Family Of Quantum Codes Research Articles

Family of Quantum Codes with Exotic Transversal Gates.

Recently, an algorithm has been constructed that shows that the binary icosahedral group 2I together with a T-like gate forms the most efficient single-qubit universal gate set. To carry out the algorithm fault tolerantly requires a code that implements 2I transversally. However, no such code has ever been demonstrated in the literature. We fill this void by constructing a family of distance d=3 codes that all implement 2I transversally. A surprising feature of this family is that the codes can be deduced entirely from symmetry considerations that only 2I affords.

Some families of optimal quantum codes derived from constacyclic codes

ABSTRACTQuantum maximum-distance-separable (MDS) codes form an important family of quantum codes. Quantum MDS codes are optimal, which can attain the quantum Singleton bound. In this paper, using some classes of constacyclic codes and Hermitian constructions, we propose the constructions of optimal quantum convolutional codes and optimal asymmetric quantum codes in the sense of the quantum Singleton bound. Compared with the codes obtained in the literature, the parameters of the constructed quantum codes are new as well as the distance of optimal asymmetric quantum codes constructed in this paper is larger.

Tensor networks and quantum error correction.

We establish several relations between quantum error correction (QEC) and tensor network (TN) methods of quantum many-body physics. We exhibit correspondences between well-known families of QEC codes and TNs, and demonstrate a formal equivalence between decoding a QEC code and contracting a TN. We build on this equivalence to propose a new family of quantum codes and decoding algorithms that generalize and improve upon quantum polar codes and successive cancellation decoding in a natural way.

On the Construction of Nonbinary Quantum BCH Codes

Four quantum code constructions generating several new families of good nonbinary quantum nonprimitive nonnarrow-sense, Bose-Chaudhuri-Hocquenghem codes, are presented in this paper. The first two are based on Calderbank-Shor-Steane (CSS) construction derived from two nonprimitive Bose-Chaudhuri-Hocquenghem codes. The third one is based on Steane's enlargement of nonbinary CSS codes applied to suitable subfamilies of nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. The fourth construction is derived from suitable subfamilies of Hermitian dual-containing nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. These constructions generate new families of quantum codes whose parameters are better than the ones available in the literature.

Quantum codes from Hadamard matrices

From each q × q unitary matrix Φ, we construct a family of quantum codes , t ≥ 1, for q-state systems which encode (2t + 1)2 q-states into one q-state. We show that such codes are capable of correcting the errors of weight up to t if and only if Φ is a complex Hadamard matrix.

On binary constructions of quantum codes

We improve estimates on the parameters of quantum codes obtained by Steane's (see ibid., vol.45, no.7, p.2492-5, 1999) construction from binary codes. This yields several new families of quantum codes.