Quantum Algorithms for Some Hidden Shift Problems

It has recently been shown that quantum computers can efficiently solve the Heisenberg hidden subgroup problem, a problem whose classical query complexity is exponential. This quantum algorithm was...

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in $\Z_{p}^{n}$, whenever $p$ is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Translating Coset in a finite solvable group $G$ is reducible to instances of Translating Coset in $G/N$ and $N$, for appropriate normal subgroups $N$ of $G$. Our self-reducibility framework combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.

In this brief chapter we present a quantum algorithm, which can be seen as a generalization of Shor’s algorithm. We can here explain, in a more structural manner, why quantum computers can speed up some computational problems. The so-called Simon’s hidden subgroup problem can be stated in a general form as follows [18]: Input: A finite abelian group G and function ρ: G → R, where R is a finite set. Promise: There exists a nontrivial subgroup H ≤ G such that ρ is constant and distinct on each coset of H. Output: A generating set for H.

Quantum computing is an extremely promising research combining theoretical and experimental quantum physics, mathematics, quantum information theory and computer science. Classical simulation of quantum computations will cover part of the gap between the theoretical mathematical formulation of quantum mechanics and the realization of quantum computers. One of the most important problems in “quantum computer science” is the development of new symbolic languages for quantum computing and the adaptation of existing symbolic languages for classical computing to quantum algorithms. The present paper is devoted to the adaptation of the Mathematica symbolic language to known quantum algorithms and corresponding simulation on the classical computer. Concretely we shall represent in the Mathematica symbolic language Simon’s algorithm, the Deutsch-Josza algorithm, Grover’s algorithm, Shor’s algorithm and quantum error-correcting codes. We shall see that the same framework can be used for all these algorithms. This framework will contain the characteristic property of the symbolic language representation of quantum computing and it will be a straightforward matter to include this framework in future algorithms.KeywordsDeutsch-Josza algorithmGrover’s algorithmQuantum computingQuantum error-correctingShor’s algorithmSimon’s algorithmSimulation of quantum algorithms

In this paper we discuss the Hidden Subgroup Problem (HSP) in relation to post-quantum cryptography. We review the relationship between HSP and other computational problems, discuss an optimal solution method, and review results about the quantum complexity of HSP. We also overview some platforms for group-based cryptosystems. Notably, efficient algorithms for solving HSP in the proposed infinite group platforms are not yet known.

Computer science will be radically transformed if ongoing efforts to build large-scale quantum computers eventually succeed and if the properties of these computers meet optimistic expectations. Nevertheless, computer scientists still lack a thorough understanding of the power of quantum computing, and it is not always clear how best to utilize the power that is understood. This dilemma exists because quantum algorithms are difficult to grasp and even more difficult to write. Despite large-scale international efforts, only a few important quantum algorithms are documented, leaving many essential questions about the potential of quantum algorithms unanswered. These unsolved problems are ideal challenges for the application of automatic programming technologies. Genetic programming techniques, in particular, have already produced several new quantum algorithms and it is reasonable to expect further discoveries in the future. These methods will help researchers to discover how additional practical problems can be solved using quantum computers, and they will also help to guide theoretical work on both the power and limits of quantum computing. This tutorial will provide an introduction to quantum computing and an introduction to the use of evolutionary computation for automatic quantum computer programming. No background in physics or in evolutionary computation will be assumed. While the primary focus of the tutorial will be on general concepts, specific results will also be presented, including human-competitive results produced by genetic programming. Follow-up material is available from the presenter's book, Automatic Quantum Computer Programming: A Genetic Programming Approach, published by Springer and Kluwer Academic Publishers.

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in $\Z_{p}^{n}$, whenever $p$ is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Translating Coset in a finite solvable group $G$ is reducible to instances of Translating Coset in $G/N$ and $N$, for appropriate normal subgroups $N$ of $G$. Our self-reducibility framework combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.

Quantum computing is emerging as a powerful computational paradigm capable of solving infeasible problems for classical computers. This review explores the intersection of quantum computing and artificial intelligence (AI), focusing on how quantum algorithms can enhance AI applications. The purpose of this review is to provide a comprehensive analysis of current advancements in quantum machine learning (QML), quantum optimization, and quantum neural networks, highlighting their implications for data processing, cryptography, and decision-making systems. The scope of the literature includes peer-reviewed articles, conference papers, and technical reports from 2020 to 2025, focusing on quantum algorithms, applications in AI, and hardware limitations. Key findings indicate that quantum computing can significantly accelerate data processing tasks, optimize complex problem-solving techniques, and improve the efficiency of AI models. However, challenges such as quantum noise, error correction, and hardware scalability remain substantial barriers to widespread adoption. This review concludes with discussing future research directions, including hybrid quantum-classical systems and advanced quantum error correction methods. This study serves as a foundational reference for researchers and practitioners aiming to leverage quantum computing for AI-driven applications. 3. Keywords Quantum Computing, Artificial Intelligence, Quantum Machine Learning, Quantum Algorithms, Data Processing, Optimization, Quantum Neural Networks

