H. Bombin

Publications (27)23.27 Total impact

• Source

  Available from: arxiv.org

  Article: Topological computation without braiding.

  H Bombin, M A Martin-Delgado
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  ABSTRACT: We show that universal quantum computation can be performed within the ground state of a topologically ordered quantum system, which is a naturally protected quantum memory. In particular, we show how this can be achieved using brane-net condensates in 3-colexes. The universal set of gates is implemented without selective addressing of physical qubits and, being fully topologically protected, it does not rely on quasiparticle excitations or their braiding.
  Physical Review Letters 05/2007; 98(16):160502. · 7.37 Impact Factor
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  Available from: arxiv.org

  Article: Optimal Resources for Topological 2D Stabilizer Codes: Comparative Study

  H. Bombin, M. A. Martin-Delgado
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  ABSTRACT: We study the resources needed to construct topological 2D stabilizer codes as a way to estimate in part their efficiency and this leads us to perform a comparative study of surface codes and color codes. This study clarifies the similarities and differences between these two types of stabilizer codes. We compute the error correcting rate $C:=n/d^2$ for surface codes $C_s$ and color codes $C_c$ in several instances. On the torus, typical values are $C_s=2$ and $C_c=3/2$, but we find that the optimal values are $C_s=1$ and $C_c=9/8$. For planar codes, a typical value is $C_s=2$, while we find that the optimal values are $C_s=1$ and $C_c=3/4$. In general, a color code encodes twice as much logical qubits as a surface code does. Comment: revtex, 6 pages, 7 figures
  03/2007;
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  Available from: arxiv.org

  Article: Topological quantum distillation.

  H Bombin, M A Martin-Delgado
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  ABSTRACT: We construct a class of topological quantum codes to perform quantum entanglement distillation. These codes implement the whole Clifford group of unitary operations in a fully topological manner and without selective addressing of qubits. This allows us to extend their application also to quantum teleportation, dense coding, and computation with magic states.
  Physical Review Letters 12/2006; 97(18):180501. · 7.37 Impact Factor
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  Available from: arxiv.org

  Article: Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates

  H. Bombin, M. A. Martin-Delgado
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  ABSTRACT: We construct an exactly solvable Hamiltonian acting on a 3-dimensional lattice of spin-$\frac 1 2$ systems that exhibits topological quantum order. The ground state is a string-net and a membrane-net condensate. Excitations appear in the form of quasiparticles and fluxes, as the boundaries of strings and membranes, respectively. The degeneracy of the ground state depends upon the homology of the 3-manifold. We generalize the system to $D\geq 4$, were different topological phases may occur. The whole construction is based on certain special complexes that we call colexes.
  08/2006;
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  Available from: arxiv.org

  Article: Homological Error Correction: Classical and Quantum Codes

  H. Bombin, M. A. Martin-Delgado
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  ABSTRACT: We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for non-orientable surfaces it is impossible to construct homological codes based on qudits of dimension $D>2$, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension $D$. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor's 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.
  06/2006;
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  Available from: arxiv.org

  Article: Topological Quantum Error Correction with Optimal Encoding Rate

  H. Bombin, M. A. Martin-Delgado
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  ABSTRACT: We prove the existence of topological quantum error correcting codes with encoding rates $k/n$ asymptotically approaching the maximum possible value. Explicit constructions of these topological codes are presented using surfaces of arbitrary genus. We find a class of regular toric codes that are optimal. For physical implementations, we present planar topological codes.
  03/2006;
• Source

  Available from: arxiv.org

  Article: Entanglement Distillation Protocols and Number Theory

  H. Bombin, M. A. Martin-Delgado
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  ABSTRACT: We show that the analysis of entanglement distillation protocols for qudits of arbitrary dimension $D$ benefits from applying basic concepts from number theory, since the set $\zdn$ associated to Bell diagonal states is a module rather than a vector space. We find that a partition of $\zdn$ into divisor classes characterizes the invariant properties of mixed Bell diagonal states under local permutations. We construct a very general class of recursion protocols by means of unitary operations implementing these local permutations. We study these distillation protocols depending on whether we use twirling operations in the intermediate steps or not, and we study them both analitically and numerically with Monte Carlo methods. In the absence of twirling operations, we construct extensions of the quantum privacy algorithms valid for secure communications with qudits of any dimension $D$. When $D$ is a prime number, we show that distillation protocols are optimal both qualitatively and quantitatively. Comment: REVTEX4 file, 7 color figures, 2 tables
  03/2005;