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arXiv:2012.07014 · Physical Review A

Variational quantum algorithm for the Poisson equation

Hailing Liu, Yusen Wu, Lin‐Chun Wan, Shi‐Jie Pan, Su‐Juan Qin, Fei Gao, Qiaoyan Wen

Published Aug 18, 2021• 127 citations
arXivHTMLPDF

Abstract

The Poisson equation has wide applications in many areas of science and engineering. Although there are some quantum algorithms that can efficiently solve the Poisson equation, they generally require a fault-tolerant quantum computer, which is beyond the current technology. We propose a variational quantum algorithm (VQA) to solve the Poisson equation, which can be executed on noisy intermediate-scale quantum devices. In detail, we first adopt the finite-difference method to transform the Poisson equation into a linear system. Then, according to the special structure of the linear system, we find an explicit tensor product decomposition, with only $(2{log}_{2}n+1)$ items, of its coefficient matrix under a specific set of simple operators, where $n$ is the dimension of the coefficient matrix. This implies that the proposed VQA needs fewer quantum measurements, which dramatically reduces the required quantum resources. Additionally, we design observables to efficiently evaluate the expectation values of the simple operators on a quantum computer. Numerical experiments demonstrate that our algorithm can solve the Poisson equation.

DOI
10.1103/physreva.104.022418

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The Poisson equation has wide applications in many areas of science and engineering. Although there are some quantum algorithms that can efficiently solve the Poisson equation, they generally require a fault-tolerant quantum computer, which is beyond the current technology. We propose a variational quantum algorithm (VQA) to solve the Poisson equation, which can be executed on noisy intermediate-scale quantum devices. In detail, we first adopt the finite-difference method to transform the Poisson equation into a linear system. Then, according to the special structure of the linear system, we find an explicit tensor product decomposition, with only (2log2n+1)(2{log}_{2}n+1)(2log2​n+1) items, of its coefficient matrix under a specific set of simple operators, where nnn is the dimension of the coefficient matrix. This implies that the proposed VQA needs fewer quantum measurements, which dramatically reduces the required quantum resources. Additionally, we design observables to efficiently evaluate the expectation values of the simple operators on a quantum computer. Numerical experiments demonstrate that our algorithm can solve the Poisson equation.
Computer-aided engineering techniques are indispensable in modern engineering developments. In particular, partial differential equations are commonly used to simulate the dynamics of physical phenomena, but very large systems are often intractable within a reasonable computation time, even when using supercomputers. To overcome the inherent limit of classical computing, we present a variational quantum algorithm for solving the Poisson equation that can be implemented in noisy intermediate-scale quantum devices. The proposed method defines the total potential energy of the Poisson equation as an expectation of certain observables, which are decomposed into a linear combination of Pauli operators and simple observables. The expectation value of these observables is then minimized with respect to a parameterized quantum state. Because the number of decomposed terms is independent of the size of the problem, this method requires relatively few quantum measurements. Numerical experiments demonstrate the faster computing speed of this method compared with classical computing methods and a previous variational quantum approach. We believe that our approach brings quantum computer-aided techniques closer to future applications in engineering developments.
Abstract Variational quantum algorithms (VQAs) have attracted attention as quantum algorithms employing on noisy intermediate-scale quantum devices. VQAs for solving the Poisson equation were recently proposed, as this method transforms the linear equation with a symmetric positive definite matrix obtained by discretizing the Poisson equation into a minimization problem. In this study, we propose a VQA for second-order linear differential equations including the Poisson equation on the basis of the referenced study, where the coefficient matrix obtained through discretization is a non-symmetric matrix. Furthermore, in our VQAs, we succeeded in decomposing the coefficient matrices arising from the periodic and the Dirichlet boundary condition into linear combinations of measurement operators and unitary matrices.
