A Shot-Efficient Differential Equation Integrator using Quantum Neural Networks
Keywords: Variational Quantum Algorithms, Physics-Informed Machine Learning, Quantum Computing
Abstract: Physics-informed regularisation on quantum neural networks provides a promising means for solving differential equations on near-term quantum computers.
However, most demonstrations of this technique assume idealised simulated quantum circuits where the respective expectations are available.
In real quantum hardware, such ideal expectations are not accessible and must be averaged over many shots, introducing additional computations, the cost of which has not been considered in the majority of the preceding studies.
The requirements of higher-order derivatives for physics-informed regularisers are especially high in terms of circuit repetitions (shots) compared to lower-order derivatives required for supervised learning.
We demonstrate how to construct a global formulation of physics-informed losses especially amenable to solve ordinary differential equations on near-term quantum computers in a shot-efficient manner.
The resulting approach can reduce the order of derivatives required to calculate a loss compared to Physics-informed Neural Networks (PINNs).
In the case of initial value problems in ordinary differential equations (ODEs) and some partial differential equations (PDEs), our method removes completely the need for higher-order automatic differentiation,
thus providing an $\mathcal{O}(N)$ improvement in shot-efficiency, where $N$ is the number of data-encodings of the quantum neural network.
Our formulation naturally incorporates boundary conditions and physics-informed losses into a single optimisation term.
Numerical experiments demonstrate favourable empirical performance, in terms of both shot-efficiency and error, on (simulated) quantum circuits compared to existing quantum methodologies.
We demonstrate that the relative performance of quantum neural network algorithms in the infinite shot limit does not necessarily correspond to relative performance in the finite shot limit.
We hope this works provides insights on how to efficiently design schemes that will reduce the shot requirements and will become the basis for further developing efficient quantum algorithms for the solution of differential equations.
Primary Area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
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Submission Number: 6201
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