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A quantum Poisson solver implementable on NISQ devices

Shengbin Wang, Zhimin Wang, Wendong Li, Lixin Fan, Guolong Cui, Zhiqiang Wei, Yongjian Gu

Published May 1, 2020• 5 citations
OpenAlex

Abstract

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.

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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.
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.
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.
Solving differential equations is one of the most promising applications of quantum computing. Recently we proposed an efficient quantum algorithm for solving one-dimensional Poisson equation avoiding the need to perform quantum arithmetic or Hamiltonian simulation. In this paper, we further develop this algorithm to make it closer to the real application on the noisy intermediate-scale quantum (NISQ) devices. To this end, we first optimize the quantum 1D-Poisson solver by developing a new way of performing the sine transformation. The circuit depth for implementing the sine transform is reduced from n2 to n without increasing the total qubit cost of the whole circuit, which is achieved by neatly reusing the additional ancillary quits. Then, we analyse the effect of common noise existing in the real quantum devices on our algorithm using the IBM Qiskit toolkit. We find that the phase damping noise has little effect on our algorithm, while the bit flip noise has the greatest impact. In addition, threshold errors of the quantum gates are obtained to make the fidelity of the circuit output being greater than 90%. The results of noise analysis will provide a good guidance for the subsequent work of error mitigation and error correction for our algorithm. The noise-analysis method developed in this work can be used for other algorithms to be executed on the NISQ devices.
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.
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.
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.