Quantum Algorithms Research Articles

Finding Zero Correlation Linear Hulls Using Quantum Bernstein-vazirani Algorithm

Abstract Facing the threat of quantum computing to cryptography schemes, it is urgent to study the applications of quantum algorithms to cryptanalysis and evaluate the security of cryptographic schemes in a quantum computing environment. To do this, it is necessary to analyze the ability of cryptanalysis methods, such as integral and linear attacks, when combined with quantum algorithms. In this article, we investigate the properties of block ciphers with SPN structure and their security against a quantum adversary and apply the Bernstein-Vazirani algorithm to zero correlation linear attacks. Based on the relation between linear approximates of SPN ciphers and the dual cipher’s linear structures, we proposed a quantum algorithm that can find linear hulls with zero correlation of SPN ciphers. We show the validity of the proposed quantum algorithm and estimate the corresponding computational complexity. Implementing our quantum algorithm only costs polynomial qubits and quantum gates.

Secure Delegated Variational Quantum Algorithms

Secure Delegated Variational Quantum Algorithms

Quantum algorithm for nonlinear Burgers' equation for high-speed compressible flows

Quantum algorithm for nonlinear Burgers' equation for high-speed compressible flows

Momentum exchange method for quantum Boltzmann methods

The past years have seen a surge in quantum algorithms for computational fluid dynamics (CFD). These algorithms have in common that whilst promising a speed-up in the performance of the algorithm, no specific method of measurement has been suggested. This means that while the algorithms presented in the literature may be promising methods for creating the quantum state that represents the final flow field, an efficient measurement strategy is not available. This paper marks the first quantum method proposed to efficiently calculate quantities of interest (QoIs) from a state vector representing the flow field. In particular, we propose a method to calculate the force acting on an object immersed in the fluid using a quantum version of the momentum exchange method (MEM) that is commonly used in lattice Boltzmann methods to determine the drag and lift coefficients. In order to achieve this we furthermore give a scheme that implements bounce back boundary conditions on a quantum computer, as those are the boundary conditions the momentum exchange method is designed for.

Quantum Algorithms for the Variational Optimization of Correlated Electronic States with Stochastic Reconfiguration and the Linear Method.

Solving the electronic Schrodinger equation for strongly correlated ground states is a long-standing challenge. We present quantum algorithms for the variational optimization of wave functions correlated by products of unitary operators, such as Local Unitary Cluster Jastrow (LUCJ) ansatzes, using stochastic reconfiguration (SR) and the linear method (LM). While an implementation on classical computing hardware would require exponentially growing compute cost, the cost (number of circuits and shots) of our quantum algorithms is polynomial in system size. We find that classical simulations of optimization with the linear method consistently find lower energy solutions than with the L-BFGS-B optimizer across the dissociation curves of the notoriously difficult N2 and C2 dimers; LUCJ predictions of the ground-state energies deviate from exact diagonalization by 1 kcal/mol or less at all points on the potential energy curve. While we do characterize the effect of shot noise on the LM optimization, these noiseless results highlight the critical but often overlooked role that optimization techniques must play in attacking the electronic structure problem (on both classical and quantum hardware), for which even mean-field optimization is formally NP hard. We also discuss the challenge of obtaining smooth curves in these strongly correlated regimes, and propose a number of quantum-friendly solutions ranging from symmetry-projected ansatz forms to a symmetry-constrained optimization algorithm.

Photonic counterdiabatic quantum optimization algorithm

One of the key applications of near-term quantum computers has been the development of quantum optimization algorithms. However, these algorithms have largely been focused on qubit-based technologies. Here, we propose a hybrid quantum-classical approximate optimization algorithm for photonic quantum computing, specifically tailored for addressing continuous-variable optimization problems. Inspired by counterdiabatic protocols, our algorithm reduces the required quantum operations for optimization compared to adiabatic protocols. This reduction enables us to tackle non-convex continuous optimization within the near-term era of quantum computing. Through illustrative benchmarking, we show that our approach can outperform existing state-of-the-art hybrid adiabatic quantum algorithms in terms of convergence and implementability. Our algorithm offers a practical and accessible experimental realization, bypassing the need for high-order operations and overcoming experimental constraints. We conduct a proof-of-principle demonstration on Xanadu’s eight-mode nanophotonic quantum chip, successfully showcasing the feasibility and potential impact of the algorithm.

