Variational Quantum Algorithms (VQAs) have emerged as promising methods for tackling complex problems on near-term quantum devices. Among these algorithms, the Variational Quantum Linear Solver (VQLS) addresses linear systems of the form $Ax=b$, aiming to prepare a quantum state $|x\rangle$ such that $A|x\rangle$ is proportional to the quantum state corresponding to $b$. A key advantage of VQLS is its use of amplitude encoding, which requires a number of qubits that scales logarithmically with the linear system size. However, the existing literature has primarily focused on linear systems of limited size or with a specific structure. In this study, we extend the application of VQLS to more general and larger problem instances, including problems where state preparation is non-trivial and problems within the real domain of fluid dynamics. Our investigation reveals some critical challenges inherent to VQLS, including the need for a sufficiently expressive ansatz, the large number of circuit executions required to estimate the cost function, and the high gate count in the circuits in the most general setting. Our analysis highlights the obstacles that need to be addressed for a broader application of VQLS and concludes that further research is necessary to fully leverage the algorithm's capabilities in addressing real-world problems.
Recently a new class of quantum algorithms that are based on the quantum computation of the connected moment expansion has been reported to find the ground and excited state energies. In particular, the Peeters-Devreese-Soldatov (PDS) formulation is found variational and bearing the potential for further combining with the existing variational quantum infrastructure. Here we find that the PDS formulation can be considered as a new energy functional of which the PDS energy gradient can be employed in a conventional variational quantum solver. In comparison with the usual variational quantum eig…
Recently, an adaptive variational algorithm termed Adaptive Derivative-Assembled Pseudo-Trotter ansatz Variational Quantum Eigensolver (ADAPT-VQE) has been proposed by Grimsley et al. (Nat. Commun. 10, 3007) while the number of measurements required to perform this algorithm scales O(N^8). In this work, we present an efficient adaptive variational quantum solver of the Schrodinger equation based on ADAPT-VQE together with the reduced density matrix reconstruction approach, which reduces the number of measurements from O(N^8) to O(N^4). This new algorithm is quite suitable for quantum simulatio…
Variational quantum algorithms have found success in the NISQ era owing to their hybrid quantum-classical approach which mitigate the problems of noise in quantum computers. In our study we introduce the dynamic ansatz in the Variational Quantum Linear Solver for a system of linear algebraic equations. In this improved algorithm, the number of layers in the hardware efficient ansatz circuit is evolved, starting from a small and gradually increasing until convergence of the solution is reached. We demonstrate the algorithm advantage in comparison to the standard, static ansatz by utilizing fewe…
To solve nonlinear partial differential equations (PDEs) is one of the most common but important tasks in not only basic sciences but also many practical industries. We here propose a quantum variational (QuVa) PDE solver with the aid of machine learning (ML) schemes to synergise two emerging technologies in mathematically hard problems. The core quantum processing in this solver is to calculate efficiently the expectation value of specially designed quantum operators. For a large quantum system, we only obtain data from measurements of few control qubits to avoid the exponential cost in the m…
Variational Hybrid Quantum Classical Algorithms (VHQCAs) are a class of quantum algorithms intended to run on noisy intermediate-scale quantum (NISQ) devices. These algorithms employ a parameterized quantum circuit (ansatz) and a quantum-classical feedback loop. A classical device is used to optimize the parameters in order to minimize a cost function that can be computed far more efficiently on a quantum device. The cost function is constructed such that finding the ansatz parameters that minimize its value, solves some problem of interest. We focus specifically on the Variational Quantum Lin…
Variational quantum algorithms offer a promising new paradigm for solving partial differential equations on near-term quantum computers. Here, we propose a variational quantum algorithm for solving a general evolution equation through implicit time-stepping of the Laplacian operator. The use of encoded source states informed by preceding solution vectors results in faster convergence compared to random re-initialization. Through statevector simulations of the heat equation, we demonstrate how the time complexity of our algorithm scales with the ansatz volume for gradient estimation and how the…
