We develop a comprehensive approach of stabilizer codes and provide a scheme generating equientangled basis interpolating between the product basis and maximally entangled basis. The key ingredient is the theory of phase states for finite-dimensional systems (qudits). In this respect, we derive entangled phase states for a multiqudit system whose dynamics is governed by a two-qudit interaction Hamiltonian. We construct the stabilizer codes for this family of entangled phase states. The stabilizer phase states are defined as the common eigenvectors of the stabilizer group generators which are explicitly specified. Furthermore, we construct equally entangled bases from bipartite as well as multipartite entangled qudit phase states.
We develop a comprehensive approach of stabilizer codes and provide a scheme generating equientangled basis interpolating between the product basis and maximally entangled basis. The key ingredient is the theory of phase states for finite-dimensional systems (qudits). In this respect, we derive entangled phase states for a multiqudit system whose dynamics is governed by a two-qudit interaction Hamiltonian. We construct the stabilizer codes for this family of entangled phase states. The stabilizer phase states are defined as the common eigenvectors of the stabilizer group generators which are explicitly specified. Furthermore, we construct equally entangled bases from bipartite as well as multipartite entangled qudit phase states.
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.
For the space of two identical systems of arbitrary dimensions, we introduce a continuous family of bases with the following properties: (i) the bases are orthonormal, (ii) in each basis, all the states have the same values of entanglement, and (iii) they continuously interpolate between the product basis and the maximally entangled basis. The states thus constructed may find applications in many areas related to the quantum information science including quantum cryptography, optimal Bell tests, and the investigation of the enhancement of channel capacity due to entanglement.
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.
We construct stabilizer states and error-correcting codes on hybrid discrete- and continuous-variable systems, generalizing the Gottesman-Kitaev-Preskill (GKP) lattice formalism. Our framework embeds qudit phase space into a hybrid phase space parameterized by continuous variables, with unit cells that grow with qudit dimension—enabling simultaneous measurement of an arbitrarily large range of noncommuting displacements. Simple hybrid states arise from applying a conditional displacement to a GKP state and a Pauli eigenstate, or by encoding qudits of a stabilizer state into a GKP code; their oscillator-qudit entanglement is non-Gaussian, distinguishing them as a resource beyond tensor products of stabilizer states. Simple hybrid codes are subsystem GKP codes whose gauge factor is entangled with a qudit. Numerical experiments suggest that they can outperform GKP codes against physical noise, with decoders tunable to either qudit or oscillator errors. We also relate stabilizer codes to noncommutative tori, showing that a general construction yields multimode multiqudit GKP extensions. We compute the codes’ logical dimension and operators via Morita equivalence.
The stabilizer formalism allows the efficient description of a sizeable class of pure as well as mixed quantum states of n -qubit systems. That same formalism has important applications in the field of quantum error correcting codes, where mixed stabilizer states correspond to projectors on subspaces associated with stabilizer codes. In this paper, we derive efficient reduction procedures to obtain various useful normal forms for stabilizer states. We explicitly prove that these procedures will always converge to the correct result and that these procedures are efficient in that they only require a polynomial number of operations on the generators of the stabilizers. On one hand, we obtain two single-party normal forms. The first, the row-reduced echelon form, is obtained using only permutations and multiplications of generators. This form is useful to calculate partial traces of stabilizer states. The second is the fully reduced form, where the reduction procedure invokes single-qubit operations and CNOT operations as well. This normal form allows for the efficient calculation of the overlap between two stabilizer states, as well as of the Uhlmann fidelity between them, and their Bures distance. On the other hand, we also find a reduction procedure of bipartite stabilizer states, where the operations involved are restricted to be local ones. The two-party normal form thus obtained lays bare a very simple bipartite entanglement structure of stabilizer states. To wit, we prove that every bipartite mixed stabilizer state is locally equivalent to a direct product of a number of maximally entangled states and, potentially, a separable state. As a consequence, using this normal form we can efficiently calculate every reasonable bipartite entanglement measure of mixed stabilizer states.
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.