Polarization and Orbital Angular Momentum Encoded Quantum Toffoli Gate Enabled by Diffractive Neural Networks.

Controlled quantum gates play a crucial role in enabling quantum universal operations by facilitating interactions between qubits. Direct implementation of three-qubit gates simplifies the design of quantum circuits, thereby being conducive to performing complex quantum algorithms. Here, we propose and present an experimental demonstration of a quantum Toffoli gate fully exploiting the polarization and orbital angular momentum of a single photon. The Toffoli gate is implemented using the polarized diffractive neural networks scheme, achieving a mean truth table visibility of 97.27±0.20%. We characterize the gate's performance through quantum state tomography on 216 different input states and quantum process tomography, which yields a process fidelity of 94.05±0.02%. Our method offers a novel approach for realizing the Toffoli gate without requiring exponential optical elements while maintaining extensibility to the implementation of other three-qubit gates.

Quantum-Enhanced Generalized Pattern Search Optimization

While the development of quantum computers promises a myriad of advantages over their classical counterparts, care must be taken when designing algorithms that substitute a classical technique with a potentially advantageous quantum method. The probabilistic nature of many quantum algorithms may result in new behavior that could negatively impact the performance of the larger algorithm. The purpose of this work is to preserve the advantages of applying quantum search methods for generalized pattern search algorithms (GPSs) without violating the convergence criteria. It is well known that quantum search methods are able to reduce the expected number of oracle calls needed for finding the solution to a search problem from O(N) to O(N) However, the number of oracle calls needed to determine that no solution exists with certainty is exceedingly high and potentially infinite. In the case of GPS, this is a significant problem since overlooking a solution during an iteration will violate a needed assumption for convergence. Here, we overcome this problem by introducing the quantum improved point search (QIPS), a classical–quantum hybrid variant of the quantum search algorithm QSearch. QIPS retains the O(N) oracle query complexity of QSearch when a solution exists. However, it is able to determine when no solution exists, with certainty, using only O(N) oracle calls.

Potential of quantum machine learning for solving the real-world problem of cancer classification

Quantum machine learning (QML) algorithms have demonstrated the power of quantum computing for solving complex problems and big data in certain tasks. In this study, we explore the capabilities of QML for the classification of real-world biological large datasets including ten different cancer types based on gene expression values. By comparing the classification results obtained from the quantum algorithm with those from classical approaches, we disclose that the QML algorithm overall achieves comparable and reliable results. Moreover, we identify novel biomarkers that can contribute to the understanding of cancer biology. Some of these biomarkers are consistent with DNA promoter methylation. Our findings highlight the potential of QML in cancer classification and biomarker discovery, paving the way for future advancements in other disease research and clinical applications.

Adaptive Variational Quantum Computing Approaches for Green's Functions and Nonlinear Susceptibilities.

We present and benchmark quantum computing approaches for calculating real-time single-particle Green's functions and nonlinear susceptibilities of Hamiltonian systems. The approaches leverage adaptive variational quantum algorithms for state preparation and propagation. Using automatically generated compact circuits, the dynamical evolution is performed over sufficiently long times to achieve adequate frequency resolution of the response functions. We showcase accurate Green's function calculations using a statevector simulator on classical hardware for Fermi-Hubbard chains of 4 and 6 sites, with maximal ansatz circuit depths of 65 and 424 layers, respectively, and for the molecule LiH with a maximal ansatz circuit depth of 81 layers. Additionally, we consider an antiferromagnetic quantum spin-1 model that incorporates the Dzyaloshinskii-Moriya interaction to illustrate calculations of the third-order nonlinear susceptibilities, which can be measured in two-dimensional coherent spectroscopy experiments. These results demonstrate that real-time approaches using adaptive parametrized circuits to evaluate linear and nonlinear response functions can be feasible with near-term quantum processors.

