In this paper, we address several problems concerning vector spaces enclosed in a given set. Let V be a vector space over a finite field of cardinality c, and let S ⊆ V be a set of vectors. A space enclosed in S is a vector subspace W of V that is also contained in S: W ⊆ S. We focus on enumeration problems, where the task is to list all solutions, and we first provide an algorithm to enumerate all spaces that are enclosed in S. Our algorithm is further adapted to solve two more problems: the enumeration of (inclusion-)maximal enclosed spaces, and the problem of finding an enclosed space of maximum dimension. The latter problem arises in the context of Boolean functions' regularity detection. It can also be seen as a dual version of the well-known linear span: indeed, the span is the minimum-dimension vector space that contains a given set of vectors S, and it is a fundamental concept in linear algebra. Our proposed algorithms are based on the binary partition paradigm, and have total time complexity e^1/2ln cln^2 n - Θ(log n loglog n), where n= ||. The first version, for enumerating all enclosed spaces, also achieves a delay (time between consecutive outputs) of O(n). Our algorithms provide a quadratic speed-up with respect to a brute-force approach, although the speed-up appears even greater in our experimental evaluation on boolean vector spaces.

Polynomial Formal Verification (PFV) ensures that a class of circuits can be verified efficiently by calculating polynomial upper bounds for the resource demands of the verification process. In this paper, we address the PFV of Boolean affine spaces represented by a 2-XOR sum of products. We show that time and space resources remain quadratic in the number of input variables during the entire verification process. Specifically, we prove that the dimensions of ROBDDs and QRBDDs representing a 2-affine space are linear. Furthermore, we prove that all ROBDDs generated during the symbolic simulation of the circuit can be computed in linear time. Finally, we provide an overall quadratic upper bound for the formal verification of QRBDD-based circuits. The experimental results confirm the given bounds.

The ERC-1155 standard introduced on the Ethereum blockchain allows for managing multiple tokens, both fungible and non-fungible, within a single contract. It also supports batch transfers, thereby reducing transaction costs and enabling a more efficient use of blockchain resources. To assess its impact and level of adoption, this paper presents a comprehensive analysis of the ERC-1155 token ecosystem. First, we examine the activity of ERC-1155 contracts and compare the evolution of transfer volumes with those of the two alternative most popular token management standards. Next, we model the economy of each ERC-1155 contract as a directed graph, where nodes represent users and edges denote token transfers. We then study the topological properties of such graphs, analyzing approximately 40,000 networks until the end of 2024. Results indicate that, within our dataset, the adoption of ERC-1155 is growing, although its functionalities are not being fully utilized. Additionally, about 60

Many Boolean functions arising in circuit applications exhibit structural regularities that can be exploited to improve logic synthesis. Among them, autosymmetry is an XOR-based property that allows a function to be studied in a lower-dimensional space through a suitable restriction. In this work, we propose an efficient incremental heuristic for testing autosymmetry directly from And-Inverter Graph representations, without requiring the preliminary construction of a BDD. The method can identify the autosymmetry degree when run to completion, or return a certified lower bound if interrupted, while exploiting algebraic properties of the output to improve performance. Experimental results show the effectiveness of the proposed approach.

Matrix chain multiplication – computing 𝒲 = M^(0)⋯ M^(K-1) where M^(k)∈ℝ^P_k × P_k+1 – arises in scientific computing, machine learning, and graph analysis. Despite the importance of this problem, for chains of distinct matrices, the classical number of operations grows linearly with the chain length K and polynomially in the matrix dimensions. We present Two-Tower Matrix Multiplication, a quantum subroutine that encodes the product 𝒲 of the K matrices into a quantum state in circuit depth 𝒪(max_kpolylog (P_k P_k+1)), which is independent of K within the QRAM-based state-preparation model, whereas the qubit count is 𝒪(∑_klog P_k ); the total gate count remains linear in K, so the gain is in the circuit depth. The construction interleaves state-preparation operators across two layers; within each layer, all operators act on disjoint registers and execute in parallel. This subroutine can be specialized for the chain-vector case, which computes the product of K-1 matrices applied to a vector. We prove the correctness of the subroutine for all K and provide two implementations using the Qiskit and QCLAB frameworks. The subroutine is applicable to any downstream quantum algorithm that operates on a matrix encoded in the statevector, including norm estimation, graph-matrix powers, linear system solving, and quantum machine learning kernels.

