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Branching programs are quite popular for studying time-space lower bounds. Bera et al. recently introduced the model of generalized quantum branching program aka. GQBP that generalized two earlier models of quantum branching programs. In this work we study a restricted version of GQBP with the motivation of proving bounds on the query-space requirement of quantum-query circuits. We show the first explicit query-space lower bound for our restricted version. We prove that the well-studied OR$_n$ decision problem, given a promise that at most one position of an $n$-sized Boolean array is a 1, satisfies the bound $Q^2 s = \Omega(n^2)$, where $Q$ denotes the number of queries and $s$ denotes the width of the GQBP. We then generalize the problem to show that the same bound holds for deciding between two strings with a constant Hamming distance; this gives us query-space lower bounds on problems such as Parity and Majority. Our results produce an alternative proof of the $\Omega(\sqrt{n})$-lower bound on the query complexity of any non-constant symmetric Boolean function., Comment: 20 pages, 3 figures

Classical branching programs are studied to understand the space complexity of computational problems. Prior to this work, Nakanishi and Ablayev had separately defined two different quantum versions of branching programs that we refer to as NQBP and AQBP. However, none of them, to our satisfaction, captures the intuitive idea of being able to query different variables in superposition in one step of a branching program traversal. Here we propose a quantum branching program model, referred to as GQBP, with that ability. To motivate our definition, we explicitly give examples of GQBP for n-bit Deutsch-Jozsa, n-bit Parity, and 3-bit Majority with optimal lengths. We the show several equivalences, namely, between GQBP and AQBP, GQBP and NQBP, and GQBP and query complexities (using either oracle gates and a QRAM to query input bits). In way this unifies the different results that we have for the two earlier branching programs, and also connects them to query complexity. We hope that GQBP can be used to prove space and space-time lower bounds for quantum solutions to combinatorial problems., Comment: 21 pages

The agnostic setting is the hardest generalization of the PAC model since it is akin to learning with adversarial noise. In this paper, we give a poly$(n,t,{\frac{1}{\varepsilon}})$ quantum algorithm for learning size $t$ decision trees with uniform marginal over instances, in the agnostic setting, without membership queries. Our algorithm is the first algorithm (classical or quantum) for learning decision trees in polynomial time without membership queries. We show how to construct a quantum agnostic weak learner by designing a quantum version of the classical Goldreich-Levin algorithm that works with strongly biased function oracles. We show how to quantize the agnostic boosting algorithm by Kalai and Kanade (NIPS 2009) to obtain the first efficient quantum agnostic boosting algorithm. Our quantum boosting algorithm has a polynomial improvement in the dependence of the bias of the weak learner over all adaptive quantum boosting algorithms while retaining the standard speedup in the VC dimension over classical boosting algorithms. We then use our quantum boosting algorithm to boost the weak quantum learner we obtained in the previous step to obtain a quantum agnostic learner for decision trees. Using the above framework, we also give quantum decision tree learning algorithms for both the realizable setting and random classification noise model, again without membership queries., Comment: Accepted at AISTATS 2024

Amplitude filtering is concerned with identifying basis-states in a superposition whose amplitudes are greater than a specified threshold; probability filtering is defined analogously for probabilities. Given the scarcity of qubits, the focus of this work is to design log-space algorithms for them. Both algorithms follow a similar pattern of estimating the amplitude (or, probability for the latter problem) of each state, in superposition, then comparing each estimate against the threshold for setting up a flag qubit upon success, finally followed by amplitude amplification of states in which the flag is set. We show how to implement each step using very few qubits by designing three subroutines. Our first algorithm performs amplitude amplification even when the "good state'' operator has a small probability of being incorrect -- here we improve upon the space complexity of the previously known algorithms. Our second algorithm performs "true amplitude estimation'' in roughly the same complexity as that of "amplitude estimation'', which actually estimates a probability instead of an amplitude. Our third algorithm is for performing amplitude estimation in parallel (superposition) which is difficult when each estimation branch involves different oracles. As an immediate reward, we observed that the above algorithms for the filtering problems directly improved the upper bounds on the space-bounded query complexity of problems such as non-linearity estimation of Boolean functions and $k$-distinctness., Comment: 35 pages

