Quantum search of partially ordered sets

Query complexity of generalized Simon's problem

A probabilistic version of the Bernstein-Vazirani problem (which is a generalization of the original Bernstein-Vazirani problem) and a quantum algorithm to solve it are proposed. The problem involves finding one or more secret keys from a set of multiple secret keys (encoded in binary form) using a quantum oracle. From a set of multiple unknown keys, the proposed quantum algorithm is capable of (a) obtaining any key (with certainty) using a single query to the probabilistic oracle and (b) finding all keys with a high probability (approaching 1 in the limiting case). In contrast, a classical algorithm will be unable to find even a single bit of a secret key with certainty (in the general case). Owing to the probabilistic nature of the oracle, a classical algorithm can only be useful in obtaining limiting probability distributions of $ 0 $ and $ 1 $ for each bit-position of secret keys (based on multiple oracle calls) and this information can further be used to infer some estimates on the distribution of secret keys based on combinatorial considerations. For comparison, it is worth noting that a classical algorithm can be used to exactly solve the original Bernstein-Vazirani problem (involving a deterministic oracle and a single hidden key containing $n$ bits) with a query complexity of $\mathcal{O}(n)$. An interesting class of problems similar to the probabilistic version of the Bernstein-Vazirani problem can be construed, where quantum algorithms can provide efficient solutions with certainty or with a high degree of confidence and classical algorithms would fail to do so.

Abstract. The aim of this paper is to describe existing approaches to automation in enterprise management systems of invoice factoring operations. The relevance of this problem is estimated by the volume and dynamics of growth of the invoice factoring market in the Russian Federation. The existing performance problems inherent in software built on the basis of classical algorithms are considered, and their optimization by using quantum algorithms is proposed. Attention is focused on the problem of parallel computing as the main feature, which allows to achieve a significant increase in computing power. Possible way of solution, based on the joint use of quantum and classical approach within one software complex is given. Application of quantum Monte Carlo method for the purpose of improvement of the automated information system of the enterprise is described. The received result allows to draw a conclusion about perspective of use of the given technique for construction of the combined quantum and classical algorithms in systems of management of factoring operations.

The computation of the diameter is one of the most central problems in distributed computation. In the standard CONGEST model, in which two adjacent nodes can exchange $O(\log n)$ bits per round (here $n$ denotes the number of nodes of the network), it is known that exact computation of the diameter requires $\tilde \Omega(n)$ rounds, even in networks with constant diameter. In this paper we investigate quantum distributed algorithms for this problem in the quantum CONGEST model, where two adjacent nodes can exchange $O(\log n)$ quantum bits per round. Our main result is a $\tilde O(\sqrt{nD})$-round quantum distributed algorithm for exact diameter computation, where $D$ denotes the diameter. This shows a separation between the computational power of quantum and classical algorithms in the CONGEST model. We also show an unconditional lower bound $\tilde \Omega(\sqrt{n})$ on the round complexity of any quantum algorithm computing the diameter, and furthermore show a tight lower bound $\tilde \Omega(\sqrt{nD})$ for any distributed quantum algorithm in which each node can use only $\textrm{poly}(\log n)$ quantum bits of memory.

Only a few classes of quantum algorithms are known which provide a speed-up over classical algorithms. However, these and any new quantum algorithms provide important motivation for the development of quantum computers. In this article new quantum algorithms are given which are based on quantum state tomography. These include an algorithm for the calculation of several quantum mechanical expectation values and an algorithm for the determination of polynomial factors. These quantum algorithms are important in their own right. However, it is remarkable that these quantum algorithms are immune to a large class of errors. We describe these algorithms and provide conditions for immunity.

