Reliable Computation by Formulas in the Presence of Noise

It is shown that if formulas are used to compute Boolean functions in the presence of randomly occurring failures then: (1) there is a limit strictly less than 1/2 to the failure probability per gate that can be tolerated, and (2) formulas that tolerate failures must be deeper (and, therefore, compute more slowly) than those that do not. The heart of the proof is an information-theoretic argument that deals with computation and errors in very general terms. The strength of this argument is that it applies with equal ease no matter what types of gate are available. Its weaknesses is that it does not seem to predict quantitatively the limiting value of the failure probability or the ratio by which computation proceeds more slowly in the presence of failures

On the maximum tolerable noise of k-input gates for reliable computation by formulas

We determine the precise threshold of component noise below which formulas composed of odd degree components can reliably compute all Boolean functions

Signal propagation and noisy circuits

The information carried by a signal decays when the signal is corrupted by random noise. This occurs when a message is transmitted over a noisy channel, as well as when a noisy component performs computation. We first study this signal decay in the context of communication and obtain a tight bound on the rate at which information decreases as a signal crosses a noisy channel. We then use this information theoretic result to obtain depth lower bounds in the noisy circuit model of computation defined by von Neumann. In this model, each component fails (produces 1 instead of 0 or vice-versa) independently with a fixed probability, and yet the output of the circuit is required to be correct with high probability. Von Neumann showed how to construct circuits in this model that reliably compute a function and are no more than a constant factor deeper than noiseless circuits for the function. We provide a lower bound on the multiplicative increase in circuit depth necessary for reliable computation, and an upper bound on the maximum level of noise at which reliable computation is possible

Noise threshold for universality of 2-input gates

Evans and Pippenger showed in 1998 that noisy gates with 2 inputs are universal for arbitrary computation (i.e. can compute any function with bounded error), if all gates fail independently with probability epsilon and epsilon<theta, where theta is roughly 8.856%. We show that formulas built from gates with 2 inputs, in which each gate fails with probability at least theta cannot be universal. Hence, there is a threshold on the tolerable noise for formulas with 2-input gates and it is theta. We conjecture that the same threshold also holds for circuits.Comment: International Symposium on Information Theory, 2007, minor corrections in v

Signal Propagation, with Application to a Lower Bound on the Depth of Noisy Formulas

We study the decay of an information signal propagating through a series of noisy channels. We obtain exact bounds on such decay, and as a result provide a new lower bound on the depth of formulas with noisy components. This improves upon previous work of N. Pippenger and significantly decreases the gap between his lower bound and the classical upper bound of von Neumann. We also discuss connections between our work and the study of mixing rates of Markov chains

Information Theory and Noisy Computation

We report on two types of results. The first is a study of the rate of decay of information carried by a signal which is being propagated over a noisy channel. The second is a series of lower bounds on the depth, size, and component reliability of noisy logic circuits which are required to compute some function reliably. The arguments used for the circuit results are information-theoretic, and in particular, the signal decay result is essential to the depth lower bound. Our first result can be viewed as a quantified version of the data processing lemma, for the case of Boolean random variables

Nonlocally-induced (quasirelativistic) bound states: Harmonic confinement and the finite well

Nonlocal Hamiltonian-type operators, like e.g. fractional and quasirelativistic, seem to be instrumental for a conceptual broadening of current quantum paradigms. However physically relevant properties of related quantum systems have not yet received due (and scientifically undisputable) coverage in the literature. In the present paper we address Schr\"{o}dinger-type eigenvalue problems for $H=T+V$, where a kinetic term $T=T_m$ is a quasirelativistic energy operator $T_m = \sqrt{-\hbar ^2c^2 \Delta + m^2c^4} - mc^2$ of mass $m\in (0,\infty)$ particle. A potential $V$ we assume to refer to the harmonic confinement or finite well of an arbitrary depth. We analyze spectral solutions of the pertinent nonlocal quantum systems with a focus on their $m$-dependence. Extremal mass $m$ regimes for eigenvalues and eigenfunctions of $H$ are investigated: (i) $m\ll 1$ spectral affinity ("closeness") with the Cauchy-eigenvalue problem ($T_m \sim T_0=\hbar c |\nabla |$) and (ii) $m \gg 1$ spectral affinity with the nonrelativistic eigenvalue problem ($T_m \sim -\hbar ^2 \Delta /2m$). To this end we generalize to nonlocal operators an efficient computer-assisted method to solve Schr\"{o}dinger eigenvalue problems, widely used in quantum physics and quantum chemistry. A resultant spectrum-generating algorithm allows to carry out all computations directly in the configuration space of the nonlocal quantum system. This allows for a proper assessment of the spatial nonlocality impact on simulation outcomes. Although the nonlocality of $H$ might seem to stay in conflict with various numerics-enforced cutoffs, this potentially serious obstacle is kept under control and effectively tamed.Comment: 23 pages, 16 figure

A 1-bit Synchronization Algorithm for a Reduced Complexity Energy Detection UWB Receiver

This work investigates the possibility of performing synchronization in a reduced complexity Energy Detection receiver. A new receiver scheme employing a single comparator only is defined and the related synchronization algorithm is presented. The possibility of synchronizing has been analyzed both for an idealized Dirac Delta input signal and for realistic UWB signals obtained through the TG4a channel model. The matlab simulations show that it is possible to obtain coarse synchronization using a simple maximum detection algorithm computed on collected energies for the ideal case of Dirac Delta pulses. For realistic UWB signals better synchronization performances are possible by employing a searchback algorithm. Due to the low complexity of the receiver scheme, the synchronization algorithm requires a long locking time

Quantum Computing with Very Noisy Devices

In theory, quantum computers can efficiently simulate quantum physics, factor large numbers and estimate integrals, thus solving otherwise intractable computational problems. In practice, quantum computers must operate with noisy devices called ``gates'' that tend to destroy the fragile quantum states needed for computation. The goal of fault-tolerant quantum computing is to compute accurately even when gates have a high probability of error each time they are used. Here we give evidence that accurate quantum computing is possible with error probabilities above 3% per gate, which is significantly higher than what was previously thought possible. However, the resources required for computing at such high error probabilities are excessive. Fortunately, they decrease rapidly with decreasing error probabilities. If we had quantum resources comparable to the considerable resources available in today's digital computers, we could implement non-trivial quantum computations at error probabilities as high as 1% per gate.Comment: 47 page