# Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits
## 4 Materials and Methods
### Multidimensional Poisson Equation
#### 4.2.4 4D Poisson Equation
2023 Available online: [[https://docs.quantum.ibm.com/api/qiskit/0.32/qiskit.aqua.operators.primitive_ops.MatrixOp.to_pauli_op](https://docs.quantum.ibm.com/api/qiskit/0.32/qiskit.aqua.operators.primitive_ops.MatrixOp.to_pauli_op)]([https://docs.quantum.ibm.com/api/qiskit/0.32/qiskit.aqua.operators.primitive_ops.MatrixOp.to_pauli_op](https://docs.quantum.ibm.com/api/qiskit/0.32/qiskit.aqua.operators.primitive_ops.MatrixOp.to_pauli_op)) (accessed on 28 February 2025). * IBM (2023) IBM. EfficientSU2 Circuit. 2023. Available online: [[https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.EfficientSU2](https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.EfficientSU2)]([https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.EfficientSU2](https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.EfficientSU2)) (accessed on 28 February 2025). * IBM (2023) IBM. ADAM Optimizer. 2023. Available online: [[https://docs.quantum.ibm.com/api/qiskit/0.28/qiskit.algorithms.optimizers](https://docs.quantum.ibm.com/api/qiskit/0.28/qiskit.algorithms.optimizers)]([https://docs.quantum.ibm.com/api/qiskit/0.28/qiskit.algorithms.optimizers](https://docs.quantum.ibm.com/api/qiskit/0.28/qiskit.algorithms.optimizers)). ADAM (accessed on 28 February 2025). * [PERSON] et al. (2023) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Quantum error mitigation. _Rev. Mod. Phys._**2023**, _95_, 045005. * IBM (2023) IBM. IBM Debuts Next-Generation Quantum Processor & IBM Quantum System Two, Extends Roadmap to Advance Era of Quantum Utility IBM Press Release, 2023. Available online: [[https://newsroom.ibm.com/2023-12-04-IBM-Debuts-Next-Generation-Quantum-Processor-IBM-Quantum-System-Two%2C-Extends-Roadmap-to-Advance-Era-of-Quantum-Utility](https://newsroom.ibm.com/2023-12-04-IBM-Debuts-Next-Generation-Quantum-Processor-IBM-Quantum-System-Two%2C-Extends-Roadmap-to-Advance-Era-of-Quantum-Utility)]([https://newsroom.ibm.com/2023-12-04-IBM-Debuts-Next-Generation-Quantum-Processor-IBM-Quantum-System-Two%2C-Extends-Roadmap-to-Advance-Era-of-Quantum-Utility](https://newsroom.ibm.com/2023-12-04-IBM-Debuts-Next-Generation-Quantum-Processor-IBM-Quantum-System-Two%2C-Extends-Roadmap-to-Advance-Era-of-Quantum-Utility)) (accessed on 28 February 2025). * [PERSON] et al. (2025) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] New fractional gates reduce circuit depth for utility-scale workloads. IBM Research Blog, 15 January 2025. Available online: https://[[https://www.ibm.com/quantum/blog/fractional-gates](https://www.ibm.com/quantum/blog/fractional-gates)]([https://www.ibm.com/quantum/blog/fractional-gates](https://www.ibm.com/quantum/blog/fractional-gates)) (accessed on 28 February 2025). * [PERSON] et al. (2022) [PERSON]; [PERSON]; [PERSON]; [PERSON] Connecting ansatz expressibility to gradient magnitudes and barren plateaus. _PRX Quantum_**2022**, \(3\), 010313. * [PERSON] et al. (2022) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Adapt-vqe is insensitive to rough parameter landscapes and barren plateaus. _arXiv_**2022**, arXiv:2204.07179. * [PERSON] and [PERSON] (2014) [PERSON]; [PERSON] Poisson solvers for self-consistent multi-particle simulations. _arXiv_**2014**, arXiv:1410.3633. * [PERSON] (2010) [PERSON] _Explorations in Quantum Computing_; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2010. * [PERSON] (2014) [PERSON] Polar decomposition of a matrix. University of Puget Sound: Tacoma, WA, USA, 2014. Available online: [[http://buzzard.ups.edu/courses/2014](http://buzzard.ups.edu/courses/2014) spring/420 projects/math420-UPS-spring-2014-buffington-polar-decomposition.pdf]([http://buzzard.ups.edu/courses/2014](http://buzzard.ups.edu/courses/2014) spring/420 projects/math420-UPS-spring-2014-buffington-polar-decomposition.pdf) (accessed on 28 February 2025). * [PERSON] (1986) [PERSON], N.J. Computing the polar decomposition--With applications. _SIAM J. Sci. Stat. Comput._**1986**, \(7\), 1160-1174. [CrossRef].