Longest common substring (LCS) is an important text processing problem, which has recently been investigated in the quantum query model. The decision version of this problem, LCS with threshold \(d\) , asks whether two length- \(n\) input strings have a common substring of length \(d\) . The two extreme cases, \(d=1\) and \(d=n\) , correspond, respectively to Element Distinctness and Unstructured Search, two fundamental problems in quantum query complexity. However, the intermediate case \(1\ll d\ll n\) was not fully understood. We show that the complexity of LCS with threshold \(d\) smoothly interpolates between the two extreme cases up to \(n^{o(1)}\) factors: — LCS with threshold \(d\) has a quantum algorithm in \(n^{2/3+o(1)}/d^{1/6}\) query complexity and time complexity, and requires at least \(\Omega(n^{2/3}/d^{1/6})\) quantum query complexity. Our result improves upon previous upper bounds \(\widetilde{O}(\min\{n/d^{1/2},n^{2/3}\})\) (Le Gall and Seddighin ITCS 2022, Akmal and Jin SODA 2022), and answers an open question of Akmal and Jin. Our main technical contribution is a quantum speed-up of the powerful String Synchronizing Set technique introduced by Kempa and Kociumaka (STOC 2019). It consistently samples \(n/\tau^{1-o(1)}\) synchronizing positions in the string depending on their length- \(\Theta(\tau)\) contexts, and each synchronizing position can be reported by a quantum algorithm in \(\widetilde{O}(\tau^{1/2+o(1)})\) time. Our quantum string synchronizing set also yields a near-optimal LCE data structure in the quantum setting. As another application of our quantum string synchronizing set, we study the \(k\) -mismatch Matching problem, which asks if the pattern has an occurrence in the text with at most \(k\) Hamming mismatches. Using a structural result of Charalampopoulos et al. (FOCS 2020), we obtain: — \(k\) -mismatch matching has a quantum algorithm with \(k^{3/4}n^{1/2+o(1)}\) query complexity and \(\widetilde{O}(kn^{1/2})\) time complexity. We also observe a non-matching quantum query lower bound of \(\Omega(\sqrt{kn})\) .

We present classical and quantum algorithms for approximating partition functions of classical Hamiltonians at a given temperature. Our work has two main contributions: first, we modify the classical algorithm of Stefankovic, Vempala and Vigoda (J. ACM, 56(3), 2009) to improve its sample complexity; second, we quantize this new algorithm, improving upon the previously fastest quantum algorithm for this problem, due to Harrow and Wei (SODA 2020). The conventional approach to estimating partition functions requires approximating the means of Gibbs distributions at a set of inverse temperatures that form the so-called cooling schedule. The length of the cooling schedule directly affects the complexity of the algorithm. Combining our improved version of the algorithm of Stefankovic, Vempala and Vigoda with the paired-product estimator of Huber (Ann. Appl. Probab., 25(2), 2015), our new quantum algorithm uses a shorter cooling schedule than previously known. This length matches the optimal length conjectured by Stefankovic, Vempala and Vigoda. The quantum algorithm also achieves a quadratic advantage in the number of required quantum samples compared to the number of random samples drawn by the best classical algorithm, and its computational complexity has quadratically better dependence on the spectral gap of the Markov chains used to produce the quantum samples.

Subset-Sum is an NP-complete problem where one must decide if a multiset of $n$ integers contains a subset whose elements sum to a target value $m$. The best-known classical and quantum algorithms run in time $\tilde{O}(2^{n/2})$ and $\tilde{O}(2^{n/3})$, respectively, based on the well-known meet-in-the-middle technique. Here we introduce a novel classical dynamic-programming-based data structure with applications to Subset-Sum and a number of variants, including Equal-Sums (where one seeks two disjoint subsets with the same sum), 2-Subset-Sum (a relaxed version of Subset-Sum where each item in the input set can be used twice in the summation), and Shifted-Sums, a generalization of both of these variants, where one seeks two disjoint subsets whose sums differ by some specified value. Given any modulus $p$, our data structure can be constructed in time $O(n^2p)$, after which queries can be made in time $O(n^2)$ to the lists of subsets summing to any value modulo $p$. We use this data structure in combination with variable-time amplitude amplification and a new quantum pair finding algorithm, extending the quantum claw finding algorithm to the multiple solutions case, to give an $O(2^{0.504n})$ quantum algorithm for Shifted-Sums, an improvement on the best-known $O(2^{0.773n})$ classical running time. Incidentally, we obtain new $\tilde{O}(2^{n/2})$ and $\tilde{O}(2^{n/3})$ classical and quantum algorithms for Subset-Sum, not based on the seminal meet-in-the-middle method. We also study Pigeonhole Equal-Sums and Pigeonhole Modular Equal-Sums, where the existence of a solution is guaranteed by the pigeonhole principle. For the former problem, we give faster classical and quantum algorithms with running time $\tilde{O}(2^{n/2})$ and $\tilde{O}(2^{2n/5})$, respectively. For the more general modular problem, we give a classical algorithm that also runs in time $\tilde{O}(2^{n/2})$.

