Classically, for many computational problems one can conclude time lower bounds conditioned on the hardness of one or more of key problems: k-SAT, 3SUM and APSP. More recently, similar results have been derived in the quantum setting conditioned on the hardness of k-SAT and 3SUM. This is done using fine-grained reductions, where the approach is to (1) select a key problem $X$ that, for some function $T$, is conjectured to not be solvable by any $O(T(n)^{1-\epsilon})$ time algorithm for any constant $\epsilon > 0$ (in a fixed model of computation), and (2) reduce $X$ in a fine-grained way to these computational problems, thus giving (mostly) tight conditional time lower bounds for them. Interestingly, for Delta-Matching Triangles and Triangle Collection, classical hardness results have been derived conditioned on hardness of all three mentioned key problems. More precisely, it is proven that an $n^{3-\epsilon}$ time classical algorithm for either of these two graph problems would imply faster classical algorithms for k-SAT, 3SUM and APSP, which makes Delta-Matching Triangles and Triangle Collection worthwhile to study. In this paper, we show that an $n^{1.5-\epsilon}$ time quantum algorithm for either of these two graph problems would imply faster quantum algorithms for k-SAT, 3SUM, and APSP. We first formulate a quantum hardness conjecture for APSP and then present quantum reductions from k-SAT, 3SUM, and APSP to Delta-Matching Triangles and Triangle Collection. Additionally, based on the quantum APSP conjecture, we are also able to prove quantum lower bounds for a matrix problem and many graph problems. The matching upper bounds follow trivially for most of them, except for Delta-Matching Triangles and Triangle Collection for which we present quantum algorithms that require careful use of data structures and Ambainis' variable time search.
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Studying the properties of stochastic noise to optimize complex non-convex functions has been an active area of research in the field of machine learning. Prior work has shown that the noise of stochastic gradient descent improves optimization by overcoming undesirable obstacles in the landscape. Moreover, injecting artificial Gaussian noise has become a popular idea to quickly escape saddle points. Indeed, in the absence of reliable gradient information, the noise is used to explore the landscape, but it is unclear what type of noise is optimal in terms of exploration ability. In order to narrow this gap in our knowledge, we study a general type of continuous-time non-Markovian process, based on fractional Brownian motion, that allows for the increments of the process to be correlated. This generalizes processes based on Brownian motion, such as the Ornstein-Uhlenbeck process. We demonstrate how to discretize such processes which gives rise to the new algorithm fPGD. This method is a generalization of the known algorithms PGD and Anti-PGD. We study the properties of fPGD both theoretically and empirically, demonstrating that it possesses exploration abilities that, in some cases, are favorable over PGD and Anti-PGD. These results open the field to novel ways to exploit noise for training machine learning models.
The input to the \emph{Triangle Evacuation} problem is a triangle $ABC$. Given a starting point $S$ on the perimeter of the triangle, a feasible solution to the problem consists of two unit-speed trajectories of mobile agents that eventually visit every point on the perimeter of $ABC$. The cost of a feasible solution (evacuation cost) is defined as the supremum over all points $T$ of the time it takes that $T$ is visited for the first time by an agent plus the distance of $T$ to the other agent at that time. Similar evacuation type problems are well studied in the literature covering the unit circle, the $\ell_p$ unit circle for $p\geq 1$, the square, and the equilateral triangle. We extend this line of research to arbitrary non-obtuse triangles. Motivated by the lack of symmetry of our search domain, we introduce 4 different algorithmic problems arising by letting the starting edge and/or the starting point $S$ on that edge to be chosen either by the algorithm or the adversary. To that end, we provide a tight analysis for the algorithm that has been proved to be optimal for the previously studied search domains, as well as we provide lower bounds for each of the problems. Both our upper and lower bounds match and extend naturally the previously known results that were established only for equilateral triangles.
The Strong Exponential Time Hypothesis (SETH) asserts that for every $\varepsilon>0$ there exists $k$ such that $k$-SAT requires time $(2-\varepsilon)^n$. The field of fine-grained complexity has leveraged SETH to prove quite tight conditional lower bounds for dozens of problems in various domains and complexity classes, including Edit Distance, Graph Diameter, Hitting Set, Independent Set, and Orthogonal Vectors. Yet, it has been repeatedly asked in the literature whether SETH-hardness results can be proven for other fundamental problems such as Hamiltonian Path, Independent Set, Chromatic Number, MAX-$k$-SAT, and Set Cover. In this paper, we show that fine-grained reductions implying even $\lambda^n$-hardness of these problems from SETH for any $\lambda>1$, would imply new circuit lower bounds: super-linear lower bounds for Boolean series-parallel circuits or polynomial lower bounds for arithmetic circuits (each of which is a four-decade open question). We also extend this barrier result to the class of parameterized problems. Namely, for every $\lambda>1$ we conditionally rule out fine-grained reductions implying SETH-based lower bounds of $\lambda^k$ for a number of problems parameterized by the solution size $k$. Our main technical tool is a new concept called polynomial formulations. In particular, we show that many problems can be represented by relatively succinct low-degree polynomials, and that any problem with such a representation cannot be proven SETH-hard (without proving new circuit lower bounds).
Group-based cryptography is a relatively young family in post-quantum cryptography. In this paper we give the first dedicated security analysis of a central problem in group-based cryptography: the so-called Semidirect Product Key Exchange (SDPKE). We present a subexponential quantum algorithm for solving SDPKE. To do this we reduce SDPKE to the Abelian Hidden Shift Problem (for which there are known quantum subexponential algorithms). We stress that this does not per se constitute a break of SDPKE; rather, the purpose of the paper is to provide a connection to known problems.
