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

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. Subsystem 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. Fault-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. Finally, 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

The model theory of Commutative Near Vector Spaces

In this paper we study near vector spaces over a commutative $F$ from a model theoretic point of view. In this context we show regular near vector spaces are in fact vector spaces. We find that near vector spaces are not first order axiomatisable, but that finite block near vector spaces are. In the latter case we establish quantifier elimination, and that the theory is controlled by which elements of the pointwise additive closure of $F$ are automorphisms of the near vector space

Narain CFTs from nonbinary stabilizer codes

We generalize the construction of Narain conformal field theories (CFTs) from qudit stabilizer codes to the construction from quantum stabilizer codes over the finite field of prime power order ($\mathbb{F}_{p^m}$ with $p$ prime and $m\geq 1$) or over the ring $\mathbb{Z}_k$ with $k>1$. Our construction results in rational CFTs, which cover a larger set of points in the moduli space of Narain CFTs than the previous one. We also propose a correspondence between a quantum stabilizer code with non-zero logical qubits and a finite set of Narain CFTs. We illustrate the correspondence with well-known stabilizer codes.Comment: 38 page

Near-linear algebra

This paper establishes some fundamental results for near-vector spaces toward extending classical linear algebra to near-linear algebra. In the present paper, we finalize the algebraic proof that any non-empty subspace stable under addition and scalar multiplication is a subspace. The first author established this result over division rings in [6], and we propose a direct generalization over any near-field. We demonstrate that any quotient of a near-vector space by a subspace is a near-vector space and the first isomorphism theorem for near-vector spaces. In doing this, we obtain fundamental descriptions of the span. Defining linear independence outside the quasi-kernel, we prove that near-vector spaces are characterized in terms of the existence of a scalar basis, and we obtain a new important notion of basis

Families of Cyclic Codes over Finite Chain Rings

A major difficulty in quantum computation and communication is preventing and correcting errors in the quantum bits. Most of the research in this area has focused on stabilizer codes derived from self-orthogonal cyclic error-correcting codes over finite fields. Our goal is to develop a similar theory for self-orthogonal cyclic codes over the class of finite chain rings which have been proven to also produce stabilizer codes. We also will discuss these restrictions on families of cyclic codes, including, but not limited to quadratic residue codes and Bose-Chaudhuri-Hocquenghem codes. Finally, we will extend the concepts of weight enumerators to the class of Frobenius rings and use them to derive bounds for our codes