Given a dataset of input states, measurements, and probabilities, is it possible to efficiently predict the measurement probabilities associated with a quantum circuit? Recent work of Caro and Datta (2020) studied the problem of PAC learning quantum circuits in an information theoretic sense, leaving open questions of computational efficiency. In particular, one candidate class of circuits for which an efficient learner might have been possible was that of Clifford circuits, since the corresponding set of states generated by such circuits, called stabilizer states, are known to be efficiently PAC learnable (Rocchetto 2018). Here we provide a negative result, showing that proper learning of CNOT circuits is hard for classical learners unless $\textsf{RP} = \textsf{NP}$. As the classical analogue and subset of Clifford circuits, this naturally leads to a hardness result for Clifford circuits as well. Additionally, we show that if $\textsf{RP} = \textsf{NP}$ then there would exist efficient proper learning algorithms for CNOT and Clifford circuits. By similar arguments, we also find that an efficient proper quantum learner for such circuits exists if and only if $\textsf{NP} \subseteq \textsf{RQP}$.
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We describe some pseudorandom properties of binary linear codes achieving capacity on the binary erasure channel under bit-MAP decoding (as shown in Kudekar et al this includes doubly transitive codes and, in particular, Reed-Muller codes). We show that for all integer $q \ge 2$ the $\ell_q$ norm of the characteristic function of such 'pseudorandom' code decreases as fast as that of any code of the same rate (and equally fast as that of a random code) under the action of the noise operator. In information-theoretic terms this means that the $q^{th}$ R\'enyi entropy of this code increases as fast as possible over the binary symmetric channel. In particular (taking $q = \infty$) this shows that such codes have the smallest asymptotic undetected error probability (equal to that of a random code) over the BSC, for a certain range of parameters. We also study the number of times a certain local pattern, a 'rhombic' $4$-tuple of codewords, appears in a linear code, and show that for a certain range of parameters this number for pseudorandom codes is similar to that for a random code.
Asymptotic properties of three estimators of probability density function of sample maximum $f_{(m)}:=mfF^{m-1}$ are derived, where $m$ is a function of sample size $n$. One of the estimators is the parametrically fitted by the approximating generalized extreme value density function. However, the parametric fitting is misspecified in finite $m$ cases. The misspecification comes from mainly the following two: the difference $m$ and the selected block size $k$, and the poor approximation $f_{(m)}$ to the generalized extreme value density which depends on the magnitude of $m$ and the extreme index $\gamma$. The convergence rate of the approximation gets slower as $\gamma$ tends to zero. As alternatives two nonparametric density estimators are proposed which are free from the misspecification. The first is a plug-in type of kernel density estimator and the second is a block-maxima-based kernel density estimator. Theoretical study clarifies the asymptotic convergence rate of the plug-in type estimator is faster than the block-maxima-based estimator when $\gamma> -1$. A numerical comparative study on the bandwidth selection shows the performances of a plug-in approach and cross-validation approach depend on $\gamma$ and are totally comparable. Numerical study demonstrates that the plug-in nonparametric estimator with the estimated bandwidth by either approach overtakes the parametrically fitting estimator especially for distributions with $\gamma$ close to zero as $m$ gets large.
Wireless sensor networks are among the most promising technologies of the current era because of their small size, lower cost, and ease of deployment. With the increasing number of wireless sensors, the probability of generating missing data also rises. This incomplete data could lead to disastrous consequences if used for decision-making. There is rich literature dealing with this problem. However, most approaches show performance degradation when a sizable amount of data is lost. Inspired by the emerging field of graph signal processing, this paper performs a new study of a Sobolev reconstruction algorithm in wireless sensor networks. Experimental comparisons on several publicly available datasets demonstrate that the algorithm surpasses multiple state-of-the-art techniques by a maximum margin of 54%. We further show that this algorithm consistently retrieves the missing data even during massive data loss situations.
Inductive logic reasoning is one of the fundamental tasks on graphs, which seeks to generalize patterns from the data. This task has been studied extensively for traditional graph datasets such as knowledge graphs (KGs), with representative techniques such as inductive logic programming (ILP). Existing ILP methods typically assume learning from KGs with static facts and binary relations. Beyond KGs, graph structures are widely present in other applications such as video instructions, scene graphs and program executions. While inductive logic reasoning is also beneficial for these applications, applying ILP to the corresponding graphs is nontrivial: they are more complex than KGs, which usually involve timestamps and n-ary relations, effectively a type of hypergraph with temporal events. In this work, we study two of such applications and propose to represent them as hypergraphs with time intervals. To reason on this graph, we propose the multi-start random B-walk that traverses this hypergraph. Combining it with a path-consistency algorithm, we propose an efficient backward-chaining ILP method that learns logic rules by generalizing from both the temporal and the relational data.
