Noise threshold for universality of 2-input gates

arXiv (Cornell University), 1996

In this paper we study noisy reversible circuits. Noisy computation and reversible computation have been studied separately, and it is known that they are equivalent in power to unrestricted computation. We study the case where both noise and reversibility are combined and show that the combined model is weaker than unrestricted computation. We consider the model of reversible computation with noise, where the value of each wire in the circuit is flipped with some fixed probability 1/2 > p > 0 each time step, and all the inputs to the circuit are present in time 0. We prove that any noisy reversible circuit must have size exponential in its depth in order to compute a function with high probability. This is tight as we show that any (not necessarily reversible or noise-resistant) circuit can be converted into a reversible one that is noise-resistant with a blow up in size which is exponential in the depth. This establishes that noisy reversible computation has the power of the complexity class N C 1. We extend the upper bound to quantum circuits, and prove that any noisy quantum circuit must have size exponential in its depth in order to compute a function with high probability. This highlight the fact that current error-correction schemes for quantum computation require constant inputs throughout the computation (and not just at time 0), and shows that this is unavoidable. As for the lower bound, we show that quasi-polynomial noisy quantum circuits are at least powerful as quantum circuits with logarithmic depth (or QN C 1). Making these bounds tight is left open in the quantum case.

2006

Abstract Most recent works on soft errors only address circuit reliability under single gate errors caused by SEUs. In this paper, we compare the probabilities of single and multiple errors. We formulate a criterion based on gate error probabilities for considering multiple gate errors in circuit reliability. Gate error probabilities are increased by technology trends such as the down-scaling of device features and process variation. The probability of multiple errors is generally higher when there is correlation between gate errors.

IEEE Transactions on Information Theory, 2015

We study noisy computation in randomly generated k-ary Boolean formulas. We establish bounds on the noise level above which the results of computation by random formulas are not reliable. This bound is saturated by formulas constructed from a single majority-like gates.

IEEE Access

Due to the reduction in device feature size and supply voltage, achieving soft error reliability in sub-micrometer digital circuits is becoming extremely challenging. We consider the problem of choosing the gate sizes in a combinational logic circuit in order to minimize the soft error rate (SER) of the circuit. This problem can be solved using the heuristic as well as the greedy-based approaches for small-size problems; however, when the circuit size increases, the computational time grows exponentially, and hence, the previous methods become impractical. This paper proposes a novel technique for soft error tolerant design of largescale combinational circuits using a cone-oriented gate sizing. Circuit partitioning is used to split the circuit into a set of small sub-circuits. The gates of sub-circuits are resized, such that the entire circuit SER is reduced based on a new soft error descriptor metric. The proposed cone-oriented gate sizing framework is used for selective gate sizing, leading up to 31% SER reduction with less than 17% area overhead when applied to large-scale benchmarks. The results also show that the proposed method is 21% more efficient and up to 292 times faster when compared with that obtained using a similar work based on the sensitive-based gate sizing scheme.

Physical Review Letters, 2009

The response of a noisy nonlinear system to deterministic input signals can be enhanced by cooperative phenomena e.g. stochastic resonance (SR); this effect has been demonstrated in many physical systems, and is thought to occur in nature. Here we show that, when one presents two square waves as input to a two-state system, the response of the system can produce a logical output (NOR/OR) with a probability controlled by the interplay between the noise intensity and the nonlinearity. As one increases the noise (for fixed threshold/nonlinearity), the probability of the output reflecting a NOR/OR operation increases to unity and then decreases. Changing the nonlinearity (or the thresholds) of the system changes the output into another logic operation (NAND/AND) whose probability displays analogous behavior. Thus, the interplay of nonlinearity and noise can yield flexible logic behavior, and the natural emergent outcome of such systems is, effectively, a logic gate. This "Logical Stochastic Resonance" (LSR) is described here, via a simple experimental realization of a two-state system underpinned by two (adjustable) thresholds.

Journal of Physics: Conference Series, 2010

Random Boolean formulae, generated by a growth process of noisy logical gates are analyzed using the generating functional methodology of statistical physics. We study the type of functions generated for different input distributions, their robustness for a given level of gate error and its dependence on the formulae depth and complexity and the gates used. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding typical-case phase transitions. Results for error-rates, function-depth and sensitivity of the generated functions are obtained for various gate-type and noise models.

2006

2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06), 2006

We show that quantum circuits cannot be made fault-tolerant against a depolarizing noise level ofθ = (6 − 2 √ 2)/7 ≈ 45%, thereby improving on a previous bound of 50% (due to Razborov [18]). 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 noiseθ. We prove that this set of gates cannot be universal for arbitrary (even classical) computation, from which the upper bound on the noise threshold for fault-tolerant quantum computation follows.

Quantum Information and Computation, 2007

We consider a model of quantum computation in which the set of operations is limited to nearest-neighbor interactions on a 2D lattice. We model movement of qubits with noisy \SWAP\ operations. For this architecture we design a fault-tolerant coding scheme using the concatenated $[[7,1,3]]$ Steane code. Our scheme is potentially applicable to ion-trap and solid-state 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 $1.85 \times 10^{-5}$ for the local setting, where memory error rates are one-tenth of the failure rates of gates, measurement, and preparation steps. For the analogous nonlocal setting, we obtain a noise threshold of $3.61 \times 10^{-5}$. Our results thus show that the additional \SWAP\ operations required to move qubits in the local model affect the noise threshold only moderately.

Algorithmica, 2013

An ongoing line of research has shown super-polynomial lower bounds for uniform and slightly-non-uniform small-depth threshold and arithmetic circuits [All99, KP09, JS11]. We give a unified framework that captures and improves each of the previous results. Our main results are that Permanent does not have threshold circuits of the following kinds.