Subspace-Search Variational Quantum Eigen solver (SSVQE) is a searching method of multiple states and relies on the unitarity of transformations to ensure the orthogonality of output states for multiple states. Therefore, this method is thought to be a promising method for quantum chemistry because ordinary Variational Quantum Eigen solver (VQE) can only calculate the excited states step by step from the ground state based on Variational Quantum deflation (VQD). We compare the advantage of VQE, SSVQE with/without the constraint term and/or Tabu search term, which is added by the Lagrange multi…
The rapid progress of noisy intermediate-scale quantum (NISQ) computing underscores the need to test and evaluate new devices and applications. Quantum chemistry is a key application area for these devices, and therefore serves as an important benchmark for current and future quantum computer performance. Previous benchmarks in this field have focused on variational methods for computing ground and excited states of various molecules, including a benchmarking suite focused on performance of computing ground states for alkali-hydrides under an array of error mitigation methods. Here, we outline…
Variational Quantum Linear Solver enhanced Quantum Support Vector Machine
Fraunhofer Institute for Industrial Mathematics ITWM（弗劳恩霍夫工业数学研究所ITWM）；RPTU Kaiserslautern-Landau（凯撒斯劳滕-兰道工业大学）；Leibniz Universität Hannover（莱布尼茨汉诺威大学）
The social worker scheduling problem is a class of combinatorial optimization problems that combines scheduling with routing issues. These types of problems with classical computing can only be solved, in the best of cases, in an approximate way and significantly when the input data does not grow considerably. Today, the focus on the quantum computer should no longer be only on its enormous computing power, but also on the use of its imperfection for this era, (Noisy Intermediate-Scale Quantum (NISQ)) to create a powerful optimization and learning device that uses variational techniques. We ha…
Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. For quantum systems, where the dimension of the problem space grows exponentially, finding the eigenvalues of certain operators is one such intractable problem and remains a fundamental challenge. The quantum phase estimation algorithm can efficiently find the eigenvalue of a given eigenvector but requires fully coherent evolution. We present an alternative approach that greatly reduces the requirements for coherent evolution and we combine this method with a new approach to state…
In this paper we discuss the utilization of Variational Quantum Solver (VQE) and recently introduced Generalized Unitary Coupled Cluster (GUCC) formalism for the diagonalization of downfolded/effective Hamiltonians in active spaces. In addition to effective Hamiltonians defined by the downfolding of a subset of virtual orbitals we also consider their form defined by freezing core orbitals, which enables us to deal with larger systems. We also consider various solvers to identify solutions of the GUCC equations. We use N$_2$, H$_2$O, and C$_2$H$_4$, and benchmark systems to illustrate the perfo…
The accurate computation of ground and excited states of many-fermion quantum systems is one of the most consequential, contemporary challenges in the physical and computational sciences whose solution stands to benefit significantly from the advent of quantum computing devices. Existing methodologies using phase estimation or variational algorithms have potential drawbacks such as deep circuits requiring substantial error correction or non-trivial high-dimensional classical optimization. Here we introduce a quantum solver of contracted eigenvalue equations, the quantum analogue of classical m…
The calculation of excited state energies of electronic structure Hamiltonians has many important applications, such as the calculation of optical spectra and reaction rates. While low-depth quantum algorithms, such as the variational quantum eigenvalue solver (VQE), have been used to determine ground state energies, methods for calculating excited states currently involve the implementation of high-depth controlled-unitaries or a large number of additional samples. Here we show how overlap estimation can be used to deflate eigenstates once they are found, enabling the calculation of excited s…
Rensselaer Polytechnic Institute（伦斯勒理工学院）；Department of Computer Science（计算机科学系）；Troy NY, USA（美国纽约州特洛伊市（此处仅为地址信息，根据提取规则应忽略，正确仅保留大学））
