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 a Poisson equation is generally reduced to solving a linear system with a coefficient matrix
A of entries
aij,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
A be decomposed into a sum of
O[poly(log2n)] simple operators in order to evaluate efficiently the loss function. A tensor product decomposition of
A with
2log2n+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
d-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) 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 for $c=0,1,...,n\ensuremath{-}1$ and $i=0,\ensuremath{\cdots},n\ensuremath{-}1\ensuremath{-}c$.
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.
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) 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.
Abstract The discretized Poisson equation matrix (DPEM) in 1D has been shown to require an exponentially large number of terms when decomposed in the Pauli basis when solving numerical linear algebra problems on a quantum computer. Additionally, traditional ansatz for variational quantum algorithms (VQAs) that are used to heuristically solve linear systems (such as the DPEM) have many parameters, making them harder to train. This research attempts to resolve these problems by utilizing quantum computers that are capable of total connectivity between qubits. We propose a decomposition of the DPEM that is based on 2- or 3-qubit entanglement gates and is shown to have O(1) terms with respect to system size, with one term having an
O(n2) O ( n 2 ) circuit depth and the rest having only an O(1) circuit depth (where n is the number of qubits defining the system size). Additionally, we introduce the Globally-Entangling Ansatz which reduces the parameter space of the quantum ansatz while maintaining enough expressibility to find the solution. To test these new improvements, we ran numerical simulations to examine how well the VQAs performed with varying system sizes, showing that the new setup offers an improved scaling of the number of iterations required for convergence compared to Hardware-Efficient Ansatz.
We present a variational quantum linear solver (VQLS) for the Poisson equation built on an exact block encoding of the discrete Laplacian, and demonstrate its performance on physically motivated benchmarks. Unlike LCU-based VQLS where the number of distinct circuits required per cost-function evaluation is
O(L2), where
L is the number of terms in the LCU decomposition of the discrete Laplacian operator, this approach requires only a single circuit for cost evaluation. We further empirically demonstrate that the choice of classical optimizer materially affects where the variational optimization ceases to make progress. The solver is benchmarked on three problems: a Poisson equation with sinusoidal forcing and a steady-state heat conduction problem with a localized Gaussian source, both with Dirichlet boundaries, and the pressure-Poisson equation of a two-dimensional lid-driven cavity flow, in which the solver is invoked once per time step under Neumann boundary conditions.
Abstract We design a variational quantum algorithm (VQA) to solve multi-dimensional Poisson equations with mixed boundary conditions that are typically required in various fields of computational science. Employing an objective function that is formulated with the concept of the minimal potential energy, we not only present in-depth discussion on the cost-efficient & noise-robust design of quantum circuits that are essential for evaluation of the objective function, but, more remarkably, employ the proposed algorithm to calculate bias-dependent spatial distributions of electric fields in semiconductor systems that are described with a two-dimensional domain and up to 10-qubit circuits. Extending the application scope to multi-dimensional problems with mixed boundary conditions for the first time, fairly solid computational results of this work clearly demonstrate the potential of VQAs to tackle Poisson equations derived from physically meaningful problems.