We present a quantum algorithm that analyzes time series data simulated by a quantum differential equation solver. The proposed algorithm is a quantum version of a dynamic mode decomposition algorithm used in diverse fields such as fluid dynamics, molecular dynamics, and epidemiology. Our quantum algorithm can also compute matrix eigenvalues and eigenvectors by analyzing the corresponding linear dynamical system. Our algorithm handles a broad range of matrices, particularly those with complex eigenvalues. The complexity of our quantum algorithm is O ( poly log N ) for an N -dimensional system. This is an exponential speedup over known classical algorithms with at least O ( N ) complexity. Thus, our quantum algorithm is expected to enable high-dimensional dynamical systems analysis and large matrix eigenvalue decomposition, intractable for classical computers. Published by the American Physical Society 2024
We present a quantum algorithm that analyzes time series data simulated by a quantum differential equation solver. The proposed algorithm is a quantum version of a dynamic mode decomposition algorithm used in diverse fields such as fluid dynamics, molecular dynamics, and epidemiology. Our quantum algorithm can also compute matrix eigenvalues and eigenvectors by analyzing the corresponding linear dynamical system. Our algorithm handles a broad range of matrices, particularly those with complex eigenvalues. The complexity of our quantum algorithm is O ( poly log N ) for an N -dimensional system. This is an exponential speedup over known classical algorithms with at least O ( N ) complexity. Thus, our quantum algorithm is expected to enable high-dimensional dynamical systems analysis and large matrix eigenvalue decomposition, intractable for classical computers. Published by the American Physical Society 2024
The dynamic mode decomposition (DMD) algorithm is a widely used factorization and dimensionality reduction technique in time series analysis. When analyzing high-dimensional time series, the DMD algorithm requires extremely large amounts of computational power. To accelerate the DMD algorithm, we propose a quantum-classical hybrid algorithm that we call the quantum dynamic mode decomposition (QDMD) algorithm. Given a time series X ∈ R n × ( m + 1) with n ≫ m , the QDMD algorithm first executes quantum singular value decomposition on a matrix related to X and obtains a quantum state containing the main singular values and singular vectors of the decomposed matrix, then performs a low-sampling-frequency process on the obtained quantum state and computes the low-dimensional projection of the DMD operator through the sampling results. Finally, the algorithm computes the DMD eigenvalues and prepares the amplitude-encoding states of the DMD modes using the obtained classical information and X . Considering the main variables, the complexity of the QDMD algorithm is O ~ M m polylog n / ϵ , where M = O ~ m 3 / ϵ 2 denotes the number of samples. Compared with the classical DMD algorithm, which has complexity O ~ n m 2 log 1 / ϵ , the QDMD algorithm provides an exponential acceleration of n , at the cost of greater dependence on M and ϵ . We test the effects of M on the QDMD algorithm in the specific application scenarios of data denoising, scene background extraction, and fluid dynamics analysis. We determined that the QDMD algorithm requires only a small number of samples M in specific applications, further demonstrating the quantum advantage of the QDMD algorithm in high-dimensional data analysis.
We present an efficient, nearly optimal quantum algorithm for solving linear matrix differential equations, with applications to the simulation of open quantum systems and beyond. For unitary or dissipative dynamics, the algorithm computes an entry of the solution matrix with query complexity O(νLt/ε), where the constant ν depends on the problem parameters, L involves a time integral of upper bounds on the norms of evolution operators, and ε is the error. In particular, νL is linear in t for unitary dynamics and can be a constant for dissipative dynamics. Our result contrasts prior quantum approaches for differential equations that typically require exponential time for this problem due to the encoding in a quantum state, which can lead to exponentially small amplitudes. We demonstrate the utility of the algorithm through an end-to-end application, namely the simulation of dissipative dynamics for non-interacting fermions, which can be extended to other quantum and classical systems. We compare with classical algorithms and give evidence of polynomial quantum speedups for systems in a lattice, which become more pronounced for systems with long-range interactions and can be shown to be exponential in general. We also provide a lower bound of Ω(νLt/ε) for unitary or dissipative dynamics that proves our algorithm is optimal up to logarithmic factors.
Eigenproblem arises in a large number of disciplines of sciences and engineering. Combining quantum algorithms to solve linear differential equations, quantum singular value estimation and quantum phase estimation, we propose quantum algorithms to compute the eigenvalues of diagonalizable matrices of two types: matrices that only have real eigenvalues and normal matrices. The quantum algorithms return a quantum state such that the first register stores eigenvalues and the second register stores the corresponding eigenvectors. For normal matrices, the complexity to obtain this quantum state is dominated by the complexity to perform quantum singular value estimation. For the other case, the complexity is determined by the solving of a linear system of differential equations. Under certain conditions (e.g. sparse), the complexities are polylog on the size n of the given matrix. If we perform measurements, then we can obtain all the eigenvalues classically. If M is s sparse, then the complexity is O(sn∥M∥maxκ2/ϵ2), where κ is the condition number of the matrix of eigenvectors.
Quantum algorithms for differential equation solving, data processing, and machine learning potentially offer an exponential speedup over all known classical algorithms. However, there also exist obstacles to obtaining this potential speedup in useful problem instances. The essential obstacle for quantum differential equation solving is that outputting useful information may require difficult postprocessing, and the essential obstacle for quantum data processing and machine learning is that inputting the data is a difficult task just by itself. In this study, we demonstrate that, when combined, these difficulties solve one another. We show how the output of quantum differential equation solving can serve as the input for quantum data processing and machine learning, allowing dynamical analysis in terms of principal components, power spectra, and wavelet decompositions. To illustrate this, we consider continuous-time Markov processes on epidemiological and social networks. These quantum algorithms provide an exponential advantage over existing classical Monte Carlo methods.
Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming [Formula: see text], where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity [Formula: see text], where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for [Formula: see text] Finally, we discuss potential applications, showing that the [Formula: see text] condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R.