Simulating photonic devices with noisy optical elements

Quantum computers are inherently affected by noise. While in the long term, error correction codes will account for noise at the cost of increasing physical qubits, in the near term, the performance of any quantum algorithm should be tested and simulated in the presence of noise. As noise acts on the hardware, the classical simulation of a quantum algorithm should not be agnostic on the platform used for the computation. In this paper, we apply the recently proposed noisy gates approach to efficiently simulate noisy optical circuits described in the dual rail framework. The evolution of the state vector is simulated directly, without requiring the mapping to the density matrix framework. Notably, we test the method on both the gate-based and measurement-based quantum computing models, showing that the approach is very versatile. We also evaluate the performance of a photonic variational quantum algorithm to solve the MAX-2-CUT problem. In particular we design and simulate an ansatz, which is resilient to photon losses up to p∼10−3 making it relevant for near-term applications. Published by the American Physical Society 2024

A linear photonic swap test circuit for quantum kernel estimation

Abstract The swap test is a quantum algorithm capable of computing the absolute value of the scalar product of two arbitrary wavefunctions. Scalar products represent a crucial ingredient to many QML methods, but their evaluation is not straightforward at all. For this reason, many research efforts have been made without achieving an efficient and robust implementation. Here, we present an integrated photonic circuit designed to implement the swap test algorithm. Our approach relies solely on linear optical integrated components and qudits, represented by single photons from an attenuated laser beam propagating through a set of waveguides. By utilizing 2$^3$ spatial degrees of freedom for the qudits, we can configure all the necessary arrangements to set any two-qubit state and perform the swap test. This simplifies the requirements on the circuitry elements and eliminates the need for non-linearity, heralding, or post-selection to achieve multi-qubit gates. Our photonic swap test circuit successfully encodes two qubits and estimates their scalar product with a measured root mean square error smaller than 0.05. This result paves the way for the development of integrated photonic architectures capable of performing Quantum Machine Learning tasks with robust devices operating at room temperature.

The Impact of Quantum Computing on Financial Risk Management: A Business Perspective

The rapid advancement of quantum computing is poised to revolutionize industries, particularly in areas requiring complex computational power. This paper explores the potential impact of quantum computing on financial risk management from a business perspective. Traditional risk management models struggle to cope with the increasing complexity of global financial systems, making quantum computing a promising solution. By leveraging quantum algorithms, financial institutions can optimize risk assessment, enhance portfolio management, and significantly improve predictive models. This paper examines key algorithms and their applications in finance, discusses the challenges and opportunities in adopting quantum solutions, and presents case studies of early adopters. The findings suggest that quantum computing can offer a competitive advantage to businesses by providing superior risk analysis capabilities. However, limitations in technology, infrastructure, and expertise remain significant hurdles. This study contributes to the growing discourse on quantum computing by providing actionable insights for businesses looking to integrate quantum solutions into their risk management strategies.

Feedback-based quantum algorithms for ground state preparation

The ground state properties of quantum many-body systems are a subject of interest across chemistry, materials science, and physics. Thus, algorithms for finding ground states can have broad impacts. Variational quantum algorithms are one class of ground state algorithms that has received significant attention in recent years. These algorithms utilize a hybrid quantum-classical computing framework to prepare ground states on quantum computers. However, this requires solving a classical optimization problem that can become prohibitively expensive in high dimensions. Here, we develop formulations of feedback-based quantum algorithms for ground state preparation that can be used to address this challenge for two broad classes of Hamiltonians: Fermi-Hubbard Hamiltonians, and molecular Hamiltonians represented in second quantization. Feedback-based quantum algorithms are optimization-free; in place of classical optimization, quantum circuit parameters are set according to a deterministic feedback law derived from quantum Lyapunov control principles. This feedback law guarantees a monotonic improvement in solution quality with respect to the depth of the quantum circuit. A variety of numerical illustrations are provided that analyze the convergence and robustness of feedback-based quantum algorithms for these problem classes. Published by the American Physical Society 2024

