Developing Quantum Algorithms Research Articles

Quantum social network analysis: Methodology, implementation, challenges, and future directions

Quantum social network analysis (QSNA) is a recent advancement in the interdisciplinary field of quantum computing and social network analysis. This manuscript comprehensively reviews QSNA, emphasizing its methodologies, implementation strategies, challenges, and potential applications. It explores the conceptual foundation of key social network analysis research problems, including link prediction, influence maximization, and community detection. The research examines how quantum algorithms can revolutionize such social network tasks by leveraging principles from quantum mechanics and information theory and highlights the advantages of quantum algorithms in handling complex social network structures. The implementation section delves into the practical aspects of QSNA, such as frameworks, experimental setups, and evaluation methods. We assess the capabilities of existing quantum programming language tools and platforms. Various case studies illustrate the potential of quantum computing to enhance the performance of social network analysis. Additionally, we identify several crucial challenges and future research directions for QSNA, including the complexity of developing quantum algorithms, the need for interdisciplinary knowledge, and the challenges of integrating quantum and classical computing resources. This paper aims to serve as a foundational resource for researchers and practitioners, providing insights into the transformative potential of quantum computing in advancing the analysis of social networks and outlining future research directions in this emerging field.

A QSimulator Quantum Programming System

In this paper the authors propose their own QSimulator quantum programming system, which is a software tool for deve­loping and debugging quantum algorithms in a high-level C++ language. The basic structure of the presented system is de­scribed, including quantum algorithms and quantum operations, available to the programmer. The components of the system and their fundamental basis are briefly outlined. This system emulates the behaviour of a quantum coprocessor in a standard computational basis and thus can be used for debugging and programming quantum algorithms in the language of quantum circuits. The system proposed by the authors also implements the Grover search algorithm and the Durr-Hoyer minimization algorithm based on it. Many tasks can be formulated in terms of the search task. In particular, the SAT problem. In this article, we also propose a variant of constructing an oracle in the form of a quantum circuit for the SAT problem, which can be integrated into the Grover algorithm implemented in the presented system. Automatic oracle generation for the SAT task will help save time in developing quantum algorithms related to this task when working with our system.

Option pricing under stochastic volatility on a quantum computer

We develop quantum algorithms for pricing Asian and barrier options under the Heston model, a popular stochastic volatility model, and estimate their costs, in terms of T-count, T-depth and number of logical qubits, on instances under typical market conditions. These algorithms are based on combining well-established numerical methods for stochastic differential equations and quantum amplitude estimation technique. In particular, we empirically show that, despite its simplicity, weak Euler method achieves the same level of accuracy as the better-known strong Euler method in this task. Furthermore, by eliminating the expensive procedure of preparing Gaussian states, the quantum algorithm based on weak Euler scheme achieves drastically better efficiency than the one based on strong Euler scheme. Our resource analysis suggests that option pricing under stochastic volatility is a promising application of quantum computers, and that our algorithms render the hardware requirement for reaching practical quantum advantage in financial applications less stringent than prior art.

Koopman Spectral Linearization vs. Carleman Linearization: A Computational Comparison Study

Nonlinearity presents a significant challenge in developing quantum algorithms involving differential equations, prompting the exploration of various linearization techniques, including the well-known Carleman Linearization. Instead, this paper introduces the Koopman Spectral Linearization method tailored for nonlinear autonomous ordinary differential equations. This innovative linearization approach harnesses the interpolation methods and the Koopman Operator Theory to yield a lifted linear system. It promises to serve as an alternative approach that can be employed in scenarios where Carleman Linearization is traditionally applied. Numerical experiments demonstrate the effectiveness of this linearization approach for several commonly used nonlinear ordinary differential equations.

