Small Success Probability Research Articles

Quantum Time–Space Tradeoff for Finding Multiple Collision Pairs

We study the problem of finding K collision pairs in a random function f : [N] → [N] by using a quantum computer. We prove that the number of queries to the function in the quantum random oracle model must increase significantly when the size of the available memory is limited. Namely, we demonstrate that any algorithm using S qubits of memory must perform a number T of queries that satisfies the tradeoff T 3 S ≥ Ω (K 3 N) . Classically, the same question has only been settled recently by Dinur [ 22 ], who showed that the Parallel Collision Search algorithm of van Oorschot and Wiener [ 36 ] achieves the optimal time–space tradeoff of T 2 S = Θ (K 2 N) . Our result limits the extent to which quantum computing may decrease this tradeoff. Our method is based on a novel application of Zhandry’s recording query technique [ 42 ] for proving lower bounds in the exponentially small success probability regime. As a second application, we give a simpler proof of the time–space tradeoff T 2 S ≥ Ω (N 3 ) for sorting N numbers on a quantum computer, which was first obtained by Klauck, Špalek, and de Wolf [ 30 ].

Verifying BQP Computations on Noisy Devices with Minimal Overhead

Recent experimental achievements motivate an ever-growing interest from companies starting to feel the limitations of classical computing. Yet, in light of ongoing privacy scandals, the future availability of quantum computing through remotely accessible servers pose peculiar challenges: Clients with quantum-limited capabilities want their data and algorithms to remain hidden, while being able to verify that their computations are performed correctly. Research in blind and verifiable delegation of quantum computing attempts to address this question. However, available techniques suffer not only from high overheads but also from over-sensitivity: When running on noisy devices, imperfections trigger the same detection mechanisms as malicious attacks, resulting in perpetually aborted computations. Hence, while malicious quantum computers are rendered harmless by blind and verifiable protocols, inherent noise severely limits their usability. We address this problem with an efficient, robust, blind, verifiable scheme to delegate deterministic quantum computations with classical inputs and outputs. We show that: 1) a malicious Server can cheat at most with an exponentially small success probability; 2) in case of sufficiently small noise, the protocol succeeds with a probability exponentially close to 1; 3) the overhead is barely a polynomial number of repetitions of the initial computation interleaved with test runs requiring the same physical resources in terms of memory and gates; 4) the amount of tolerable noise, measured by the probability of failing a test run, can be as high as 25% for some computations and will be generally bounded by 12.5% when using a planar graph resource state. The key points are that security can be provided without universal computation graphs and that, in our setting, full fault-tolerance is not needed to amplify the confidence level exponentially close to 1.

Feature subset selection for predicting the success of crowdfunding project campaigns

Statistics from crowdfunding platforms show that a small percent of crowdfunding projects succeed in securing funds. This makes project creators eager to know the probability of success of their campaign and the features that contribute to its success before launching it on crowdfunding platforms. The existing literature focuses on examining success probability using the entire list of identified projects features. For situations for which project creators have limited resources to invest on the required project features, the list suggested by previous researchers is somewhat large and gives a small success probability. A minimal number of features that predict success with a higher probability can benefit project creators by providing them with insight and guidance in investing their limited resources. This paper presents a metaheuristic whale optimization algorithm (WOA) in the crowdfunding context to perform a complete search of a subset of features that have a high success contribution power. Experiments were conducted using WOA with the K-Nearest Neighbor (KNN) classifier on a Kickstarter dataset. Our approach obtains a subset of 9 features that predict the success of project campaigns with an accuracy (F-score) of 90.28% (90.11%), which is an increase (F-score) of 22.23% (21.61%) than when a complete set of features is used. The findings of this study contribute knowledge to various crowdfunding stakeholders, as they will provide new insights regarding a subset of essential features th4at influence the success of project campaigns with high accuracy.