Decentralized applications, the driving force behind the new Web3 paradigm, require continuous access to blockchain data. Their adoption, however, is hindered by the constantly increasing size of blockchains and the sequential scan nature of their read operations, which introduce a clear inefficiency bottleneck. Also, the growing amount of data recorded on the blockchain makes resource-constrained light nodes dependent on untrusted full nodes for fetching information, with a consequent need for query authentication protocols ensuring result integrity. Motivated by these reasons, in this paper we propose the skip index, an indexing data structure that allows users to quickly retrieve information simultaneously from multiple blocks of a blockchain. Our solution is also designed to be used as an authenticated data structure to guarantee the integrity of query results for light nodes. We discuss the theoretical properties of skip indices, propose efficient algorithms for their construction and querying, and detail their computational complexity. Finally, we assess the effectiveness of our proposal through an experimental evaluation on the Ethereum blockchain. As a reference use case, we focus on the popular CryptoKitties application and simulate a scenario where users seek to retrieve the events generated by the service. Our experimental results suggest that the use of skip indices offers a constant multiplicative speedup, thanks to search times that are at most logarithmic within a chosen search window. This allows to reduce the number of visited blocks by up to two orders of magnitude if compared to the naive sequential approach currently in use.

This work explores the potential of Quantum Matrix Multiplication (QMM) to accelerate several computational tasks, demonstrating substantial speedups. We present three distinct applications showcasing QMM’s versatility and efficiency. We introduce a novel Hybrid-Quantum Angle Based Outlier Detection (H-QABOD) algorithm, leveraging QMM to efficiently identify outliers in numerical datasets. We evaluate H-QABOD’s performance on both synthetic and real-world datasets, benchmarking it against its classical counterpart. Furthermore, we investigate QMM’s application within quantum linear algebra, specifically for multitrace estimation and Frobenius norm computation.

Bi-decomposition rewrites logic functions as the composition of simpler components. It is related to Boolean division, where a given function is rewritten as the product of a divisor and a quotient, but bi-decomposition can be defined for any Boolean operation of two operands. The key questions are how to find a good divisor and then how to compute the quotient. In this paper we select the divisor by approximation of the original function and then characterize by an incompletely specified function the full flexibility of the quotient for each binary operator. We target area-driven exact bi-decomposition and we apply it to the bi-decomposition of SOP forms. We report experiments that exhibit significant gains in literals of SOP forms when rewritten as bi-decompositions with respect to the product operator. This suggests the application of this framework to other logic forms and binary operations, both for exact and approximate implementations.

The ERC-1155 standard has introduced significant innovations in the management of fungible and non-fungible tokens in Ethereum, enabling batch transfers and the management of multiple tokens within a single contract. This lowers transaction costs and allows for a more efficient use of blockchain resources. To understand its impact and adoption, in this paper we conduct a comprehensive analysis of the ERC-1155 token ecosystem. To this end, we represent the economy of each ERC-1155 contract as a network where nodes represent users and edges symbolize token transfers. First, we study the activity of contracts by number of tokens managed and transfers produced, differentiating between single and batch exchanges. Then, we assess how contract activity has evolved over time, also comparing the overall transfer volume with that of the ERC-20 and ERC-721 standards. Finally, we analyze the topological properties of the networks associated with the top 100 largest ERC-1155 economies, for a total of 1.2 million nodes and 3.5 million edges. We characterize network topologies using five numerical features and study the degree distributions to identify the potential presence of a scale-free property. Results show that, within the analyzed time frame, ERC-1155 adoption is growing, but its functionalities are not being fully exploited and most networks lack both small-world and scale-free properties.