Categorical attributes are those that can take a discrete set of values, e.g., colours. This work is about compressing vectors over categorical attributes to low-dimension discrete vectors. The current hash-based methods compressing vectors over categorical attributes to low-dimension discrete vectors do not provide any guarantee on the Hamming distances between the compressed representations. Here we present FSketch to create sketches for sparse categorical data and an estimator to estimate the pairwise Hamming distances among the uncompressed data only from their sketches. We claim that these sketches can be used in the usual data mining tasks in place of the original data without compromising the quality of the task. For that, we ensure that the sketches also are categorical, sparse, and the Hamming distance estimates are reasonably precise. Both the sketch construction and the Hamming distance estimation algorithms require just a single-pass; furthermore, changes to a data point can be incorporated into its sketch in an efficient manner. The compressibility depends upon how sparse the data is and is independent of the original dimension -- making our algorithm attractive for many real-life scenarios. Our claims are backed by rigorous theoretical analysis of the properties of FSketch and supplemented by extensive comparative evaluations with related algorithms on some real-world datasets. We show that FSketch is significantly faster, and the accuracy obtained by using its sketches are among the top for the standard unsupervised tasks of RMSE, clustering and similarity search., Comment: Accepted in IEEE Transactions on Knowledge and Data Engineering. Copyright IEEE, 1969

In this work, we present a dimensionality reduction algorithm, aka. sketching, for categorical datasets. Our proposed sketching algorithm Cabin constructs low-dimensional binary sketches from high-dimensional categorical vectors, and our distance estimation algorithm Cham computes a close approximation of the Hamming distance between any two original vectors only from their sketches. The minimum dimension of the sketches required by Cham to ensure a good estimation theoretically depends only on the sparsity of the data points - making it useful for many real-life scenarios involving sparse datasets. We present a rigorous theoretical analysis of our approach and supplement it with extensive experiments on several high-dimensional real-world data sets, including one with over a million dimensions. We show that the Cabin and Cham duo is a significantly fast and accurate approach for tasks such as RMSE, all-pairs similarity, and clustering when compared to working with the full dataset and other dimensionality reduction techniques.

Boosting is an ensemble learning method that converts a weak learner into a strong learner in the PAC learning framework. Freund and Schapire designed the Godel prize-winning algorithm named AdaBoost that can boost learners, which output binary hypotheses. Recently, Arunachalam and Maity presented the first quantum boosting algorithm with similar theoretical guarantees. Their algorithm, which we refer to as QAdaBoost henceforth, is a quantum adaptation of AdaBoost and only works for the binary hypothesis case. QAdaBoost is quadratically faster than AdaBoost in terms of the VC-dimension of the hypothesis class of the weak learner but polynomially worse in the bias of the weak learner. Izdebski et al. posed an open question on whether we can boost quantum weak learners that output non-binary hypothesis. In this work, we address this open question by developing the QRealBoost algorithm which was motivated by the classical RealBoost algorithm. The main technical challenge was to provide provable guarantees for convergence, generalization bounds, and quantum speedup, given that quantum subroutines are noisy and probabilistic. We prove that QRealBoost retains the quadratic speedup of QAdaBoost over AdaBoost and further achieves a polynomial speedup over QAdaBoost in terms of both the bias of the learner and the time taken by the learner to learn the target concept class. Finally, we perform empirical evaluations on QRealBoost and report encouraging observations on quantum simulators by benchmarking the convergence performance of QRealBoost against QAdaBoost, AdaBoost, and RealBoost on a subset of the MNIST dataset and Breast Cancer Wisconsin dataset., Comment: 24 pages, 2 figures, 1 table