Applicability of measurement-based quantum computation towards physically-driven variational quantum eigensolver

Variational quantum algorithms are considered one of the most promising methods for obtaining near-term quantum advantages; however, most of these algorithms are only expressed in the conventional quantum circuit scheme. The roadblock to developing quantum algorithms with the measurement-based quantum computation (MBQC) scheme is resource cost. Recently, we discovered that the realization of multi-qubit rotation operations only requires a constant number of single-qubit measurements with the MBQC scheme, providing a potential advantage in terms of resource cost. The structure of the Hamiltonian variational ansatz aligns well with this characteristic. Thus, we propose an efficient measurement-based quantum algorithm for quantum many-body system simulation tasks, called measurement-based Hamiltonian variational ansatz (MBHVA). We then demonstrate its effectiveness, efficiency, and advantages with the two-dimensional Heisenberg model and the Fermi–Hubbard chain. Numerical experiments show that MBHVA can have similar performance as circuit-based ansatz, and is expected to reduce operation counts during execution compared to quantum circuits, bringing the advantage of running time. We conclude that the MBQC scheme is potentially feasible for achieving near-term quantum advantages in the noisy intermediate-scale quantum era, especially in the presence of large multi-qubit rotation operations.

Two-Particle Scattering on Non-Translation Invariant Line Lattices

Quantum walks have been used to develop quantum algorithms since their inception, and can be seen as an alternative to the usual circuit model; combining single-particle quantum walks on sparse graphs with two-particle scattering on a line lattice is sufficient to perform universal quantum computation. In this work we solve the problem of two-particle scattering on the line lattice for a family of interactions without translation invariance, recovering the Bose-Hubbard interaction as the limiting case. Due to its generality, our systematic approach lays the groundwork to solve the more general problem of multi-particle scattering on general graphs, which in turn can enable design of different or simpler quantum gates and gadgets. As a consequence of this work, we show that a CPHASE gate can be achieved with high fidelity when the interaction acts only on a small portion of the line graph.

Adaptive variational simulation for open quantum systems

Emerging quantum hardware provides new possibilities for quantum simulation. While much of the research has focused on simulating closed quantum systems, the real-world quantum systems are mostly open. Therefore, it is essential to develop quantum algorithms that can effectively simulate open quantum systems. Here we present an adaptive variational quantum algorithm for simulating open quantum system dynamics described by the Lindblad equation. The algorithm is designed to build resource-efficient ansatze through the dynamical addition of operators by maintaining the simulation accuracy. We validate the effectiveness of our algorithm on both noiseless simulators and IBM quantum processors and observe good quantitative and qualitative agreement with the exact solution. We also investigate the scaling of the required resources with system size and accuracy and find polynomial behavior. Our results demonstrate that near-future quantum processors are capable of simulating open quantum systems.

Roadmap for Simulating Quantum Circuits Utilising IBM’s Qiskit Library: Programming Approach

This paper explains the roadmap for running quantum circuit programs based on the Qiskit library in quantum simulators as well as real cloud-based IBM quantum computers. Qiskit is a free and open-source software development platform based on the Python programming language that is used in quantum programming. Qiskit acts as a link between quantum computing’s theoretical foundations and the practical aspects of programming and experimentation. It also allows users to experiment with and develop quantum algorithms, as well as simulate and execute them on simulators and real-world cloud-based quantum devices. It also simplifies the quantum programming process and allows a diverse range of people to participate in the exciting world of quantum computing. The paper, on the other hand, provides the mathematical foundation for analyzing quantum circuits and algorithms using linear algebra principles, as they provide the tools needed to describe and manipulate quantum states and operations. Furthermore, the paper shows quantum circuit design and implementation using real Qiskit codes.

Towards practical and massively parallel quantum computing emulation for quantum chemistry

Quantum computing is moving beyond its early stage and seeking for commercial applications in chemical and biomedical sciences. In the current noisy intermediate-scale quantum computing era, the quantum resource is too scarce to support these explorations. Therefore, it is valuable to emulate quantum computing on classical computers for developing quantum algorithms and validating quantum hardware. However, existing simulators mostly suffer from the memory bottleneck so developing the approaches for large-scale quantum chemistry calculations remains challenging. Here we demonstrate a high-performance and massively parallel variational quantum eigensolver (VQE) simulator based on matrix product states, combined with embedding theory for solving large-scale quantum computing emulation for quantum chemistry on HPC platforms. We apply this method to study the torsional barrier of ethane and the quantification of the protein–ligand interactions. Our largest simulation reaches 1000 qubits, and a performance of 216.9 PFLOP/s is achieved on a new Sunway supercomputer, which sets the state-of-the-art for quantum computing emulation for quantum chemistry.

