Secret Sharing Schemes with a large number of players from Toric Varieties

A general theory for constructing linear secret sharing schemes over a finite field \Fq from toric varieties is introduced. The number of players can be as large as $(q-1)^r-1$ for $r\geq 1$. We present general methods for obtaining the reconstruction and privacy thresholds as well as conditions for multiplication on the associated secret sharing schemes. In particular we apply the method on certain toric surfaces. The main results are ideal linear secret sharing schemes where the number of players can be as large as $(q-1)^2-1$. We determine bounds for the reconstruction and privacy thresholds and conditions for strong multiplication using the cohomology and the intersection theory on toric surfaces.Comment: 15 pages, 4 figures. arXiv admin note: text overlap with arXiv:1203.454

Quantum Codes from Toric Surfaces

A theory for constructing quantum error correcting codes from Toric surfaces by the Calderbank-Shor-Steane method is presented. In particular we study the method on toric Hirzebruch surfaces. The results are obtained by constructing a dualizing differential form for the toric surface and by using the cohomology and the intersection theory of toric varieties. In earlier work the author developed methods to construct linear error correcting codes from toric varieties and derive the code parameters using the cohomology and the intersection theory on toric varieties. This method is generalized in section to construct linear codes suitable for constructing quantum codes by the Calderbank-Shor-Steane method. Essential for the theory is the existence and the application of a dualizing differential form on the toric surface. A.R. Calderbank, P.W. Shor and A.M. Steane produced stabilizer codes from linear codes containing their dual codes. These two constructions are merged to obtain results for toric surfaces. Similar merging has been done for algebraic curves with different methods by A. Ashikhmin, S. Litsyn and M.A. Tsfasman.Comment: IEEE copyrigh

Unfolding the color code

The topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. Here we show that the color code on a $d$-dimensional closed manifold is equivalent to multiple decoupled copies of the $d$-dimensional toric code up to local unitary transformations and adding or removing ancilla qubits. Our result not only generalizes the proven equivalence for $d=2$, but also provides an explicit recipe of how to decouple independent components of the color code, highlighting the importance of colorability in the construction of the code. Moreover, for the $d$-dimensional color code with $d+1$ boundaries of $d+1$ distinct colors, we find that the code is equivalent to multiple copies of the $d$-dimensional toric code which are attached along a $(d-1)$-dimensional boundary. In particular, for $d=2$, we show that the (triangular) color code with boundaries is equivalent to the (folded) toric code with boundaries. We also find that the $d$-dimensional toric code admits logical non-Pauli gates from the $d$-th level of the Clifford hierarchy, and thus saturates the bound by Bravyi and K\"{o}nig. In particular, we show that the $d$-qubit control-$Z$ logical gate can be fault-tolerantly implemented on the stack of $d$ copies of the toric code by a local unitary transformation.Comment: 46 pages, 15 figure

On the structure of generalized toric codes

Toric codes are obtained by evaluating rational functions of a nonsingular toric variety at the algebraic torus. One can extend toric codes to the so called generalized toric codes. This extension consists on evaluating elements of an arbitrary polynomial algebra at the algebraic torus instead of a linear combination of monomials whose exponents are rational points of a convex polytope. We study their multicyclic and metric structure, and we use them to express their dual and to estimate their minimum distance