Quantum Algorithms Research Articles

Quantum circuit mutants: Empirical analysis and recommendations

As a new research area, quantum software testing lacks systematic testing benchmarks to assess testing techniques’ effectiveness. Recently, some open-source benchmarks and mutation analysis tools have emerged. However, there is insufficient evidence on how various quantum circuit characteristics (e.g., circuit depth, number of quantum gates), algorithms (e.g., Quantum Approximate Optimization Algorithm), and mutation characteristics (e.g., mutation operators) affect the detection of mutants in quantum circuits. Studying such relations is important to systematically design faulty benchmarks with varied attributes (e.g., the difficulty in detecting a seeded fault) to facilitate assessing the cost-effectiveness of quantum software testing techniques efficiently. To this end, we present a large-scale empirical evaluation with more than 700K faulty benchmarks (quantum circuits) generated by mutating 382 real-world quantum circuits. Based on the results, we provide valuable insights for researchers to define systematic quantum mutation analysis techniques. We also provide a tool to recommend mutants to users based on chosen characteristics (e.g., a quantum algorithm type) and the required difficulty of detecting mutants. Finally, we also provide faulty benchmarks that can already be used to assess the cost-effectiveness of quantum software testing techniques.

Research on Key Technologies for Analyzing Quantum Computing with Deep Learning

In the global field of computational exploration, quantum computing stands as an enigmatic force, poised to transcend classical limitations in certain tasks. Recent years have borne witness to an unprecedented surge in quantum computing research, fostering a vibrant ecosystem of inquiry and innovation. This study delves into the complex fabric of quantum computing, revealing its evolution from theoretical inception to blooming practical applications. It defines the evolution of quantum computing and paves a roadmap for its potential future through a detailed examination of cutting-edge research initiatives. This study takes readers on a tour through the history of quantum computing, revealing the rich fabric of technological advances and theoretical achievements. From the early stages of quantum hardware creation to the refinement of quantum algorithms and error mitigation measures, the story unfolds with utmost clarity. Furthermore, it deciphers the mysterious mechanics of found quantum algorithms, contrasting their abilities with regular computational paradigms. This paper aims to cast light on a largely untrod territory of interdisciplinary cooperation: the intersection of quantum computing and deep learning. It paves the way to a new generation of computational advantage through the incorporation of quantum rules and deep learning platforms where its application can redefine model enhancement, algorithms formulation and data modeling. Concepts such as quantum neural networks and quantum-enhanced optimization, paint the picture of potentials yet to be imagined inside this mutually beneficial relationship. Moving from conjecture consideration, this paper aims to discuss how quantum computing supports deep learning concretely in numerous application areas. Deep learning utilizing quantum computing has the possibility of transcending the previous boundaries in such areas as image identification, drug design and additional optimization. This investigation maps out the way forward for pragmatic realization by deconstructing existing applications and speculatively constructing new ones. At the end of the voyage, the paper looks forward and presents a roadmap to the next phase of voyaging and inventing. The requirement for a cross-disciplinary approach is then presented, and quantum computing scientists and deep learners are encouraged to collaborate. Through this process of collaboration and creativity the unison of quantum computers with deep learning represents the pursuit of further boundaryless violations as the next step to the advancement of tremendously significant scientific accomplishment in the future.

Quantum simulation for time-dependent Hamiltonians—with applications to non-autonomous ordinary and partial differential equations