Tight and Efficient Gradient Bounds for Parameterized Quantum Circuits

The training of a parameterized model largely depends on the landscape of the underlying loss function. In particular, vanishing gradients are a central bottleneck in the scalability of variational quantum algorithms (VQAs), and are known to arise in various ways. However, a caveat of most existing gradient bound results is the requirement of t-design circuit assumptions that are typically not satisfied in practice. In this work, we loosen these assumptions altogether and derive tight upper and lower bounds on loss and gradient concentration for a large class of parameterized quantum circuits and arbitrary observables, which are significantly stronger than prior work. Moreover, we show that these bounds, as well as the variance of the loss itself, can be estimated efficiently and classically-providing practical tools to study the loss landscapes of VQA models, including verifying whether or not a circuit/observable induces barren plateaus. In particular, our results can readily be leveraged to rule out barren plateaus for a realistic class of ansätze and mixed observables, namely, observables containing a non-vanishing local term. This insight has direct implications for hybrid Quantum Generative Adversarial Networks (qGANs). We prove that designing the discriminator appropriately leads to 1-local weights that stay constant in the number of qubits, regardless of discriminator depth. This implies that qGANs with appropriately chosen generators do not suffer from barren plateaus even at scale-making them a promising candidate for applications in generative quantum machine learning. We demonstrate this result by training a qGAN to learn a 2D mixture of Gaussian distributions with up to 16 qubits, and provide numerical evidence that global contributions to the gradient, while initially exponentially small, may kick in substantially over the course of training.

Simulating the quantum Fourier transform, Grover's algorithm, and the quantum counting algorithm with limited entanglement using tensor networks

Quantum algorithms reformulate computational problems as quantum evolutions in a large Hilbert space. Most quantum algorithms assume that the time evolution is perfectly unitary and that the full Hilbert space is available. However, in practice, the available entanglement may be limited, leading to a reduced fidelity of the quantum algorithms. To simulate the execution of quantum algorithms with limited entanglement, tensor-network methods provide a useful framework, since they allow us to restrict the entanglement in a quantum circuit. Thus, we here use tensor networks to analyze the fidelity of the quantum Fourier transform, Grover's algorithm, and the quantum counting algorithm as the entanglement is reduced, and we map out the entanglement that is generated during the execution of each algorithm. In all three cases, we find that the algorithms can be executed with high fidelity even if the entanglement is somewhat reduced. For example, the quantum counting algorithm can be executed with a high fidelity beyond a certain threshold level of entanglement. Our results provide an understanding of the entanglement that is required to execute the three algorithms, and our simulations based on tensor networks can easily be applied to other algorithms. Published by the American Physical Society 2024

Multimodal Deep Representation Learning for Quantum Cross-Platform Verification.

Cross-platform verification, a critical undertaking in the realm of early-stage quantum computing, endeavors to characterize the similarity of two imperfect quantum devices executing identical algorithms, utilizing minimal measurements. While the random measurement approach has been instrumental in this context, the quasiexponential computational demand with increasing qubit count hurdles its feasibility in large-qubit scenarios. To bridge this knowledge gap, here we introduce an innovative multimodal learning approach, recognizing that the formalism of data in this task embodies two distinct modalities: measurement outcomes and classical description of compiled circuits on explored quantum devices, both containing unique information about the quantum devices. Building upon this insight, we devise a multimodal neural network to independently extract knowledge from these modalities, followed by a fusion operation to create a comprehensive data representation. The learned representation can effectively characterize the similarity between the explored quantum devices when executing new quantum algorithms not present in the training data. We evaluate our proposal on platforms featuring diverse noise models, encompassing system sizes up to 50 qubits. The achieved results demonstrate an improvement of 3 orders of magnitude in prediction accuracy compared to the random measurements and offer compelling evidence of the complementary roles played by each modality in cross-platform verification. These findings pave the way for harnessing the power of multimodal learning to overcome challenges in wider quantum system learning tasks.