Simple Optimal Hitting Sets for Small-Success RL

We give a simple explicit hitting set generator for read-once branching programs of width $w$ and length $r$ with known variable order and acceptance probability at least $\epsilon$. When $r = w$, our generator has seed length $O(\log^2 r + \log(1/\epsilon))$. When $r = \text{polylog } w$, our generator has optimal seed length $O(\log w + \log(1/\epsilon))$. For intermediate values of $r$, our generator's seed length smoothly interpolates between these two extremes. Our generator's seed length improves on recent work by Braverman, Cohen, and Garg [SIAM J. Comput., (2020), doi:10.1137/18M1197734]. In addition, our generator and its analysis are dramatically simpler than the work by Braverman et al. When $\epsilon$ is small, our generator's seed length improves on all the classic generators for space-bounded computation [N. Nisan, Combinatorica, 12 (1992), pp. 449--461; R. Impagliazzo, N. Nisan, and A. Wigderson, in Proceedings of the 26th Annual ACM Symposium on Theory of Computing, ACM, 1994, pp. 356--364; N. Nisan and D. Zuckerman, J. Comput. System Sci., 52 (1996), pp. 43--52]. However, all of these other works construct more general objects than we do. As a corollary of our construction, we show that every ${RL}$ algorithm that uses $r$ random bits can be simulated by an ${NL}$ algorithm that uses only $O(r/\log^c n)$ nondeterministic bits, where $c$ is an arbitrarily large constant. Finally, we show that any ${RL}$ algorithm with small success probability $\epsilon$ can be simulated deterministically in space $O(\log^{3/2} n + \log n \log \log(1/\epsilon))$. This space bound improves on work by Saks and Zhou [J. Comput. System Sci., 58 (1999), pp. 376--403], who gave an algorithm for the more general “two-sided” problem that runs in space $O(\log^{3/2} n + \sqrt{\log n} \log(1/\epsilon))$.

Gaussian postselection and virtual noiseless amplification in continuous-variable quantum key distribution

The noiseless amplification or attenuation are two heralded filtering operations that enable respectively to increase or decrease the mean field of any quantum state of light with no added noise, at the cost of a small success probability. We show that inserting such noiseless operations in a transmission line improves the performance of continuous-variable quantum key distribution over this line. Remarkably, these noiseless operations do not need to be physically implemented but can simply be simulated in the data post-processing stage. Hence, virtual noiseless amplification or attenuation amounts to perform a Gaussian post-selection, which enhances the secure range or tolerable excess noise while keeping the benefits of Gaussian security proofs.

New Direct-Product Testers and 2-Query PCPs

The “direct-product code” of a function $f$ gives its values on all $k$-tuples $(f(x_1),\dots, f(x_k))$. This basic construct underlies “hardness amplification” in cryptography, circuit complexity, and probabilistically checkable proofs (PCPs). Goldreich and Safra [SIAM J. Comput., 29 (2000), pp. 1132--1154] pioneered its local testing and its PCP application. A recent result by Dinur and Goldenberg [Proceedings of the Forty-Ninth Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 613--622] enabled for the first time testing proximity to this important code in the “list-decoding” regime. In particular, they give a $2$-query test which works for polynomially small success probability $1/k^{\alpha}$ and show that no such test works below success probability $1/k$. Our main result is a $3$-query test which works for exponentially small success probability $\exp({-k^{\alpha}})$. Our techniques (based on recent simplified decoding algorithms for the same code [R. Impagliazzo et al., Proceeding...

Bi-directional mapping between polarization and spatially encoded photonic qutrits

Qutrits, the triple level quantum systems in various forms, have been proposed for quantum information processing recently. By the methods presented in this paper, a biphotonic qutrit, which is encoded with the polarizations of two photons in the same spatial-temporal mode, can be mapped to a single-photon qutrit in spatial modes. It will make arbitrary unitary operation on such biphotonic qutrit possible if we can also realize the inverse map to polarization space. Among the two schemes proposed in this paper, the one based only on linear optics realizes an arbitrary U(3) operation with a very small success probability. However, if added with weak nonlinearity, the success probability can be greatly improved. These schemes are feasible with the current experimental technology.

Scheme for Efficient Construction of 2-Dimensional Qutrit Cluster State

Utilizing quantum entangling gates with an arbitrary small success probability p, we propose a three-step scheme for conclusively constructing 2-dimensional (2D) lattice qutrit cluster states (QTCS). We first prepare one-dimensional (1D) QTCS, then use the 1D states to generate cross-shaped QTCS, and finally take the cross-shaped QTCS as “building blocks” to construct 2D lattice QTCS. In our scheme parallel operations are executed so that the construction process is significantly accelerated. Our scheme provides a potential efficient way to implement realistic one-way quantum computers.

Influence of primary recoil energy spectrum on microstructural evolution

In order to decompose the effect on microstructure evolution from a wide recoil energy spectrum of neutron irradiation, self-ion irradiation with variable energy is performed. Binary collision simulation of subcascade formation is made in parallel with experiments. Microstructure formed by 14 MeV neutron irradiation is reconstructed both from self-ion irradiation data and from collision simulation data. The concept of subcascade formation is reconsidered from the viewpoint of microstructural evolution. Examination of the extremely small success probability of cascade collision induced interstitial cluster nuclei to grow into dislocation loops leads to a proposal of the important role of stochastic fluctuation of point defect reaction. Various sources which cause the unbalanced reaction of vacancies and interstitials are examined: interstitial predominance by vacancy cluster formation, vacancy predominance by interstitial cluster formation, and a particular emphasis on the cascade localisation induced bias effect.