Kerdock Codes Determine Unitary 2-Designs

The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length $N = 2^m$ over $\mathbb{Z}_4$. We show that exponentiating these $\mathbb{Z}_4$-valued codewords by $\imath \triangleq \sqrt{-1}$ produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organize the stabilizer states to form $N+1$ mutually unbiased bases and prove that automorphisms of the Kerdock code permute their corresponding MCS, thereby forming a subgroup of the Clifford group. When represented as symplectic matrices, this subgroup is isomorphic to the projective special linear group PSL($2,N$). We show that this automorphism group acts transitively on the Pauli matrices, which implies that the ensemble is Pauli mixing and hence forms a unitary $2$-design. The Kerdock design described here was originally discovered by Cleve et al. (arXiv:1501.04592), but the connection to classical codes is new which simplifies its description and translation to circuits significantly. Sampling from the design is straightforward, the translation to circuits uses only Clifford gates, and the process does not require ancillary qubits. Finally, we also develop algorithms for optimizing the synthesis of unitary $2$-designs on encoded qubits, i.e., to construct logical unitary $2$-designs. Software implementations are available at https://github.com/nrenga/symplectic-arxiv18a, which we use to provide empirical gate complexities for up to $16$ qubits.Comment: 16 pages double-column, 4 figures, and some circuits. Accepted to 2019 Intl. Symp. Inf. Theory (ISIT), and PDF of the 5-page ISIT version is included in the arXiv packag

Kerdock Codes Determine Unitary 2-Designs

The binary non-linear Kerdock codes are Gray images of ℤ_4-linear Kerdock codes of length N =2^m . We show that exponentiating ı=−√-1 by these ℤ_4-valued codewords produces stabilizer states, which are the common eigenvectors of maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the proof of the classical weight distribution of Kerdock codes. Next, we partition stabilizer states into N +1 mutually unbiased bases and prove that automorphisms of the Kerdock code permute the associated MCS. This automorphism group, represented as symplectic matrices, is isomorphic to the projective special linear group PSL(2,N) and forms a unitary 2-design. The design described here was originally discovered by Cleve et al. (2016), but the connection to classical codes is new. This significantly simplifies the description of the design and its translation to circuits

Particle-Based Sampling and Meshing of Surfaces in Multimaterial Volumes

Methods that faithfully and robustly capture the geometry of complex material interfaces in labeled volume data are important for generating realistic and accurate visualizations and simulations of real-world objects. The generation of such multimaterial models from measured data poses two unique challenges: first, the surfaces must be well-sampled with regular, efficient tessellations that are consistent across material boundaries; and second, the resulting meshes must respect the nonmanifold geometry of the multimaterial interfaces. This paper proposes a strategy for sampling and meshing multimaterial volumes using dynamic particle systems, including a novel, differentiable representation of the material junctions that allows the particle system to explicitly sample corners, edges, and surfaces of material intersections. The distributions of particles are controlled by fundamental sampling constraints, allowing Delaunay-based meshing algorithms to reliably extract watertight meshes of consistently high-quality.Engineering and Applied Science

Reinforcement learning in populations of spiking neurons

Population coding is widely regarded as a key mechanism for achieving reliable behavioral responses in the face of neuronal variability. But in standard reinforcement learning a flip-side becomes apparent. Learning slows down with increasing population size since the global reinforcement becomes less and less related to the performance of any single neuron. We show that, in contrast, learning speeds up with increasing population size if feedback about the populationresponse modulates synaptic plasticity in addition to global reinforcement. The two feedback signals (reinforcement and population-response signal) can be encoded by ambient neurotransmitter concentrations which vary slowly, yielding a fully online plasticity rule where the learning of a stimulus is interleaved with the processing of the subsequent one. The assumption of a single additional feedback mechanism therefore reconciles biological plausibility with efficient learning

Kerdock Codes Determine Unitary 2-Designs

The binary non-linear Kerdock codes are Gray images of ℤ_4-linear Kerdock codes of length N =2^m . We show that exponentiating ı=−√-1 by these ℤ_4-valued codewords produces stabilizer states, which are the common eigenvectors of maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the proof of the classical weight distribution of Kerdock codes. Next, we partition stabilizer states into N +1 mutually unbiased bases and prove that automorphisms of the Kerdock code permute the associated MCS. This automorphism group, represented as symplectic matrices, is isomorphic to the projective special linear group PSL(2,N) and forms a unitary 2-design. The design described here was originally discovered by Cleve et al. (2016), but the connection to classical codes is new. This significantly simplifies the description of the design and its translation to circuits

Gentle Perturbations of the Free Bose Gas I

It is demonstrated that the thermal structure of the noncritical free Bose Gas is completely described by certain periodic generalized Gaussian stochastic process or equivalently by certain periodic generalized Gaussian random field. Elementary properties of this Gaussian stochastic thermal structure have been established. Gentle perturbations of several types of the free thermal stochastic structure are studied. In particular new models of non-Gaussian thermal structures have been constructed and a new functional integral representation of the corresponding euclidean-time Green functions have been obtained rigorously.Comment: 51 pages, LaTeX fil