### Citations

226 | Fault-tolerant quantum computation by anyons
- Kitaev
- 2003
(Show Context)
Citation Context ...separately on suitable subsystems, in the simplest case independently on each qubit. Unfortunately, no code admits a universal transversal set of gates [2]. Topological quantum error correcting codes =-=[3]-=- introduce a richer notion of locality by considering the spatial location of the physical qubits, which are assumed to be arranged on a lattice. They come in families parametrized with a lattice size... |

30 |
Threshold accuracy for quantum computation. quantph /9610011
- Knill, Laflamme, et al.
- 1996
(Show Context)
Citation Context ... also very interesting: CNot and R3 gates are transversal, so that the Hadamard gate suffices to complete a universal gate set. The latter in turn can be recovered if |+〉 encoded states are available =-=[12]-=-, but these can be easily created since color codes are CSS codes. From a practical perspective, however, 3D color codes pose two difficulties. The main one is that the measurements for error-correcti... |

10 | Restrictions on transversal encoded quantum gate sets
- Eastin, Knill
- 2009
(Show Context)
Citation Context ...erators that transform encoded states by acting separately on suitable subsystems, in the simplest case independently on each qubit. Unfortunately, no code admits a universal transversal set of gates =-=[2]-=-. Topological quantum error correcting codes [3] introduce a richer notion of locality by considering the spatial location of the physical qubits, which are assumed to be arranged on a lattice. They c... |

9 |
Topological Quantum Distillation,” Phys
- Bombin, Martin-Delgado
- 2006
(Show Context)
Citation Context ...des that saturate these bounds can be constructed for every D, in the sense that they admit at least a transversal gate in PD (not belonging to PD−1). In particular, the result applies to color codes =-=[7, 8, 9, 10]-=-, where the single-qubit gate RD := ( 1 0 0 e2pii/2 D ) is transversal (in the more strict non-topological sense). 1 It was already known that color codes could have this property as long as lattices ... |

7 | Vacuum degeneracy of chiral spin states in compactified space,” Phys
- Wen
- 1989
(Show Context)
Citation Context ...interpretation. Topological stabilizer codes can be regarded as ground states of gapped local Hamiltonians: they give examples of a form of quantum order in condensed matter, namely topological order =-=[11]-=-. Different topological orders are characterized by the topological interactions of their excitations. Given the (topological) notion of transversal gate, it is natural to expect that each kind of top... |

6 |
Universal topological phase of two-dimensional stabilizer codes
- Bombin, Duclos-Cianci, et al.
- 2012
(Show Context)
Citation Context ...the system size. Rather than sticking to the above definition of transversality, for topological codes it is natural to consider instead quantum circuits of fixed depth with geometrically local gates =-=[4, 5]-=-. Remarkably, it has been recently shown that spatial dimension imposes constraints on the transversal gates that are allowed [5]. In particular, this is true in the case of topological stabilizer cod... |

5 |
Topological Computation without Braiding,” Phys
- Bombin, Martin-Delgado
- 2007
(Show Context)
Citation Context ...des that saturate these bounds can be constructed for every D, in the sense that they admit at least a transversal gate in PD (not belonging to PD−1). In particular, the result applies to color codes =-=[7, 8, 9, 10]-=-, where the single-qubit gate RD := ( 1 0 0 e2pii/2 D ) is transversal (in the more strict non-topological sense). 1 It was already known that color codes could have this property as long as lattices ... |

4 |
Classification of topologically protected gates for local stabilizer codes
- Bravyi, Koenig
- 2012
(Show Context)
Citation Context ...the system size. Rather than sticking to the above definition of transversality, for topological codes it is natural to consider instead quantum circuits of fixed depth with geometrically local gates =-=[4, 5]-=-. Remarkably, it has been recently shown that spatial dimension imposes constraints on the transversal gates that are allowed [5]. In particular, this is true in the case of topological stabilizer cod... |

4 |
Exact topological quantum order in D= 3 and beyond: Branyons and brane-net condensates,” Phys
- Bombin, Martin-Delgado
- 2007
(Show Context)
Citation Context ...des that saturate these bounds can be constructed for every D, in the sense that they admit at least a transversal gate in PD (not belonging to PD−1). In particular, the result applies to color codes =-=[7, 8, 9, 10]-=-, where the single-qubit gate RD := ( 1 0 0 e2pii/2 D ) is transversal (in the more strict non-topological sense). 1 It was already known that color codes could have this property as long as lattices ... |

4 |
Universal fault-tolerant quantum computation with only transversal gates and error correction
- Paetznick, Reichardt
(Show Context)
Citation Context ...xing the gauge degrees of freedom (gauge → conventional) or doing nothing (conventional → gauge). Thus a universal set of transversal gates becomes available! All thanks to the gauge fixing technique =-=[14]-=-, which elegantly dodges the no-go theorem on transversal gates. As anticipated above, the problem of measuring operators involving dozens of qubits also disappears. The reason is that instead of meas... |

3 |
Self-correcting quantum computers
- Bombin, Chhajlany, et al.
(Show Context)
Citation Context |

2 |
Gauge color codes.” eprint arXiv:1311.0879
- Bombin
(Show Context)
Citation Context ...ar, this is true in the case of topological stabilizer codes, where it has been found that transversal gates on D-dimensional codes have to belong to certain sets PD. A first main result of this work =-=[6]-=- is that codes that saturate these bounds can be constructed for every D, in the sense that they admit at least a transversal gate in PD (not belonging to PD−1). In particular, the result applies to c... |

2 |
Single-shot fault-tolerant quantum error correction.” eprint arXiv:1404.5504
- Bombin
(Show Context)
Citation Context ...atively one can measure gauge operators that only involve 4 or 6 qubits each. As a byproduct, such measurements give redundant information that can be put to use in the presence of measurement errors =-=[15]-=-. Summary Local codes receive increasing attention because of their potential practical applications. Among them 3D gauge color code are quite singular: they allow a universal transversal set of gates... |