Representation learning using network embedding has received tremendous attention due to its efficacy to solve downstream tasks. Popular embedding methods (such as deepwalk, node2vec, LINE) are based on a neural architecture, thus unable to scale on large networks both in terms of time and space usage. Recently, we proposed BinSketch, a sketching technique for compressing binary vectors to binary vectors. In this paper, we show how to extend BinSketch and use it for network hashing. Our proposal named QUINT is built upon BinSketch, and it embeds nodes of a sparse network onto a low-dimensional space using simple bi-wise operations. QUINT is the first of its kind that provides tremendous gain in terms of speed and space usage without compromising much on the accuracy of the downstream tasks. Extensive experiments are conducted to compare QUINT with seven state-of-the-art network embedding methods for two end tasks - link prediction and node classification. We observe huge performance gain for QUINT in terms of speedup (up to 7000x) and space saving (up to 80x) due to its bit-wise nature to obtain node embedding. Moreover, QUINT is a consistent top-performer for both the tasks among the baselines across all the datasets. Our empirical observations are backed by rigorous theoretical analysis to justify the effectiveness of QUINT. In particular, we prove that QUINT retains enough structural information which can be used further to approximate many topological properties of networks with high confidence., Comment: Accepted in IEEE TKDE

We introduce the problem of determining if the mode of the output distribution of a quantum circuit (given as a black-box) is larger than a given threshold, named HighDist, and a similar problem based on the absolute values of the amplitudes, named HighAmp. We design quantum algorithms for promised versions of these problems whose space complexities are logarithmic in the size of the domain of the distribution, but the query complexities are independent. Using these, we further design algorithms to estimate the largest probability and the largest amplitude among the output distribution of a quantum black-box. All of these allow us to improve the query complexity of a few recently studied problems, namely, $k$-distinctness and its gapped version, estimating the largest frequency in an array, estimating the min-entropy of a distribution, and the non-linearity of a Boolean function, in the $\tilde{O}(1)$-qubits scenario. The time-complexities of almost all of our algorithms have a small overhead over their query complexities making them efficiently implementable on currently available quantum backends., Comment: 34 Pages, 3 figures

Non-linearity of a Boolean function indicates how far it is from any linear function. Despite there being several strong results about identifying a linear function and distinguishing one from a sufficiently non-linear function, we found a surprising lack of work on computing the non-linearity of a function. The non-linearity is related to the Walsh coefficient with the largest absolute value; however, the naive attempt of picking the maximum after constructing a Walsh spectrum requires $\Theta(2^n)$ queries to an $n$-bit function. We improve the scenario by designing highly efficient quantum and randomised algorithms to approximate the non-linearity allowing additive error, denoted $\lambda$, with query complexities that depend polynomially on $\lambda$. We prove lower bounds to show that these are not very far from the optimal ones. The number of queries made by our randomised algorithm is linear in $n$, already an exponential improvement, and the number of queries made by our quantum algorithm is surprisingly independent of $n$. Our randomised algorithm uses a Goldreich-Levin style of navigating all Walsh coefficients and our quantum algorithm uses a clever combination of Deutsch-Jozsa, amplitude amplification and amplitude estimation to improve upon the existing quantum versions of the Goldreich-Levin technique., Comment: Accepted in ACM Transactions on Quantum Computing

The Quantum Alternating Operator Ansatz (QAOA+) framework has recently gained attention due to its ability to solve discrete optimization problems on noisy intermediate-scale quantum (NISQ) devices in a manner that is amenable to derivation of worst-case guarantees. We design a technique in this framework to tackle a few problems over maximal matchings in graphs. Even though maximum matching is polynomial-time solvable, most counting and sampling versions are #P-hard. We design a few algorithms that generates superpositions over matchings allowing us to sample from them. In particular, we get a superposition over all possible matchings when given the empty state as input and a superposition over all maximal matchings when given the W -states as input. Our main result is that the expected size of the matchings corresponding to the output states of our QAOA+ algorithm when ran on a 2-regular graph is greater than the expected matching size obtained from a uniform distribution over all matchings. This algorithm uses a W -state as input and we prove that this input state is better compared to using the empty matching as the input state., Comment: 9 pages, 3 figures, 3 charts

