Resilient Quantum Computation: Error Models and Thresholds

Recent research has demonstrated that quantum computers can solve certain types of problems substantially faster than the known classical algorithms. These problems include factoring integers and certain physics simulations. Practical quantum computation requires overcoming the problems of environmental noise and operational errors, problems which appear to be much more severe than in classical computation due to the inherent fragility of quantum superpositions involving many degrees of freedom. Here we show that arbitrarily accurate quantum computations are possible provided that the error per operation is below a threshold value. The result is obtained by combining quantum error-correction, fault tolerant state recovery, fault tolerant encoding of operations and concatenation. It holds under physically realistic assumptions on the errors.Comment: 19 pages in RevTex, many figures, the paper is also avalaible at http://qso.lanl.gov/qc

Experimental Implementation of a Codeword Stabilized Quantum Code

A five-qubit codeword stabilized quantum code is implemented in a seven-qubit system using nuclear magnetic resonance (NMR). Our experiment implements a good nonadditive quantum code which encodes a larger Hilbert space than any stabilizer code with the same length and capable of correcting the same kind of errors. The experimentally measured quantum coherence is shown to be robust against artificially introduced errors, benchmarking the success in implementing the quantum error correction code. Given the typical decoherence time of the system, our experiment illustrates the ability of coherent control to implement complex quantum circuits for demonstrating interesting results in spin qubits for quantum computing

Bromophenyl functionalization of carbon nanotubes : an ab initio study

We study the thermodynamics of bromophenyl functionalization of carbon nanotubes with respect to diameter and metallic/insulating character using density-functional theory (DFT). On one hand, we show that the activation energy for the grafting of a bromophenyl molecule onto a semiconducting zigzag nanotube ranges from 0.73 eV to 0.76 eV without any clear trend with respect to diameter within numerical accuracy. On the other hand, the binding energy of a single bromophenyl molecule shows a clear diameter dependence and ranges from 1.51 eV for a (8,0) zigzag nanotube to 0.83 eV for a (20,0) zigzag nanotube. This is in part explained by the transition from sp2 to sp3 bonding occurring to a carbon atom of a nanotube when a phenyl is grafted to it and the fact that smaller nanotubes are closer to a sp3 hybridization than larger ones due to increased curvature. Since a second bromophenyl unit can attach without energy barrier next to an isolated grafted unit, they are assumed to exist in pairs. The para configuration is found to be favored for the pairs and their binding energy decreases with increasing diameter, ranging from 4.34 eV for a (7,0) nanotube to 2.27 eV for a (29,0) nanotube. An analytic form for this radius dependence is derived using a tight binding hamiltonian and first order perturbation theory. The 1/R^2 dependance obtained (where R is the nanotube radius) is verified by our DFT results within numerical accuracy. Finally, metallic nanotubes are found to be more reactive than semiconducting nanotubes, a feature that can be explained by a non-zero density of states at the Fermi level for metallic nanotubes.Comment: 7 pages, 5 figures and 3 table

Experimental detection of non-classical correlations in mixed state quantum computation

We report on an experiment to detect non-classical correlations in a highly mixed state. The correlations are characterized by the quantum discord and are observed using four qubits in a liquid state nuclear magnetic resonance quantum information processor. The state analyzed is the output of a DQC1 computation, whose input is a single quantum bit accompanied by n maximally mixed qubits. This model of computation outperforms the best known classical algorithms, and although it contains vanishing entanglement it is known to have quantum correlations characterized by the quantum discord. This experiment detects non-vanishing quantum discord, ensuring the existence of non-classical correlations as measured by the quantum discord.Comment: The second version more closely resembles that of the published manuscrip

Three path interference using nuclear magnetic resonance: a test of the consistency of Born's rule

The Born rule is at the foundation of quantum mechanics and transforms our classical way of understanding probabilities by predicting that interference occurs between pairs of independent paths of a single object. One consequence of the Born rule is that three way (or three paths) quantum interference does not exist. In order to test the consistency of the Born rule, we examine detection probabilities in three path intereference using an ensemble of spin-1/2 quantum registers in liquid state nuclear magnetic resonance (LSNMR). As a measure of the consistency, we evaluate the ratio of three way interference to two way interference. Our experiment bounded the ratio to the order of $10^{-3} \pm 10^{-3}$, and hence it is consistent with Born's rule.Comment: 11 pages, 4 figures; Improved presentation of figures 1 and 4, changes made in section 2 to better describe the experiment, minor changes throughout, and added several reference

Experimental implementation of encoded logical qubit operations in a perfect quantum error correcting code

Large-scale universal quantum computing requires the implementation of quantum error correction (QEC). While the implementation of QEC has already been demonstrated for quantum memories, reliable quantum computing requires also the application of nontrivial logical gate operations to the encoded qubits. Here, we present examples of such operations by implementing, in addition to the identity operation, the NOT and the Hadamard gate to a logical qubit encoded in a five qubit system that allows correction of arbitrary single qubit errors. We perform quantum process tomography of the encoded gate operations, demonstrate the successful correction of all possible single qubit errors and measure the fidelity of the encoded logical gate operations

Optimizing the discrete time quantum walk using a SU(2) coin

We present a generalized version of the discrete time quantum walk, using the SU(2) operation as the quantum coin. By varying the coin parameters, the quantum walk can be optimized for maximum variance subject to the functional form $\sigma^2 \approx N^2$ and the probability distribution in the position space can be biased. We also discuss the variation in measurement entropy with the variation of the parameters in the SU(2) coin. Exploiting this we show how quantum walk can be optimized for improving mixing time in an $n$-cycle and for quantum walk search.Comment: 6 pages, 6 figure

Multiqubit Spin

It is proposed that the state space of a quantum object with a complicated discrete spectrum can be used as a basis for multiqubit recording and processing of information in a quantum computer. As an example, nuclear spin 3/2 is considered. The possibilities of writing and reading two quantum bits of information, preparation of the initial state, implementation of the "rotation" and "controlled negation" operations, which are sufficient for constructing any algorithms, are demonstrated.Comment: 7 pages, PostScript, no figures; translation of Pis'ma Zh. Eksp. Teor. Fiz. 70, No. 1, pp. 59-63, 10 July 1999; (Submitted 29 April 1999; resubmitted 2 June 1999