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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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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.
An introduction to quantum error correction and faulttolerant quantum computation
, 2009
"... Abstract. Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. ..."
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Abstract. Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers.
Noise Threshold for a FaultTolerant TwoDimensional Lattice Architecture
 Quant. Inf. Comp
"... We consider a model of quantum computation in which the set of operations is limited to nearestneighbor interactions on a 2D lattice. We model movement of qubits with noisy SWAP operations. For this architecture we design a faulttolerant coding scheme using the concatenated [[7, 1, 3]] Steane code ..."
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Cited by 24 (2 self)
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We consider a model of quantum computation in which the set of operations is limited to nearestneighbor interactions on a 2D lattice. We model movement of qubits with noisy SWAP operations. For this architecture we design a faulttolerant coding scheme using the concatenated [[7, 1, 3]] Steane code. Our scheme is potentially applicable to iontrap and solidstate quantum technologies. We calculate a lower bound on the noise threshold for our local model using a detailed failure probability analysis. We obtain a threshold of 1.85×10 −5 for the local setting, where memory error rates are onetenth of the failure rates of gates, measurement, and preparation steps. For the analogous nonlocal setting, we obtain a noise threshold of 3.61×10 −5. Our results thus show that the additional SWAP operations required to move qubits in the local model affect the noise threshold only moderately.
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
Sophisticated Quantum Search Without Entanglement
, 2000
"... Although entanglement is widely considered to be necessary for quantum algorithms to improve on classical ones, Lloyd has observed recently that Grover's quantum search algorithm can be implemented without entanglement, by replacing multiple particles with a single particle having exponentially ..."
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Cited by 23 (6 self)
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Although entanglement is widely considered to be necessary for quantum algorithms to improve on classical ones, Lloyd has observed recently that Grover's quantum search algorithm can be implemented without entanglement, by replacing multiple particles with a single particle having exponentially many states. We explain that this maneuver removes entanglement from any quantum algorithm. But all physical resources must be accounted for to quantify algorithm complexity, and this scheme typically incurs exponential costs in some other resource(s). In particular, we demonstrate that a recent experimental realization requires exponentially increasing precision. There is, however, a quantum algorithm which searches a `sophisticated' database (not unlike a Web search engine) with a single query, but which we show does not require entanglement even for multiparticle implementations. 1999 Physics and Astronomy Classification Scheme: 03.67.Lx, 32.80.Rm. 2000 American Mathematical Society Subject ...
An introduction to quantum complexity theory
 Collected Papers on Quantum Computation and Quantum Information Theory
, 2000
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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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Cited by 22 (3 self)
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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.
Bounds for Kac’s Master Equation
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
"... Mark Kac considered a Markov Chain on the n–sphere based on random rotations in randomly chosen coordinate planes. This same walk was used by Hastings on the orthogonal group. We show that the walk has spectral gap bounded below by c/n³. This and curvature information are used to bound the rate of ..."
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Cited by 21 (0 self)
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Mark Kac considered a Markov Chain on the n–sphere based on random rotations in randomly chosen coordinate planes. This same walk was used by Hastings on the orthogonal group. We show that the walk has spectral gap bounded below by c/n³. This and curvature information are used to bound the rate of convergence to stationarity.
A simple proof that Toffoli and Hadamard are quantum universal
 IN QUANTPH/0301040
, 2003
"... Recently Shi [15] proved that Toffoli and Hadamard are universal for quantum computation. This is perhaps the simplest universal set of gates that one can hope for, conceptually; It shows that one only needs to add the Hadamard gate to make a ’classical ’ set of gates quantum universal. In this note ..."
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Recently Shi [15] proved that Toffoli and Hadamard are universal for quantum computation. This is perhaps the simplest universal set of gates that one can hope for, conceptually; It shows that one only needs to add the Hadamard gate to make a ’classical ’ set of gates quantum universal. In this note we give a few lines proof of this fact relying on Kitaev’s universal set of gates [11], and discuss the meaning of the result.
Secure multiparty quantum computation with (only) a strict honest majority.
 In FOCS,
, 2006
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