Quantum computing (QC) is not a futuristic notion in agriculture, though its full potential has yet to be realized. QC is an emerging field at the intersection of physics and computer science that holds immense potential to revolutionize various sectors, including agriculture production and artificial intelligence (AI) modeling. While QC is still in the early stages of development and practical applications within agriculture are not yet widespread, researchers are actively exploring its potential benefits in various agricultural domains, including crop optimization, livestock breeding, and environmental monitoring. QC harnesses the principles of quantum mechanics to perform computations using quantum bits or qubits, which can exist in multiple states simultaneously. Unlike classical computers, which rely on binary bits representing 0 or 1, quantum computers exploit phenomena such as superposition and entanglement to process information in parallel, potentially offering exponential speedup for certain types of problems. In agriculture production, particularly in animal science, QC offers promising avenues for optimizing processes and enhancing productivity. Quantum algorithms can analyze vast amounts of genomic data to improve breeding programs, leading to the development of more resilient and productive livestock breeds. Furthermore, QC can facilitate precision farming techniques by modeling complex environmental factors and animal behavior to optimize feeding strategies, disease management, and overall farm management practices. Moreover, QC can significantly benefit AI modeling by accelerating computations and enabling more efficient training of AI models. Quantum algorithms can enhance the performance of AI algorithms in various tasks, including pattern recognition, natural language processing, and predictive analytics. By leveraging quantum-enhanced optimization algorithms, AI models can achieve better convergence and accuracy, leading to more effective decision-making and problem-solving capabilities. While hybrid intelligent models also represent a novel frontier in agriculture, QC has the potential to expedite the merging of mechanistic and AI modeling paradigms, facilitating a more holistic understanding of complex systems in agriculture and beyond. By integrating mechanistic models, which describe the underlying physical processes, with AI models, which learn patterns from data, quantum computing can enable comprehensive simulations and predictions of agricultural systems. This fusion of modeling paradigms can lead to more accurate and robust predictions of crop yields, livestock performance, and environmental impacts, facilitating informed decision-making for farmers and policymakers. The application of QC in agriculture, however, requires interdisciplinary collaborations between physicists, computer scientists, agronomists/animal scientists, and AI researchers. These collaborations can drive the development of quantum algorithms tailored to agricultural applications, the integration of quantum-enhanced AI techniques into existing modeling frameworks, and the deployment of QC resources in real-world agricultural systems. Ultimately, harnessing the power of QC holds the potential to revolutionize agriculture production practices, including regenerative agriculture, and advance AI modeling capabilities, paving the way for a more sustainable and efficient agricultural industry.

Circuit-based quantum computing (CBQC), also known as the gate model, is currently regarded as the leading paradigm for performing quantum computation and has garnered widespread interest due to its relatively better-established implementation and close intuitive resemblance to classical computation. However, an alternative model that solely utilizes single-qubit measurements on entangled states to perform computation, namely the measurement-based quantum computing (MBQC), has started to attract further research interest as it is deemed more naturally suited for certain quantum hardware architectures, though its actual implementation remains largely underexplored. Recently, there have been preliminary efforts to characterize the specific regimes in which MBQC may potentially be more advantageous compared to CBQC. Nevertheless, a more systematic assessment is necessary to assess the suitability of each model across a broader range of more practical cases. In this work, we present a detailed comparative analysis of CBQC and MBQC, beginning with reviewing the theoretical foundations of each model before highlighting its distinguishing features that set the two paradigms apart. Furthermore, building on previous work, we propose a generalized regime analysis that more accurately captures the conditions under which a given quantum algorithm is more favorably executed in MBQC or CBQC. Subsequently, we identify critical parameters that likely determine whether a quantum algorithm is better suited to CBQC or MBQC, categorizing detailed key factors and performance metrics into layers of quantum computation. Next, we propose a decision flow that, given specific hardware constraints and algorithm requirements, guides the selection to the most appropriate model, discussing suitability in two representative worked examples of quantum algorithm applications: Quantum Fourier Transform (QFT) and Variational Quantum Algorithm (VQE). Last, we discuss the potential to combine both models as a hybrid approach to leverage the best of both worlds and explore its benefits for practical quantum computing applications.

Chapter 5 - Quantum Algorithms and Methods

The Hidden Subgroup Problem (HSP) generalises many problems that are candidates to be NP-intermediate. It was shown that the decision version of HSP belongs to the zero-knowledge complexity class HVPZK and that, if the size of the group is known, it also belongs to NISZK. We show that whenever we can sample uniformly at random elements of the group and of a set, with the same size of the group, that contains the image of the function that hides the subgroup, the problem is in NIPZK1 (i.e. NIPZK with perfect completeness). As a second contribution, we show that NIPZK1 has a complete promise problem that is a restricted version of a complete promise problem for the NIPZK class.

Besides the Hidden Subgroup Problem, the second large class of quantum speed-ups is for functions with constant-sized 1-certificates. This includes the OR function, solvable by the Grover algorithm, the distinctness, the triangle and other problems. The usual way to solve them is by quantum walk on the Johnson graph. We propose a solution for the same problems using span programs. The span program is a computational model equivalent to the quantum query algorithm in its strength, and yet very different in its outfit. We prove the power of our approach by designing a quantum algorithm for the triangle problem with query complexity $O(n^{35/27})$ that is better than $O(n^{13/10})$ of the best previously known algorithm by Magniez et al.

As we begin to reach the limits of classical computing, quantum computing has emerged as a technology that has captured the imagination of the scientific world. While for many years, the ability to execute quantum algorithms was only a theoretical possibility, recent advances in hardware mean that quantum computing devices now exist that can carry out quantum computation on a limited scale. Thus, it is now a real possibility, and of central importance at this time, to assess the potential impact of quantum computers on real problems of interest. One of the earliest and most compelling applications for quantum computers is Feynman's idea of simulating quantum systems with many degrees of freedom. Such systems are found across chemistry, physics, and materials science. The particular way in which quantum computing extends classical computing means that one cannot expect arbitrary simulations to be sped up by a quantum computer, thus one must carefully identify areas where quantum advantage may be achieved. In this review, we briefly describe central problems in chemistry and materials science, in areas of electronic structure, quantum statistical mechanics, and quantum dynamics that are of potential interest for solution on a quantum computer. We then take a detailed snapshot of current progress in quantum algorithms for ground-state, dynamics, and thermal-state simulation and analyze their strengths and weaknesses for future developments.

Low rank approximation in simulations of quantum algorithms