Solving a Poisson equation is generally reduced to solving a linear system with a coefficient matrix AAA of entries aij,i,j=1,2,...,n{a}_{ij}, i,j=1,2,...,naij​,i,j=1,2,...,n, from the discretized Poisson equation. Although the variational quantum algorithms are promising algorithms to solve the discretized Poisson equation, they generally require that AAA be decomposed into a sum of O[poly(log2n)]O[\text{poly}({\text{log}}_{2}n)]O[poly(log2​n)] simple operators in order to evaluate efficiently the loss function. A tensor product decomposition of AAA with 2log2n+12{\text{log}}_{2}n+12log2​n+1 terms has been explored in previous works. In this paper, based on the decomposition of sparse Hamiltonians we greatly reduce the number of terms. We first write the loss function in terms of the operator ${\ensuremath{\sigma}}_{x}\ensuremath{\bigotimes}A$ with ${\ensuremath{\sigma}}_{x}$ denoting the standard Pauli operator. Then for the one-dimensional Poisson equations with different boundary conditions and for the ddd-dimensional Poisson equations with Dirichlet boundary conditions, we decompose ${\ensuremath{\sigma}}_{x}\ensuremath{\bigotimes}A$ into a sum of at most 7 and (4d+1)(4d+1)(4d+1) Hermitian, one-sparse, and self-inverse operators, respectively. We design explicitly the quantum circuits to evaluate efficiently the loss function. The decomposition method and the design of quantum circuits can also be easily extended to linear systems with Hermitian and sparse coefficient matrices satisfying ai,i+c=ac{a}_{i,i+c}={a}_{c}ai,i+c​=ac​ for $c=0,1,...,n\ensuremath{-}1$ and $i=0,\ensuremath{\cdots},n\ensuremath{-}1\ensuremath{-}c$.
Abstract To solve the Poisson equation it is usually possible to discretize it into solving the corresponding linear system Ax = b. Variational quantum algorithms (VQAs) for the discretized Poisson equation have been studied before. We present a VQA based on the banded Toeplitz systems for solving the Poisson equation with respect to the structural features of matrix A. In detail, we decompose the matrices A and A 2 into a linear combination of the corresponding banded Toeplitz matrix and sparse matrices with only a few non-zero elements. For the one-dimensional Poisson equation with different boundary conditions and the d-dimensional Poisson equation with Dirichlet boundary conditions, the number of decomposition terms is less than that reported in [Phys. Rev. A 2023 108, 032 418 ]. Based on the decomposition of the matrix, we design quantum circuits that efficiently evaluate the cost function. Additionally, numerical simulation verifies the feasibility of the proposed algorithm. Finally, the VQAs for linear systems of equations and matrix-vector multiplications with the K-banded Toeplitz matrix T n K are given, where T n K ∈ R n × n and K ∈ O ( ploylog n ) .
Solving differential equations is one of the most compelling applications of quantum computing. Most existing quantum algorithms addressing general ordinary and partial differential equations are thought to be too expensive to execute successfully on Noisy Intermediate-Scale Quantum (NISQ) devices. Here we propose a compact quantum algorithm for solving one-dimensional Poisson equation based on simple Ry rotation. The major operations are performed on probability amplitudes. Therefore, the present algorithm avoids the need to do phase estimation, Hamiltonian simulation and arithmetic. The solution error comes only from the finite difference approximation of the Poisson equation. Our quantum Poisson solver (QPS) has gate-complexity of 3n in qubits and 4n^3 in one- and two-qubit gates, where n is the logarithmic of the dimension of the linear system of equations. In terms of solution error ε, the complexity is log(1/ε) in qubits and poly(log(1/ε)) in operations, which is consist with the best known results. The present QPS may represent a potential application on NISQ devices.
Recent advances in quantum computing and their increased availability has led to a growing interest in possible applications. Among those is the solution of partial differential equations (PDEs) for, e.g., material or flow simulation. Currently, the most promising route to useful deployment of quantum processors in the short to near term are so-called hybrid variational quantum algorithms (VQAs). Thus, variational methods for PDEs have been proposed as a candidate for quantum advantage in the noisy intermediate-scale quantum (NISQ) era. In this work, we conduct an extensive study of utilizing VQAs on real quantum devices to solve the simplest prototype of a PDE---the Poisson equation. Although results on noiseless simulators for small problem sizes may seem deceivingly promising, the performance on quantum computers is very poor. We argue that direct resolution of PDEs via an amplitude encoding of the solution is not a good use case within reach of today's quantum devices---especially when considering large system sizes and more complicated nonlinear PDEs that are required in order to be competitive with classical high-end solvers.