Abstract Non-autonomous dynamical systems appear in a very wide range of interesting applications, both in classical and quantum dynamics, where in the latter case, it corresponds to having a time-dependent Hamiltonian. Designing quantum algorithms for time-dependent Hamiltonian is generally much more complicated than time-independent Hamiltonian problems, due to the challenge in handling the time-ordering; moreover, almost all existing such algorithms are developed for qubit-based quantum devices. Here we propose an alternative formalism based on Sambe-Howland’s clock that turns any non-autonomous unitary dynamical system into an autonomous unitary system, i.e. quantum system with a time-independent Hamiltonian, in one higher dimension, while keeping time continuous. This makes the simulation with time-dependent Hamiltonians not much more difficult than that of time-independent Hamiltonians, and can also be framed in terms of an analogue quantum system evolving continuously in time. We show how our new quantum protocol for time-dependent Hamiltonians can be performed in a resource-efficient way and without measurements, and can be made possible on either continuous-variable, qubit or hybrid systems. Combined with a technique called Schrödingerisation, this dilation technique can be applied to the quantum simulation of any linear ODEs and PDEs, and nonlinear ODEs and certain nonlinear PDEs, with time-dependent coefficients.

Distributed Quantum Algorithm for the NISQ Era: A Novel Approach to Solving Simon's Problem with Reduced Resources

AbstractDistributed quantum computation has gained significant interest in the noisy intermediate‐scale quantum (NISQ) era. This paradigm requires each computing node to possess a reduced number of qubits and quantum gates. In this study, a Distributed Simon's Algorithm (DSA) is designed to tackle Simon's problem, which entails the discovery of a hidden string of a promised Boolean function , where if and only if or . Specifically, 1) our algorithm is capable of being partitioned into any nodes, where ; 2) the number of queries required by the DSA is , while that of the original Simon's algorithm (SA) is ; 3) the maximum number of qubits required by our approach at a single node is , which is fewer than the qubits required by both the SA and existing distributed Simon's algorithm. Here, denotes the number of computing qubits needed for the ‐th node and satisfies ; 4) the optimal circuit depth of the DSA is , which is reduced compared to the circuit depth of the SA, ; 5) in contrast to currently distributed schemes, the DSA eliminates the need for classical queries; 6) how the DSA solves a specific Simon's problem (e.g., ) is also simulated using MindSpore Quantum, a quantum simulation software. The simulation results show that the DSA features a shallower quantum circuit, thereby demonstrating enhanced resistance to circuit noise. This characteristic makes it more feasible for implementation in the NISQ era.

Towards a Multiqudit Quantum Processor Based on a 171Yb Ion String: Realizing Basic Quantum Algorithms+

We demonstrate a quantum processor based on a 3D linear Paul trap that uses 171Yb+ ions with eight individually controllable four-level qudits (ququarts), which is computationally equivalent to a sixteen-qubit quantum processor. The design of the developed ion trap provides high secular frequencies and a low heating rate, which, together with individual addressing and readout optical systems, allows executing quantum algorithms. In each of the eight ions, we use four electronic levels coupled by E2 optical transition at 435 nm for qudit encoding. We present the results of single- and two-qubit operations benchmarking and realizing basic quantum algorithms, including the Bernstein–Vazirani and Grover’s search algorithms as well as H2 and LiH molecular simulations. Our results pave the way to scalable qudit-based quantum processors using trapped ions.

AI-Powered Algorithm-Centric Quantum Processor Topology Design

Quantum computing promises to revolutionize various fields, yet the execution of quantum programs necessitates an effective compilation process. This involves strategically mapping quantum circuits onto the physical qubits of a quantum processor. The qubits' arrangement, or topology, is pivotal to the circuit's performance, a factor that often defies traditional heuristic or manual optimization methods due to its complexity. In this study, we introduce a novel approach leveraging reinforcement learning to dynamically tailor qubit topologies to the unique specifications of individual quantum circuits, guiding algorithm-driven quantum processor topology design for reducing the depth of mapped circuit, which is particularly critical for the output accuracy on noisy quantum processors. Our method marks a significant departure from previous methods that have been constrained to mapping circuits onto a fixed processor topology. Experiments demonstrate that we have achieved notable enhancements in circuit performance, with a minimum of 20% reduction in circuit depth in 60% of the cases examined, and a maximum enhancement of up to 46%. Furthermore, the pronounced benefits of our approach in reducing circuit depth become increasingly evident as the scale of the quantum circuits increases, exhibiting the scalability of our method in terms of problem size. This work advances the co-design of quantum processor architecture and algorithm mapping, offering a promising avenue for future research and development in the field.

