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 $n=11$ qubits, this method reveals the exponential speedup.
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=11 qubits, this method reveals the exponential speedup.
A variational quantum algorithm for numerically solving partial differential equations (PDEs) on a quantum computer was proposed by Lubasch et al. [Phys. Rev. A 101, 010301(R) (2020)]. In this paper we generalize the method introduced by Lubasch et al. to cover a broader class of nonlinear PDEs as well as multidimensional PDEs and study the performance of the variational quantum algorithm on several example equations. Specifically, we show via numerical simulations that the algorithm can solve instances of the single-asset Black-Scholes equation with a nontrivial nonlinear volatility model, the double-asset Black-Scholes equation, the Buckmaster equation, and the deterministic Kardar-Parisi-Zhang equation. Our simulations use up to n=12 Ansatz qubits, computing PDE solutions with 2n grid points. We also perform proof-of-concept demonstrations with a trapped-ion quantum processor from IonQ [Nat. Commun. 10, 5464 (2019)], showing accurate computation of two representative expectation values needed for the calculation of a single time step of the nonlinear Black-Scholes equation. Through our classical simulations and demonstrations on quantum hardware, we identify and discuss several open challenges for using quantum variational methods to solve PDEs in a regime with a large number (much greater than 220) of grid points, but also a practical number of gates per circuit and circuit shots.
Quantum computers can be used for the solution of various problems of mathematical physics. In the present paper, we consider a discretized version of the heat equation and address its solution on quantum computer using variational Anzats tree approach (ATA). We extend this method originally proposed for the system of linear equations to tackle full time dependent heat equation. The key ingredients of our method are (i) special probabilistic quantum circuit in order to add heat sources to temperature distribution, (ii) limiting auxiliary register in the preparation of quantum state, (iii) utilizing a robust cluster of repetitive nodes in the anzats tree structure. We suggest that our procedure provides an exponential speedup compared to the classical algorithms in the case of time dependent heat equation.
The prosperous development of both hardware and algorithms for quantum computing (QC) potentially prompts a paradigm shift in scientific computing in various fields. As an increasingly active topic in QC, the variational quantum algorithm leads a promising tool for solving partial differential equations on noisy intermediate scale quantum devices. Although a clear perspective on the advantages of QC over classical computing techniques for specific mathematical and physical problems exists, applications of QC in computational fluid dynamics to solve practical flow problems, though promising, are still at the early stage of development. To explore QC in practical simulation of flow problems, this work applies a variational hybrid quantum-classical algorithm, namely the variational quantum linear solver (VQLS), to resolve the heat conduction equation through finite difference discretization of the Laplacian operator. Details of the VQLS implementation are discussed by various test instances of linear systems. The effect of the number of shots on the accuracy is studied, which reveals a logarithmic relationship. Furthermore, the heuristic scaling of the VQLS with the precision ε, the number of qubits n and the condition number k validates its time complexity reported in the literature. In addition, the successful state vector simulations of the heat conduction equation in one and two dimensions demonstrate the validity of the present VQLS-based algorithm by proof-of-concept results. Finally, the heuristic scaling for the heat conduction problem indicates that the time complexity of the present approach is logarithmically dependent on the precision ε and linearly dependent on the number of qubits n.
Partial differential equations (PDEs) play a central role in scientific and engineering analysis, with applications spanning fluid dynamics, heat and mass transfer, electromagnetism, quantum mechanics, and financial modeling, where they are used to describe diffusion processes, wave propagation, and the evolution of complex systems over space and time. Solving multidimensional partial differential equations (PDEs) is a computationally challenging problem, even for the most advanced classical systems. Over the past decade, quantum computing has attracted significant interest as a potential approach for solving complex computational problems, including multidimensional PDEs. Although a variety of approaches have been proposed for solving PDEs, most of the existing techniques are based on variational quantum algorithms (VQAs). Despite being promising, these VQA-based approaches suffer from low accuracy, long execution times, and limited scalability. In this work, we propose a scalable and efficient quantum algorithm for solving multidimensional PDEs. Our algorithm has two variants. One variant is based on the finite difference method (FDM), classical-to-quantum (C2Q) encoding, and numerical instantiation, whereas the other is based on FDM, C2Q, and column-by-column decomposition (CCD). We have also evaluated our algorithm using several practical case studies; namely, Poisson, heat, Black–Scholes, and Navier–Stokes equations. The results show that our proposed approach achieves higher accuracy, greater scalability, and faster execution time than the VQA-based approaches. We validated these findings on both noise-free and noisy simulators, as well as on a hardware emulator and real IBM quantum hardware.
Quantum computing has the potential to transform computational problem-solving by leveraging quantum mechanical principles of superposition and entanglement. This capability is particularly important for the numerical solution of complex and/or multidimensional partial differential equations (PDEs). There are currently many quantum techniques available for solving PDEs, which are mainly based on variational quantum algorithms (VQAs). However, the existing quantum PDE solvers, particularly those based on VQAs suffer from limitations such as low accuracy, high execution times, and low scalability on both simulators as well as on real quantum devices. In this work, we propose an efficient and scalable algorithm for solving multidimensional PDEs. We present two variants of our algorithm: the first leverages finite-difference method (FDM), classical-to-quantum (C2Q) encoding, and numerical instantiation, while the second employs FDM, C2Q, and column-by-column decomposition (CCD). Both variants are designed to enhance accuracy and scalability while reducing execution times. We have validated our proposed algorithm by solving several practically useful PDEs such as Poisson, heat, Black-Scholes, and Navier-Stokes equations. Our results demonstrate higher accuracy, higher scalability, and faster execution times compared to VQA-based solvers on noise-free and noisy quantum simulators from IBM and achieved promising results on real quantum hardware.
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.