Journal Impact Factor is a popular metric for determining the quality of a journal in academia. The number of citations received by a journal is a crucial factor in determining the impact factor, which may be misused in multiple ways. Therefore, it is crucial to detect citation anomalies for further identifying manipulation and inflation of impact factor. Citation network models the citation relationship between journals in terms of a directed graph. Detecting anomalies in the citation network is a challenging task which has several applications in spotting citation cartels and citation stack and understanding the intentions behind the citations. In this paper, we present a novel approach to detect the anomalies in a journal-level scientific citation network, and compare the results with the existing graph anomaly detection algorithms. Due to the lack of proper ground-truth, we introduce a journal-level citation anomaly dataset which consists of synthetically injected citation anomalies and use it to evaluate our methodology. Our method is able to predict the anomalous citation pairs with a precision of 100\% and an F1-score of 86%. We further categorize the detected anomalies into various types and reason out possible causes. We also analyze our model on the Microsoft Academic Search dataset - a real-world citation dataset and interpret our results using a case study, wherein our results resemble the citations and SCImago Journal Rank (SJR) rating-change charts, thus indicating the usefulness of our method. We further design `Journal Citation Analysis Tool', an interactive web portal which, given the citation network as an input, shows the journal-level anomalous citation patterns and helps users analyze citation patterns of a given journal over the years., Comment: Accepted: ACM/IEEE Joint Conference on Digital Libraries (JCDL)

The ubiquity of GPS enabled devices result into the generation of an enormous amount of user movement data consisting of a sequence of locations together with activities performed at those locations. Such data, commonly known as {\it activity-trajectory data}, can be analysed to find common user movements and preferred activities, which will have tremendous business value. However, various countries have now introduced data privacy regulations that make it mandatory to anonymize any data before releasing it. This makes it difficult to look for patterns as the existing mining techniques may not be directly applicable over anonymized data. User locations in an activity-trajectory dataset are anonymized to regions of different shapes and sizes making them uncomparable; therefore, it is unclear how to define suitable patterns over those regions. In this paper, we propose a top-k pattern mining technique called TopKMintra that employs a pre-processing step to transform anonymized activity-trajectory into an intermediate representation that address the above problem. Unlike usual sequential data, activity-trajectory data is 2-dimensional that can lead to generation of duplicate patterns. To handle this, TopKMintra restricts arbitrary extensions in the two dimensions by imposing an order over the search space; this also helps in addressing the common problem of updating the threshold in top-k pattern mining algorithms during various stages. We perform extensive experiments on real datasets to demonstrate the efficiency and the effectiveness of TopKMintra. Our results show that even after anonymization, certain patterns may remain in a dataset and those could be mined efficiently.

High-utility itemset mining finds itemsets from a transaction database with utility no less than a fixed user-defined threshold. The utility of an itemset is defined as the sum of the utilities of its item. Several algorithms were proposed to mine high-utility itemsets. However, no state-of-the-art algorithm performs consistently good across dense and sparse datasets. In this paper, we propose a novel data structure called Utility-Tree, and a tree-based algorithm called UT-Miner that mines high-utility itemsets in one-phase only without generating any candidates and uses a lightweight construction method to reduce the cost of creating projected databases during the search space exploration. The transaction information is stored compactly with every node of the Utility-Tree, and the information is computed efficiently during the recursive invocation of the algorithm. Experimental results on several real-life dense and sparse datasets reveal that UT-Miner is among the top-performing efficient algorithms across different datasets.