Much of the literature on optimal design of bandit algorithms is based on minimization of expected regret. It is well known that designs that are optimal over certain exponential families can achieve expected regret that grows logarithmically in the number of arm plays, at a rate governed by the Lai-Robbins lower bound. In this paper, we show that when one uses such optimized designs, the regret distribution of the associated algorithms necessarily has a very heavy tail, specifically, that of a truncated Cauchy distribution. Furthermore, for $p>1$, the $p$'th moment of the regret distribution grows much faster than poly-logarithmically, in particular as a power of the total number of arm plays. We show that optimized UCB bandit designs are also fragile in an additional sense, namely when the problem is even slightly mis-specified, the regret can grow much faster than the conventional theory suggests. Our arguments are based on standard change-of-measure ideas, and indicate that the most likely way that regret becomes larger than expected is when the optimal arm returns below-average rewards in the first few arm plays, thereby causing the algorithm to believe that the arm is sub-optimal. To alleviate the fragility issues exposed, we show that UCB algorithms can be modified so as to ensure a desired degree of robustness to mis-specification. In doing so, we also provide a sharp trade-off between the amount of UCB exploration and the tail exponent of the resulting regret distribution.
In this paper we provide a practical demonstration of how the modularity in a Behavior Tree (BT) decreases the effort in programming a robot task when compared to a Finite State Machine (FSM). In recent years the way to represent a task plan to control an autonomous agent has been shifting from the standard FSM towards BTs. Many works in the literature have highlighted and proven the benefits of such design compared to standard approaches, especially in terms of modularity, reactivity and human readability. However, these works have often failed in providing a tangible comparison in the implementation of those policies and the programming effort required to modify them. This is a relevant aspect in many robotic applications, where the design choice is dictated both by the robustness of the policy and by the time required to program it. In this work, we compare backward chained BTs with a fault-tolerant design of FSMs by evaluating the cost to modify them. We validate the analysis with a set of experiments in a simulation environment where a mobile manipulator solves an item fetching task.
We have formalised Szemer\'edi's Regularity Lemma and Roth's Theorem on Arithmetic Progressions, two major results in extremal graph theory and additive combinatorics, using the proof assistant Isabelle/HOL. For the latter formalisation, we used the former to first show the Triangle Counting Lemma and the Triangle Removal Lemma: themselves important technical results. Here, in addition to showcasing the main formalised statements and definitions, we focus on sensitive points in the proofs, describing how we overcame the difficulties that we encountered.
Emerging quantum algorithms for problems such as element distinctness, subset sum, and closest pair demonstrate computational advantages by relying on abstract data structures. Practically realizing such an algorithm as a program for a quantum computer requires an efficient implementation of the data structure whose operations correspond to unitary operators that manipulate quantum superpositions of data. To correctly operate in superposition, an implementation must satisfy three properties -- reversibility, history independence, and bounded-time execution. Standard implementations, such as the representation of an abstract set as a hash table, fail these properties, calling for tools to develop specialized implementations. In this work, we present Core Tower, the first language for quantum programming with random-access memory. Core Tower enables the developer to implement data structures as pointer-based, linked data. It features a reversible semantics enabling every valid program to be translated to a unitary quantum circuit. We present Boson, the first memory allocator that supports reversible, history-independent, and constant-time dynamic memory allocation in quantum superposition. We also present Tower, a language for quantum programming with recursively defined data structures. Tower features a type system that bounds all recursion using classical parameters as is necessary for a program to execute on a quantum computer. Using Tower, we implement Ground, the first quantum library of data structures, including lists, stacks, queues, strings, and sets. We provide the first executable implementation of sets that satisfies all three mandated properties of reversibility, history independence, and bounded-time execution.
I survey, for a general scientific audience, three decades of research into which sorts of problems admit exponential speedups via quantum computers -- from the classics (like the algorithms of Simon and Shor), to the breakthrough of Yamakawa and Zhandry from April 2022. I discuss both the quantum circuit model, which is what we ultimately care about in practice but where our knowledge is radically incomplete, and the so-called oracle or black-box or query complexity model, where we've managed to achieve a much more thorough understanding that then informs our conjectures about the circuit model. I discuss the strengths and weaknesses of switching attention to sampling tasks, as was done in the recent quantum supremacy experiments. I make some skeptical remarks about widely-repeated claims of exponential quantum speedups for practical machine learning and optimization problems. Through many examples, I try to convey the "law of conservation of weirdness," according to which every problem admitting an exponential quantum speedup must have some unusual property to allow the amplitude to be concentrated on the unknown right answer(s).
Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. Here we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Runge-Kutta methods in time. Using a convergence framework we developed in previous work, we prove for a subclass of these discretizations that, if using the standard approach of rediscretizing the fine-grid problem on the coarse grid, robust MGRIT convergence with respect to CFL number and coarsening factor is not possible. This poor convergence and non-robustness is caused, at least in part, by an inadequate coarse-grid correction for smooth Fourier modes known as characteristic components.We propose an alternative coarse-grid that provides a better correction of these modes. This coarse-grid operator is related to previous work and uses a semi-Lagrangian discretization combined with an implicitly treated truncation error correction. Theory and numerical experiments show the coarse-grid operator yields fast MGRIT convergence for many of the method-of-lines discretizations considered, including for both implicit and explicit discretizations of high order.
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