The Development of an Automated Approach for Designing Quantum Algorithms Using Circuits Generated By Generative Adversarial Networks (Gans)

The advent of quantum computing has inaugurated a novel epoch of computational prowess, offering the potential to tackle intricate problems with unparalleled speed (Zoufal et al., 2019). Quantum circuits, which are essential components of quantum computation, serve as representations of sequences of quantum gates designed for specific quantum processes (Zoufal et al., 2019). Nevertheless, the task of creating efficient quantum circuits continues to be a formidable and labor-intensive undertaking. This research proposal presents a unique methodology that utilizes Generative Adversarial Networks (GANs) to automate the process of generating quantum circuits that are specifically designed for particular quantum gates and operations (Zoufal et al., 2019). The main goal of this study is to create a model based on Generative Adversarial Networks (GANs) that can generate quantum circuits by leveraging the collaborative efforts of the generator and discriminator networks (Zoufal et al., 2019). The GAN model will be trained using a carefully selected dataset that includes established quantum circuits and their corresponding required quantum operations (Zoufal et al., 2019). This dataset will form the basis for the training process. Following this, the quantum circuits that are produced will be thoroughly assessed in terms of fidelity, efficiency, and resource allocation (Zoufal et al., 2019). Additionally, the objective of this work is to refine and optimize the circuits that are formed by employing reinforcement learning and gradient-based techniques (Zoufal et al., 2019). In addition to investigating circuit production, this research will delve into the practical implications and consequences of quantum circuits formed by Generative Adversarial Networks (GANs) on the development of quantum algorithms (Zoufal et al., 2019). The study's value is in its capacity to accelerate the creation of quantum algorithms through the automation of circuit design (Zoufal et al., 2019). This research makes a valuable contribution to the field of quantum computing by improving the efficiency and resource utilization of quantum circuits (Zoufal et al., 2019). These developments are crucial for the development of practical quantum computing applications and will play a significant role in the evolution of quantum algorithms and processing capabilities in the future (Zoufal et al., 2019).

Review of medical image processing using quantum-enabled algorithms

Efficient and reliable storage, analysis, and transmission of medical images are imperative for accurate diagnosis, treatment, and management of various diseases. Since quantum computing can revolutionize big data analytics by providing faster solutions and security tactics, numerous studies in this field have focused on the use of quantum and quantum-inspired algorithms to enhance the performance of traditional medical image processing approaches. This review aims to provide readers with a succinct yet adequate compendium of the advances in medical image processing combined with quantum behaviors for disease diagnosis and medical image security. Some open challenges are outlined, identifying the performance limitations of current quantum technology in their applications, while addressing the short-, medium-, and long-term development plans of this field in designing future quantum healthcare systems. We hope that this review will provide full guidance for upcoming researchers interested in this area and will stimulate further appetite of experts already active in this area aimed at the pursuit of more advanced quantum paradigms in medical image processing applications.

Simulating Noisy Variational Quantum Algorithms: A Polynomial Approach.

Large-scale variational quantum algorithms are widely recognized as a potential pathway to achieve practical quantum advantages. However, the presence of quantum noise might suppress and undermine these advantages, which blurs the boundaries of classical simulability. To gain further clarity on this matter, we present a novel polynomial-scale method based on the path integral of observable's backpropagation on Pauli paths (OBPPP). This method efficiently approximates expectation values of operators in variational quantum algorithms with bounded truncation error in the presence of single-qubit Pauli noise. Theoretically, we rigorously prove: (i)For a constant minimal nonzero noise rate γ, OBPPP's time and space complexity exhibit a polynomial relationship with the number of qubits n, the circuit depth L. (ii)For variable γ, in scenarios where more than two nonzero noise factors exist, the complexity remains Poly(n,L) if γ exceeds 1/logL, but grows exponential with L when γ falls below 1/L. Numerically, we conduct classical simulations of IBM's zero-noise extrapolated experimental results on the 127-qubit Eagle processor [Y. Kim et al., Evidence for the utility of quantum computing before fault tolerance, Nature (London) 618, 500 (2023).NATUAS0028-083610.1038/s41586-023-06096-3]. Our method attains higher accuracy and faster runtime compared to the quantum device. Furthermore, our approach allows us to simulate noisy outcomes, enabling accurate reproduction of IBM's unmitigated results that directly correspond to raw experimental observations. Our research reveals the vital role of noise in classical simulations and the derived method is general in computing the expected value for a broad class of quantum circuits and can be applied in the verification of quantum computers.