Electric‐Circuit Simulation of Quantum Fast Hitting with Exponential Speedup

AbstractThe ultimate goal of developing quantum algorithms and constructing quantum computers is to achieve faster information processing than using current classical computers. Quantum walks are powerful kernels in quantum computing protocols and possess strong capabilities in speeding up various simulation and optimization tasks. One striking example is provided by quantum walkers evolving on unbalanced trees, which demonstrate faster hitting performances than classical random walk. However, direct experimental construction of unbalanced trees to prove quantum advantage with exponential speedup remains a great challenge due to the highly complex arrangements of the structure. This study attempts to simulate quantum algorithm by classical circuit. Inspired by the quantum algorithm, the classical circuit networks are designed and fabricated with unbalanced tree structures. It is then demonstrated, both theoretically and experimentally, that the quantum algorithm for the fast hitting problem can be simulated in the structure. It is shown that the hitting efficiency of electric signals in the circuit networks with unbalanced tree structures is exponentially faster than the corresponding cases of classical random walks. Because classical circuit networks possess good scalability and stability, the results open up a scalable new path toward quantum speedup in complex problems.

Q<sup>2</sup>Chemistry: A quantum computation platform for quantum chemistry

Quantum computers provide new opportunities for quantum chemistry. In this article,we present a versatile, extensible, and efficient software package, named Q<sup>2</sup>Chemistry, for developing quantum algorithms and quantum inspired classical algorithms in the field of quantum chemistry. In Q<sup>2</sup>Chemistry, the wave function and Hamiltonian can be conveniently mapped into the qubit space, then quantum circuits can be generated corresponding to a specific quantum algorithm already implemented in the package or newly developed by the users. The generated circuits can be dispatched to either a physical quantum computer, if available, or to the internal virtual quantum computer realized by simulating quantum circuits on classical computers. As demonstrated by our benchmark simulations, Q<sup>2</sup>Chemistry achieves excellent performance in simulating medium scale quantum circuits using the matrix product state algorithm. Applications of Q<sup>2</sup>Chemistry to simulate molecules and periodic systems are given with performance analysis.

Quantum circuits for the realization of equivalent forms of one-dimensional discrete-time quantum walks on near-term quantum hardware

Quantum walks are a promising framework for developing quantum algorithms and quantum simulations. They represent an important test case for the application of quantum computers. Here we present different forms of discrete-time quantum walks (DTQWs) and show their equivalence for physical realizations. Using an appropriate digital mapping of the position space on which a walker evolves to the multiqubit states of a quantum processor, we present different configurations of quantum circuits for the implementation of DTQWs in one-dimensional position space. We provide example circuits for a five-qubit processor and address scalability to higher dimensions as well as larger quantum processors.

High-precision quantum algorithms for partial differential equations

Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity poly(1/ϵ), where ϵ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be poly(d,log⁡(1/ϵ)), where d is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations.

Determining quantum phase diagrams of topological Kitaev-inspired models on NISQ quantum hardware

Topological protection is employed in fault-tolerant error correction and in developing quantum algorithms with topological qubits. But, topological protection intrinsic to models being simulated, also robustly protects calculations, even on NISQ hardware. We leverage it by simulating Kitaev-inspired models on IBM quantum computers and accurately determining their phase diagrams. This requires constructing conventional quantum circuits for Majorana braiding to prepare the ground states of Kitaev-inspired models. The entanglement entropy is then measured to calculate the quantum phase boundaries. We show how maintaining particle-hole symmetry when sampling through the Brillouin zone is critical to obtaining high accuracy. This work illustrates how topological protection intrinsic to a quantum model can be employed to perform robust calculations on NISQ hardware, when one measures the appropriate protected quantum properties. It opens the door for further simulation of topological quantum models on quantum hardware available today.

Hamiltonian simulation algorithms for near-term quantum hardware

The quantum circuit model is the de-facto way of designing quantum algorithms. Yet any level of abstraction away from the underlying hardware incurs overhead. In this work, we develop quantum algorithms for Hamiltonian simulation "one level below” the circuit model, exploiting the underlying control over qubit interactions available in most quantum hardware and deriving analytic circuit identities for synthesising multi-qubit evolutions from two-qubit interactions. We then analyse the impact of these techniques under the standard error model where errors occur per gate, and an error model with a constant error rate per unit time. To quantify the benefits of this approach, we apply it to time-dynamics simulation of the 2D spin Fermi-Hubbard model. Combined with new error bounds for Trotter product formulas tailored to the non-asymptotic regime and an analysis of error propagation, we find that e.g. for a 5 × 5 Fermi-Hubbard lattice we reduce the circuit depth from 1, 243, 586 using the best previous fermion encoding and error bounds in the literature, to 3, 209 in the per-gate error model, or the circuit-depth-equivalent to 259 in the per-time error model. This brings Hamiltonian simulation, previously beyond reach of current hardware for non-trivial examples, significantly closer to being feasible in the NISQ era.