Quantum Variational Circuits (QVCs) are often claimed as one of the most potent uses of both near term and long term quantum hardware. The standard approaches to optimizing these circuits rely on a classical system to compute the new parameters at every optimization step. However, this process can be extremely challenging, due to the nature of navigating the exponentially scaling complex Hilbert space, barren plateaus, and the noise present in all foreseeable quantum hardware. Although a variety of optimization algorithms are employed in practice, there is often a lack of theoretical or empiri…
Quantum solver for single-impurity Anderson models with particle-hole symmetry
Middle Tennessee State University（中田纳西州立大学）；Quantum Research Interdisciplinary Science and Education (QRISE) Center, Middle Tennessee State University（中田纳西州立大学量子研究交叉科学与教育中心）
College of Science, China University of Petroleum（中国石油大学理学院）
In this work, we present a quantum neighborhood preserving embedding and a quantum local discriminant embedding for dimensionality reduction and classification. We demonstrate that these two algorithms have an exponential speedup over their respectively classical counterparts. Along the way, we propose a variational quantum generalized eigenvalue solver that finds the generalized eigenvalues and eigenstates of a matrix pencil $(\mathcal{G},\mathcal{S})$. As a proof-of-principle, we implement our algorithm to solve $2^5\times2^5$ generalized eigenvalue problems. Finally, our results offer two o…
The simulation of strongly correlated many-electron systems is one of the most promising applications for near-term quantum devices. Here we use a class of eigenvalue solvers (presented in Phys. Rev. Lett. 126, 070504 (2021)) in which a contraction of the Schrödinger equation is solved for the two-electron reduced density matrix (2-RDM) to resolve the energy splittings of ortho-, meta-, and para-isomers of benzyne ${\textrm C_6} {\textrm H_4}$. In contrast to the traditional variational quantum eigensolver, the contracted quantum eigensolver solves an integration (or contraction) of the many-e…
The density matrix renormalization group (DMRG) is a celebrated tensor network algorithm, which computes the ground states of one-dimensional quantum many-body systems very efficiently. Here we propose an improved formulation of continuous tensor network algorithms, which we name a matrix product renormalization group (MPRG). MPRG is a universal quantum many-body solver, which potentially works at both zero and finite temperatures, in two and higher dimensions, and is even applicable to open quantum systems. Furthermore, MPRG does not rely on any variational principles and thus supports any ki…
University of Minho（米尼奥大学）；INESC TEC（葡萄牙系统与计算机工程研究所（INESC TEC））；International Nanotechnology Laboratory (INL)（国际纳米技术实验室（INL））
Variational Quantum Circuits are being used as versatile Quantum Machine Learning models. Some empirical results exhibit an advantage in supervised and generative learning tasks. However, when applied to Reinforcement Learning, less is known. In this work, we considered a Variational Quantum Circuit composed of a low-depth hardware-efficient ansatz as the parameterized policy of a Reinforcement Learning agent. We show that an $ε$-approximation of the policy gradient can be obtained using a logarithmic number of samples concerning the total number of parameters. We empirically verify that such …
Quantum power flow (QPF) provides inspiring directions for tackling power flow's computational burdens leveraging quantum computing. However, existing QPF methods are mainly based on noise-sensitive quantum algorithms, whose practical utilization is significantly hindered by the limited capability of today's noisy-intermediate-scale quantum (NISQ) devices. This paper devises a NISQ-QPF algorithm, which enables power flow calculation on noisy quantum computers. The main contributions include: (1) a variational quantum circuit (VQC)-based AC power flow formulation, which enables QPF using short-…
University of Michigan（密歇根大学）；Flatiron Institute（熨斗研究院）
Variational quantum algorithms (VQAs) utilize a hybrid quantum-classical architecture to recast problems of high-dimensional linear algebra as ones of stochastic optimization. Despite the promise of leveraging near- to intermediate-term quantum resources to accelerate this task, the computational advantage of VQAs over wholly classical algorithms has not been firmly established. For instance, while the variational quantum eigensolver (VQE) has been developed to approximate low-lying eigenmodes of high-dimensional sparse linear operators, analogous classical optimization algorithms exist in the…