Quantum Best Arm Identification with Quantum Oracles

Best arm identification (BAI) is a key problem in stochastic multi-armed bandits, where K arms each has an associated reward distribution, and the objective is to minimize the number of queries needed to identify the best arm with high confidence. In this paper, we explore BAI using quantum oracles. For the case where each query probes only one arm (m=1), we devise a quantum algorithm with a query complexity upper bound of O((K/Delta)log(1/delta)), where delta is the confidence parameter and Delta is the reward gap between best and second best arms. This improves on the classical bound by a factor of 1/Delta. For the general case where a single query can probe m arms (1

Q-MAML: Quantum Model-Agnostic Meta-Learning for Variational Quantum Algorithms

In the Noisy Intermediate-Scale Quantum (NISQ) era, using variational quantum algorithms (VQAs) to solve optimization problems has become a key application. However, these algorithms face significant challenges, such as choosing an effective initial set of parameters and the limited quantum processing time that restricts the number of optimization iterations. In this study, we introduce a new framework for optimizing parameterized quantum circuits (PQCs) that employs a classical optimizer, inspired by Model-Agnostic Meta-Learning (MAML) technique. This approach aim to achieve better parameter initialization that ensures fast convergence. Our framework features a classical neural network, called Learner, which interacts with a PQC using the output of Learner as an initial parameter. During the pre-training phase, Learner is trained with the meta objective function. In the adaptation phase, the framework requires only a few PQC updates to converge to a more accurate value, while the learner remains unchanged. This method is highly adaptable and is effectively extended to various Hamiltonian optimization problems. We validate our approach through experiments, including distribution function mapping and optimization of the Heisenberg XYZ Hamiltonian. The result implies that the Learner successfully estimates initial parameters that generalize across the problem space, enabling fast adaptation.

Speedup of high-order unconstrained binary optimization using quantum Z2 lattice gauge theory

An important and difficult problem in optimization is the high-order unconstrained binary optimization, which can represent many optimization problems more efficiently than quadratic unconstrained binary optimization, but how to quickly solve it has remained difficult. Here, we present an approach by mapping the high-order unconstrained binary optimization to quantum Z2 lattice gauge theory and propose the gauged local quantum annealing, which is the local quantum annealing protected by the gauge symmetry. We present the quantum algorithm and its corresponding quantum-inspired classical algorithm for this problem and achieve algorithmic speedup by using gauge symmetry. By running the quantum-inspired classical algorithm, we demonstrate that the gauged local quantum annealing reduces the computational time by one order of magnitude from that of the local quantum annealing.

Exploring Quantum Supremacy: A Review Of Quantum Technologies And Algorithms

This review paper provides an in-depth exploration of quantum computing and the algorithms that drive its potential. It begins by introducing quantum computers and explores various types of quantum computers, focusing on the most prominent architectures. This paper also explains the concept of qubits, highlighting their unique properties such as superposition and entanglement, which differentiate quantum systems from classical computing. It discusses the specific properties of quantum computers that make them particularly suited to solving certain classes of problems, such as optimization, cryptography, and simulation. Additionally, the paper provides a detailed overview of key quantum algorithms, including Shor’s algorithm for integer factorization, Grover’s algorithm for unstructured search, and the use of quantum simulation in modelling complex physical systems. Through this review, we aim to showcase the evolving landscape of quantum computing, its capabilities, and the algorithms driving its future impact across various fields.

