Quantum-PDE-Benchmark

Benchmark near-term quantum algorithms for Partial Differential Equations (PDEs).

Solving PDE efficiently is important in scientific research and engineering. Many quantum algorithms have been proposed to solve PDE on near-term quantum computers. However, most algorithms are verified on simulators, ignoring the effect of quantum hardware noise. This leaves the practical performance and utility of those algorithms an open question. We propose an effort toward filling this gap via extensive benchmarks.

Also, see this website for a brief survey of solving various PDEs with quantum algorithms.

Install

Download all the files and finish the installation locally,

git clone https://github.com/comp-physics/Quantum-PDE-Benchmark
cd qpde-benchmark
pip install -e.

Introduction

We consider the following PDEs in this benchmark with various boundary conditions.

PDE Library

The Poisson equation (elliptic)

$$ \Delta \phi = f, $$

wave equation (hyperbolic)

$$ \partial_t^2 \phi = c^2 \Delta \phi, $$

heat equation (parabolic)

$$ \partial_t \phi = \Delta \phi, $$

1D Reaction-diffusion equation (linear, mixed)

$$ \partial_t \phi = D\partial_x^2\phi + f(\phi). $$

and 1D Burgers Equation (nonlinear, mixed)

$$ \partial_t \phi = \nu \partial_x^2\phi - \phi \partial_x \phi + f. $$

Quantum Solvers

1. Variational Quantum Eigensolver (VQE)

2. Variational Quantum Linear Solver (VQLS)

3. Hamiltonian Simulation

4. Quantum Spectral Method

Quantum Hardware

Hardware	qubits	Quantum Volume	Speed	1q gate error	2q gate error	Measurement error	Vendor
ibmq-guadalupe	16	32	2.4k CLOPS	3.176e-4	1.037e-2	1.795e-2	IBM
ibmq-toronto	27	32	1.8k CLOPS	3.064e-4	1.191e-2	2.930e-2	IBM
ibmq-montreal	27	128	2k CLOPS	2.986e-4	1.168e-2	1.569e-2	IBM
ibmq-washington	127	64	850 CLOPS	2.923e-4	1.305e-2	1.170e-2	IBM

Tutorials

We provide tutorials to solve PDEs on IBM's superconducting quantum hardware:

1. 1D Poisson Equation using VQE
2. 1D Poisson Equation using VQLS

Copyright and License

Quantum PDE Benchmark uses Apache-2.0 license.

Reference

[1] Peruzzo, Alberto, et al. "A variational eigenvalue solver on a photonic quantum processor." Nature communications 5.1 (2014): 1-7.
[2] Bravo-Prieto, Carlos, et al. "Variational quantum linear solver." arXiv preprint arXiv:1909.05820 (2019).
[3] Huang, Hsin-Yuan, Kishor Bharti, and Patrick Rebentrost. "Near-term quantum algorithms for linear systems of equations." arXiv preprint arXiv:1909.07344 (2019).