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arXiv:2210.16668

Advancing Algorithm to Scale and Accurately Solve Quantum Poisson Equation on Near-term Quantum Hardware

Kamal Krishna Saha, Walter Robson, Connor Howington, In‐Saeng Suh, Zhimin Wang, Jarosław Nabrzyski

Published Oct 29, 2022• 1 citation
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Abstract

The Poisson equation has many applications across the broad areas of science and engineering. Most quantum algorithms for the Poisson solver presented so far either suffer from lack of accuracy and/or are limited to very small sizes of the problem, and thus have no practical usage. Here we present an advanced quantum algorithm for solving the Poisson equation with high accuracy and dynamically tunable problem size. After converting the Poisson equation to a linear system through the finite difference method, we adopt the HHL algorithm as the basic framework. Particularly, in this work we present an advanced circuit that ensures the accuracy of the solution by implementing non-truncated eigenvalues through eigenvalue amplification, as well as by increasing the accuracy of the controlled rotation angular coefficients, which are the critical factors in the HHL algorithm. Consequently, we are able to drastically reduce the relative error in the solution while achieving higher success probability as the amplification level is increased. We show that our algorithm not only increases the accuracy of the solutions but also composes more practical and scalable circuits by dynamically controlling problem size in NISQ devices. We present both simulated and experimental results and discuss the sources of errors. Finally, we conclude that though overall results on the existing NISQ hardware are dominated by the error in the CNOT gates, this work opens a path to realizing a multidimensional Poisson solver on near-term quantum hardware.