Learning Density Functionals from Noisy Quantum Data

Abstract The search for useful applications of noisy intermediate-scale quantum (NISQ) devices in quantum simulation has been hindered by their intrinsic noise and the high costs associated with achieving high accuracy. A promising approach to finding utility despite these challenges involves using quantum devices to generate training data for classical machine learning (ML) models.
In this study, we explore the use of noisy data generated by quantum algorithms in training an ML model to learn a density functional for the Fermi-Hubbard model. We benchmark various ML models against exact solutions, demonstrating that a neural-network ML model can successfully generalize from small datasets subject to noise typical of NISQ algorithms. The learning procedure can effectively filter out unbiased sampling noise, resulting in a trained model that outperforms any individual training data point. Conversely, when trained on data with expressibility and optimization error typical of the variational quantum eigensolver, the model replicates the biases present in the training data. The trained models can be applied to solving new problem instances in a Kohn-Sham-like density optimization scheme, benefiting from automatic differentiability and achieving reasonably accurate solutions on most problem instances. Our findings suggest a promising pathway for leveraging NISQ devices in practical quantum simulations, highlighting both the potential benefits and the challenges that need to be addressed for successful integration of quantum computing and ML techniques.

Black-box security of stream ciphers under the quantum algorithm for linear systems of equations

In this paper, we analyse the impact of the HHL quantum algorithm on stream ciphers in a black-box setting. We assume a black-box access to an oracle MS defining the output of the stream cipher. For a state k encapsulating the key material, the value MS·k is the keystream (k1′,k2′,...,kN′) generated by some stream cipher S. We translate this scenario into the quantum setting and describe how the HHL algorithm could be used to attack this construction. Further, we give simple and verifiable criteria under which a black-box attack on stream ciphers with the HHL algorithm is not efficiently feasible. Usually, these criteria follow from already known design principles for symmetric ciphers and should apply to the ciphers used today. We complement the criteria with a simple test, which confirms the resistance of said cipher. Moreover, we use our technique to test the currently used stream ciphers: Trivium, HC-128, and Salsa20.

Multiset Variational Quantum Dynamics Algorithm for Simulating Nonadiabatic Dynamics on Quantum Computers.

Accelerating quantum dynamical simulations with quantum computing has received considerable attention but remains a significant challenge. In variational quantum algorithms for quantum dynamics, designing an expressive and shallow-depth parametrized quantum circuit (PQC) is a key difficulty. Here, we propose a multiset variational quantum dynamics algorithm (MS-VQD) tailored for nonadiabatic dynamics involving multiple electronic states. The MS-VQD employs multiple PQCs to represent the electronic-nuclear coupled wave function, with each circuit adapting to the motion of the nuclear wavepacket on a specific potential energy surface. By simulating excitation energy transfer dynamics in molecular aggregates described by the Frenkel-Holstein model, we demonstrate that the MS-VQD achieves the same accuracy as the traditional VQD while requiring significantly shallower PQCs. Notably, its advantage increases with the number of electronic states, making it suitable for simulating nonadiabatic quantum dynamics in complex molecular systems.

Quantum-Enhanced AI and Machine Learning: Transforming Predictive Analytics

Background: Artificial Intelligence (AI) and Machine Learning (ML) have significantly transformed predictive analytics across domains such as healthcare, finance, and environmental sciences. However, traditional ML models face limitations when dealing with large-scale, high-dimensional datasets, particularly due to computational inefficiencies and the "curse of dimensionality." Quantum computing offers promising solutions by leveraging superposition and entanglement to perform complex computations at exponential speeds. Objective: This study aims to explore how quantum computing can enhance AI and ML models, particularly in the domain of predictive analytics. It proposes a comprehensive Quantum-AI framework that integrates quantum technologies into existing predictive systems to overcome the challenges posed by classical approaches. Methods: The study employed hybrid quantum-classical models using frameworks like Qiskit, TensorFlow Quantum, and Pennylane. Datasets were sourced from real-world applications in finance, healthcare, and climate science. Experiments were designed to compare classical and quantum- enhanced models on metrics such as accuracy, scalability, and computational time. Specific algorithms used include Quantum Support Vector Machines (QSVM) and Quantum Neural Networks (QNN). Result: The proposed quantum-enhanced models showed significant improvements: • Accuracy increased from 85% (classical) to 92% (quantum) in finance-related predictions. • Diagnostic models in healthcare achieved a 94% accuracy rate and reduced training time by 50%. • Climate modeling tasks achieved 90% improved accuracy and 20% faster simulation times. The results validate the effectiveness of quantum models in processing large, complex datasets more efficiently. Conclusion: Quantum-enhanced AI presents a transformative approach to predictive analytics by addressing the core limitations of classical ML systems. This study demonstrates the practical feasibility and industrial relevance of integrating quantum computing into AI workflows. It opens avenues for further research in developing scalable, domain-specific quantum algorithms and hybrid AI models for real-world applications.