Variational quantum algorithms (VQAs) are promising hybrid quantum-classical methods designed to leverage the computational advantages of quantum computing while mitigating the limitations of current noisy intermediate-scale quantum (NISQ) hardware. Although VQAs have been demonstrated as proofs of concept, their practical utility in solving real-world problems—and whether quantum-inspired classical algorithms can match their performance—remains an open question. We present a novel application of the variational quantum linear solver (VQLS) and its classical neural quantum states-based counterpart, the variational neural linear solver (VNLS), as key components within a minimum map Newton solver for a complementarity-based rigid-body contact model. We demonstrate using the VNLS that our solver accurately simulates the dynamics of rigid spherical bodies during collision events. These results suggest that quantum and quantum-inspired linear algebra algorithms can serve as viable alternatives to standard linear algebra solvers for modelling certain physical systems.This article is part of the theme issue ‘Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms’.

A quantum computer encodes information in quantum states and runs quantum algorithms to surpass the classical counterparts by exploiting quantum superposition and quantum correlation. Grover's quantum search algorithm is a typical quantum algorithm that proves the superiority of quantum computing over classical computing. It has a quadratic reduction in the query complexity of a database search and is known to be optimal when no a priori information about the elements of the database is provided. In this work we consider a probabilistic Grover search algorithm allowing nonzero probability of failure for a database with a general a priori probability distribution of the elements and minimize the number of oracle calls by optimizing the initial state of the quantum system and the reflection axis of the diffusion operator. The initial state and the reflection axis are allowed to not coincide and thus the quantum search algorithm rotates the quantum system in a three-dimensional subspace spanned by the initial state, the reflection axis, and the search target state in general. The number of oracle calls is minimized by a variational method, and formal results are obtained with the assumption of low failure probability. The results show that for a nonuniform a priori distribution of the database elements, the number of oracle calls can be significantly reduced given a small decrease in the success probability of the quantum search algorithm, leading to a lower average query complexity to find the solution of the search problem. The results are applied to a simple but nontrivial database model with two-value a priori probabilities to show the power of the optimized quantum search algorithm. The paper concludes with a discussion about the generalization to higher-order results that allows for a larger failure probability for the quantum search algorithm.

In this paper, we consider two versions of the Text Assembling problem. We are given a sequence of strings $s^1,\dots,s^n$ of total length $L$ that is a dictionary, and a string $t$ of length $m$ that is a text. The first version of the problem is assembling $t$ from the dictionary. The second version is the ``Shortest Superstring Problem''(SSP) or the ``Shortest Common Superstring Problem''(SCS). In this case, $t$ is not given, and we should construct the shortest string (we call it superstring) that contains each string from the given sequence as a substring.These problems are connected with the sequence assembly method for reconstructing a long DNA sequence from small fragments. For both problems, we suggest new quantum algorithms that work better than their classical counterparts. In the first case, we present a quantum algorithm with $O(m+\log m\sqrt{nL})$ query complexity. In the case of SSP, we present a quantum algorithm with $\tilde{O}(n^3 1.728^n +L +n^{1.5}\sqrt{L})$ query complexity. Here $\tilde{O}$ hides not only constants but logarithms of $L$ and $n$ also.

Many classes of algorithms that suffer from large complexities when implemented on conventional computers may be reformulated resulting in greatly reduced complexity when implemented on quantum computers. The dramatic reductions in complexity for certain types of quantum algorithms coupled with the computationally challenging problems in some bioinformatics problems motivates researchers to devise efficient quantum algorithms for sequence (DNA, RNA, protein) analysis. This chapter shows that the important sequence classification problem in bioinformatics is suitable for formulation as a quantum algorithm. This chapter leverages earlier research for sequence classification based on Extensible Markov Model (EMM) and proposes a quantum computing alternative. The authors utilize sequence family profiles built using EMM methodology which is based on using pre-counted word data for each sequence. Then a new method termed quantum seeding is proposed for generating a key based on high frequency words. The key is applied in a quantum search based on Grover algorithm to determine a candidate set of models resulting in a significantly reduced search space. Given Z as a function of M models of size N, the quantum version of the seeding algorithm has a time complexity in the order of as opposed to O(Z) for the standard classic version for large values of Z.