Most of the trace-checking tools only yield a Boolean verdict. However, when a property is violated by a trace, engineers usually inspect the trace to understand the cause of the violation; such manual diagnostic is time-consuming and error-prone. Existing approaches that complement trace-checking tools with diagnostic capabilities either produce low-level explanations that are hardly comprehensible by engineers or do not support complex signal-based temporal properties. In this paper, we propose TD-SB-TemPsy, a trace-diagnostic approach for properties expressed using SB-TemPsy-DSL. Given a property and a trace that violates the property, TD-SB-TemPsy determines the root cause of the property violation. TD-SB-TemPsy relies on the concepts of violation cause, which characterizes one of the behaviors of the system that may lead to a property violation, and diagnoses, which are associated with violation causes and provide additional information to help engineers understand the violation cause. As part of TD-SB-TemPsy, we propose a language-agnostic methodology to define violation causes and diagnoses. In our context, its application resulted in a catalog of 34 violation causes, each associated with one diagnosis, tailored to properties expressed in SB-TemPsy-DSL. We assessed the applicability of TD-SB-TemPsy using an industrial case study from the satellite domain. The results show that TD-SB-TemPsy could finish within a timeout of 1 min for ~83:66% of the trace-property combinations in our dataset, yielding a diagnosis in ~99:84% of these cases; these results suggest that our tool is applicable and efficient in most cases.
Rotations and poses are ubiquitous throughout many fields of science and engineering such as robotics, aerospace, computer vision and graphics. In this paper, we provide a complete characterization of rotations and poses in terms of the eigenstructure of their matrix Lie group representations, SO(3), SE(3) and Ad(SE(3)). An eigendecomposition of the pose representations reveals that they can be cast into a form very similar to that of rotations although the structure of the former can vary depending on the relative nature of the translation and rotation involved. Understanding the eigenstructure of these important quantities has merit in and of itself but it is also essential to appreciating such practical results as the minimal polynomial for rotations and poses and the calculation of Jacobians; moreover, we can speak of a principal-axis pose in much the same manner that we can of a principal-axis rotation.
The fusion of multi-modal sensors has become increasingly popular in autonomous driving and intelligent robots since it can provide richer information than any single sensor, enhance reliability in complex environments. Multi-sensor extrinsic calibration is one of the key factors of sensor fusion. However, such calibration is difficult due to the variety of sensor modalities and the requirement of calibration targets and human labor. In this paper, we demonstrate a new targetless cross-modal calibration framework by focusing on the extrinsic transformations among stereo cameras, thermal cameras, and laser sensors. Specifically, the calibration between stereo and laser is conducted in 3D space by minimizing the registration error, while the thermal extrinsic to the other two sensors is estimated by optimizing the alignment of the edge features. Our method requires no dedicated targets and performs the multi-sensor calibration in a single shot without human interaction. Experimental results show that the calibration framework is accurate and applicable in general scenes.
In this paper, we work uniform error bounds for proper orthogonal decomposition reduced order modeling (POD-ROM) of Burgers equation, considering difference quotients (DQs), introduced in [23]. In particular, we study the optimality behavior of the DQ ROM error bounds by considering $L^2(\Omega)$ and $H^1_0(\Omega)$ POD spaces. We present some meaningful numerical tests checking the optimality error behavior. Based on our numerical observations, noDQ POD-ROM errors have an optimal behavior, while DQ POD-ROM errors demonstrate an optimality/super-optimality behavior. It is conjectured that this possibly occurs because the DQ inner products allow the time dependency in the ROM spaces to take into account.
We examine the following problem: given a collection of Clifford gates, describe the set of unitaries generated by circuits composed of those gates. Specifically, we allow the standard circuit operations of composition and tensor product, as well as ancillary workspace qubits as long as they start and end in states uncorrelated with the input, which rule out common "magic state injection" techniques that make Clifford circuits universal. We show that there are exactly 57 classes of Clifford unitaries and present a full classification characterizing the gate sets which generate them. This is the first attempt at a quantum extension of the classification of reversible classical gates introduced by Aaronson et al., another part of an ambitious program to classify all quantum gate sets. The classification uses, at its center, a reinterpretation of the tableau representation of Clifford gates to give circuit decompositions, from which elementary generators can easily be extracted. The 57 different classes are generated in this way, 30 of which arise from the single-qubit subgroups of the Clifford group. At a high level, the remaining classes are arranged according to the bases they preserve. For instance, the CNOT gate preserves the X and Z bases because it maps X-basis elements to X-basis elements and Z-basis elements to Z-basis elements. The remaining classes are characterized by more subtle tableau invariants; for instance, the T_4 and phase gate generate a proper subclass of Z-preserving gates.
Estimation of a conditional mean (linking a set of features to an outcome of interest) is a fundamental statistical task. While there is an appeal to flexible nonparametric procedures, effective estimation in many classical nonparametric function spaces (e.g., multivariate Sobolev spaces) can be prohibitively difficult -- both statistically and computationally -- especially when the number of features is large. In this paper, we present (penalized) sieve estimators for regression in nonparametric tensor product spaces: These spaces are more amenable to multivariate regression, and allow us to, in-part, avoid the curse of dimensionality. Our estimators can be easily applied to multivariate nonparametric problems and have appealing statistical and computational properties. Moreover, they can effectively leverage additional structures such as feature sparsity. In this manuscript, we give theoretical guarantees, indicating that the predictive performance of our estimators scale favorably in dimension. In addition, we also present numerical examples to compare the finite-sample performance of the proposed estimators with several popular machine learning methods.
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