## New limits on fault-tolerant quantum computation (2006)

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Venue: | In Proceedings of 47th IEEE FOCS |

Citations: | 7 - 2 self |

### BibTeX

@INPROCEEDINGS{Buhrman06newlimits,

author = {Harry Buhrman and Monique Laurent and Noah Linden and Alexander Schrijver},

title = {New limits on fault-tolerant quantum computation},

booktitle = {In Proceedings of 47th IEEE FOCS},

year = {2006},

pages = {411--419}

}

### OpenURL

### Abstract

### Citations

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Numerical Optimization
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(Show Context)
Citation Context ...zer (a, b) to the program (16). Then, the Karush-Kuhn-Tucker conditions have to be satisfied, since the gradient vectors {rgi(a, b) | i = 1, 2, 3} are linearly independent; see, e.g., Theorem 12.1 in =-=[13]-=-. (Here, the gradient vector rgi(a, b) consists of the partial derivatives with respect to the six variables a1, . . . , b3.) That is, there exist scalars y"1, y"2, y"3 for which rf (a, b) + X i=1,2,3... |

1368 |
Quantum Computation and Quantum Information
- Nielsen, Chuang
- 2000
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Citation Context ...epresentation In our further analysis it will be convenient to use the Bloch-sphere representation of 1-qubit states and 1-qubit operations, which we review now (see for example Chapters 4.2 and 8 in =-=[12]-=-). For r 2 R3 define r * ? = rxX + ryY + rzZ, where ? = (X, Y, Z) is the vector of Pauli-matrices X = t, 0 11 0 u"" Y = t, 0 -ii 0 u"" Z = t, 1 00 -1 u"" . (2) Then, all 1-qubit density matrices ! can... |

262 | Good quantum error-correcting codes exist
- Calderbank, Shor
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Citation Context ... IRC in Quantum Information Processing. 1sresults have progressively improved on the first fault-tolerant schemes, proving better and better bounds on the noise tolerable in quantum computation (e.g. =-=[4, 3]-=-). Recent results suggest that fault-tolerant quantum computation is possible with gates that have as much as 3% of depolarizing errors [9]. In this paper we will concentrate on the opposite task of p... |

204 | Fault-tolerant quantum computation
- Shor
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Citation Context ...tes major difficulties in storing and operating on quantum states reliably. A key advance was the realization that quantum error correcting codes [16, 18] exist and fault-tolerant quantum computation =-=[15]-=- is possible for a number of apparently reasonable error models. Subsequent *Supported by the EU project RESQ IST-2001-37559 and the NWO vici project 2004-2009.# Supported in part by Canada's NSERC, C... |

192 | Fault-Tolerant Quantum Computation With Constant Error Rate,” ArXiv e-prints
- Aharonov, Ben-Or
- 1999
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Citation Context ... IRC in Quantum Information Processing. 1sresults have progressively improved on the first fault-tolerant schemes, proving better and better bounds on the noise tolerable in quantum computation (e.g. =-=[4, 3]-=-). Recent results suggest that fault-tolerant quantum computation is possible with gates that have as much as 3% of depolarizing errors [9]. In this paper we will concentrate on the opposite task of p... |

89 | Quantum entanglement and the communication complexity of the inner product function
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Citation Context ...hese bits, Alice and Bob obtain the correct representation for qubits of C as in Lemma 1. Since the inner product function has communication complexity \Omega (n) even in the presence of entanglement =-=[6]-=- we see that Theorem 1 is also true for quantum inputs. Note however, that now Bob can no longer compute the result of the circuit with certainty, because he does not even know his input state. The co... |

84 |
Scheme for reducing decoherence in quantum memory
- Shor
- 1995
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Citation Context ... fundamental problem is to cope with noise, which creates major difficulties in storing and operating on quantum states reliably. A key advance was the realization that quantum error correcting codes =-=[16, 18]-=- exist and fault-tolerant quantum computation [15] is possible for a number of apparently reasonable error models. Subsequent *Supported by the EU project RESQ IST-2001-37559 and the NWO vici project ... |

51 | Fast parallel circuits for the quantum Fourier transform
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- 2004
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Citation Context ... qubit per time step) will be overwhelmed by the noise and produce a statistically meaningless outcome. (Thus the complexity theoretic assumption BQP 6= QN C1 is invoked.) We note that it is shown in =-=[5]-=- that in fact log-depth quantum circuits can perform interesting feats, including efficient integer factorization (if combined by classical polynomial-time pre- and post-processing). Our error model i... |

