scholarly journals Upper bounds on the noise threshold for fault-tolerant quantum computing

2010 ◽  
Vol 10 (5&6) ◽  
pp. 361-376
Author(s):  
J. Kempe ◽  
O. Regev ◽  
F. Unger ◽  
R. de Wolf
Keyword(s):  
Fault Tolerant ◽  
Quantum Circuit ◽  
Upper Bounds ◽  
Final State ◽  
Important Special Case ◽  
Main Technique ◽  
Noise Threshold ◽  
Special Case ◽  
Tolerable Level ◽  
Better Than

We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary $k$-qubit gates each of whose input wires is subject to depolarizing noise of strength $p$, as well as arbitrary one-qubit gates that are essentially noise-free. We assume that the output of the circuit is the result of measuring some designated qubit in the final state. Our main result is that for $p>1-\Theta(1/\sqrt{k})$, the output of any such circuit of large enough depth is essentially independent of its input, thereby making the circuit useless. For the important special case of $k=2$, our bound is $p>35.7\%$. Moreover, if the only allowed gate on more than one qubit is the two-qubit CNOT gate, then our bound becomes $29.3\%$. These bounds on $p$ are numerically better than previous bounds, yet are incomparable because of the somewhat different circuit model that we are using. Our main technique is the use of a Pauli basis decomposition, in which the effects of depolarizing noise are very easy to describe.

ROUTING BALANCED COMMUNICATIONS ON HAMILTON DECOMPOSABLE NETWORKS

2004 ◽  
Vol 14 (03n04) ◽  
pp. 377-385 ◽  
Author(s):  
LADISLAV STACHO ◽  
JOZEF ŠIRÁŇ ◽  
SANMING ZHOU
Keyword(s):  
Wave Length ◽  
Upper Bounds ◽  
Ring Network ◽  
Special Case ◽  
Better Than

In [10] the authors proved upper bounds for the arc-congestion and wave-length number of any permutation demand on a bidirected ring. In this note, we give generalizations of their results in two directions. The first one is that instead of considering only permutation demands we consider any balanced demand, and the second one is that instead of the ring network we consider any Hamilton decomposable network. Thus, we obtain upper bounds (which are best possible in general) for the arc-congestion and wavelength number of any balanced demand on a Hamilton decomposable network. As a special case, we obtain upper bounds on arc- and edge-forwarding indices of Hamilton decomposable networks that are in many cases better than the known ones.

scholarly journals Clifford recompilation for faster classical simulation of quantum circuits

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 170
Author(s):  
Hammam Qassim ◽  
Joel J. Wallman ◽  
Joseph Emerson
Keyword(s):  
Fault Tolerant ◽  
State Of The Art ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Monte Carlo Algorithm ◽  
Quantum Devices ◽  
Output State ◽  
Classical Simulation ◽  
Main Technique ◽  
Near Term

Simulating quantum circuits classically is an important area of research in quantum information, with applications in computational complexity and validation of quantum devices. One of the state-of-the-art simulators, that of Bravyi et al, utilizes a randomized sparsification technique to approximate the output state of a quantum circuit by a stabilizer sum with a reduced number of terms. In this paper, we describe an improved Monte Carlo algorithm for performing randomized sparsification. This algorithm reduces the runtime of computing the approximate state by the factorℓ/m, whereℓandmare respectively the total and non-Clifford gate counts. The main technique is a circuit recompilation routine based on manipulating exponentiated Pauli operators. The recompilation routine also facilitates numerical search for Clifford decompositions of products of non-Clifford gates, which can further reduce the runtime in certain cases by reducing the 1-norm of the vector of expansion,‖a‖1. It may additionally lead to a framework for optimizing circuit implementations over a gate-set, reducing the overhead for state-injection in fault-tolerant implementations. We provide a concise exposition of randomized sparsification, and describe how to use it to estimate circuit amplitudes in a way which can be generalized to a broader class of gates and states. This latter method can be used to obtain additive error estimates of circuit probabilities with a faster runtime than the full techniques of Bravyi et al. Such estimates are useful for validating near-term quantum devices provided that the target probability is not exponentially small.

SOME NEW RELIABILITY BOUNDS FOR SUMS OF NBUE RANDOM VARIABLES

2010 ◽  
Vol 25 (1) ◽  
pp. 83-102
Author(s):  
Steven G. From
Keyword(s):  
Lower Bounds ◽  
Random Variables ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Important Special Case ◽  
Lifetime Distributions ◽  
Survivor Function ◽  
Numerical Studies ◽  
Special Case ◽  
Better Than

In this article, we discuss some new upper and lower bounds for the survivor function of the sum of n independent random variables each of which has an NBUE (new better than used in expectation) distribution. In some cases, only the means of the random variables are assumed known. These bounds are compared to the sharp bounds given in Cheng and Lam [6], which requires both means and variances known. Although the new bounds are not sharp, they often produce better upper bounds for the survivor function in the extreme right tail of many NBUE lifetime distributions, an important special case in applications. Moreover, a lower bound exists in one case not handled by the lower bounds of Theorem 3 in Cheng and Lam [6]. Numerical studies are presented along with theoretical discussions.

ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

2004 ◽  
Vol 04 (01) ◽  
pp. 63-76 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Hausdorff Dimension ◽  
Finite Subset ◽  
Continued Fraction ◽  
Cantor Set ◽  
Computational Method ◽  
Accurate Approximation ◽  
Natural Numbers ◽  
Important Special Case ◽  
The One ◽  
Special Case

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

scholarly journals Bounding the Size and Probability of Epidemics on Networks

2008 ◽  
Vol 45 (2) ◽  
pp. 498-512 ◽  
Author(s):  
Joel C. Miller
Keyword(s):  
Infectious Disease ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Marginal Probability ◽  
Transmission Properties ◽  
Disease Spreading ◽  
Special Case

We consider an infectious disease spreading along the edges of a network which may have significant clustering. The individuals in the population have heterogeneous infectiousness and/or susceptibility. We define the out-transmissibility of a node to be the marginal probability that it would infect a randomly chosen neighbor given its infectiousness and the distribution of susceptibility. For a given distribution of out-transmissibility, we find the conditions which give the upper (or lower) bounds on the size and probability of an epidemic, under weak assumptions on the transmission properties, but very general assumptions on the network. We find similar bounds for a given distribution of in-transmissibility (the marginal probability of being infected by a neighbor). We also find conditions giving global upper bounds on the size and probability. The distributions leading to these bounds are network independent. In the special case of networks with high girth (locally tree-like), we are able to prove stronger results. In general, the probability and size of epidemics are maximal when the population is homogeneous and minimal when the variance of in- or out-transmissibility is maximal.

Optimization Design of Electromagnetic Devices Using an Enhanced Salp Swarm Algorithm

2021 ◽  
Vol 35 (12) ◽  
pp. 1471-1476
Author(s):  
Houssem Bouchekara ◽  
Mostafa Smail ◽  
Mohamed Javaid ◽  
Sami Shamsah
Keyword(s):  
Optimization Design ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Design Problems ◽  
Salp Swarm Algorithm ◽  
Electromagnetic Devices ◽  
Swarm Algorithm ◽  
Better Than

An Enhanced version of the Salp Swarm Algorithm (SSA) referred to as (ESSA) is proposed in this paper for the optimization design of electromagnetic devices. The ESSA has the same structure as of the SSA with some modifications in order to enhance its performance for the optimization design of EMDs. In the ESSA, the leader salp does not move around the best position with a fraction of the distance between the lower and upper bounds as in the SAA; rather, a modified mechanism is used. The performance of the proposed algorithm is tested on the widely used Loney’s solenoid and TEAM Workshop Problem 22 design problems. The obtained results show that the proposed algorithm is much better than the initial one. Furthermore, a comparison with other well-known algorithms revealed that the proposed algorithm is very competitive for the optimization design of electromagnetic devices.

An Elementary Proof of Suslin Reciprocity

2005 ◽  
Vol 48 (2) ◽  
pp. 221-236 ◽  
Author(s):  
Matt Kerr
Keyword(s):  
Elementary Proof ◽  
Function Fields ◽  
Algebraic Cycles ◽  
Important Special Case ◽  
Special Case

AbstractWe state and prove an important special case of Suslin reciprocity that has found significant use in the study of algebraic cycles. An introductory account is provided of the regulator and norm maps on Milnor K2-groups (for function fields) employed in the proof.

Modeling of Nonlinear Viscoelastic Creep of Polycarbonate

e-Polymers ◽  
2007 ◽  
Vol 7 (1) ◽  
Author(s):  
Wenbo Luo ◽  
Said Jazouli ◽  
Toan Vu-Khanh
Keyword(s):  
Creep Behavior ◽  
Room Temperature ◽  
Multiple Integral ◽  
Model Fit ◽  
Viscoelastic Creep ◽  
Commercial Grade ◽  
Nonlinear Viscoelastic ◽  
Schapery Model ◽  
Special Case ◽  
Better Than

AbstractThe creep behavior of a commercial grade polycarbonate was investigated in this study. 10 different constant stresses ranging from 8 MPa to 50 MPa were applied to the specimen, and the resultant creep strains were measured at room temperature. It was found that the creep could be modeled linearly below 15 MPa, and nonlinearly above 15 MPa. Different nonlinear viscoelastic models have been briefly reviewed and used to fit the test data. It is shown that the Findley model is a special case of the Schapery model, and both the Findley model and the simplified multiple integral representation are suitable for properly describing the creep behavior of the polycarbonate investigated in this paper; however, the Findley model fit the data better than the simplified multiple integral with three terms.

FAULT TOLERANT ROUTING IN HYPERCUBES AND STAR GRAPHS

1996 ◽  
Vol 06 (01) ◽  
pp. 127-136 ◽  
Author(s):  
QIAN-PING GU ◽  
SHIETUNG PENG
Keyword(s):  
Free Path ◽  
Fault Tolerant ◽  
Linear Time ◽  
Time Efficiency ◽  
Routing Problem ◽  
Star Graphs ◽  
Linear Time Algorithms ◽  
Better Than

In this paper, we give two linear time algorithms for node-to-node fault tolerant routing problem in n-dimensional hypercubes Hn and star graphs Gn. The first algorithm, given at most n−1 arbitrary fault nodes and two non-fault nodes s and t in Hn, finds a fault-free path s→t of length at most [Formula: see text] in O(n) time, where d(s, t) is the distance between s and t. Our second algorithm, given at most n−2 fault nodes and two non-fault nodes s and t in Gn, finds a fault-free path s→t of length at most d(Gn)+3 in O(n) time, where [Formula: see text] is the diameter of Gn. When the time efficiency of finding the routing path is more important than the length of the path, the algorithms in this paper are better than the previous ones.

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