A dominating set of a graph G = (V, E) is a subset U of its vertices V, such that any vertex of G is either in U, or has a neighbor in U. The dominating-set problem is to find a minimum dominating set in G. Dominating sets are of critical importance for various types of networks/graphs, and find therefore potential applications in many fields. Particularly, in the area of communication, dominating sets are prominently used in the efficient organization of large-scale wireless ad hoc and sensor networks. However, the dominating set problem is also a hard optimization problem and thus currently is not efficiently solvable on classical computers. Here, we propose a biomolecular and a quantum algorithm for this problem, where the quantum algorithm provides a quadratic speedup over any classical algorithm. We show that the dominating set problem can be solved in O(2^{n/2}) queries by our proposed quantum algorithm, where n is the number of vertices in G. We also demonstrate that our quantum algorithm is the best known procedure to date for this problem. We confirm the correctness of our algorithm by executing it on IBM Quantum’s qasm simulator and the Brooklyn superconducting quantum device. And lastly, we show that molecular solutions obtained from solving the dominating set problem are represented in terms of a unit vector in a finite-dimensional Hilbert space.

Quantum Machine Learning (QML) is emerging as a transformative technology in cybersecurity, particularly in anomaly detection for cyber security audits. Traditional machine learning models are effective but face scalability and efficiency limitations as cyber threats grow more sophisticated. QML, leveraging quantum computing’s ability to process and analyze large datasets in parallel, offers potential breakthroughs in identifying anomalous patterns that could signify cyber threats such as data breaches, insider threats, or unauthorized access. Content Analysis Research Methodology used in this research work. This paper explores the integration of QML into anomaly detection systems for cyber security audits, where detecting deviations from normal behavior is crucial. Quantum algorithms, particularly those based on Quantum Support Vector Machines (QSVM), Quantum Neural Networks (QNN), and Quantum Principal Component Analysis (QPCA) can enhance the detection of subtle anomalies that classical algorithms may overlook due to noise or the complex, high-dimensional nature of cyber data. The inherent properties of quantum computing, such as superposition and entanglement, allow for more efficient feature selection and optimization, potentially leading to faster and more accurate anomaly detection. The impact of implementing QML in cyber security audits is profound. First, it enhances detection capabilities by identifying anomalies with greater precision, reducing false positives, and improving response times to cyber incidents. Second, quantum algorithms’ ability to manage exponentially large datasets makes them ideal for environments with extensive data logs, such as enterprise networks and cloud infrastructures. Third, as cyber threats become increasingly adaptive and stealthy, QML offers a dynamic solution that evolves alongside these threats by continuously learning from new patterns of attack. However, practical challenges remain, including the need for quantum hardware advancements, the development of hybrid quantum-classical models, and ensuring the interpretability of quantum models in audit scenarios. Despite these challenges, early research and experimental implementations demonstrate the potential of QML to revolutionize anomaly detection in cybersecurity audits. This paper concludes that while QML is still in its early stages, its application to anomaly detection holds promise for significantly enhancing the effectiveness of cyber security audits. The impact of this technology, when fully realized, could redefine how organizations protect their networks and data from ever-evolving cyber threats, making QML a critical area for future research and development in cybersecurity.

Shor’s quantum algorithm is very important for cryptography, because it can factor large numbers much faster than classical algorithms. In this study, we implement a simulator for Shor’s quantum algorithm on graphic processor units (GPU) and compare our results with Liquid, which is a Microsoft quantum simulation platform, and two classical CPU implementations. We evaluate 10 benchmarks for comparing our GPU implementation with Liquid and single-core implementation. The analysis shows that GPU vector operations are more suitable for Shor’s quantum algorithm. Our GPU kernel function is compute-bound, due to all threads in a block reaching the same element of the state vector. Our implementation has 52.5× speedup over single-core algorithm and 20.5× speedup over Liquid.