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arXiv:2406.10156 · Quantum Information Processing

High-entanglement capabilities for variational quantum algorithms: the Poisson equation case

Fouad Ayoub, James D. Baeder

Published Jul 21, 2025• 1 citation
arXivHTMLPDF

Abstract

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 $$O(n^2)$$ 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.

DOI
10.1007/s11128-025-04846-y

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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)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.
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 ) .
Different hybrid quantum-classical algorithms have recently been developed as a near-term way to solve linear systems of equations on quantum devices. However, the focus has so far been mostly on the methods, rather than the problems that they need to tackle. In fact, these algorithms have been run on real hardware only for problems in quantum physics, such as Hamiltonians of a few qubits systems. These problems are particularly favorable for quantum hardware, since their matrices are the sum of just a few unitary terms and since only shallow quantum circuits are required to estimate the cost function. However, for many interesting problems in linear algebra, it appears far less trivial to find an efficient decomposition and to trade it off with the depth of the cost quantum circuits. A first simple yet interesting instance to consider are tridiagonal systems of equations. These arise, for instance, in the discretization of one-dimensional finite element analyses. This work presents a method to solve a class of tridiagonal systems of equations with the variational quantum linear solver (VQLS), a recently proposed variational hybrid algorithm for solving linear systems. In particular, we present a new decomposition for this class of matrices based on both Pauli strings and multi--qubit gates, resulting in less terms than those obtained by just using Pauli gates. Based on this decomposition, we discuss the tradeoff between the number of terms and the near-term implementability of the quantum circuits. Furthermore, we present the first simulated and real-hardware results obtained by solving tridiagonal linear systems with VQLS, using the decomposition proposed.
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 $i=0,\ensuremath{\cdots},n\ensuremath{-}1\ensuremath{-}c$.
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.
Variational quantum algorithms are a promising tool for solving partial differential equations. The standard approach for its numerical solution is finite-difference schemes, which can be reduced to the linear algebra problem. We consider three approaches to solve the heat equation on a quantum computer. Using the direct variational method we minimize the expectation value of a Hamiltonian with its ground state being the solution of the problem under study. Typically, an exponential number of Pauli products in the Hamiltonian decomposition does not allow for the quantum speedup to be achieved. The Hadamard-test-based approach solves this problem, however, the performed simulations do not evidently prove that the Ansatz circuit has a polynomial depth with respect to the number of qubits. The Ansatz tree approach exploits an explicit form of the matrix that makes it possible to achieve an advantage over classical algorithms. In our numerical simulations with up to n=11n=11n=11 qubits, this method reveals the exponential speedup.