Recent advancement of the WWW, IOT, social network, e-commerce, etc. have generated a large volume of data. These datasets are mostly represented by high dimensional and sparse datasets. Many fundamental subroutines of common data analytic tasks such as clustering, classification, ranking, nearest neighbour search, etc. scale poorly with the dimension of the dataset. In this work, we address this problem and propose a sketching (alternatively, dimensionality reduction) algorithm -- $\binsketch$ (Binary Data Sketch) -- for sparse binary datasets. $\binsketch$ preserves the binary version of the dataset after sketching and maintains estimates for multiple similarity measures such as Jaccard, Cosine, Inner-Product similarities, and Hamming distance, on the same sketch. We present a theoretical analysis of our algorithm and complement it with extensive experimentation on several real-world datasets. We compare the performance of our algorithm with the state-of-the-art algorithms on the task of mean-square-error and ranking. Our proposed algorithm offers a comparable accuracy while suggesting a significant speedup in the dimensionality reduction time, with respect to the other candidate algorithms. Our proposal is simple, easy to implement, and therefore can be adopted in practice.

We give a technique to reduce the error probability of quantum algorithms that determine whether its input has a specified property of interest. The standard process of reducing this error is statistical processing of the results of multiple independent executions of an algorithm. Denoting by $\rho$ an upper bound of this probability (wlog., assume $\rho \le \frac{1}{2}$), classical techniques require $O(\frac{\rho}{[(1-\rho) - \rho]^2})$ executions to reduce the error to a negligible constant. We investigated when and how quantum algorithmic techniques like amplitude amplification and estimation may reduce the number of executions. On one hand, the former idea does not directly benefit algorithms that can err on both yes and no answers and the number of executions in the latter approach is $O(\frac{1}{(1-\rho) - \rho})$. We propose a novel approach named as {\em Amplitude Separation} that combines both these approaches and achieves $O(\frac{1}{\sqrt{1-\rho} - \sqrt{\rho}})$ executions that betters existing approaches when the errors are high. In the Multiple-Weight Decision Problem, the input is an $n$-bit Boolean function $f()$ given as a black-box and the objective is to determine the number of $x$ for which $f(x)=1$, denoted as $wt(f)$, given some possible values $\{w_1, \ldots, w_k\}$ for $wt(f)$. When our technique is applied to this problem, we obtain the correct answer, maybe with a negligible error, using $O(\log_2 k \sqrt{2^n})$ calls to $f()$ that shows a quadratic speedup over classical approaches and currently known quantum algorithms.

High-utility Itemset Mining (HUIM) finds itemsets from a transaction database with utility no less than a user-defined threshold where the utility of an itemset is defined as the sum of the item-wise utilities. In this paper, we generalize this notion to utility functions that need not be a simple sum of individual utilities. In particular, we study generalized utility functions that are subadditive and monotone (SM). We also describe a novel function that allows us to include external information in the form of a relationship graph for computing utility. Next, we focus on algorithms for HUIM problems with SM utility functions. We note that the existing HUIM algorithms use upper-bounds like "Transaction Weighted Utility" and "Exact-Utility, Remaining-Utility" for efficient search-space exploration. We derive analogous and tighter upper-bounds for SM utility functions. We design a novel inverted-list data structure called SMI-list and a new algorithm called SM-Miner to mine HUIs for SM functions. We explain how existing tree-based and projection-based HUIM algorithms can be adapted using these bounds. We experimentally compare adaptations of some of the latest HUIM algorithms and point out some caveats that should be kept in mind while handling utility functions that allow integration of domain knowledge with a transaction database., Comment: Pre-print of our paper

In this paper, we propose efficient probabilistic algorithms for several problems regarding the autocorrelation spectrum. First, we present a quantum algorithm that samples from the Walsh spectrum of any derivative of $f()$. Informally, the autocorrelation coefficient of a Boolean function $f()$ at some point $a$ measures the average correlation among the values $f(x)$ and $f(x \oplus a)$. The derivative of a Boolean function is an extension of autocorrelation to correlation among multiple values of $f()$. The Walsh spectrum is well-studied primarily due to its connection to the quantum circuit for the Deutsch-Jozsa problem. We extend the idea to "Higher-order Deutsch-Jozsa" quantum algorithm to obtain points corresponding to large absolute values in the Walsh spectrum of a certain derivative of $f()$. Further, we design an algorithm to sample the input points according to squares of the autocorrelation coefficients. Finally we provide a different set of algorithms for estimating the square of a particular coefficient or cumulative sum of their squares., Comment: Accepted in Indocrypt 2019