Quantum computing, based on quantum theory, holds great promise as an advanced computational paradigm for achieving fast computations. Quantum algorithms are expected to surpass their classical counterparts in terms of computational complexity for certain tasks, including machine learning. In this paper, we design, implement, and evaluate three hybrid quantum k -Means algorithms, exploiting different degrees of parallelism. Indeed, each algorithm incrementally leverages quantum parallelism to reduce the complexity of the cluster assignment step up to a constant cost. In particular, we exploit quantum phenomena to speed up the computation of distances. The core idea is that the computation of distances between records and centroids can be executed simultaneously, thus saving time, especially for big datasets. We show that our hybrid quantum k -Means algorithms are theoretically faster than the classical algorithm, while experiments suggest that it is possible to obtain comparable clustering results.

Over the past few years, we observed a rethinking of classical artificial intelligence algorithms from a quantum computing perspective. This trend is driven by the peculiar properties of quantum mechanics, which offer the potential to enhance artificial intelligence capabilities, enabling it to surpass the constraints of classical computing. However, redesigning classical algorithms into their quantum equivalents is not straightforward and poses numerous challenges. In this study, we analyze in-depth two orthogonal designs of the quantum K-nearest neighbor classifier. In particular, we show two solutions based on amplitude encoding and basis encoding of data, respectively. These two types of encoding impact the overall structure of the respective algorithms, which employ different distance metrics and show different performances. By breaking down each quantum algorithm, we clarify and compare implementation aspects ranging from data preparation to classification. Eventually, we discuss the difficulties associated with data preparation, the theoretical advantage of quantum algorithms, and their impact on performance with respect to the classical counterpart.

In quantum computing, state preparation techniques for loading classical data into quantum states can be resource-intensive and error-prone, especially on Noisy Intermediate-Scale Quantum (NISQ) devices. This work applies variational learning to compress quantum circuits for state preparation, transforming resource-heavy techniques into hardware-efficient ansatzes suitable for current quantum architectures. This approach enhances the efficiency of state preparation, broadening the applicability of quantum algorithms. We demonstrate significant reductions in circuit size and depth compared to the explicit circuit of the FF-QRAM state preparation. Additionally, we show that the learned ansatzes perform better in noisy environments, with significant fidelity improvements over the explicit state preparation technique under analysis.

We propose an efficient quantum subroutine for matrix multiplication that computes a state vector encoding the entries of the product of two matrices in superposition. The subroutine exploits efficient state preparation techniques and shows a potential speed-up with respect to classical methods. The most important benefit of our subroutine is that it encodes the entries of the matrix product directly in the state vector, which can be used for further computations within the same quantum circuit. All scenarios involving the computation of non-homomorphic functions of the product of two matrices can benefit from our technique. As a possible application, we discuss the computation of the variance of the entries of a matrix product, which can be a useful tool for some machine learning algorithms.

The increasing adoption of tokens on the Ethereum blockchain has given rise to many distinct economic communities whose activity history is publicly accessible. In this paper we study the communities of Ethereum fungible and non-fungible tokens, regulated, respectively, by the ERC-20 and ERC-721 standards. In particular, we focus on token transfers and consider the top 100 largest ERC-20 and ERC-721 ecosystems by number of transfers, modeling them as networks where nodes correspond to participants and edges represent token transfers. We analyze their main topological properties and conduct a clustering-based study to identify groups of graphs with similar topologies. Subsequently, we classify the networks based on the application domain of their corresponding token and investigate whether graphs with similar topologies correspond to tokens within the same domain. We also conduct a temporal analysis of token popularity based on the historical transfer activity. Our findings highlight the existence of common topological properties (e.g., absence of small world effect) across both types of tokens. In contrast, the clustering analysis indicates no evident connection between the token application domain and the structure of the induced transfer networks, with the exception of non-fungible tokens associated with spamming activities.

