers. This work presents the first systematic study of efficient verification of stabilizer code subspaces with local measurements.
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 a 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.