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Faulttolerant quantum computation by anyons
, 2003
"... A twodimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation ..."
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A twodimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation is faulttolerant by its physical nature.
Reliable quantum computers
 Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
, 1998
"... The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mist ..."
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The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mistakes occur during the recovery procedure. Furthermore, encoded quantum information can be processed without serious propagation of errors. Hence, an arbitrarily long quantum computation can be performed reliably, provided that the average probability of error per quantum gate is less than a certain critical value, the accuracy threshold. A quantum computer storing about 106 qubits, with a probability of error per quantum gate of order 106, would be a formidable factoring engine. Even a smaller lessaccurate quantum computer would be able to perform many useful tasks. This paper is based on a talk presented at the ITP Conference on Quantum Coherence
Fast versions of Shor’s quantum factoring algorithm
, 2002
"... We present fast and highly parallelized versions of Shor’s algorithm. With a sizable quantum computer it would then be possible to factor numbers with millions of digits. The main algorithm presented here uses FFTbased fast integer multiplication. The quick reader can just read the introduction and ..."
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We present fast and highly parallelized versions of Shor’s algorithm. With a sizable quantum computer it would then be possible to factor numbers with millions of digits. The main algorithm presented here uses FFTbased fast integer multiplication. The quick reader can just read the introduction and the “Results” section.
Threshold estimate for fault tolerant quantum computing
 LANL eprint quantph/9612028) 19 P γ ǫ (5n + 4)/k (10−14) (10−6) (10−6
"... It has been show [2, 3] that once elementary unitary operations (“gates”) in a quantum computer can be carried out with more than some threshold accuracy, then it is possible to carry out arbitrary precision operations on suitably encoded “computational ” qubits. In this paper I will take a more pra ..."
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It has been show [2, 3] that once elementary unitary operations (“gates”) in a quantum computer can be carried out with more than some threshold accuracy, then it is possible to carry out arbitrary precision operations on suitably encoded “computational ” qubits. In this paper I will take a more practical, engineerlike point of view. I will follow P.Shor [1] for fault tolerant error correction (FTEC) and the fault tolerant implementation of elementary operations on states encoded by the 7bit code. I will use (and try to justify) the most simple and natural error model. Computer simulation of an optimized version of Shors techniques gives an (astonishingly high) threshold of ǫ≈1/300 on the tolerable error probability. For comparison I also provide a very rough calculation by hand. 1
Quantum Dynamics of Cold Trapped Ions With Application to Quantum Computation
"... . The theory of interactions between lasers and cold trapped ions as it pertains to the design of CiracZoller quantum computers is discussed. The mean positions of the trapped ions, the eigenvalues and eigenmodes of the ions' oscillations, the magnitude of the Rabi frequencies for both allowe ..."
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Cited by 27 (3 self)
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. The theory of interactions between lasers and cold trapped ions as it pertains to the design of CiracZoller quantum computers is discussed. The mean positions of the trapped ions, the eigenvalues and eigenmodes of the ions' oscillations, the magnitude of the Rabi frequencies for both allowed and forbidden internal transitions of the ions, and the validity criterion for the required Hamiltonian are calculated. Energy level data for a variety of ion species are also presented. PACS: 32.80.Qk; 42.50.Vk; 89.80.+h A quantum computer is a device in which data can be stored in a network of quantum mechanical twolevel systems, such as spin1/2 particles or twolevel atoms. The quantum mechanical nature of such systems allows the possibility of a powerful new feature to be incorporated into data processing, namely, the capability of performing logical operations upon quantum mechanical superpositions of numbers. Thus in a conventional digital computer each data register is, throughout an...
Efficient faulttolerant quantum computing
 Nature
, 1999
"... Fault tolerant quantum computing methods which work with efficient quantum error correcting codes are discussed. Several new techniques are introduced to restrict accumulation of errors before or during the recovery. Classes of eligible quantum codes are obtained, and good candidates exhibited. This ..."
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Cited by 26 (4 self)
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Fault tolerant quantum computing methods which work with efficient quantum error correcting codes are discussed. Several new techniques are introduced to restrict accumulation of errors before or during the recovery. Classes of eligible quantum codes are obtained, and good candidates exhibited. This permits a new analysis of the permissible error rates and minimum overheads for robust quantum computing. It is found that, under the standard noise model of ubiquitous stochastic, uncorrelated errors, a quantum computer need be only an order of magnitude larger than the logical machine contained within it in order to be reliable. For example, a scaleup by a factor of 22, with gate error rate of order 10 −5, is sufficient to permit large quantum algorithms such as factorization of thousanddigit numbers.
Quantum algorithms for algebraic problems
, 2008
"... Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational pro ..."
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Cited by 24 (2 self)
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Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum
Decoherence limits to quantum computation using trapped ions
 Online preprint quantph/9610015), Proc.Roy.Soc.Lond. A453
, 1997
"... We investigate the problem of factorization of large numbers on a quantum computer which we imagine to be realized within a linear ion trap. We derive upper bounds on the size of the numbers that can be factorized on such a quantum computer. These upper bounds are independent of the power of the app ..."
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We investigate the problem of factorization of large numbers on a quantum computer which we imagine to be realized within a linear ion trap. We derive upper bounds on the size of the numbers that can be factorized on such a quantum computer. These upper bounds are independent of the power of the applied laser. We investigate two possible ways to implement qubits, in metastable optical transitions and in Zeeman sublevels of a stable ground state, and show that in both cases the numbers that can be factorized are not large enough to be of practical interest. We also investigate the effect of quantum error correction on our estimates and show that in realistic systems the impact of quantum error correction is much smaller than expected. Again no number of practical interest can be factorized.