Quantum-walk-based search algorithms with multiple marked vertices

The quantum walk is a powerful tool to develop quantum algorithms, which usually are based on searching for a vertex in a graph with multiple marked vertices, Ambainis's quantum algorithm for solving the element distinctness problem being the most shining example. In this work, we address the problem of calculating analytical expressions of the time complexity of finding a marked vertex using quantum walk-based search algorithms with multiple marked vertices on arbitrary graphs, extending previous analytical methods based on Szegedy's quantum walk, which can be applied only to bipartite graphs. Two examples based on the coined quantum walk on two-dimensional lattices and hypercubes show the details of our method.

A quantum circuit simulator and its applications on Sunway TaihuLight supercomputer

Classical simulation of quantum computation is vital for verifying quantum devices and assessing quantum algorithms. We present a new quantum circuit simulator developed on the Sunway TaihuLight supercomputer. Compared with other simulators, the present one is distinguished in two aspects. First, our simulator is more versatile. The simulator consists of three mutually independent parts to compute the full, partial and single amplitudes of a quantum state with different methods. It has the function of emulating the effect of noise and support more kinds of quantum operations. Second, our simulator is of high efficiency. The simulator is designed in a two-level parallel structure to be implemented efficiently on the distributed many-core Sunway TaihuLight supercomputer. Random quantum circuits can be simulated with 40, 75 and 200 qubits on the full, partial and single amplitude, respectively. As illustrative applications of the simulator, we present a quantum fast Poisson solver and an algorithm for quantum arithmetic of evaluating transcendental functions. Our simulator is expected to have broader applications in developing quantum algorithms in various fields.

One-Dimensional Lazy Quantum Walk in Ternary System

Quantum walks play an important role for developing quantum algorithms and quantum simulations. Here we present one dimensional three-state quantum walk(lazy quantum walk) and show its equivalence for circuit realization in ternary quantum logic for the first of its kind. Using an appropriate logical mapping of the position space on which a walker evolves onto the multi-qutrit states, we present efficient quantum circuits considering the nearest neighbour position space for the implementation of lazy quantum walks in one-dimensional position space in ternary quantum system. We also address scalability in terms of $n$-qutrit ternary system with example circuits for a three qutrit state space.

Development of a technique for modeling entangled quantum computations that are applicable in Simon’s quantum algorithm

This paper describes the basics of developing quantum algorithms and modeling entangled quantum computations applicable in quantum algorithms. Quantum algorithms involve the use of vector and matrix algebra. The basic tasks of the simulation proposed in the work are determined within the framework of the algorithm for executing quantum algorithms, taking into account entanglement. A technique has been developed for modeling entangled quantum calculations applicable in the Simon quantum algorithm, which helps to predict the behavior of the quantum algorithm (or any other computing process that proceeds as part of the work of a quantum computer system) with partial entanglement.

A quantum algorithm for evolving open quantum dynamics on quantum computing devices

Designing quantum algorithms for simulating quantum systems has seen enormous progress, yet few studies have been done to develop quantum algorithms for open quantum dynamics despite its importance in modeling the system-environment interaction found in most realistic physical models. In this work we propose and demonstrate a general quantum algorithm to evolve open quantum dynamics on quantum computing devices. The Kraus operators governing the time evolution can be converted into unitary matrices with minimal dilation guaranteed by the Sz.-Nagy theorem. This allows the evolution of the initial state through unitary quantum gates, while using significantly less resource than required by the conventional Stinespring dilation. We demonstrate the algorithm on an amplitude damping channel using the IBM Qiskit quantum simulator and the IBM Q 5 Tenerife quantum device. The proposed algorithm does not require particular models of dynamics or decomposition of the quantum channel, and thus can be easily generalized to other open quantum dynamical models.