The importance of geo-spatial data in critical applications such as emergency response, transportation, agriculture etc., has prompted the adoption of recent GeoSPARQL standard in many RDF processing engines. In addition to large repositories of geo-spatial data -- e.g., LinkedGeoData, OpenStreetMap, etc. -- spatial data is also routinely found in automatically constructed knowledgebases such as Yago and WikiData. While there have been research efforts for efficient processing of spatial data in RDF/SPARQL, very little effort has gone into building end-to-end systems that can holistically handle complex SPARQL queries along with spatial filters. In this paper, we present Streak, a RDF data management system that is designed to support a wide-range of queries with spatial filters including complex joins, top-k, higher-order relationships over spatially enriched databases. Streak introduces various novel features such as a careful identifier encoding strategy for spatial and non-spatial entities, the use of a semantics-aware Quad-tree index that allows for early-termination and a clever use of adaptive query processing with zero plan-switch cost. We show that Streak can scale to some of the largest publicly available semantic data resources such as Yago3 and LinkedGeoData which contain spatial entities and quantifiable predicates useful for result ranking. For experimental evaluations, we focus on top-k distance join queries and demonstrate that Streak outperforms popular spatial join algorithms as well as state of the art end-to-end systems like Virtuoso and PostgreSQL.

Amplitude amplification is a central tool used in Grover's quantum search algorithm and has been used in various forms in numerous quantum algorithms since then. It has been shown to completely eliminate one-sided error of quantum search algorithms where input is accessed in the form of black-box queries. We generalize amplitude amplification for two-sided error quantum algorithm for decision problems in the familiar form where input is accessed in the form of initial states of quantum circuits and where arbitrary projective measurements may be used to ascertain success or failure. This generalization allows us to derive interesting applications of amplitude amplification for distinguishing between two given distributions based on their samples, detection of faults in quantum circuits and eliminating error of one and two-sided quantum algorithms with exact errors.

The Apriori algorithm is a classical algorithm for the frequent itemset mining problem. A significant bottleneck in Apriori is the number of I/O operation involved, and the number of candidates it generates. We investigate the role of LSH techniques to overcome these problems, without adding much computational overhead. We propose randomized variations of Apriori that are based on asymmetric LSH defined over Hamming distance and Jaccard similarity.

Detecting and isolating faults is crucial for synthesis of quantum circuits. Under the single fault assumption that is now routinely accepted in circuit fault analysis, we show that the behaviour of faulty quantum circuits can be fully characterized by the single faulty gate and the corresponding fault model. This allows us to efficiently determine test input states as well as measurement strategy that can be used to detect every single-gate fault using very few test cases and with minimal probability of error; in fact we demonstrate that most commonly used quantum gates can be isolated without any error under the single missing gate fault (SMGF) model. We crucially exploit the quantum nature of circuits to show vast improvement upon the existing works of automatic test pattern generation (ATPG) for quantum circuits.

"Reverse Nearest Neighbor" query finds applications in decision support systems, profile-based marketing, emergency services etc. In this paper, we point out a few flaws in the branch and bound algorithms proposed earlier for computing monochromatic RkNN queries over data points stored in hierarchical index. We give suitable counter examples to validate our claims and propose a correct algorithm for the corresponding problem. We show that our algorithm is correct by identifying necessary conditions behind correctness of algorithms for this problem.

We define and construct efficient depth-universal and almost-size-universal quantum circuits. Such circuits can be viewed as general-purpose simulators for central classes of quantum circuits and can be used to capture the computational power of the circuit class being simulated. For depth we construct universal circuits whose depth is the same order as the circuits being simulated. For size, there is a log factor blow-up in the universal circuits constructed here. We prove that this construction is nearly optimal., Comment: 13 pages