Bias-field digitized counterdiabatic quantum optimization

We introduce a method for solving combinatorial optimization problems on digital quantum computers, where we incorporate auxiliary counterdiabatic (CD) terms into the adiabatic Hamiltonian, while integrating bias terms derived from an iterative digitized counterdiabatic quantum algorithm. We call this protocol bias-field digitized counterdiabatic quantum optimization (BF-DCQO). Designed to effectively tackle large-scale combinatorial optimization problems, BF-DCQO demonstrates resilience against the limitations posed by the restricted coherence times of current quantum processors and shows clear enhancement even in the presence of noise. Additionally, our purely quantum approach eliminates the dependency on classical optimization required in hybrid classical-quantum schemes, thereby circumventing the trainability issues often associated with variational quantum algorithms. Through the analysis of an all-to-all connected general Ising spin-glass problem, we exhibit a polynomial scaling enhancement in ground-state success probability compared to traditional DCQO and finite-time adiabatic quantum optimization methods. Furthermore, it achieves scaling improvements in ground-state success probabilities, increasing by up to two orders of magnitude, and offers an average 1.3× better approximation ratio than the quantum approximate optimization algorithm for the problem sizes studied. We validate these findings through experimental implementations on both trapped-ion quantum computers and superconducting processors, tackling a maximum weighted independent set problem with 36 qubits and a spin glass on a heavy-hexagonal lattice with 100 qubits, respectively. These results mark a significant advancement in gate-based quantum computing, employing a fully quantum algorithmic approach. Published by the American Physical Society 2025

Quantum metamaterials: Applications in quantum information science

Metamaterials are a class of artificially engineered materials with periodic structures possessing exceptional properties not found in conventional materials. This definition can be extended when we introduce a degree of freedom by adding quantum elements such as quantum dots, cold atoms, Josephson junctions, and molecules, making metamaterials highly valuable for various quantum applications. Metamaterials have been used to achieve invisibility cloaking, super-resolution, energy harvesting, and sensing, among other applications. Most of these applications are performed in the classical regime. Metamaterials have gradually made their way into the quantum regime since the advent of quantum computing and quantum sensing and imaging. Quantum metamaterials are a relatively new technology, and their use in quantum information processing has proliferated. We restrict this study to quantum state manipulation and control, quantum entanglement, single photon generation, quantum state switching, quantum state engineering, quantum key distribution, quantum algorithms, orbital angular momentum, and quantum imaging. Considering these developments, we examine the theory, fabrication, and applications contributing to quantum information processing and how quantum metamaterials contribute to this field. We find that the ability to harness the unique properties of metamaterials to drive these applications is of great importance, as they have the potential to unlock new possibilities for revolutionizing quantum information processing, bringing the world closer to practical quantum technologies with unprecedented capabilities. We conclude by suggesting possible future research directions.