DOI
10.48550/arxiv.2210.16668

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The Poisson equation has many applications across the broad areas of science and engineering. Most quantum algorithms for the Poisson solver presented so far either suffer from lack of accuracy and/or are limited to very small sizes of the problem, and thus have no practical usage. Here we present an advanced quantum algorithm for solving the Poisson equation with high accuracy and dynamically tunable problem size. After converting the Poisson equation to a linear system through the finite difference method, we adopt the HHL algorithm as the basic framework. Particularly, in this work we present an advanced circuit that ensures the accuracy of the solution by implementing non-truncated eigenvalues through eigenvalue amplification, as well as by increasing the accuracy of the controlled rotation angular coefficients, which are the critical factors in the HHL algorithm. Consequently, we are able to drastically reduce the relative error in the solution while achieving higher success probability as the amplification level is increased. We show that our algorithm not only increases the accuracy of the solutions but also composes more practical and scalable circuits by dynamically controlling problem size in NISQ devices. We present both simulated and experimental results and discuss the sources of errors. Finally, we conclude that though overall results on the existing NISQ hardware are dominated by the error in the CNOT gates, this work opens a path to realizing a multidimensional Poisson solver on near-term quantum hardware.
The Poisson equation has many applications across the broad areas of science and engineering. Most quantum algorithms for the Poisson solver presented so far, either suffer from lack of accuracy and/or are limited to very small sizes of the problem, and thus have no practical usage. Here we present an advanced quantum algorithm for solving the Poisson equation with high accuracy and dynamically tunable problem size. After converting the Poisson equation to the linear systems through the finite difference method, we adopt the Harrow-Hassidim-Lloyd (HHL) algorithm as the basic framework. Particularly, in this work we present an advanced circuit that ensures the accuracy of the solution by implementing non-truncated eigenvalues through eigenvalue amplification as well as by increasing the accuracy of the controlled rotation angular coefficients, which are the critical factors in the HHL algorithm. We show that our algorithm not only increases the accuracy of the solutions, but also composes more practical and scalable circuits by dynamically controlling problem size in the NISQ devices. We present both simulated and experimental results, and discuss the sources of errors. Finally, we conclude that overall results on the quantum hardware are dominated by the error in the CNOT gates.
Abstract The Poisson equation has many applications across the broad areas of science and engineering. Most quantum algorithms for the Poisson solver presented so far either suffer from lack of accuracy and/or are limited to very small sizes of the problem and thus have no practical usage. In this regard, our previous work (Robson in 2022 IEEE International Conference on Quantum Computing and Engineering (QCE), 2022) showed a proof-of-concept demonstration in advancing quantum Poisson solver algorithm and validated preliminary results for a simple case of 3×33\times 33×3 3 × 3 problem. In this work, we delve into comprehensive research details, presenting the results on up to 15×1515\times 1515×15 15 × 15 problems that include step-by-step improvements in Poisson equation solutions, scaling performance, and experimental exploration. In particular, we demonstrate the implementation of eigenvalue amplification by a factor of up to 282^828 2 8 , achieving a significant improvement in the accuracy of our quantum Poisson solver and comparing that to the exact solution. Additionally, we present success probability results, highlighting the reliability of our quantum Poisson solver. Moreover, we explore the scaling performance of our algorithm against the circuit depth and width, demonstrating how our approach scales with larger problem sizes and thus further solidifies the practicality of easy adaptation of this algorithm in real-world applications. We also discuss a multilevel strategy for how this algorithm might be further improved to explore much larger problems with greater performance. Finally, through our experiments on the IBM quantum hardware, we conclude that though overall results on the existing NISQ hardware are dominated by the error in the CNOT gates, this work opens a path to realizing a multidimensional Poisson solver on near-term quantum hardware.
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.
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 {\epsilon}, the complexity is log(1/{\epsilon}) in qubits and poly(log(1/{\epsilon})) in operations, which is consist with the best known results. The present QPS may represent a potential application on NISQ devices.
Abstract Solving differential equations is one of the most promising applications of quantum computing. The Poisson equation has applications in various domains of physics and engineering, including the simulation of ocean current dynamics. Here, we propose an efficient quantum algorithm for solving the one-dimensional Poisson equation based on the controlled Ry rotations. Our quantum Poisson solver (QPS) removes the need for expensive routines such as phase estimation, quantum arithmetic or Hamiltonian simulation. The computational cost of our QPS is 3n in qubits and 5/3n3 in one- and two-qubit gates, where n is the logarithmic of the number of discrete points. An overwhelming reduction of the constant factors of the big-O complexity is achieved, which is critical to evaluate the practicality of implementing the algorithm on a quantum computer. In terms of the error ε, the complexity is log(1/ε) in qubits and poly(log(1/ε)) in operations. The algorithms are demonstrated using a quantum virtual computing system, and the circuits are executed successfully on the IBM real quantum computers. The present QPS could exhibit a potential real-world application for solving differential equations on noisy intermediate-scale quantum (NISQ) devices.
This paper presents a quantum algorithm for solving the fractional Poisson equation (−Δ)su=f(-Δ)^s u = f(−Δ)su=f with s∈(0,1)s \in (0,1)s∈(0,1) on bounded domains. The proposed approach combines rational approximation techniques with quantum linear system solvers to achieve exponential quantum advantage. The rational approximation represents the inverse fractional Laplacian as a weighted sum of standard resolvents, transforming the original nonlocal problem into a collection of shifted integer-order partial differential equations. These equations are consolidated into a single large linear system through a modified right-hand side construction that simplifies the quantum implementation. To enable practical implementation, we develop explicit quantum circuits via the Schrödingerization technique, which converts the non-unitary dynamics of the linear system into a higher-dimensional Schrödinger-type equation, allowing the use of standard Hamiltonian simulation. The circuit construction leverages the decomposition of shift operators to realize the discrete Laplacian and employs controlled operations to implement the select oracle. Under finite difference discretization, we provide detailed algorithmic procedures utilizing block-encoding techniques for the coefficient matrices. A comprehensive complexity analysis demonstrates that the quantum algorithm achieves a dependence on the inverse mesh size h−1h^{-1}h−1 that is independent of the spatial dimension ddd, in stark contrast to classical methods which suffer from exponential growth in high dimensions. This establishes an exponential quantum advantage for high-dimensional fractional problems, effectively overcoming the curse of dimensionality that limits classical approaches.