The transparent nature of public blockchain systems allows for unprecedented access to economic community data. Examples of such communities are the fungible token networks created by the ERC-20 standard on the Ethereum protocol. In this paper we study ERC-20 token networks, where nodes represent users and edges represent fungible token transfers between them. We focus our analysis on the top 100 largest networks, including a total of about 160 million edges and 60 million nodes. After a global analysis of the size and temporal evolution of such networks, we define and study seven features describing their main topological properties. In an attempt to characterize the networks by their topologies, we use the introduced features to cluster the networks together. To evaluate our results, we manually classify each network depending on the application domain of the corresponding contract and measure the homogeneity of the obtained clusterings. Overall, the results appear to indicate a lack of relationship between the scope of a contract and the topological features of the induced networks.

In the classical CMOS technology, EXOR gates are considered expensive and impractical. Recently, the growing relevance of cryptography-related applications and emerging technologies has revived the interest in EXOR gates. In this contexts, it is therefore important to consider network representations that assume EXOR gates explicitly, since the non-EXOR gates have, in general, a higher cost than EXOR ones. As a result, the widely adopted And-Inverter Graph (AIG) logic networks, have recently evolved into the new XOR-AND Graph (XAG) multi-level logic representation, and the logic synthesis on XAGs mainly aims at reducing the number of AND nodes. In general, we call EXOR-based synthesis the synthesis methods that minimize the non-EXOR gates in a Boolean circuit. Projected Sum Of Product, PSOP, decomposition is an EXOR based technique that can be applied to any Boolean function as a very fast pre-processing step for further minimizing the circuit area in standard logic synthesis. In this paper, we exploit PSOP decomposition in EXOR-based synthesis. The experimental results validate the proposed pre-processing method in EXOR-based synthesis, showing an interesting gain in area, within the same time limit.

Quantum computing sets the foundation for new ways of designing algorithms, thanks to the peculiar properties inherited by quantum mechanics. The exploration of this new paradigm faces new challenges concerning which field quantum speedup can be achieved. Toward finding solutions, looking for the design of quantum subroutines that are more efficient than their classical counterpart poses solid pillars to new powerful quantum algorithms. Herewith, we delve into a grounding subroutine, the computation of the variance, whose usefulness spaces across different fields of application, particularly the artificial intelligence (AI) one. Indeed, the finding of the quantum counterpart of these building blocks impacts vertically those algorithms that leverage this metric. In this work, we propose QVAR, a quantum subroutine, to compute the variance that exhibits a logarithmic complexity both in the circuit depth and width, excluding the state preparation cost. With the vision of showing the use of QVAR as a subroutine for new quantum algorithms, we tackle two tasks from the AI domain: feature selection and outlier detection. In particular, we showcase two AI hybrid quantum algorithms that leverage QVAR: the hybrid quantum feature selection (HQFS) algorithm and the quantum outlier detection algorithm (QODA). In this manuscript, we describe the implementation of QVAR, HQFS, and QODA, providing their correctness and complexities and showing the effectiveness of these hybrid quantum algorithms with respect to their classical counterpart.

We propose a new three-level XOR-AND-XOR form for autosymmetric functions, called XORAX expression. In general, a Boolean function $f$ over $n$ variables is $k$ -autosymmetric if it can be projected onto a smaller function $f_{k}$ , which depends on $n-k$ variables only. We show that XORAX expressions can ease the reversible synthesis of autosymmetric functions, producing compact reversible networks, without inserting additional new input lines. Autosymmetry occurs especially for functions that exhibit a regular structure, as for instance arithmetic functions. For this reason, compact reversible networks for autosymmetric functions might be interesting for quantum computing. Experimental results validate the proposed approach.

The classical synthesis method for quantum oracles generally requires a reversible logic synthesis and a quantum compilation step. In the reversible logic synthesis it is important to obtain a compact reversible circuit in order to minimize the quantum cost of the final quantum circuit. In this paper, we exploit function regularities for enabling efficient reversible syn-thesis. In particular, we propose and implement a new method for the quantum synthesis of Dimension reducible Boolean functions. The experimental results validate the proposed approach showing relevant gains in area.