45 | Improved simulation of stabilizer circuits
- Aaronson, Gottesman
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Citation Context ..., our approach will yield stronger bounds on the tolerable level. Our results also yield a simple proof that not all classical functions can be computed using Clifford gates (complementing results in =-=[1]-=-). In particular, in Corollary 1, we show that a boolean function which can be computed by Clifford circuits can be written as the parity of a subset of input bits. The main idea of our approach is as... |

35 |
Quantum computing with realistically noisy devices,” Nature 434
- Knill
- 2005
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Citation Context ...ounds on the noise tolerable in quantum computation (e.g. [4, 3]). Recent results suggest that fault-tolerant quantum computation is possible with gates that have as much as 3% of depolarizing errors =-=[9]-=-. In this paper we will concentrate on the opposite task of proving that, for certain noise levels, quantum computation is impossible. Our main result is as follows: Let Clifford be the set of all (no... |

23 | The invariants of the Clifford groups
- Nebe, Rains, et al.
(Show Context)
Citation Context ...p2 t, 1 11 -1 u"" S = t, 1 00 i u"" (1) It is known that Clifford together with any other one-qubit gate, not generated by the gates in Clifford, form a universal set of gates for quantum computation =-=[17, 11]-=-. We show however, that such additional one-qubit gates should not be too noisy. More precisely, let Clifford* be Clifford augmented with arbitrary one-qubit gates with depolarizing error at least ^t,... |

22 | Polynomial simulations of decohered quantum computers
- Aharonov, Ben-Or
- 1996
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Citation Context ...coherence rate were obtained by showing that quantum computers with faulty gates can be simulated efficiently on a classical computer. The first to prove one of these results were Aharonov and Ben-Or =-=[2]-=-, with the value 97% for the noise. Later Harrow and Nielsen [8] showed that if 74% of depolarizing noise is applied to each output qubit of each gate, then (faulty) two-qubit gates cannot produce ent... |

12 |
An upper bound on the threshold quantum decoherence rate
- Razborov
(Show Context)
Citation Context ... 2005 Abstract We show that quantum circuits cannot be made fault-tolerant against a depolarizing noise level of t, = (6 - 2p2)/7 ij 45%, thereby improving on a previous bound of 1/2 (due to Razborov =-=[14]-=-). Our precise quantum circuit model enables perfect gates from the Clifford group (CNOT, Hadamard, S, X, Y , Z) and arbitrary additional one-qubit gates that are subject to depolarizing noise t,. We ... |

7 |
How robust is a quantum gate in the presence of noise? Phys
- Harrow, Nielsen
(Show Context)
Citation Context ...ith faulty gates can be simulated efficiently on a classical computer. The first to prove one of these results were Aharonov and Ben-Or [2], with the value 97% for the noise. Later Harrow and Nielsen =-=[8]-=- showed that if 74% of depolarizing noise is applied to each output qubit of each gate, then (faulty) two-qubit gates cannot produce entanglement. They concluded that circuits containing only one- and... |

4 | Quantum universality from Magic States Distillation applied to CSS codes - Reichardt |

2 |
Software cdd+. available from http://www.ifor.math.ethz.ch∼fukuda
- Fukuda
(Show Context)
Citation Context ...2 R3*3 | hF, Si <= 1 for all F 2 F l/ , (10) where F := l^C1BC2|C1, C2 2 C, B 2 {B1, BT1 , B2}l/ , (11) B1 := 0@ 1 0 0 1 0 0 1 0 0 1A , B2 := 0@ 1 -1 0 1 1 0 0 0 -1 1A . One can use the software from =-=[7]-=- for computing the facet description (10); we will give a direct proof in the Appendix. In view of (10), our claim (9) is equivalent to (1 - ^t,)hF, Si <= 1 for all S 2 SO(3), F 2 F . (12) Let F 2 F o... |

2 |
talk at mathematical sciences research institute
- Solovay
(Show Context)
Citation Context ...p2 t, 1 11 -1 u"" S = t, 1 00 i u"" (1) It is known that Clifford together with any other one-qubit gate, not generated by the gates in Clifford, form a universal set of gates for quantum computation =-=[17, 11]-=-. We show however, that such additional one-qubit gates should not be too noisy. More precisely, let Clifford* be Clifford augmented with arbitrary one-qubit gates with depolarizing error at least ^t,... |

2 |
Classical simulatability, entanglement breaking, and quantum computation thresholds
- Virmani, Huelga, et al.
- 2004
(Show Context)
Citation Context ... circuits containing only one- and two-qubit gates with depolarizing noise at least 74% can be simulated efficiently on a classical computer. Another result along these lines is due to Virmani et al. =-=[19]-=-, however they use a different noise model (dephasing noise) that is not quantitatively comparable to depolarizing noise. Note that all these results do not exclude the possibility that quantum circui... |