Variational quantum algorithm for the Poisson equation

Liu, Hai-Ling; Wu, Yusen; Wan, Lin‐Chun; Pan, Shi‐Jie; Qin, Su‐Juan; Gao, Fei; Wen, Qiaoyan

doi:10.1103/physreva.104.022418

Cited by 71 publications

(52 citation statements)

References 49 publications

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“…Due to mild requirements on the gate noise and the circuit connectivity, variational quantum algorithms (VQAs) [4] become one of the most promising frameworks for achieving practical quantum advantages on NISQ devices. Specifically, different VQAs have been proposed for many topics, e.g., quantum chemistry [5,6,7,8,9,10,11,12,13], quantum simulations [14,15,16,17,18,19,20,21,22,23], machine learning [24,25,26,27,28,29,30,31], numerical analysis [32,33,34,35,36], and linear algebra problems [37,38,39]. Recently, various small-scale VQAs have been implemented on real quantum computers for tasks such as finding the ground state of molecules [8,11,12] and exploring promising applications in supervised learning [25], generative learning [30] and reinforcement learning [29].…”

Section: Introductionmentioning

confidence: 99%

Gaussian initializations help deep variational quantum circuits escape from the barren plateau

Zhang¹,

Hsieh²,

Liu³

et al. 2022

Preprint

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Variational quantum circuits have been widely employed in quantum simulation and quantum machine learning in recent years. However, quantum circuits with random structures have poor trainability due to the exponentially vanishing gradient with respect to the circuit depth and the qubit number. This result leads to a general belief that deep quantum circuits will not be feasible for practical tasks. In this work, we propose an initialization strategy with theoretical guarantees for the vanishing gradient problem in general deep circuits. Specifically, we prove that under proper Gaussian initialized parameters, the norm of the gradient decays at most polynomially when the qubit number and the circuit depth increase. Our theoretical results hold for both the local and the global observable cases, where the latter was believed to have vanishing gradients even for shallow circuits. Experimental results verify our theoretical findings in the quantum simulation and quantum chemistry.

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Section: Introductionmentioning

confidence: 99%

Gaussian initializations help deep variational quantum circuits escape from the barren plateau

Zhang¹,

Hsieh²,

Liu³

et al. 2022

Preprint

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show abstract

“…where A k , B l are unitaries which can be easily implemented on an NISQ computer [35]. We also assume that the number of the terms L and K scale polynomially with the number of qubits, O(poly log N ), and there are r different generalized eigenvalues ordered in increasing order,…”

Section: A Theoretical Basis Of Vqgementioning

confidence: 99%

“…Since the number of Pauli basis is O(4 n ), A has at most O(4 n ) terms. However, under some special structure of A, the number of terms can be reduced to O(2n + 1) [35].…”

Section: Appendix A: Proof Of Theoremmentioning

confidence: 99%

Quantum algorithms for the generalized eigenvalue problem

Liang,

Shen,

et al. 2021

Preprint

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The generalized eigenvalue (GE) problems are of particular importance in various areas of science engineering and machine learning. We present a variational quantum algorithm for finding the desired generalized eigenvalue of the GE problem, A|ψ = λB|ψ , by choosing suitable loss functions. Our approach imposes the superposition of the trial state and the obtained eigenvectors with respect to the weighting matrix B on the Rayleigh-quotient. Furthermore, both the values and derivatives of the loss functions can be calculated on near-term quantum devices with shallow quantum circuit. Finally, we propose a full quantum generalized eigensolver (FQGE) to calculate the minimal generalized eigenvalue with quantum gradient descent algorithm. As a demonstration of the principle, we numerically implement our algorithms to conduct a 2-qubit simulation and successfully find the generalized eigenvalues of the matrix pencil (A, B). The numerically experimental result indicates that FQGE is robust under Gaussian noise.

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“…Recently, quantum computing has exhibited potential acceleration advantages over classical computing by exploiting the unique properties of supposition and entanglement in quantum mechanics in solving certain problems, such as factoring integers [7], unstructured database searching [8], solving equations [9][10][11], regression [12][13][14], dimensionality reduction [15][16][17], anomaly detection [18,19] and neural network [20]. Therefore, some scholars have proposed employing quantum algorithm to solve eigenproblem of the Laplacian matrix.…”

Section: Introductionmentioning

confidence: 99%

A quantum algorithm for solving eigenproblem of the Laplacian matrix of a fully connected weighted graph

Liu¹,

Qin²,

Wan³

et al. 2022

Preprint

Self Cite

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Solving eigenproblem of the Laplacian matrix of a fully connected graph has wide applications in data science, machine learning, and image processing, etc. However, it is very challenging because it involves expensive matrix operations. Here, we propose an efficient quantum algorithm to solve it based on a general assumption that the element of each vertex and its norms can be accessed via a quantum random access data structure. Specifically, we propose in detail how to construct the block-encodings of operators containing the information of the weight matrix and the degree matrix respectively, and further obtain the block-encoding of the Laplacian matrix. This enables us to implement the quantum simulation of the Laplacian matrix by employing the optimal Hamiltonian simulation technique based on the block-encoding framework. Afterwards, the eigenvalues and eigenvectors of the Laplacian matrix are extracted by the quantum phase estimation algorithm. Compared with its classical counterpart, our algorithm has a polynomial speedup on the number of vertices and an exponential speedup on the dimension of each vertex over its classical counterpart. We also show that our algorithm can be extended to solve the eigenproblem of symmetric (nonsymmetric) normalized Laplacian matrix.

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Variational quantum algorithm for the Poisson equation

Cited by 71 publications

References 49 publications

Gaussian initializations help deep variational quantum circuits escape from the barren plateau

Gaussian initializations help deep variational quantum circuits escape from the barren plateau

Quantum algorithms for the generalized eigenvalue problem

A quantum algorithm for solving eigenproblem of the Laplacian matrix of a fully connected weighted graph

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