QbC: Quantum Correctness by Construction

Thanks to the rapid progress and growing complexity of quantum algorithms, correctness of quantum programs has become a major concern. Pioneering research over the past years has proposed various approaches to formally verify quantum programs using proof systems such as quantum Hoare logic. All these prior approaches are post-hoc: one first implements a program and only then verifies its correctness. Here we propose Quantum Correctness by Construction (QbC): an approach to constructing quantum programs from their specification in a way that ensures correctness. We use pre- and postconditions to specify program properties, and propose sound and complete refinement rules for constructing programs in a quantum while language from their specification. We validate QbC by constructing quantum programs for idiomatic problems and patterns. We find that the approach naturally suggests how to derive program details, highlighting key design choices along the way. As such, we believe that QbC can play a role in supporting the design and taxonomization of quantum algorithms and software.

Quantum state preparation for multivariate functions

A fundamental step of any quantum algorithm is the preparation of qubit registers in a suitable initial state. Often qubit registers represent a discretization of continuous variables and the initial state is defined by a multivariate function. We develop protocols for preparing quantum states whose amplitudes encode multivariate functions by linearly combining block-encodings of Fourier and Chebyshev basis functions. Without relying on arithmetic circuits, quantum Fourier transforms, or multivariate quantum signal processing, our algorithms are simpler and more effective than previous proposals. We analyze requirements both asymptotically and pragmatically in terms of near/medium-term resources. Numerically, we prepare bivariate Student's t-distributions, 2D Ricker wavelets and electron wavefunctions in a 3D Coulomb potential, which are initial states with potential applications in finance, physics and chemistry simulations. Finally, we prepare bivariate Gaussian distributions on the Quantinuum H2-1 trapped-ion quantum processor using 24 qubits and up to 237 two-qubit gates.

Gate teleportation-assisted routing for quantumalgorithms

Abstract The limited qubit connectivity of quantum processors poses a significant challenge in deploying practical algorithms and logical gates, necessitating efficient qubit mapping and routing strategies. When implementing a gate that requires additional connectivity beyond the native connectivity, the qubit state must be moved to a nearby connected qubit to execute the desired gate locally. This is typically achieved using a series of SWAP gates creating a SWAP path. However, routing methods relying on SWAP gates often lead to increased circuit depth and gate count, motivating the need for alternative approaches.
This work explores the potential of teleported gates to improve qubit routing efficiency, focusing on implementation within specific hardware topologies and benchmark quantum algorithms. We propose a routing method that is assisted by gate teleportation. It establishes additional connectivity using gate teleportation paths through available unused qubits, termed auxiliary qubits, within the topology. To optimize this approach, we have developed an algorithm to identify the best gate teleportation connections, considering their potential to reduce the depth of the circuit and address possible errors that may arise from the teleportation paths. Finally, we demonstrate depth reduction with gate teleportation-assisted routing in various benchmark
algorithms, including case studies on the compilation of the Deutsch-Jozsa algorithm
and the Quantum Approximation Optimization Algorithm (QAOA) for heavy-hexagon
topology used in IBM 127-qubit Eagle r3 processors. Our benchmark results show a 10-25 % depth reduction in the routing of selected algorithms compared to regular routing without using the teleported gate.

Quantum Amplitude‐Phase Judgment Circuits for Full Quantization Algorithms

AbstractQuantum algorithms are a crucial component of quantum computing. One key open question in this field is whether quantum algorithms (QAs) can be fully executed on a quantum computer. Many QAs currently rely on classical computers to evaluate conditional statements that quantum systems alone cannot assess. This dependency necessitates information transmission between quantum and classical systems, thus imposing performance limitations. To enable autonomous conditional evaluations on quantum systems, a quantum amplitude‐phase judgment circuit (QAPJC), comprising primarily quantum inequality judgment circuits and equality judgment logic blocks, is proposed. This configuration achieves conditional judgment on quantum computers while imparting physical significance to the judgment process. The feasibility of the circuits is verified on a superconducting quantum computer. Comparative experiments confirm that QAPJC preserves the original performance of QAs while demonstrating the inherent advantages of quantum computing. This circuit can implement logic judgment functions akin to classical circuits and serve as a subroutine for various QAs, promoting their implementation in the noisy intermediate‐scale quantum (NISQ) era.